Lecture Notes in Mathematics Editors: J.-M. Morel, Cachan F. Takens, Groningen B. Teissier, Paris
2002
Christian P¨otzsche
Geometric Theory of Discrete Nonautonomous Dynamical Systems
123
Christian P¨otzsche Munich University of Technology Centre for Mathematical Sciences Boltzmannstraße 3 85748 Garching Germany
[email protected]
ISBN: 978-3-642-14257-4 e-ISBN: 978-3-642-14258-1 DOI: 10.1007/978-3-642-14258-1 Springer Heidelberg Dordrecht London New York Lecture Notes in Mathematics ISSN print edition: 0075-8434 ISSN electronic edition: 1617-9692 Library of Congress Control Number: 2010933515 c Springer-Verlag Berlin Heidelberg 2010 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Cover design: SPi Publisher Services Printed on acid-free paper springer.com
Dedicated to my teacher Bernd Aulbach (1947–2005)
Preface
Πα ντ α χωρ ˆι και‘ oυδ ‘ν μ ν ι Heracleitus, 502 BC The monograph in your hands deals with difference equations, or in a terminology equivalent for us, with recursions, iterations and discrete dynamical systems. Such iterative procedures are omnipresent in mathematics, as well as in its related sciences – for approximation as well as for modelling purposes. Their history can be traced back to Pythagoras (triangular numbers, ∼500 BC), Euclid (continued fractions, ∼250 BC) and Archimedes (computation of π, ∼200 BC), that is, the beginning of mathematics as we know it today. Early systematic approaches to difference equations as independent mathematical discipline appeared in the 1920– 1950s in form of classical monographs, like for instance [42, 166, 179, 303]. These early contributions are basically concerned with a linear theory and connections to the field of functional equations. After that, corresponding research stagnated somehow and difference equations found themselves in the shadow of their continuous counterpart, namely evolutionary differential equations of various kind. However, differing from classical results obtained in the 1950s and before, in recent decades nonlinear problems and phenomena reentered the center of interest and finally led to an extensive theory of discrete dynamical systems. One reason for their popularity is definitely that already very simple equations show a surprisingly complex dynamical behavior, like, e.g., the tent-map, the logistic equation or Smale’s horseshoe map. Fields like “chaos theory” draw a strong motivation from such examples which additionally serve as prototypes to understand more complex phenomena. Indeed, over the past 20 years the mathematical community observed a renaissance of difference equations. Several new journals have been successfully introduced,1 conference series are established and various new monographs appeared (e.g., [3,4,84,103,133,175,248,276,281,289,294,297,334,425]). In the course of this revival also somewhat philosophical arguments to support discrete dynamics have occurred. Actually, many laws of nature are intrinsically 1 Discrete and Continuous Dynamical Systems, Journal of Difference Equations and Applications, Advances in Difference Equations, International Journal of Difference Equations, etc.
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discrete (cf. [132, 225, 302, 466]), providing the insight that a “correct” description of our world might be a discrete one. As a conclusion one can state that difference equations form a theory of its own right and are worth to investigate. Continuous to discrete: There is admittedly a strong analogy between the theories of discrete and of continuous dynamical systems, which even led to a unifying calculus (cf. [204]). Yet, particularly in low dimensions discrete models tend to have a more complex behavior due to the fact of nonexistent backward solutions, or missing topological constraints like connected solution curves. For that reason alone, it is unjustified that the continuous theory is usually preferred when it comes to a rigorous presentation in the literature, while its discrete counterpart is labeled as “analogous” or proofs are attributed to work “along the lines”. As a matter of course, a key application for difference equations and discrete dynamical systems comes from various discretizations of (evolutionary) differential equations. Here, “discretization” can have different meanings and discrete approaches are quite beneficial for the dynamical systems theory as a whole: •
For various problems it is convenient to study the (discrete) time-h-map ϕ(h), instead of a (continuous) semiflow (ϕ(t))t≥0 itself – for example in topological linearizations (cf. [200]) and to construct invariant manifolds (cf. [83, 285, 343], or [169] dealing with invariant manifolds for PDEs on unbounded domains). Another source for such applications are abstract functional differential equations; here, in certain cases no variation of constants formula for the continuous problem is known and one has to work with the corresponding time-h-map of the generated semiflow to obtain invariant manifolds for the continuous flow (cf. [285, Sect. 4]). • Poincar´e (or return) maps are a popular tool to study the behavior of periodic continuous motions, in particular since they offer a possibility to reduce the dimension of a problem by 1 (cf., for instance, [9, p. 320ff], [227, pp. 17–25] or [319, pp. 56–62]). • The asymptotic behavior of abstract nonautonomous (linear) evolutionary equations is often studied using difference equations, where the continuous evolution operator is restricted to the integers. Using the resulting discrete equation, it is more convenient to deduce results on the long term behavior, and then to show that they extend to the continuous problem (see [421] for stability results, [88, 201, 299, 300, 369, 372] for exponential dichotomies or [301] for a Fredholm theory). • Last but not least, numerical schemes applied to differential equations canonically lead to difference equations and it is important to have a sufficiently rich discrete theory at our disposal (cf. [447]). In conclusion, even within the field of dynamical systems it seems legitimate to claim that the continuous theory benefits more from the discrete one than the other way around. As a consequence, discrete dynamical systems and difference equation require an adequate presentation.
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Even beyond that, from a modeling and simulation perspective, it is frequently more reasonable and sometimes plainly honest to work with discrete models right from the beginning, instead of enforcing a continuous model and then to discretize it in order to make it solvable on a computer. Autonomous to nonautonomous: Beyond our above considerations, the recent years have seen a growing interest in nonautonomous problems, i.e., equations whose right-hand sides explicitly depend on time or chance (see, e.g., the upcoming monographs [79, 266]). Indeed, nonautonomous equations allow more realistic models, since they enable us to include seasonal influences, as well as regulation, controlling, modulating or random effects. In concrete situations this is realized in a way that constant parameters are replaced by time-dependent sequences (parametric perturbations) or driven by external (decoupled) equations (driven equations). Moreover, in contrast to an already stochastic approach, the advantage of deterministic nonautonomous models is that their results are easier to interpret (cf. [454]) and to tackle, because they require only point estimation of constants instead to specify complete distributions for random variables as in the case of stochastic models. Further reasons illustrating the importance of a nonautonomous deterministic theory are as follows: •
It canonically appears in a seemingly autonomous setting, like, e.g., to study the behavior near nonconstant reference solutions or in the construction of invariant foliations (see, for example, [33,83,89,157]). So why not considering nonautonomous equations right from the beginning? • Time-adaptive discretization schemes lead to nonautonomous problems (cf., e.g., [55, 173, 267, 268]). In fact, so far analytical discretization theory essentially never leaves the framework of autonomous dynamical systems. Thus, often schemes with constant stepsizes are considered, which from an applied point of view and referring to adaptive schemes is a rather artificial point of view. • Results from the deterministic theory of difference equations are applicable to random difference equations on a path-wise basis (cf. [12, pp. 50, Sect. 2.1] or [459]), i.e., by considering concrete realizations of random variables. Our approach to nonautonomous dynamical systems is based on 2-parameter semigroups (or discrete processes) rather than on skew product dynamics – a notion coined in a series of papers by Sacker and Sell (see, e.g., [415–417, 419] or the memoirs [418]) during the 1970s. In a skew product framework, one enlarges the state space by encoding the time-dependence using a flow on the so-called base space (cf. [429]). Hence, one is in the convenient position to apply methods from classical autonomous dynamical systems. Skew product dynamics is motivated by re-capturing the geometric flavor that is inherent to autonomous dynamics and also various hierarchical or triangular systems fall into the abstract skew product category. Nevertheless, in contrast to the admittedly elegant skew product setting, we avoid the resulting topologically subtle questions and assumptions on the particular time dependencies, which guarantee that the corresponding base space becomes compact.
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Geometric theory to discretizations: A central motivation for this work is to bring together ideas and results from three related, yet different areas of applied mathematics mentioned above: Difference Equations, (Nonautonomous) Dynamical Systems and (Theoretical) Numerical Analysis. They are obviously related in the sense that iterations are of central importance. But on the other hand, unfortunately they rarely rely on each other, the corresponding scientific communities hardly overlap and chances for synergetic effects are missed. We intend to introduce some modern concepts from the recent theory of nonautonomous dynamical systems into the seemingly classical field of difference equations. Within this broad field, we restrict on certain aspects of what is commonly known as “qualitative” or more precisely as “geometric theory”. This area was essentially initiated by Poincar´e and Lyapunov over a century ago. It aims to identify certain invariant subsets of the state space, which “prescribe” the long-term behavior of a system. First, it deals with questions of the existence of special solutions (equilibria, periodic, almost-periodic or complete bounded solutions, etc.) or collections of solutions (invariant manifolds) with a particular growth behavior, as well as their stability and domain of attraction. Second, it intends to identify prototype system which are particularly simple but share the essential dynamics (topological conjugation and structural stability). Third, also addressed are related global questions, like starting from an “arbitrary” initial value, what can be said about the long-term dynamics (or the (global) attractor). For a broader overview, we refer to, for instance, [12, 192, 198, 201, 211, 245, 253, 348, 432, 434, 462]. To a minor extent, we are interested in discretization theory or what is nowadays known as numerical dynamics. The essential goals in this field are (1) to investigate and determine features of continuous dynamical systems which persist under discretization, and (2) to obtain convergence results for small stepsizes or spatial discretization meshes. For a survey, see [54, 172, 193, 222, 313, 445–447]. This monograph aims to extend the above complex of questions and to provide a consistent reference. In doing so, we throughout deal with nonautonomous discrete equations. In order to possess stability properties required in discretization theory, they are allowed to be implicit. Furthermore, their state spaces can be infinite-dimensional and time-dependent. This set-up allows immediate applications to various temporal and full discretizations of evolutionary differential equations, i.e., to address the aspect (1) above. However, we clearly point out to focus on the persistence aspect of numerical dynamics and totally neglect the crucial convergence questions addressed in aspect (2). Yet, we hope to lay down the basics for future applications towards convergence issues. At hand is particularly a rather complete approach to invariant manifold theory for implicit nonautonomous difference equations in Banach spaces. Here, differing from various approaches in the literature, fully implicit numerical schemes fit into our set-up. In detail, our contents can be summarized as follows: •
The first chapter introduces 2-parameter semigroups acting on the extended state space – our notion to describe nonautonomous dynamics. We consistently use the concept of pullback convergence. Accordingly, the corresponding limit sets and attractors are sequences of sets rather than single sets as in the classical
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autonomous situation. Under various compactness assumptions, we provide criteria for their existence and derive basic properties. Moreover, we illustrate how these objects simplify to known-ones for the periodic or autonomous case. A quite flexible notion for difference equations is discussed in Chap. 2, which includes implicit discretization methods. We investigate conditions for them to generate 2-parameter semigroups, to be dissipative or to possess (global) attractors; in doing so, we particularly address one-step methods. Surely, the nonautonomous stability theory is in part classical, but understandably more complex than in the autonomous (or periodic) situation. Yet, we present and relate it to attraction and stability notions based on pullback convergence. Finally, simplifications in the periodic and autonomous case are illuminated. The theory of linear difference equations in Chap. 3 serves as foundation for our following perturbation arguments. Here, stability is a property of the whole system and not only of single solutions. After that we briefly touch periodic equations and Floquet theory. Exponential dichotomies and more general splittings turn out to be an appropriate hyperbolicity notion in our nonautonomous setting. In addition, we provide several results discussing the behavior of splittings under perturbation. Our time-dependent counterpart to classical invariant manifolds are so-called invariant fiber bundles. We provide an abstract approach, which as application, yields bundles associated to given reference solutions (local theory), as well as a discrete version of inertial manifolds (global theory). In doing so, we prove results on invariant foliations and asymptotic phases. Smoothness issues are tackled as well, using an elementary approach which is essentially based on the contraction mapping principle. This allows us to obtain Taylor approximations of local invariant fiber bundles. We also describe a numerical scheme to compute global approximations. Finally, our achievements from the previous chapter, allow to deduce results on topological decoupling and linearization. They include a generalized Hartman– Grobman theorem for invertible nonautonomous problems with nonhyperbolic spectrum. We can get rid of the invertibility assumption when shifting to the concept of solution conjugacy. The latter is still sufficient to deduce smoothness properties of invariant fiber bundles.
Every chapter is supplemented by an illustrating section dealing with applications. It extends our so far theoretical approach and illustrates that the previous results and methods are applicable to discretizations of various evolutionary differential equations, like for example of functional differential-, reaction-diffusion- or abstract type. Moreover, a final concluding section points out the relevant literature, provides historical context and indicates directions for further research. The appendix collects a number of helpful results needed in the text. It addresses discrete inequalities, various fixed point and global inversion theorems, as well as explanations on smooth functions. In particular, we provide a survey on smooth norms, which are important to construct global extensions of differentiable mappings and locally invariant fiber bundles.
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The monograph is linearly written with the exception of some references to the appendix and that certain applications in Sect. 2.6 require a lookahead to independent results from Sect. 3.7. As a general philosophy behind these notes, it is our intention to provide explicit estimates and constants to a large extent. This might lead to a technical appearance, but enables us to obtain quantitative results on, e.g., growth rates of solutions, the radius of absorbing sets or the dimension of (attractive) invariant manifolds. Understandably, the references have bias on discrete dynamics. Acknowledgements: The idea for this monograph appeared for the first time while hiking with my then-girlfriend Kristine in Yosemite National Park midway between Tenaya Lake and Clouds Rest. During the following years she continuously supported me – on a personal and mathematical level. Moreover, without my family this would have not been possible. This work started as part of a grant by the Deutsche Forschungsgemeinschaft on the topic Inertial Manifolds of Nonautonomous Evolutionary Equations: Smoothness and Discretization. The appropriate excellent research environment was the School of Mathematics at the University of Minnesota in Minneapolis, USA. In particular, I am grateful to my advisor there, Professor George R. Sell, for many stimulating discussions while being in Minneapolis, as well as during various later occasions. Prof. Dr. Peter E. Kloeden was always an inspiration, fortunately interested in things I am doing and helpful in many ways. After my return to Germany, Prof. Dr. Rupert Lasser guaranteed financial background and scientific freedom at the Munich University of Technology, and equally important, always served as an interested partner discussing various dynamical issues. For mathematical input, feedback, remarks, and criticism I want to thank Prof. Robert J. Sacker. Locally, here in Munich I am indebted to Prof. Dr. Johannes M¨uller for interest and discussions. Jos´e Langa and Peter E. Kloeden kept me up-to-date with their respective monographs. Finally, this book descents from a circle of questions related to the working group of the late Prof. Dr. Bernd Aulbach at the University of Augsburg, Germany. Among this group, I particularly like to thank PD Dr. Stefan Hilger, Prof. Dr. Thomas Wanner, Prof. Dr. Stefan Siegmund and Dr. Martin Rasmussen. Besides being a friend and loyal colleague, Dr. Martin Rasmussen was very helpful, suggestive and critical in proofreading. Beyond that I am grateful to Dr. Kristine Ey for reading parts of the manuscript. Munich, July 23, 2010
Christian P¨otzsche
Contents
Preface . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . vii Notation .. . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . xvii 1 Nonautonomous Dynamical Systems . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . 1.1 Nonautonomous Sets and 2-Parameter Semigroups . .. . . . . . . . . . . . . . . . 1.2 Invariant and Limit Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . 1.3 Attractors and Global Attractors . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . 1.4 Periodic and Autonomous 2-Parameter Semigroups ... . . . . . . . . . . . . . . . 1.5 Applications: Discretized Semiflows . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . 1.5.1 Retarded Functional Differential Equations.. .. . . . . . . . . . . . . . . . 1.5.2 Abstract Evolution Equations .. . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . 1.5.3 Reaction-Diffusion Equations . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . 1.5.4 Doubly Nonlinear Equations.. . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . 1.6 Remarks .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . .
1 2 5 16 21 24 24 26 29 31 33
2 Nonautonomous Difference Equations . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . 2.1 Basics and Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . 2.2 Existence and Boundedness of Solutions .. . . . . . . . . . . . .. . . . . . . . . . . . . . . . 2.3 Difference Equations and 2-Parameter Semigroups . .. . . . . . . . . . . . . . . . 2.4 Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . 2.5 Periodic and Autonomous Difference Equations.. . . . .. . . . . . . . . . . . . . . . 2.6 Applications.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . 2.6.1 Fully Discretized Functional Differential Equations . . . . . . . . . 2.6.2 Time-Discretized Abstract Evolution Equations .. . . . . . . . . . . . . 2.6.3 Fully Discretized Reaction-Diffusion Equations . . . . . . . . . . . . . 2.6.4 Fully Discretized Finite Difference Ginzburg–Landau Equation .. . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . 2.6.5 Time-Discretized Doubly Nonlinear Equations.. . . . . . . . . . . . . . 2.7 Remarks .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . .
37 39 44 51 60 68 70 71 73 75 80 88 89
3 Linear Difference Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . 95 3.1 Basics. .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . 96 3.2 Periodic Linear Equations .. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . 108 xiii
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3.3 3.4 3.5 3.6 3.7
Invariant Splittings and Exponential Growth .. . . . . . . . .. . . . . . . . . . . . . . . . Dichotomies and Splittings. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . Admissibility .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . Roughness.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . Applications.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . 3.7.1 Discretized Linear Functional Differential Equations . . . . . . . . 3.7.2 Time-Discretized Linear Abstract Evolution Equations .. . . . . 3.7.3 Time-Discretized Linear Parabolic Equations . . . . . . . . . . . . . . . . 3.7.4 Fully Discretized Diffusion Equations . . . . . . . .. . . . . . . . . . . . . . . . Remarks .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . .
114 128 151 161 170 170 173 174 175 180
4 Invariant Fiber Bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . 4.1 Semilinear Difference Equations.. . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . 4.2 Existence of Invariant Fiber Bundles . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . 4.3 Invariant Foliations and Asymptotic Phase . . . . . . . . . . . .. . . . . . . . . . . . . . . . 4.4 Smoothness of Fiber Bundles and Foliations. . . . . . . . . .. . . . . . . . . . . . . . . . 4.5 Normal Hyperbolicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . 4.6 Pseudo-stable and Pseudo-unstable Fiber Bundles. . . .. . . . . . . . . . . . . . . . 4.7 Inertial Fiber Bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . 4.8 Approximation of Invariant Fiber Bundles . . . . . . . . . . . .. . . . . . . . . . . . . . . . 4.9 Applications.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . 4.9.1 Discretized Functional Differential Equations . . . . . . . . . . . . . . . . 4.9.2 Time-Discretized Abstract Evolution Equations .. . . . . . . . . . . . . 4.9.3 Time-Discretized Parabolic Evolution Equations .. . . . . . . . . . . . 4.9.4 Fully Discretized Reaction-Diffusion Equations . . . . . . . . . . . . . 4.9.5 Fully Discretized Finite Difference Ginzburg–Landau Equation .. . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . 4.10 Remarks .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . .
187 189 194 214 230 250 256 274 279 287 287 289 293 295
5 Linearization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . 5.1 Topological Conjugation and Decoupling .. . . . . . . . . . . .. . . . . . . . . . . . . . . . 5.2 Generalized Hartman–Grobman Theorem . . . . . . . . . . . .. . . . . . . . . . . . . . . . 5.3 Solution Conjugation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . 5.4 Applications.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . 5.4.1 Time-Discretized Abstract Evolution Equations .. . . . . . . . . . . . . 5.4.2 Time-Discretized Parabolic Evolution Equations .. . . . . . . . . . . . 5.5 Remarks .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . .
317 318 326 334 338 339 340 341
A
345 345 348 350
3.8
Discrete Inequalities.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . A.1 Generalized Exponential Function .. . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . A.2 Gronwall Inequalities .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . A.3 Remarks .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . .
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B Fixed Point and Inversion Theorems .. . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . B.1 Contractive Mappings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . B.2 Compact Mappings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . B.3 Global Inverse Function Theorems . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . B.4 Remarks .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . .
351 352 355 356 360
C Smooth Mappings and Extensions . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . C.1 Differentiability .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . C.2 Smooth Norms and Extensions . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . C.3 Remarks .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . .
363 363 364 371
References .. . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . 373 Index . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . 393
Notation
Sets, mappings, numbers: Let X, Y be sets. The empty set is denoted by the symbol ∅, 2X is the power set (the set of all subsets) of X, and if X is finite, #X is its cardinality. For a mapping f : X → Y the restriction to a subset A ⊆ X is denoted by f |A . If Y ⊆ X we define the iterates f n : X → Y of f , where n is a nonnegative integer, recursively as f 0 (x) := x,
f n (x) := f ◦ f n−1 (x)
for all n > 0, x ∈ X.
As frequently used in the mathematical literature, we write N Z R C C∞ F
for the natural numbers {1, 2, 3, . . .}, for the integers {0, ±1, ±2, . . .}, for the field of real numbers, for the field of complex numbers, for the extended complex plane, and for one of the fields R or C.
In the latter sets, |·| is the real or complex absolute value. The complex-conjugate of ζ ∈ C is denoted by ζ; we write ζ for its real and ζ for its imaginary part. For given k, n ∈ N we write k N ∪ . . . ∪ Nk = {1, . . . , n} , Pk (n) := (N1 , . . . , Nk ) ∈ 2{1,...,n} 1 Ni ∩ Nj = ∅ for i = j for the set of all partitions of {1,. . . , n} with length k and indexing a nonempty subset Ni ⊆ {1, . . . , n} by Ni = ni1 , . . . , ni#Ni , we define ⎧ ⎨
⎫ #Ni ≥ 1 for 1 ≤ i ≤ k, ⎬ Pk< (n) := (N1 , . . . , Nk ) ∈ Pk (n) ni1 < . . . < ni#Ni for 1 ≤ i ≤ k, ⎩ ⎭ max N < max N i i+1 for 1 ≤ i < k as the set of ordered partitions of {1, . . . , n} with length k. Note that a tuple from a partition can possess the empty set as component, whereas a tuple from an ordered partition has nonempty components. xvii
xviii
Notation
Topological spaces: We require the following definitions and terminology for a subset A ⊆ X of a topological space X. The interior of A is intX A, the closure is clX A and bdX A the boundary. If no confusion can arise, we drop the notational dependence on the space X and simply write int A, cl A or bd A, respectively. We follow this convention in our further notation. Metric spaces: For a metric space X, its metric is denoted by dX (or simply d); we sometimes write (X, dX ) to emphasize the dependence on the metric dX . For x ∈ X and r > 0 we introduce the open, pointed and closed balls in X, resp., Br (x, X) := {y ∈ X : dX (x, y) < r} , ¯r (x, X) := {y ∈ X : dX (x, y) ≤ r} ; B
B˙ r (x, X) := Br (x, X) \ {x} ,
¯ r (x), respectively. in absence of confusion we write Br (x), B˙ r (x) or B The diameter of a set A ⊆ X is diamX A := supa,b∈A dX (a, b), the distance of a point a ∈ A from B ⊆ X is defined by distX (a, B) := inf b∈B dX (a, b) and the Hausdorff separation of A and B is given by hX (A, B) := supa∈A distX (a, B); with a further set C ⊆ A one has the monotonicity relations hX (C, B) ≤ hX (A, B),
hX (B, A) ≤ hX (B, C).
If X1 , . . . , Xn , n ∈ N, are metric spaces, then their Cartesian product is always equipped with the product metric dX1 ×...×Xn ((x1 , . . . , xn ), (y1 , . . . , yn )) :=
×n
max
k∈{1,...,n}
dXk (xk , yk )
(0.0a)
×nk=1 Xk (0.0b)
for (x1 , . . . , xn ), (y1 , . . . , yn ) ∈ k=1 Xk . In [119, p. 72, (3.20.16)] it is shown n that topological properties of X1 , . . . , Xn carry over to the product k=1 Xk . Let Y be another metric space. Then a mapping f : X → Y is called bounded, if it maps bounded subsets of X into bounded sets in Y , and completely continuous, if it is continuous and compact (f maps bounded subsets of X into relatively compact subsets of Y ). We say a subspace X0 ⊆ X is continuously embedded into X, if the embedding operator J : X0 → X, Jx := x, is continuous, and we write X0 → X. Similarly, X0 is compactly embedded into X, in symbols X0 X, if J : X0 → X is compact. A Lipeomorphism is a bijective Lipschitz mapping with Lipschitz inverse. For a mapping F : X × Z → Y , where Z is a nonempty set, we define the Lipschitz constants
×
lip F (·, z) := inf {L ≥ 0 : dY (F (x, z), F (¯ x, z)) ≤ LdX (x, x ¯) for all x, x ¯ ∈ X} , lip1 F := sup lip F (·, z), z∈Z
supplemented with inf ∅ = ∞. If a set Z has a metric structure, one defines lip2 F analogously and proceeds correspondingly, if F depends on more variables.
Notation
xix
Linear spaces and mappings: Linear spaces X and Y of this work are always real (F = R) or complex (F = C). An n-tuple (x, . . . , x) ∈ X n , n ∈ N, of the same vector x ∈ X is abbreviated by xn . The space of linear mappings between X and Y is Hom(X, Y ), Hom(X) := Hom(X, X) and Iso(X, Y ) stands for the subset of linear bijections from X to Y ; Iso(X) := Iso(X, X). The identity on X is denoted by IX . For S ∈ Hom(X, Y ), we define kernel (or nullspace) and image (or range) of S, resp., ker S := {x ∈ X : Sx = 0} ,
im S := {Sx ∈ Y : x ∈ X} .
The point spectrum of S, T ∈ Hom(X, Y ) reads as σp (S, T ) := {λ ∈ C : ker(S − λT ) = {0}} and in the special case X = Y and T = IX we abbreviate σp (S) := σ(S, IX ). Suppose that X1 , . . . , Xn , n ∈ N, stand for further linear spaces. The values n of an n-linear mapping T : Xk → Y are abbreviated as T x1 · · · xn := k=1 T (x1 , . . . , xn ) and we write Hom(X1 , . . . , Xn ; Y ) for the linear space of all such mappings; in addition, Homn (X, Y ) := Hom(X, . . . , X; Y ). If all Xk are subspaces of X, then X1 ⊕ . . . ⊕ Xn is their direct sum. Topological linear spaces: Let X, Y be topological linear spaces. Then coX A is the convex hull of A ⊆ X, and we use the notation coX A for the closure clX coX A. We write L(X, Y ) for the linear space of continuous maps in Hom(X, Y ), L(X) := L(X, X) and X := L(X, F) for the dual space of X; the corresponding duality pairing is x, x := x (x). Given a homomorphism T : X → Y , its dual mapping T : Y → X is defined via T x, x = x, T x for all x ∈ X, x ∈ Y . The space of continuous isomorphisms T : X → Y with inverse T −1 ∈ L(Y, X) is denoted by GL(X, Y ). Similarly, L(X1 , . . . , Xn ; Y ) is the linear subspace of Hom(X1 , . . . , Xn ; Y ) consisting of continuous maps; Ln (X, Y ) := L(X, . . . , X; Y ). It is convenient to write L0 (X, Y ) := Y . The spectrum and spectral radius of S, T ∈ L(X, Y ), resp., are given by
×
σ(S, T ) := {λ ∈ C : S − λT ∈ GL(X, Y )} ,
ρ(S, T ) :=
sup
λ∈σ(S,T )
|λ| ;
for X = Y and T = IX we abbreviate σ(S) := σ(S, IX ) and ρ(S) := ρ(S, IX ). Sometimes one is interested in the absolute value of the spectral points, and it is convenient to write |σ(S, T )| := {r ∈ [0, ∞) : r = |λ| for some λ ∈ σ(S, T )} . Normed spaces: For normed linear spaces X, Y , their norm is denoted by ·X , ·Y , respectively, or simply ·. Then L(X, Y ) is a normed space w.r.t. the norm SL(X,Y ) := supxX =1 SxY . Moreover, L(X1 , . . . , Xn ; Y ) is a normed space canonically equipped with the norm T L(X1 ,...,Xn ;Y ) :=
sup
x1 X ,...,xn Xn ≤1 1
T x1 · · · xn Y ;
xx
Notation
it is a Banach space, if Y is complete (cf. [295, pp. 67–68]). Higher order derivatives in normed spaces lead to spaces L(X1 , L(X2 , . . . , L(Xn , Y ) . . .)) which are normisomorphic to L(X1 , . . . , Xn ; Y ) by virtue of [(x1 , . . . , xn ) → T x1 · · · xn ] → x1 → [x2 → . . . [xn → T x1 · · · xn ] . . .] (cf. [295, p. 68]). Thus, we can identify the two norm-isomorphic linear spaces L(X1 , . . . , Xn ; Y ) and L(X1 , L(X2 , . . . , L(Xn , Y ) . . .)) from now on. When all spaces X1 , . . . , Xn are equal to X we write Ln (X; Y ) := Ln (X, . . . , X; Y )). With a closed subspace X1 ⊆ X and P ∈ L(X1 ; X) we define TP ∈ Ln (X1 ; Y ), TP x1 · · · xn := T (P x1 , . . . , P x2 ) for all x1 , . . . , xn ∈ X1
(0.0c)
and obtain the norm estimate (cf. [295, p. 68]) n
TP ≤ P T
for all n ∈ N.
(0.0d)
Throughout the whole book, differentiability is always understood in the sense of Fr´echet differentiability (cf. [295, p. 333]). For Banach spaces X, Y and an open A ⊆ X, the mapping f : A → Y is said to be of class C m , m ∈ N, if it is m-times continuously differentiable. The derivative of a differentiable function f is denoted by Df : A → L(X, Y ). For mappings f : (x1 , . . . , xn ) → f (x1 , . . . , xn ) depending differentiably on several variables, Dk f is the partial derivative w.r.t. the k-th argument, k ∈ {1, . . . , n}. Inductively one defines higher order derivatives Dl f : A → Ll (X, Y ) and Dkl f . Inner product spaces: On an inner product space X we denote the inner product by ·, ·X or, if thereis no danger of confusion, by ·, ·. Then X is canonically normed by xX := x, xX and one has the inequality 2
2
±2 x, yX ≤ xX + yX
for all x, y ∈ X.
(0.0e)
Complexification: In order to use spectral theory it is often handy to work in a complex linear space. This can be achieved using the subsequent considerations. For a real linear space X the complexification XC is the cartesian product X × X equipped with the component-wise addition and the scalar multiplication ζ(x, y) := (ζx − ζy, ζx + ζy) for all ζ ∈ C, x, y ∈ X. Thus, XC is a linear space over C. We identify X with the set X × {0} ⊆ XC . If a real linear space Y is given, in addition, the complexification of S ∈ Hom(X, Y ) is the linear mapping SC ∈ Hom(XC , YC ) defined by SC (x, y) := (Sx, Sy) for x, y ∈ X. With normed spaces X, Y one defines the norm on XC by (x, y)XC :=
sup cos θ x + sin θ yX
θ∈[0,2π)
for all x, y ∈ X
Notation
xxi
and obtains for S, T ∈ L(X, Y ) (cf. [9, p. 161ff]) that SC L(XC ,YC ) = SL(X,Y ) , σ(SC , TC ) = σ(S, T ), ρ(SC , TC ) = ρ(S, T ). Function spaces: With X, Y being topological spaces, C(X, Y ) is the set of continuous functions between X and Y . Moreover, for Banach spaces X, Y and an appropriate set Ω ⊆ X, C m (Ω, Y ) denotes the set of m-times continuously differentiable functions; in case Y = F we briefly write C m (Ω) and proceed similarly with function spaces defined below. Cbm (Ω, Y ) consists of C m -functions which are bounded together with its derivatives up to order m. Lebesgue spaces: Let p ∈ [1, ∞) and we assume (Ω, Σ, μ) is a positive measure space. We equip the collection Lp (Ω) of μ-measurable functions u : Ω → F satisfying Ω |u(x)|p dμ(x) < ∞ with its canonical norm
p1 |u(x)| dμ(x) , p
uLp (Ω) :=
Ω
and u, vL2 (Ω) := Ω u(x)v(x)dμ(x) makes L2 (Ω) a Hilbert space. Sobolev spaces: The collection of all functions u : Ω → F whose weak derivatives dα u exist and satisfy dα u ∈ Lp (Ω) for multi-indices α, |α| ≤ m, is denoted by W m,p (Ω). It becomes a separable Banach space w.r.t. the norm uW m,p (Ω) :=
p1
p
|d u(x)| dμ(x) α
,
Ω |α|≤m
and u, vH m (Ω) := Ω |α|≤m u(x)v(x)dμ(x) makes H m (Ω) := W m,2 (Ω) a Hilbert space. Finally, we define (cf. [2]) W0m,p (Ω) := clW m,p (Ω) C0∞ (Ω),
H0m (Ω) := W0m,2 (Ω).
Miscellaneous: Following an established convention, “empty” sums and products are defined as follows: l
xn := 0,
n=k
l
xn := 1
n=k
for k > l and elements xk , . . . , xl of an additive resp., multiplicative monoid. Notations from this monograph: In this book, we are concerned with functions defined on the integers and need an appropriate notation. A discrete interval I is the intersection of a real interval with Z and, in particular, Z+ κ := {k ∈ Z : κ ≤ k} , + I+ κ := I ∩ Zκ ,
Z− κ := {k ∈ Z : k ≤ κ} , − I− κ := I ∩ Zκ
for all κ ∈ Z;
xxii
Notation
we write IZ := I ∩ Z for the intersection of the integers with a real interval. Frequently it is convenient to have the notation I :=
I \ {max I}
if I is bounded above,
I
else.
Let F (A, B) be an expression, e.g., a term, a formula, an equation, an inequality or a logical statement depending on “variables” A and B. Whenever convenient we use the space saving abbreviation F (A± , B ∓ ) either for F (A+ , B − ) or F (A− , B + ). Chapter 1 Nonautonomous sets, p. 2 k-Fiber of a nonautonomous set S, p. 2 Union of fibers, p. 2 p. 3 ε-Neighborhood of S, p. 3 Forward shift, p. 3 2-Parameter (semi-)group, p. 3 Generator of a 2-parameter (semi-)group, p. 4 Truncated forward orbit, p. 6 ω-Limit set, p. 8 Chapter 2 ϕ(k; κ, ξ) General (forward, backward) solution, p. 51 φ(k, p), φ(k, p) Row or column notation, p. 54 ± χ± Upper and lower characteristic exponents, p. 61 u (ψ), χl (ψ) ± ± λu (φ, ξ), λl (φ, ξ) Upper and lower characteristic exponents, p. 61 Cr,N Piecewise affine continuous functions over [−r, 0], p. 71 2 LN Discrete Lebesgue space, p. 75, p. 178 Chapter 3 LJ (g) Solution space with inhomogeneity g, p. 97 Φ(k, κ) Evolution operator, p. 98 Generator of a linear 2-parameter (semi-)group, p. 99 Φˆk U0+ Super-stable vector bundle, p. 99 + ΣL (A, B) Forward Lyapunov spectrum, p. 107 P, Q Complementary projectors, p. 115 P, Q Invariant vector bundles, p. 115 n Pm , Qnm Complementary projectors, p. 121 n Pm , Qnm Invariant vector bundles, p. 121 ± Xκ,c c± -Bounded sequences, p. 124 ± ·κ,c (Semi-)norm on X± κ,c , p. 126 Xc,d (c, d)-Bounded sequences, p. 124 ·κ,c,d Norm on Xc,d, p. 126 S, X , . . . S(k) X, Y, . ⎪ .. ⎪ ⎪ ⎪ ⎪ ⎪S(k)⎪ ⎪ Bε (S) φ ϕ(k, κ) ϕˆk γBm ωB
Notation
xxiii
Xc ·κ,c X± κ,c,B ·± κ,c,B Xc,d,B ·κ,c,d,B Xc,B ·κ,c,B Xm,± κ,c Σf (A, B) Σ(A, B) j − ∼+ n , ∼n , ∼i + − [·]n , [·]n , [·]ji Uij Us , Ucs Uu , Ucu Uc Δh δh± Dh , Dh+ 2s HN ΣB (A, B) Γ¯i Wi± Wij j − ∼+ i , ∼i , ∼i + − [·]i , [·]i , [·]ji Vi± (ξ) πi± Wφ±∗ Ws , Wcs Wu , Wcu Wc X± κ,c (K) Tk Ψ a < b, a ≤ b ab
c-Bounded sequences, p. 124 Norm on Xc , p. 126 B-weighted c± -bounded sequences, p. 124 (Semi-)norm on X± κ,c,B , p. 126 B-weighted (c, d)-bounded sequences, p. 124 Norm on Xc,d,B , p. 126 B-weighted c-bounded sequences, p. 124 Norm on Xc,B , p. 126 c± -Bounded sequences in Lm (Xκ , Xk ), p. 127 Forward dichotomy spectrum, p. 130 Dichotomy spectrum, p. 130 Equivalence relations, p. 138, p. 139 and p. 139 Equivalence classes, p. 138, p. 139 and p. 139 Intersection of invariant vector bundles, p. 123 Stable and center-stable vector bundle, p. 138 Unstable and center-unstable vector bundle, p. 139 Center vector bundle, p. 139 Discrete Laplacian, p. 176 Difference operators, p. 176, p. 177 Forward difference operator, p. 178 Discrete Sobolev space, p. 179 Bohl spectrum, p. 182 Chapter 4 Interval for growth rates, p. 195 Invariant fiber bundles, p. 200 Intersections of invariant fiber bundles, p. 208 Equivalence relations, p. 201, p. 202 and p. 209 Equivalence classes, p. 201, p. 202 and p. 209 Invariant fibers, p. 221 Asymptotic phases, p. 225 Stable resp. unstable set of a solution φ∗ , p. 257 Stable and center-stable fiber bundle, p. 260 Unstable and center-unstable fiber bundle, p. 260 Center fiber bundle, p. 266 Finite sequences, p. 280 Chapter 5 Topological conjugation, p. 320 Solution conjugation, p. 334 Appendix Order relations for sequences, p. 345 Uniform order relation for sequences, p. 345
xxiv
Notation
b − a a rX (x) Fρ χρ dar f
Point-wise infimum between sequences, p. 345 Point-wise supremum of a sequence a, p. 345 Radial retraction on normed space X, p. 364 Lipschitz resp. C m -extension of F , p. 365, p. 370 C m -cut-off function, p. 369 Darbo constant of f , p. 353
Chapter 1
Nonautonomous Dynamical Systems
Structures are the weapons of the mathematician. Bourbaki
An understanding of the asymptotic behavior of dynamical systems is probably one of the most relevant problems in sciences based on mathematical modeling. In our framework, these dynamical systems are discrete beforehand, or have to be discretized in order to simulate them numerically. Moreover, they are nonautonomous and therefore crucially differ from the classical autonomous case, since the initial time of a nonautonomous dynamical process is as important as the elapsed time since starting. This first chapter contains fundamental preliminaries and notions for all our further analysis. In particular, we investigate basic properties of attractors for general nonautonomous problems and provide a rigorous approach to certain “folklore” results in this context. Throughout, our approach is axiomatic in nature, in the sense that applications and examples are postponed at least to the end of the present and each following chapter: •
Our first section introduces elementary concepts to describe the behavior of nonautonomous dynamical systems, namely 2-parameter semigroups; we point out that their continuity is not assumed beforehand. Additionally, for such problems it is proposed to work with state spaces varying in time. This degree of generality is motivated from possible applications as time-adaptive spatial discretizations of evolutionary partial differential equations (PDEs, for short), or the analysis of dynamical systems restricted to invariant nonautonomous sets such as global attractors. • Invariant and ω-limit sets are an essential tool for a qualitative analysis. We show that attractive invariant sets are topologically connected and introduce three compactness notions in order to deduce properties of limit sets. They are sufficiently flexible to capture temporal discretizations of nonautonomous flows generated by reaction-diffusion equations (on bounded domains; i.e., compactness), retarded functional differential equations (with bounded delays, i.e., eventual compactness), and problems on unbounded domains (using asymptotic compactness). A sufficient criterion for asymptotic compactness is given in terms of set-contracting 2-parameter semigroups. C. P¨otzsche, Geometric Theory of Discrete Nonautonomous Dynamical Systems, Lecture Notes in Mathematics 2002, DOI 10.1007/978-3-642-14258-1 1, c Springer-Verlag Berlin Heidelberg 2010
1
2
1 Nonautonomous Dynamical Systems
•
In Sect. 1.3 we investigate the properties of (global) attractors under ambient dissipativity properties. They can be shown to consist of bounded complete motions, and we characterize them as ω-limit sets under the existence of an absorbing set. Furthermore, we address the upper-semicontinuous dependence of attractors on parameters – a crucial property for discretization. • Finally, in Sect. 1.4 we demonstrate that the above notions reduce to the wellknown concepts for time-periodic or autonomous problems. Moreover, based on compactness properties of the period map we provide two existence results for periodic motions. • As illustration, our results are applied to an appropriate nonautonomous counterpart of the time-h-map, h > 0, for flows of various time-dependent evolutionary differential equations – including retarded functional differential equations (with bounded delay) and in particular ordinary differential equations, reactiondiffusion equations on bounded domains, and doubly nonlinear parabolic PDEs. Throughout the chapter, let I be a discrete interval and (Xk )k∈I be a sequence of nonempty sets.
1.1 Nonautonomous Sets and 2-Parameter Semigroups We consistently use the notation X :=
Xk ,
X := {(k, x) ∈ I × X : x ∈ Xk }
k∈I
and a subset S ⊆ I × X is called nonautonomous set, if the k-fibers S(k) := {x ∈ X : (k, x) ∈ S} satisfy S(k) ⊆ Xk for all k ∈ I. As a general philosophy, calligraphic letters (S, X and so on) denote nonautonomous sets, while fractured letters (X, Y and so forth) stand for the union of a set sequence. The cartesian product of two nonautonomous sets S1 , S2 ⊆ I × X is defined to be the following set of triples S1 × S2 := {(k, x1 , x2 ) ∈ I × X × X : x1 ∈ S1 (k), x2 ∈ S2 (k)} . Accordingly, inclusions, intersections or unions of nonautonomous sets are defined fiber-wise. We conveniently use the phrase “X consists of topological (metric, normed, etc.) spaces,” if each fiber X (k), k ∈ I, is a corresponding space. If each Xk is a topological space, then S is said to be open, closed, compact (or another topological property), if every fiber S(k) is open, closed or compact, resp., w.r.t. the topology given on Xk . We write cl S := {(k, x) ∈ X : x ∈ clXk S(k)} for the closure of S and proceed analogously with the interior int S or
1.1 Nonautonomous Sets and 2-Parameter Semigroups
3
the boundary bd S. A nonautonomous set S in metric spaces Xk is called bounded, if each fiber S(k), k ∈ I, is bounded. In metric linear spaces Xk we denote S as uniformly bounded, if there exists a R > 0 so that one has ⎪ ⎪ ⎪ ⎪ ⎪ ⎪S(k)⎪ ⎪ := sup dXk (x, 0) ≤ R
for all k ∈ I.
x∈S(k)
For finite discrete intervals I the notions of a bounded and a uniformly bounded nonautonomous set coincide. A neighborhood of S is a nonautonomous set containing a so-called ε-neighborhood Bε (S) := {(κ, ξ) ∈ X : distXκ (ξ, S(κ)) < ε} with some given ε > 0; in metric linear spaces we abbreviate Bρ := Bρ (0). A vector bundle1 is a nonautonomous set consisting of linear spaces. For vector bundles X1 , X2 ⊆ X we define X1 + X2 := {(k, x) ∈ X : x ∈ X1 (k) + X2 (k)} , X1 ⊕ X2 := {(k, x) ∈ X : x ∈ X1 (k) ⊕ X2 (k)} and the latter expression is denoted as Whitney sum of X1 and X2 . Taking a dynamical systems point of view, the variables k and x are usually interpreted as “time” and “state”, respectively; the nonautonomous set X is the extended state space and each fiber X (k) = Xk , k ∈ I, is a state space. Whenever convenient for us, we identify a sequence φ : I → X with its graph, i.e., the nonautonomous set {(κ, φ(κ)) ∈ X : κ ∈ I}. Accordingly, we say φ is a sequence in X , if φ ⊆ X or equivalently φ(k) ∈ Xk holds for all k ∈ I. The shifted sequence φ : I → X is defined by φ (k) := φ(k + 1) ∈ Xk+1
for all k ∈ I .
Our first definition introduces a basic and quite general concept to describe temporally evolving dynamical processes. For this, let S ⊆ X be nonempty. Definition 1.1.1. A family of mappings ϕ(k, κ) : S(κ) → S(k)
with k, κ ∈ I, κ ≤ k,
(1.1a)
is a 2-parameter semigroup on S, if the following holds for all κ ≤ l ≤ k: (i) ϕ(κ, κ) = IXκ . (ii) One has the semigroup property: ϕ(k, l)ϕ(l, κ) = ϕ(k, κ).
(1.1b)
1 We refer to [1, p. 166, Definition 3.4.1] for the general notion of a vector bundle in differential topology.
4
1 Nonautonomous Dynamical Systems ϕ(l, κ) ϕ(k, l)
S(κ)
S(k)
S(l)
I
ϕ(k, κ) Xκ
Xl
Xk
Fig. 1.1 2-Parameter semigroup property of ϕ(k, κ) : S(κ) → S(k), κ ≤ k
In case each S(k), k ∈ I, is a topological space, we say ϕ is continuous, if every mapping (1.1a) is continuous. In case each S(k), k ∈ I, is a linear space, we say ϕ is linear, if every mapping (1.1a) is linear. In case each S(k), k ∈ I, is a metric space, we say ϕ is bounded, if every mapping (1.1a) is bounded, i.e., maps bounded subsets of S(κ) to bounded subsets of S(k). A dynamical interpretation of the semigroup property (1.1b) is given in Fig. 1.1. Note in relation (1.1b) that for the sake of a convenient notation, we consistently abbreviate ϕ(k, l)ϕ(l, κ) = ϕ(k, l) ◦ ϕ(l, κ). Remark 1.1.2. (1) It might happen that for each κ < k the mapping ϕ(k, κ) is a one-to-one mapping of S(κ) onto S(k). For such a situation, we set ϕ(κ, k) := ϕ(k, κ)−1 : S(k) → S(κ), the semigroup property (1.1b) holds for all k, l, κ ∈ I and is called group property; we speak of a 2-parameter group ϕ(k, κ), κ, k ∈ I, on S. Such a group induces an equivalence relation on S via (k1 , x1 ) ∼ (k2 , x2 ) :⇔ x2 = ϕ(k2 , k1 )x1 with equivalence classes [(κ, ξ)] = {(k, ϕ(k, κ)ξ) ∈ S : k ∈ I} for (κ, ξ) ∈ S. (2) A family of mappings ϕ(k, κ) : S(κ) → S(k) satisfying the conditions of Definition 1.1.1 for k ≤ l ≤ κ (as opposed to κ ≤ l ≤ k) is called backward 2-parameter semigroup. Such objects typically occur when dealing with adjoint equations. Trivially, 2-parameter groups are (backward) 2-parameter semigroups. (3) A continuous 2-parameter semigroup ϕ becomes a continuous 2-parameter group, if each ϕ(k, κ) : S(κ) → S(k), κ < k, is a homeomorphism. (4) In applications one encounters 2-parameter semigroups depending on parameters p from a set P (the parameter space); we indicate this by the notation ϕ(·; p) and speak of a parameter-dependent 2-parameter semigroup. For such objects, continuity means that the mapping (x, p) → ϕ(k, κ; p)x, κ ≤ k, is continuous in the product topology, provided P is a topological space. It is clear that successive compositions of mappings ϕˆk : S(k) → S (k) generate 2-parameter semigroups. We also have the converse:
1.2 Invariant and Limit Sets
5
Proposition 1.1.3. A family of mappings ϕ(k, κ), κ ≤ k, is a 2-parameter semigroup on the nonautonomous set S, if and only if there exists a sequence of mappings ϕˆk : S(k) → S (k), k ∈ I , such that ϕ(k, κ) = ϕˆk−1 ◦ . . . ◦ ϕˆκ
for all κ ≤ k.
We denote (ϕˆk )k∈I as generator of ϕ. Moreover, if S consists of topological spaces, ϕ is continuous, if and only if each ϕˆk is continuous. Proof. The assertion is obvious, if we define ϕˆk := ϕ(k + 1, k) for k ∈ I .
Remark 1.1.4. A 2-parameter semigroup extends to a group, if and only if its generator is such that each ϕˆk : S(k) → S (k), k ∈ I , is bijective. For homeomorphisms ϕˆk : S(k) → S (k) the group becomes continuous.
1.2 Invariant and Limit Sets In this section we introduce a number of notions which are important to describe and understand the asymptotic behavior of 2-parameter semigroups. Such notions include invariant and limit sets. Let (Xk )k∈I denote a sequence of nonempty sets with associated extended state space X , and suppose S ⊆ X is nonempty. Throughout the section we assume that ϕ is a 2-parameter semigroup on S. Definition 1.2.1. Let (ϕ) ˆ k∈I be the generator of ϕ. A nonautonomous set A ⊆ S is called: (i) Forward invariant, if ϕˆk (A(k)) ⊆ A (k) for all k ∈ I . (ii) Backward invariant, if A (k) ⊆ ϕˆk (A(k)) for all k ∈ I . (iii) Invariant, if ϕˆk (A(k)) = A (k) for all k ∈ I . Example 1.2.2. (1) The empty set I × ∅ and S are (forward, backward) invariant. (2) A sequence φ : I → S in S is called complete motion (for ϕ), if one has the relation φ(k) = ϕ(k, κ)φ(κ) for all k ≥ κ. Then the nonautonomous set φ is invariant. Furthermore, suppose (κ, ξ) ∈ S is given. Since we did not assume ϕ(k, l) to be onto, there might not exist a complete motion satisfying φ(κ) = ξ. Moreover, without the assumption that ϕ(k, l) is one-to-one, there can exist more than one complete motion with φ(κ) = ξ. Proposition 1.2.3. A nonautonomous set A ⊆ S is: (a) Forward invariant, if and only if ϕ(k, κ)A(κ) ⊆ A(k) for all κ ≤ k. (b) Backward invariant, if and only if A(k) ⊆ ϕ(k, κ)A(κ) for all κ ≤ k. (c) Invariant, if and only if ϕ(k, κ)A(κ) = A(k) for all κ ≤ k. Proof. The assertion follows inductively from Proposition 1.1.3.
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1 Nonautonomous Dynamical Systems
Example 1.2.4. Let m ∈ Z+ 0 , I unbounded below and A ⊆ S be a nonautonomous m set. The nonautonomous set γA ⊆ S given by the fibers m (k) := γA
ϕ(k, k − n)A(k − n) ⊆ S(k)
for all k ∈ I
n≥m 0 is called m-truncated orbit and γA the orbit of A. One has the embedding 0 , A ⊆ γA
m+1 m γA ⊆ γA
for all m ∈ Z+ 0,
m and it is easily seen using Proposition 1.2.3 that γA is (forward, backward) invariant, if A has the corresponding property.
The next result states that invariant sets consist of complete motions, which is basically due to the fact that 2-parameter semigroups are onto between the fibers of an invariant set. These motions, however, need not to be uniquely determined, since the semigroups are not assumed to be one-to-one. Proposition 1.2.5. The following assertions are equivalent: (a) A nonautonomous set A ⊆ S is invariant. (b) A is forward and backward invariant. (c) For every pair (κ, ξ) ∈ A there exists a complete motion φ : I → S for ϕ such that φ(κ) = ξ and φ ⊆ A. The complete motion φ is uniquely determined, provided every generator ϕˆk : S(k) → S (k) is one-to-one for all k ∈ I .
(1.2a)
Proof. The implication (b)⇒(a) is clear from Definition 1.2.1. (a)⇒(c) Let A be invariant and choose (κ, ξ) ∈ A. For k ≥ κ we define the sequence φ(k) := ϕ(k, κ)ξ and the invariance of A yields φ(k) ∈ A(k). On the other hand, for κ ≥ k we have A(κ) = ϕ(κ, k)A(k) and consequently there exists a sequence xk ∈ A(k) with ξ = ϕ(κ, k)xk . Thus, we define φ(k) := xk for k < κ and φ : I → S is a complete motion with the desired properties. Under (1.2a) the sequence xk is uniquely given. (c)⇒(b) For pairs (κ, ξ) ∈ A there is a complete motion φ : I → S with φ(κ) = ξ in A. Hence, one has (k, ϕ(k, κ)ξ) = (k, ϕ(k, κ)φ(κ)) = (k, φ(k)) ∈ A and thus the inclusion ϕ(k, κ)ξ ∈ A(k) for k ≥ κ. So A is forward invariant. The backward invariance of A follows from ξ = ϕ(κ, k)φ(k) ∈ ϕ(κ, k)A(k) for k ≤ κ. Proposition 1.2.6. Let {Ai }i∈I be a family of nonautonomous sets Ai ⊆ S, where I is an index set: (a) If each Ai , i∈ I, is forward invariant, then also the union i∈I Ai and the intersection i∈I Ai are forward invariant. (b) If each Ai , i ∈ I, is backward invariant, then also i∈I Ai is backward invari ant; moreover, under (1.2a) also i∈I Ai is backward invariant.
1.2 Invariant and Limit Sets
7
Proof. The whole proof is based on the elementary relations ϕˆk
Ai (k)
i∈I
⊆
ϕˆk (Ai (k)),
ϕˆk
i∈I
Ai (k)
=
i∈I
ϕˆk (Ai (k))
i∈I
for all k ∈ I , with equality in the first case, if ϕˆk is one-to-one.
Proposition 1.2.7. Let A, B ⊆ S be nonautonomous sets: (a) If A is forward invariant and B is backward invariant, then B \ A is backward invariant. (b) If A, B are invariant and (1.2a) holds, then B \ A is invariant. Proof. Directly from Definition 1.2.1 one has the relations B (k) \ A (k) ⊆ B (k) \ ϕˆk (A(k)) ⊆ ϕˆk (B(k)) \ ϕk (A(k)) ⊆ ϕˆk (B(k) \ A(k)) for all k ∈ I , with equality, if ϕˆk is one-to-one and A, B are invariant.
We postulate the nonautonomous set S ⊆ X consists of metric spaces. Moreover, which might be surprising at first reading, the discrete interval I is assumed to be unbounded below, i.e., of the form I = Z− κ with some κ ∈ Z or I = Z. Theorem 1.2.8. Suppose that S consists of complete metric spaces, ϕ is continuous and let A ⊆ S be nonempty compact. If A is forward or backward invariant, then there exists a maximal nonempty compact and invariant subset A∗ ⊆ A. Proof. We subdivide the proof into two steps and keep k ∈ I fixed: (I) Let A be forward invariant. Since A is compact, the continuity of ϕ shows that the images ϕ(k, l)A(l), l ≤ k, are compact. Moreover, thanks to (cf. Proposition 1.2.3(a)) (1.1b)
ϕ(k, κ)A(κ) = ϕ(k, l)ϕ(l, κ)A(κ) ⊆ ϕ(k, l)A(l) ⊆ A(k) for all κ ≤ l ≤ k the sets ϕ(k, k − n)A(k − n), n ≥ 0, form a nested family of nonempty compact subsets of A(k).By [465, p. 495, Proposition 11.4] their intersection fiber-wise given by A∗ (k) := n≥0 ϕ(k, k − n)A(k − n) is nonempty and compact as well, and it remains to show that A∗ is also invariant: (⊆) For l ≤ k we choose a ∈ A∗ (l) and get a ∈ ϕ(l, l − n)A(l − n), n ≥ 0. Thus, (1.1b)
ϕ(k, l) {a} ⊆ ϕ(k, l)ϕ(l, l − n)A(l − n) = ϕ(k, l − n)A(l − n) for all n ≥ 0 implies the inclusion ϕ(k, l) {a} ⊆
0≤n
ϕ(k, l − n)A(l − n) ⊆
0≤n
ϕ(k, l − n)A(l − n) = A∗ (k)
8
1 Nonautonomous Dynamical Systems
for all l ≤ k and we deduce the desired relation ϕ(k, l)A∗ (l) ⊆ A∗ (k) for all l ≤ k. (⊇) Given an arbitrary a ∈ A(k) and l ≤ k there exists a sequence ln ∈ Z− l with limn→∞ ln = −∞ with a ∈ ϕ(k, ln )A(ln ) = ϕ(k, l)ϕ(l, ln )A(ln ). Therefore, we can choose bn ∈ ϕ(l, ln )A(ln ) ⊆ A(l) such that ϕ(k, l)bn = a. Since A(l) is compact there exists a convergent subsequence (bnj )j∈N with limit b ∈ A(l). By dist(b, A∗ (l)) ≤ dist(b, bnj ) + dist(ϕ(l, lnj )A∗ (lnj ), A∗ (l)) −−−→ 0 j→∞
we actually deduce b ∈ A∗ (l) and the continuity of ϕ shows the limit relation a = limj→∞ ϕ(k, l)bnj = ϕ(k, l)b. This simply means A∗ (k) ⊆ ϕ(k, l)A∗ (l). (II) For the case of a backward invariant A we refer to [265, Proposition 6]. Now we are interested in the asymptotic behavior of 2-parameter semigroups. Definition 1.2.9. Let A ⊆ S be nonempty. The ω-limit set ωA of A is the nonautonomous set given by the fibers m clXk ϕ(k, k − n)A(k − n) = clXk γA (k) for all k ∈ I. ωA (k) := m≥0
n≥m
m≥0
Remark 1.2.10. (1) Using a set-theoretical terminology, each fiber ωA (k), k ∈ I, is the limit superior for the sequence (ϕ(k, k − n)A(k − n))n≥0 of sets (cf., for instance, [15, p. 16ff, Sect. 1.1]). It thus allows the characterization
ωA = (k, x) ∈ S : lim inf distS(k) (x, ϕ(k, k − n)A(k − n)) = 0 . n→∞
(2) In a dual fashion using pre-images, the α-limit set αA ⊆ S of A can be defined as nonautonomous set given by the fibers clXk ϕ(k + n, k)−1 A(k + n) for all k ∈ I αA (k) := m≥0
n≥m
with ϕ(k + n, k)−1 A(k + n) = {x ∈ S(k) : ϕ(k + n, k)x ∈ A(k + n)}. Without surjectivity assumptions on ϕ(k, l), α-limit sets can be empty. (3) If A is forward invariant (backward invariant resp. invariant), then ωA ⊆ cl A (cl A ⊆ ωA resp. ωA = cl A). Particularly, for a complete motion φ one has φ = ωφ . Proposition 1.2.11. Let A ⊆ S. If a nonautonomous set K ⊆ S fulfills that for every neighborhood U ⊆ S of K there exists an N ≥ 0 with ϕ(k, k − n)A(k − n) ⊆ U(k)
for all n ≥ N, k ∈ I,
then ωA ⊆ cl K. Conversely, if S is compact, then ωA is the smallest closed subset of S satisfying the above property of K. Proof. It suffices to show the assertions for each fiber ωA (k), k ∈ I. With this in mind, the claim directly follows from [15, p. 21, Theorem 1.1.4].
1.2 Invariant and Limit Sets
9
The following characterization is helpful to derive topological and dynamical properties of ωA . Namely, each point of an ω-limit set can be approximated by points on orbits starting in their defining sets A. Lemma 1.2.12. Let A ⊆ S. Then (k, x) ∈ ωA if and only if there exist sequences (kn )n≥0 in Z+ 0 and xn ∈ A(k − kn ) such that limn→∞ kn = ∞ and lim ϕ(k, k − kn )xn = x.
n→∞
Proof. Let k ∈ I. We have to show two directions: N (⇒) Assume first that x ∈ ωA (k). Then for all N ≥ 0 one has x ∈ cl γA (k) N and there exist sequences xN in γ (k) ⊆ S(k) converging to x as k → ∞. This A k means, for each ε > 0, N ≥ 0 there exists an integer K = K(ε, N ) > 0 such that 1 d(x, xN k ) < ε for all k ≥ K. In particular, choosing ε = m for some m ∈ N and 1 setting ym := xm K(1/m,m) this yields d(x, ym ) ≤ m and, thus, x = limm→∞ ym . Now one has m y m ∈ γA (k) = ϕ(k, k − n)A(k − n) for all m ∈ N n≥m
and there exist sequences km ≥ m in N and xm ∈ A(k − km ) such that one has ym = ϕ(k, k −km )xm . Thus, limm→∞ ϕ(k, k −km )xm = limm→∞ ym = x holds and km ≥ m → ∞ for m → ∞. (⇐) Conversely, let x ∈ S(k) be the limit of a sequence ϕ(k, k − kn )xn as above. By assumption, for every N ≥ 0 there exists an m ∈ N such that km ≥ N N N and ϕ(k, k − kn )xn ∈ ϕ(k, k − kn )A(k − kn ) ⊆ γA (k) and x ∈ cl γA (k). Since N N was arbitrary, we deduce x ∈ N ≥0 cl γA (k) = ωA (k). Proposition 1.2.13. Let A ⊆ S. If {Ai }i∈I is a family of nonautonomous sets with Ai ⊆ S, where I is a finite index set, then the following holds: (a) For subsets B ⊆ A one has ωB ⊆ ωA . (b) ω i∈I Ai = i∈I ωAi and ω i∈I Ai ⊆ i∈I ωAi . Proof. Let k ∈ I and it suffices to prove the assertion for each k-fiber. (a) The claim readily follows from Lemma 1.2.12, since for (k, x) ∈ ωB the corresponding sequence xn ∈ B(k − kn ) also satisfies xn ∈ A(k − kn ). (b) One clearly has the relations ϕ(k, k − n)
Ai (k − n) =
i∈I
ϕ(k, k − n)
i∈I
ϕ(k, k − n)Ai (k − n),
i∈I
Ai (k − n) ⊆
ϕ(k, k − n)Ai (k − n) for all n ≥ 0;
i∈I
since ωAi (k), i ∈ I, is the limit superior of the sequence (ϕ(k, k−n)Ai (k−n))n≥0 of sets, the claim follows from [15, p. 27, Proposition 1.2.1].
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1 Nonautonomous Dynamical Systems
Theorem 1.2.14. For every A ⊆ S the ω-limit set ωA is closed. Moreover, if ϕ is continuous, then ωA is forward invariant. Proof. Let κ ∈ I. As intersection of closed sets, the fibers ωA (κ) are closed. Now assume ϕ is continuous and k ∈ I+ κ . In order to show that ωA is forward invariant, we pick (κ, y) ∈ ωA and show the inclusion ϕ(k, κ)y ∈ ωA (k). From y ∈ ωA (κ) we know by Lemma 1.2.12 that there exist sequences kn → ∞, xn ∈ A(κ − kn ) such that y = limn→∞ ϕ(κ, κ − kn )xn . By continuity of ϕ, (1.1b)
ϕ(k, κ − kn )xn = ϕ(k, κ)ϕ(κ, κ − kn )xn −−−−→ ϕ(k, κ)y n→∞
and Lemma 1.2.12 implies ϕ(k, κ)y ∈ ωA (k).
For the remainder of this section we suppose that Bˆ is a family of nonempty nonautonomous subsets of S. The following attraction concept for nonautonomous sets A essentially means that the fibers A(k), k ∈ I, attract particular nonautonomous sets from Bˆ coming from −∞. ˆ Definition 1.2.15. A nonautonomous set A ⊆ S is called B-attracting, if lim hXk (ϕ(k, k − n)B(k − n), A(k)) = 0 for all k ∈ I, B ∈ Bˆ
n→∞
(1.2b)
and we denote the family Bˆ as attraction universe. Remark 1.2.16. From elementary properties of the Hausdorff separation one can deduce the following properties of attracting sets: ˆ ˆ (1) Supersets of B-attracting sets are B-attracting (cf. (0.0a)). ˆ ˆ (2) Finite unions of B-attracting sets are B-attracting. ˆ a B-attracting ˆ (3) Given B1 , B2 ∈ B, set is also Bˆ ∪ {B1 ∪ B2 }-attracting. At first glance, the above attraction concept seems counter-intuitive, since it not necessarily implies the familiar forward convergence: Example 1.2.17. Suppose S = Z × R and define the 2-parameter semigroup ⎧ k−κ ⎪ for 0 ≤ κ ≤ k, ⎪ ⎨α+ −κ k ϕ(k, κ) = α+ α− for κ ≤ 0 ≤ k, ⎪ ⎪ ⎩αk−κ for κ ≤ k ≤ 0 −
with α− , α+ ∈ R satisfying 0 < |α− | < 1 < |α+ |. Obviously, the nonautonomous set A := Z × {0} is invariant and due to lim dist(ϕ(k, κ)ξ, A(k)) = 0
k→∞
⇔
ξ=0
for all κ ∈ Z
we have no pointwise forward convergence towards A. Nevertheless, A attracts all uniformly bounded subsets of S. In order to show this, suppose B ⊆ S is uniformly bounded with B(k) ⊆ [−R, R], k ∈ Z, for some R > 0. The claim follows from
1.2 Invariant and Limit Sets
11
0 ≤ h(ϕ(k, k − n)B(k − n), A(k − n)) ≤ R |ϕ(k, k − n)| ⎧ k ⎨ α+ for k ≥ 0, α− n ≤ R |α− | −−−−→ 0 for all k ∈ Z. n→∞ ⎩1 for k < 0 The motions of ϕ are illustrated in Fig. 2.1 (bottom, left). ˆ Proposition 1.2.18. For B-attracting sets A ⊆ S the following holds: ˆ (a) If A is closed, then ωB ⊆ A for all nonempty B ∈ B. ˆ (b) If B ∈ B is backward invariant, then cl A = cl B. Proof. Let B ∈ Bˆ and k ∈ I. (a) For every point x ∈ ωB (k) we obtain from Lemma 1.2.12 that there exist sequences kn → ∞, xn ∈ B(k − kn ) such that x = limn→∞ ϕ(k, k − kn )xn . If we assume x ∈ A(k), then ε := dist(x, A(k)) > 0 since A(k) is closed. Hence, for sufficiently large n we deduce dist(ϕ(k, k − kn )xn , A(k)) ≥ 2ε and therefore the contradiction h(ϕ(k, k − kn )B(k − kn ), A(k)) ≥ 2ε to (1.2b). (b) Since B is backward invariant, using Proposition 1.2.3(b) one has the inequality 0 ≤ h (B(k), A(k)) ≤ h (ϕ(k, k − n)B(k − n), A(k)) and in the limit n → ∞ we obtain the assertion from (1.2b). Proposition 1.2.19. Let ϕ be continuous and suppose S consists of complete locally ˆ convex metric linear spaces. A compact, invariant and B-attracting nonautonomous set A ⊆ S is connected, if one of the conditions holds: (a) Bˆ contains all compact nonautonomous sets. (b) A is uniformly bounded and Bˆ consists of all uniformly bounded compact nonautonomous sets. Proof. (a) For every k ∈ I we know from Mazur’s theorem (cf. [6, p. 175, Theorem 5.20]) that the closed convex hull coA(k) ⊆ S(k) of each fiber A(k) is compact and connected. Thus, A attracts coA. Suppose A is not connected. Then there exists a time k0 ∈ I and open disjoint sets U, V ⊆ S(k0 ) such that A(k0 ) ⊆ U ∪ V,
U ∩ A(k0 ) = ∅,
V ∩ A(k0 ) = ∅
hold. Yet, by continuity of ϕ we know that Cn := ϕ(k0 , k0 − n)coA(k0 − n) is a connected set for all n ≥ 0. From the invariance of A one can deduce the relation A(k0 ) = ϕ(k0 , k0 − n)A(k0 − n) ⊆ Cn and therefore U ∩ ϕ(k0 , k0 − n)A(k0 − n) = ∅,
V ∩ ϕ(k0 , k0 − n)A(k0 − n) = ∅
for all n ≥ 0. Because each subset Cn ⊆ S(k0 ) is connected, we can choose a sequence xn ∈ Cn \ (U ∪V ). Since A(k0 ) attracts every compact Cn in the sense of (1.2b), xn ∈ Cn implies that the set {xn }n≥0 is relatively compact and there exists a convergent subsequence (xmn )n≥0 with limit x ∈ S(k0 ). Moreover, because the difference Cn \ (U ∪ V ) is closed, one has x ∈ U ∪ V .
12
1 Nonautonomous Dynamical Systems
On the other hand, by construction there exist ymn ∈ coA(k0 − mn ) satisfying xmn = ϕ(k0 , k0 − mn )ymn and since A attracts compact sets, we get (1.2b)
0 ≤ h ({xmn } , A(k0 )) = h (ϕ(k0 , k − mn ) {ymn } , A(k0 )) −−−−→ 0. n→∞
Therefore, x ∈ A(k0 ) ⊆ U ∪ V . This is a contradiction. (b) It is easy to see that the uniform boundedness of A carries over to the closed convex hull coA. Then the assertion follows as above. Asymptotically Compact 2-Parameter Semigroups The general theory of topological dynamics deals with semigroups on metric spaces and most results are based only on continuity properties of ϕ(k, κ). Particularly in an infinite-dimensional setting it is natural to discuss additional features that may be obtained, if we assume some degree of compactness. For instance, 2-parameter semigroups with relatively compact orbits yield nonempty limit sets (cf. Definition 1.2.9). Indeed, many of the applications for our abstract theory to discretizations of partial or functional differential equations occur in a setting wherein the given semigroup has a smoothing property. In the following we examine three key concepts related to compactness and playing a pivotal role in the theory of attractors. For concrete applications, we refer to the survey given at the end of Sect. 1.6. Definition 1.2.20. A 2-parameter semigroup ϕ is called: ˆ (i) B-compact, if there exists a so-called compactification time N ∈ Z+ 0 such that ˆ for all B ∈ B the orbit γBN is relatively compact. ˆ (ii) B-eventually compact, if for all k ∈ I, B ∈ Bˆ there exists an N = Nk (B) ∈ Z+ 0 such that the set γBN (k) is relatively compact. ˆ ˆ and all sequences kn → ∞ (iii) B-asymptotically compact, if for all k ∈ I, B ∈ B, + in Z0 , xn ∈ B(k − kn ), the sequence (ϕ(k, k − kn )xn )n≥0 in S(k) possesses a convergent subsequence. Remark 1.2.21. (1) If S consists of compact metric spaces, then 2-parameter semiˆ groups are B-compact with compactification time 0. (2) Suppose S consists of normed spaces and the family Bˆ contains a uniformly ˆ bounded nonautonomous set. If a B-compact 2-parameter semigroup ϕ is linear, then ϕ(k, k − n), k ∈ I, is continuous for n ≥ N . (3) Often the following assumptions enable us to verify compactness properties of a 2-parameter semigroup. Here, a nonautonomous set S0 ⊆ S is compactly imbedded into S, i.e., S0 S fiberwise. Then the following holds true: + ˆ k ∈ I one has • If there exists an N ∈ Z such that for every B ∈ B, 0
ϕ(k, k − n)B(k − n) ⊆ S0 (k) for all n ≥ N, ˆ and γBN (k) is bounded in S0 (k), then ϕ is B-compact.
(1.2c)
1.2 Invariant and Limit Sets •
13
If for all k ∈ I, B ∈ Bˆ there exists an N = Nk (B) ∈ Z+ 0 such that (1.2c) holds N ˆ and γB (k) is bounded in S0 (k), then ϕ is B-eventually compact.
Typical examples of such compact embeddings S0 S are given in form of Sobolev imbedding theorems (cf. [2], [149, p. 271ff] or [432, p. 604ff]) and under (1.2c) a 2-parameter semigroup is said to fulfill a smoothing property. ˆ ˆ Corollary 1.2.22. A B-compact 2-parameter semigroup ϕ is B-eventually compact, ˆ ˆ and a B-eventually compact ϕ is B-asymptotically compact. ˆ Since the first assertion is clear by definition we Proof. Let k ∈ I and B ∈ B. N restrict to the second one. We know that there exists an N ∈ Z+ 0 such that γB (k) is relatively compact. Now choose sequences kn → ∞, xn ∈ B(k − kn ), where w.l.o.g. we can assume kn ≥ N . Then un := ϕ(k, k − kn )xn defines a sequence in the relatively compact fiber γBN (k) and consequently there exists a convergent ˆ compact. subsequence of (un )n∈N ; thus, ϕ is B-asymptotically ˆ Proposition 1.2.23. For a B-asymptotically compact ϕ one has: (a) If for every B ∈ Bˆ there exists an N = N (B) ∈ N such that B(k) ⊆ ϕ(k, k − n)B(k − n) for all n ≥ N, k ∈ I,
(1.2d)
ˆ compact. then γBN is relatively compact, i.e., ϕ is B-eventually ˆ (b) ϕ is B-eventually bounded, i.e., for every k ∈ I, B ∈ Bˆ there exists an integer N N = Nk (B) ∈ Z+ 0 such that γB (k) is bounded. ˆ Remark 1.2.24. Conversely to assertion (b), a B-eventually bounded 2-parameter ˆ semigroup is B-eventually compact, if S consists of finite-dimensional linear spaces. ˆ Proof. Let k ∈ I and B ∈ B. (a) Foran integer N ∈ N satisfying (1.2d) we have to show that the k-fiber γBN (k) = n≥N ϕ(k, k − n)B(k − n) is relatively compact. For this, let un be a sequence in γBN (k) and for each n ≥ N we choose integers kn , xn ∈ B(k − kn ) such that un = ϕ(k, k − kn )xn . The weakened backward invariance relation (1.2d) implies ϕ(k, k − kn )B(k − kn ) ⊆ ϕ(k, k − kn − l)B(k − kn − l) for all l ≥ N and therefore it is possible to construct the sequence kn such that kn+1 > kn ; hence ˆ we have kn → ∞. Since ϕ is B-asymptotically compact, there exists a convergent N subsequence umn and γB is relatively compact. (b) It it were not the case, then there exist k ∈ I and a nonautonomous set B ∈ Bˆ such that γBN (k) is unbounded for all N ≥ 0. Thus, we can find sequences kn → ∞ and xn ∈ B(k − kn ) with limn→∞ d(u0 , ϕ(k, k − kn )xn ) = ∞ for all u0 ∈ B(k). However, this contradicts the fact that ϕ(k, k − kn )xn contains a convergent subsequence.
14
1 Nonautonomous Dynamical Systems
We arrive at the main result of this section: ˆ If ϕ is B-asymptotically ˆ Theorem 1.2.25. Let A ∈ B. compact, then: (a) ωA is nonempty. (b) ωA is compact. (c) ωA is {A}-attracting. Moreover, provided ϕ is continuous, then ωA is invariant. Proof. Let A ∈ Bˆ and k ∈ I be fixed. ˆ (a) Since ϕ is B-asymptotically compact, for arbitrary sequences kn → ∞ and xn ∈ A(k − n), we can extract a convergent subsequence (ϕ(k, k − knl )xknl )l≥0 from (ϕ(k, k − kn )xn )n≥0 . By construction, the limit of ϕ(k, k − knl )xnl belongs to the fiber ωA (k) and we have ωA (k) = ∅. (b) We already know from Theorem 1.2.14 that ωA (k) is closed. Thus, in order to show its compactness, it suffices to see that for any sequence (yn )n≥0 in ωA (k) we can extract a convergent subsequence. Due to yn ∈ ωA (k) there exist kn ≥ n and xn ∈ A(k − kn ) such that d(ϕ(k, k − kn )xn , yn ) ≤
1 n
for all n ∈ N.
(1.2e)
ˆ Keeping in mind that ϕ is B-asymptotically compact, we obtain a subsequence (knl )l≥0 such that y := liml→∞ ϕ(k, k − knl )xknl exists and thus (1.2e)
d(y, yknl ) ≤ d(y, ϕ(k, k − knl )xknl ) + d(ϕ(k, k − knl )xknl , yknl ) −−−→ 0. l→∞
Hence, ωA has compact fibers. (c) For our given nonautonomous set A ∈ Bˆ we have to show the limit relation limn→∞ h(ϕ(k, k − n)A(k − n), ωA (k)) = 0 (cf. (1.2b)). We proceed indirectly and suppose this relation does not hold. Then there exists an ε > 0 and sequences kn → ∞, xn ∈ A(k − n) so that dist(ϕ(k, k − kn )xn , ωA (k)) ≥ ε.
(1.2f)
However, from the sequence (ϕ(k, k − kn )xn )n≥0 we can extract a convergent subsequence with limit y ∈ ωA (k); this contradicts (1.2f). Let ϕ be continuous and we obtain from Theorem 1.2.14 that ωA is forward invariant. It remains to prove the inclusion ωA (k) ⊆ ϕ(k, κ)ωA (κ) for integers κ < k. For (k, y) ∈ ωA there exist sequences kn → ∞, xn ∈ A(k − kn ) with y = limn→∞ ϕ(k, k − kn )xn (cf. Lemma 1.2.12). For integers kn ≥ k − κ we have (1.1b)
ϕ(k, k − kn )xn = ϕ(k, κ)ϕ(κ, k − kn )xn
(1.2g)
ˆ and as the 2-parameter semigroup ϕ is B-asymptotically compact, k − kn → −∞ and xn ∈ A(k−kn ), there exist subsequences knm → ∞, xnm ∈ A(k − knm ) with
1.2 Invariant and Limit Sets
15
z := limm→∞ ϕ(κ, k − knm )xnm ∈ ωA (κ). Hence, by (1.2g) and the continuity of ϕ we get y = ϕ(k, κ)z ∈ ϕ(k, κ)ωA (k). Set-Contracting 2-Parameter Semigroups Now we introduce another important class of semigroups. It is based on measures of noncompactness as introduced in Appendix B. For this, let χk be a measure of noncompactness on the complete metric spaces Xk , k ∈ I, and for the remaining section we suppose every B ∈ Bˆ is bounded. Definition 1.2.26. A bounded 2-parameter semigroup ϕ is called: ˆ ˆ (i) B-contracting, if limn→∞ χk (ϕ(k, k − n)B(k − n)) = 0 for all k ∈ I, B ∈ B. + ˆ (ii) B-uniformly contracting, if there exists a sequence γ : Z0 → [0, ∞) with limit 0 and an N ∈ Z+ 0 such that for all n ≥ N one has χk (ϕ(k, k − n)B(k − n)) ≤ γ(n)χk−n (B(k − n))
ˆ for all k ∈ I, B ∈ B.
Remark 1.2.27. (1) Thanks to (B.0a) it is irrelevant in Definition 1.2.26 whether one uses the Hausdorff, the Kuratowski or the separation measure of noncompactness. ˆ ˆ (2) A B-uniformly contracting 2-parameter semigroup is B-contracting, provided ˆ for every nonautonomous set B ∈ B one has supk∈I χk (B(k)) < ∞. The latter condition is fulfilled for uniformly bounded nonautonomous sets B and measures of noncompactness α, β, γ as introduced in Example B.0.1 (cf. (B.0b)). ˆ Corollary 1.2.28. Let ϕ be a bounded B-compact 2-parameter semigroup. If a nonˆ autonomous set B ∈ B is backward invariant, then B is relatively compact. Proof. Let k ∈ I. Since B is backward invariant, the monotonicity of χk yields (c4 )
χk (B(k)) ≤ χk (ϕ(k, k − n)B(k − n)) −−−−→ 0 n→∞
and consequently χk (B(k)) = 0. Due to the property (c1 ) of measures of noncompactness, the set B(k) is relatively compact. Our next result uses the generalized exponential function eq from Sect. A.1, as well as the Darbo constants introduced in Sect. B.1: Corollary 1.2.29. Let ϕ be a bounded 2-parameter semigroup with generator (ϕˆk )k∈I . If each mapping ϕˆk : S(k) → S (k), satisfies a Darbo condition with q(k) := dar ϕˆk < ∞, k ∈ I , then the following holds: ˆ (a) ϕ is B-contracting for every family Bˆ ⊆ B ⊆ S : lim eq (k, k − n)χk−n (B(k − n)) = 0 n→∞
for all k ∈ I .
ˆ (b) If limn→∞ supk∈I eq (k, k − n) = 0, then ϕ is B-uniformly contracting.
16
1 Nonautonomous Dynamical Systems
ˆ Thanks to ϕ(k, k − n) = ϕˆk−1 ◦ · · · ◦ ϕˆk−n we obtain Proof. Let k ∈ I and B ∈ B. from [35, p. 39, Proposition 5.3(b)] by mathematical induction k−1 (B.1b) χk ϕ(k, k − n)B(k − n) ≤ q(l) χk−n (B(k − n)) l=k−n
= eq (k, k − n)χk−n (B(k − n)) for all n ∈ Z+ 0 . Passing over to the limit n → ∞ yields our claim (a), whereas setting γ(n) := supk∈I eq (k, k−n) in Definition 1.2.26(ii) implies assertion (b). ˆ ˆ Prototypes of B-contracting semigroups are those of (eventually) B-compact ˆ type. Their relationship to B-asymptotically compact 2-parameter semigroups is ˆ Proposition 1.2.30. Let ϕ be a B-contracting 2-parameter semigroup. If for every ˆ then ϕ is B-asymptotiˆ B ∈ Bˆ there exists an N = N (B) ∈ Z+ such that γBN ∈ B, 0 cally compact. Proof. For k ∈ I, B ∈ Bˆ choose sequences kn → ∞, xn ∈ B(k − kn ). We define points un := ϕ(k, k − kn )xn and bounded sets An := cl {um : m ≥ n}, which obviously satisfy the inclusion An ⊆ Am ⊆ S(k) for m ≤ n. By properties of a measure of noncompactness χk (see Appendix B) one has χk (An ) = χk {ϕ(k, k − km )xm } ≤ χk ϕ(k, k − km )B(k − km ) . m≥n
m≥n
By assumption, there exists an N ≥ 0 such that γBN ∈ Bˆ and w.l.o.g. we can suppose kn ≥ N . This yields (1.1b) χk (An ) ≤ χk ϕ(k, k − (kn − N )) ϕ(k − (kn − N ), k − km )B(k − km )
m≥n
≤ χk ϕ(k, k − (kn − N ))γBN (k − (kn − N )) −−−−→ 0 n→∞
and there exist subsequences, which we relabel as kn → ∞ and xn such that ˆ ϕ(k, k − kn )xn converges; in conclusion ϕ is B-asymptotically compact.
1.3 Attractors and Global Attractors We continue our studies of the asymptotic behavior of 2-parameter semigroups. Our primary interest is the nonautonomous set which consists of all bounded complete motions, the so-called global attractor. In this section we present results that construct attractors as ω-limit sets of absorbing sets. This construction requires some degree of regularity of the 2-parameter semigroup, for instance asymptotical compactness or set contractivity.
1.3 Attractors and Global Attractors
17
In general, the global attractor has a complicated geometry reflecting the complexity of the longtime behavior of a given system. Yet, in our abstract set-up we desist from tackling such delicate issues and focus on existence issues. For this, assume the discrete interval I is unbounded below and the nonautonomous set S ⊆ X consists of metric spaces. In addition, suppose throughout that Bˆ is a family of nonempty subsets of S and ϕ is a 2-parameter semigroup on S. ˆ Definition 1.3.1. A compact nonautonomous set A∗ ⊆ S is called B-attractor, ˆ ˆ if it is invariant and B-attracting. In case the attraction universe B consists of all ˆ uniformly bounded subsets of S, a B-attractor is called global attractor. Remark 1.3.2. For a global attractor A∗ the elements of Bˆ form a cover of the extended state space S. We speak of a local attractor, whenever Bˆ consists only of neighborhoods of A∗ . Proposition 1.3.3. A global attractor A∗ ⊆ S admits the (incomplete) dynamical characterization A∗ ⊆ {(κ, ξ) ∈ S : there exists a complete motion through (κ, ξ)} , {(κ, ξ) ∈ S : there exists a bounded complete motion through (κ, ξ)} ⊆ A∗ . Proof. Concerning the first inclusion, pick (κ, ξ) ∈ A∗ arbitrarily. Due to the invariance of the global attractor A∗ , Proposition 1.2.5 yields that there exists a complete motion φ through (κ, ξ). Concerning the second inclusion, if there exists a bounded complete motion φ for ϕ, then the nonautonomous set φ is invariant and uniformly bounded. Since A∗ attracts uniformly bounded nonautonomous sets, we have 0 = lim h(ϕ(k, k − n) {φ(k − n)} , A∗ (k)) = h({φ(k)} , A∗ (k)) for all k ∈ I, n→∞
thus φ(k) ∈ cl A∗ (k). Hence, the closedness of A∗ implies φ ⊆ A∗ .
Corollary 1.3.4. A uniformly bounded global attractor A∗ admits the complete dynamical characterization A∗ = {(κ, ξ) ∈ S : there exists a complete bounded motion through (κ, ξ)} and is therefore uniquely determined. Proof. Since A∗ is uniformly bounded, the complete solution φ from the first inclusion in the above proof of Proposition 1.3.3 is clearly bounded. As demonstrated by Example 1.2.17, in general one cannot expect forward convergence towards global attractors. The following example shows that attractors need not to be unique, and without uniform boundedness assumptions also global attractors are not uniquely determined.
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1 Nonautonomous Dynamical Systems
Example 1.3.5. Suppose S = I × R and consider the 2-parameter group ϕ from the above Example 1.2.17, where 0 ∈ I. Given an arbitrary γ ∈ R, it is easy to see that αk+ for k ≥ 0, φγ (k) := γ αk− for k < 0 defines a complete motion for ϕ. Hence, the nonautonomous sets I × {0} and φγ are invariant, as well as (cf. Proposition 1.2.6) their union Aγ := (I × {0}) ∪ φγ . In Example 1.2.17 we have shown that I × {0} attracts uniformly bounded subsets of S and Remark 1.2.16(1) guarantees that also Aγ has this property. Since all the fibers Aγ (κ), κ ∈ I, are compact (in fact finite), each Aγ is a global attractor for ϕ, among which A0 is the unique uniformly bounded global attractor. An illustration is given in Fig. 2.1 (bottom, left). Definition 1.3.6. A nonempty nonautonomous set A ⊆ S is called: ˆ (a) B-absorbing, if for every k ∈ I, B ∈ Bˆ there exists an N = Nk (B) ∈ Z+ 0 with ϕ(k, k − n)B(k − n) ⊆ A(k)
for all n ≥ N.
ˆ (b) B-uniformly absorbing, if for every B ∈ Bˆ there exists an N = N (B) ∈ Z+ 0 with ϕ(k, k − n)B(k − n) ⊆ A(k) for all k ∈ I, n ≥ N and we denote the family Bˆ as absorption universe. A 2-parameter semigroup is ˆ ˆ ˆ called B-dissipative, if it has a bounded B-absorbing set. Moreover, a B-dissipative 2-parameter semigroup is called uniformly bounded (bounded, compact) dissipative, if Bˆ consists of all uniformly bounded (bounded, compact) subsets of S. ˆ ˆ Remark 1.3.7. (1) Any B-uniformly absorbing set is B-absorbing. (2) Bounded dissipative 2-parameter semigroups are uniformly bounded dissipative and compact dissipative. ˆ Lemma 1.3.8. If A is a B-absorbing nonautonomous set, then ωB ⊆ ωA for all ˆ B ∈ B holds and in case A ∈ Bˆ one additionally obtains ωA ⊆ cl A, clXk ωB (k) = ωA (k) for all k ∈ I. (1.3a) B∈Bˆ
Proof. We arbitrarily fix B ∈ Bˆ and a pair (k, y) ∈ ωB . By Lemma 1.2.12 there exist sequences kn → ∞ and points xn ∈ B(k − kn ) such that lim ϕ(k, k − kn )xn = y.
n→∞
(1.3b)
ˆ Since the nonautonomous set A is B-absorbing, for each m ∈ N there exists a knm ≥ m such that one has ϕ(k − m, k − knm )B(k − knm ) ⊆ A(k − m) and we define points ym := ϕ(k − m, k − knm )xnm ∈ A(k − m). This yields (1.1b)
ϕ(k, k − knm )xnm = ϕ(k, k − m)ϕ(k − m, k − knm )xnm = ϕ(k, k − m)ym
1.3 Attractors and Global Attractors
19
and thus limm→∞ ϕ(k, k − m)ym = y by (1.3b). By the inclusion ym ∈ A(k − m), our Lemma 1.2.12 implies y ∈ ωA (k); since (k, y) ∈ ωB was arbitrary, we have established ωB ⊆ ωA . ˆ Since A is B-absorbing, ˆ For the remaining part of the proof suppose A ∈ B. there exists an integer N = Nk (B) ≥ 0 with ϕ(k, k − nl )xl ∈ A(k) for all l ∈ N such that l ≥ N and therefore we have y ∈ cl A(k). This guarantees ωB (k) ⊆ cl A(k) and the remaining relation is a straight forward consequence of the above. Under the assumption of continuity, we arrive at the first main theorem. ˆ Theorem 1.3.9. Let a 2-parameter semigroup be continuous and B-asymptotically ˆ ˆ compact with B-absorbing set A. If A ∈ B, then: ˆ (a) The nonautonomous set A∗ := ωA is a B-attractor. ˆ then A∗ ⊆ cl A. (b) If A ∈ B, ˆ (c) A∗ is minimal in the following sense: Every closed and B-attracting set A∗ ⊆ S satisfies A∗ ⊆ A∗ . (d) If A ∈ Bˆ is closed, then each fiber of A∗ allows the characterization A∗ (k) = clXk
ωB (k) for all k ∈ I.
B∈Bˆ
Remark 1.3.10. If a 2-parameter semigroup is uniformly bounded dissipative with uniformly bounded absorbing set, then its global attractor A∗ ⊆ S is uniformly bounded and has the complete characterization from Corollary 1.3.4. Proof. Let k ∈ I and we define A∗ := ωA . (a) The compactness of A∗ follows readily from Theorem 1.2.25(b). Thanks to the inclusion B∈Bˆ ωB (k) ⊆ A∗ (k) (see (1.3a) in Lemma 1.3.8) we obtain (0.0a)
h(ϕ(k, k − n)B(k − n), A∗ (k)) ≤ h(ϕ(k, k − n)B(k − n), ωB (k)) for n ≥ 0, where the right-hand side tends to 0 for n → ∞ by Theorem 1.2.25(c), ˆ It reand the relation limn→∞ h(ϕ(k, k − n)B(k − n), A∗ (k)) = 0 for all B ∈ B. mains to show the invariance of attractors. For this, we observe from Theorem 1.2.25 that (1.3a) A∗ (k) = cl ωB (k) = cl ϕ(k, κ)ωB (κ) = cl ϕ(k, κ) ωB (κ) B∈Bˆ
B∈Bˆ
for all κ ≤ k and, since ϕ is continuous and cl cl ϕ(k, κ)
B∈Bˆ
B∈Bˆ ωB (κ)
ωB (κ) = ϕ(k, κ) cl
B∈Bˆ
thus, we have ϕ(k, κ)A∗ (κ) = A∗ (k) for all κ ≤ k.
B∈Bˆ
is compact,
ωB (κ);
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1 Nonautonomous Dynamical Systems
ˆ (b) We give another proof of the left relation in (1.3a). Since A is B-absorbing ˆ there exists an Nk (A) ∈ Z+ with ϕ(k, k − n)A(k − n) ⊆ A(k) for all and A ∈ B, 0 n ≥ Nk (A) and consequently, from Definition 1.3.6 and assertion (a) follows cl ϕ(k, k − n)A(k − n) A∗ (k) = ωA (k) = m≥0
⊆
n≥m
m≥Nk (A)
⊆
cl
ϕ(k, k − n)A(k − n)
n≥m
cl A(k) = cl A(k).
m≥Nk (A)
ˆ (c) Now, let A∗ ⊆ S be a closed and B-attracting nonautonomous set, i.e., in particular limn→∞ h(ϕ(k, k − n)B(k − n), A∗ (k)) = 0. For each pair (k, y) ∈ A∗ one has y = limn→∞ ϕ(k, k−kn )xn for some sequences kn → ∞, xn ∈ B(k−kn ) (cf. Lemma 1.2.12), and consequently we derive y ∈ cl A∗ (k) = A∗ (k). This yields the inclusion A∗ (k) ⊆ A∗ (k). (d) See Lemma 1.3.8. Uniformly bounded global attractors are topologically connected: Corollary 1.3.11. Suppose that S consists of complete locally convex metric linear spaces. Then A∗ ⊆ S is connected, if one of the conditions (a) or (b) from Proposition 1.2.19 holds. Proof. On the basis of Theorem 1.3.9 we can deduce that ωA = A∗ is compact, ˆ invariant and also B-attracting. Then Proposition 1.2.19 implies our claim. We round off this section with another central result on the attractors of parameter-dependent 2-parameter semigroups. Theorem 1.3.12 (upper-semicontinuity of attractors). Let S consist of complete metric spaces and let P be a metric space. If ϕ(·; p), p ∈ P , is a continuous ˆ parameter-dependent 2-parameter semigroup being B-asymptotically compact and ˆ ˆ B-dissipative with B-absorbing set A ∈ Bˆ (uniformly in p ∈ P ), then: ˆ (a) Every ϕ(·; p) has a B-attractor A∗p . (b) If A ∈ Bˆ is uniformly bounded and Bˆ contains all uniformly bounded finite sets, then A∗p is upper-semicontinuous, i.e. lim hS(k) (A∗p (k), A∗p0 (k)) = 0
p→p0
for all k ∈ I, p0 ∈ P.
Proof. Let k ∈ I and p ∈ P . ˆ (a) The existence of B-attractors A∗p follows from Theorem 1.3.9(a). Here, A∗p is given as ω-limit set of A w.r.t. the 2-parameter semigroup ϕ(·; p). (b) W.l.o.g. we can assume that A ⊆ S is closed. From Theorem 1.3.9(b) we obtain A∗p ⊆ A and consequently the nonautonomous set V given by the fibers A∗p (k) for all k ∈ I V(k) := p∈P
1.4 Periodic and Autonomous 2-Parameter Semigroups
21
is bounded. Due to the compactness of each A∗p (k) (cf. Theorem 1.2.25(b)) the bounded union V is relatively compact. In order to prove the upper semicontinuity of a fiber A∗p (k), we must show the following: Provided (pj )j∈N , (xj )j∈N denote convergent sequences in the spaces P , S(k), resp., with xj ∈ A∗pj and respective limits p0 ∈ P , x0 ∈ S(k), then one has x0 ∈ A∗p0 (k). Since the nonautonomous set A∗pj is invariant (cf. Theorem 1.2.25), there exists a point yj1 ∈ A∗pj (k − 1) with xj = ϕ(k, k − 1; pj )yj1 (cf. Proposition 1.2.5). Because the fiber V(k − 1) is relatively compact and A(k − 1) is closed, we may assume that the bounded sequence (yj1 )j∈N has a limit y01 ∈ A(k − 1); the continuity of ϕ guarantees x0 ∈ ϕ(k, k − 1; p0 )y01 . In the same fashion, for all n ∈ N there exists a convergent sequence (yjn )j∈N in A∗pj (k − n) with limit y0n ∈ A(k − n) and ϕ(k, k − n; p0 )y0n = x0 . Having this at hand, introduce a sequence φ in S, φ(k + n) := y0n ∈ A(k − n) for all n < 0, φ(k + n) := ϕ(k + n, k; p0 )x0
for all n ≥ 0;
by definition, φ represents a complete motion for ϕ(·; p0 ). Due to the fact that A is + ˆ a B-absorbing set, for every m ∈ Z+ 0 there exists an N (m) ∈ Z0 with φ(k + m) = ϕ(k + m, k − n; p0 )y0n ∈ A(k + m) for all n ≥ N (m). Therefore, the sequence φ is a bounded complete motion for ϕ(·; p0 ). Because, by assumption, A∗p0 attracts uniformly bounded finite sets, we can show as in the proof of Proposition 1.3.3 that φ(k) ∈ A∗p0 (k), i.e., x0 ∈ A∗p0 (k).
1.4 Periodic and Autonomous 2-Parameter Semigroups Throughout this section, w.l.o.g. we can replace the discrete interval I by the integers Z, since otherwise it is possible to extend the spaces Xk periodically in time. Moreover, let S ⊆ X be nonempty. Definition 1.4.1. Let p ∈ N. A 2-parameter semigroup ϕ on S is said to be p-periodic (or simply periodic), if S(k + p) = S(k),
ϕ(k + p, κ + p) = ϕ(k, κ) for all κ ∈ Z, k ∈ Z+ κ.
In case p = 1 we say ϕ is autonomous. Remark 1.4.2. (1) The generator (ϕˆk )k∈Z of a p-periodic 2-parameter semigroup is also p-periodic, i.e., one has ϕˆk+p = ϕˆk for all k ∈ Z. (2) In particular, autonomous 2-parameter semigroups have constant generators ϕˆk ≡ ϕ and state spaces S(k) ≡ S on I. They are always of the form ϕ(k, κ) = ϕˆk−κ for all k ≥ κ with the iterates ϕˆk−κ of a mapping ϕˆ : S → S on a nonempty set S. Note that the common terminology for continuous autonomous
22
1 Nonautonomous Dynamical Systems
2-parameter semigroups is semidynamical system; accordingly a continuous autonomous 2-parameter group is called dynamical system. (3) For an autonomous 2-parameter semigroup ϕ one has the relation ϕ(k, k − n) = ϕˆn
for all k ∈ I, n ≥ 0 with k − n ∈ I
and therefore the above notions of limit or attracting sets reduce to the well-known concepts from the classical autonomous theory. Definition 1.4.3. Let κ ∈ Z and ϕ be a p-periodic 2-parameter semigroup. Then Πκ : S(κ) → S(κ), Πκ := ϕ(p + κ, κ) is called period map. Proposition 1.4.4. If ϕ is a p-periodic 2-parameter semigroup with period map Πκ , then the following holds: (a) Πκ+p = Πκ . (b) Πκn = ϕ(κ + np, κ) for all n ∈ Z+ 0. (c) If an iterate Πκn , n ∈ N, has a fixed point ξ ∈ S(κ), then there exists a complete np-periodic motion φ : Z → X with φ(κ) = ξ. Proof. Let κ ∈ Z. The claim (a) is clear by definition and (b) follows from an induction argument. Concerning the proof of (c), let ξ be a fixed point of Πκn . First of all, it is not difficult to see that each ϕ(k, κ)ξ, κ ≤ k ≤ κ + p is a fixed point of the period map Πkn . We define φ(k) := ϕ(k, κ)ξ for k ≥ κ and obtain (b)
(1.1b)
φ(k + np) = ϕ(k + np, k)ϕ(k, κ)ξ = Πkn ϕ(k, κ)ξ = ϕ(k, κ)ξ = φ(k), i.e., φ : Z+ κ → X is np-periodic. In order to obtain a complete np-periodic motion we extend φ np-periodically in backward time. Proposition 1.4.5. If both the 2-parameter semigroup ϕ and the nonautonomous set B ⊆ S are p-periodic, then also the ω-limit set ωB is p-periodic. Proof. By the periodicity of ϕ and B one has ωB (k + p) =
cl
m≥0
=
m≥0
ϕ(k + p, k + p − n)B(k + p − n)
n≥m
cl
ϕ(k, k − n)B(k − n) = ωB (k) for all k ∈ Z
n≥m
and the nonautonomous set ωB is p-periodic.
Proposition 1.4.6. Let S consist of normed spaces. We suppose that ϕ is a continuous p-periodic 2-parameter semigroup admitting a forward invariant set B with a bounded closed and convex fiber B(κ + p) = B(κ). If a period map Πκ , κ ∈ Z, is compact, then there exists a p-periodic complete motion.
1.4 Periodic and Autonomous 2-Parameter Semigroups
23
Proof. Since ϕ is continuous, the period map Πκ is completely continuous. The forward invariance of B yields Πκ (B(κ)) = ϕ(κ + p, κ)B(κ) ⊆ B(κ) and Theorem B.2.2 guarantees a fixed point of Πκ . The claim follows with Proposition 1.4.4(c). Due to the dynamical characterization in Corollary 1.3.4, periodic motions are contained in global attractors. Next we state an existence result for such a situation, which involves the Kuratowski measure of noncompactness β from Example B.0.1. Proposition 1.4.7. Let S consist of Banach spaces and κ ∈ Z. If a continuous ˆ bounded p-periodic 2-parameter semigroup ϕ is B-dissipative having a p-periodic ˆ B-uniformly absorbing set A, then the following holds: (a) If Πκ is β-condensing and Bˆ contains the family of p-periodic compact subsets of S, then there exists a p-periodic complete motion. (b) If Πκ is a β-contraction and Bˆ is the family of uniformly bounded nonautonomous subsets of S, then there exists a p-periodic global attractor containing a p-periodic complete motion. ˆ Proof. Let k ∈ Z and A ⊆ S be as above. Since ϕ is B-dissipative, the absorbing set A is bounded and due to its p-periodicity indeed uniformly bounded. In addition, for each B ∈ Bˆ there exists an N = N (B) ∈ Z+ 0 with ϕ(k, k − n)B(k − n) ⊆ A(k) (a) Consequently, if we choose an integer M ≥
for all n ≥ N. N (B) p ,
(1.4a)
one has
Πκm B(κ) = ϕ(κ + mp, κ)B(κ) ⊆ A(κ + mp) = A(κ)
for all m ≥ M
and Πκ is compact dissipative as defined in Theorem B.2.4. Since Πκ is β-condensing, this implies that Πκ has a fixed point and the claim (a) follows from Proposition 1.4.4(c). ˆ We first show that ϕ is B-asymptotically ˆ (b) Above all, one has A ∈ B. compact. Keeping k ∈ Z fixed, we define the integers n+ := max {l ∈ Z : κ + lp ≤ k} ,
n− (n) := min {l ∈ Z : k − n ≤ κ + lp}
with limn→∞ n− (n) = −∞. Since ϕ is bounded and p-periodic, the real constants D := max0≤m−l
n→∞
ˆ for all B ∈ B;
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1 Nonautonomous Dynamical Systems
ˆ ˆ Proposition 1.2.30 thus, ϕ is B-contracting. Since (1.4a) implies γBn ⊆ A ∈ B, ˆ guarantees that ϕ is B-asymptotically compact. Moreover, Theorem 1.3.9 enˆ sures that ϕ possesses a B-attractor ωA . It is clearly the global attractor and Proposition 1.4.5 implies that ωA is p-periodic. In the set-up of (b) the assumptions of assertion (a) are fulfilled and consequently there exists a complete p-periodic motion φ∗ . Eventually, thanks to Corollary 1.3.4 it is φ∗ ⊆ A∗ .
1.5 Applications: Discretized Semiflows In applied sciences, investigations on the dynamical behavior of nonlinear evolutionary equations such as ordinary, functional or partial differential equations, are mainly computational. For that purpose problems are discretized. As a general philosophy, our results are intended to hold for discrete dynamical systems given by time-h-maps of evolutionary equations. Note that information on the time-h-map is of crucial importance in discretization theory, since it enables us to relate approximate solutions (obtained, e.g., by numerical schemes) to the original solution of the differential equation. In a first step, one obtains a straight forward time discretization which, despite being a discrete object defined on I, captures the behavior of the continuous system (cf. (1.5e)). In such a time discretization, suppose that the so-called discretization mesh (tk )k∈I is a real sequence with hk := tk+1 − tk ∈ [τ, T ]
for all k ∈ I
(1.5a)
and some given bounds 0 < τ ≤ T . This assumption ensures that (tk )k∈I is unbounded above or below, provided I has the respective property.
1.5.1 Retarded Functional Differential Equations Let d ∈ N, r ≥ 0, we abbreviate Cr := C([−r, 0], Rd ) and equip this function space with the norm |ψ|r := supθ∈[−r,0] ψ(θ). Given a function φ, define φt ∈ Cr via φt (s) := φ(t+s) for s ∈ [−r, 0]. We consider a nonautonomous retarded functional differential equation (FDE, for short) u(t) ˙ = f (t, ut )
(FDE)
under the assumptions: Hypothesis 1.5.1. The right-hand side f : R × Cr → Rd satisfies: (i) f (·, ψ) : R → Rd is continuous for all ψ ∈ Cr and for every compact subset K ⊆ R × Cr there exists a real L = L(K) ≥ 0 such that ¯ ∈ K. ¯ ≤ L ψ − ψ¯ f (t, ψ) − f (t, ψ) for all (t, ψ), (t, ψ) r
1.5 Applications: Discretized Semiflows
25
(ii) (Dissipativity condition) there exist reals α > 0, β ≥ 0 such that f (t, ψ), ψ(0) ≤ β − α ψ(0)
2
for all t ∈ R, ψ ∈ Cr .
It is clear that equations of the form (FDE) include ordinary differential equations (ODEs, for short) in the delay-free situation r = 0. Due to assumption Hypothesis 1.5.1(i), the mapping f is continuous by Lemma B.1.3, and [198, p. 44, Theorem 2.2] implies that for arbitrary initial values s ∈ R, ψ ∈ Cr there exists a unique forward solution φ(·, s, ψ) of (FDE) satisfying φs (·, s, ψ) = ψ. Having this at hand, we can define a solution oper¯ s) : Cr → Cr for (FDE) which gives the solution (in Cr ) at time t, ator φ(t, ¯ s)ψ := φt (·, s, ψ) for all s ≤ t; note that the assumption when xs = ψ via φ(t, ¯ s) is defined for times t ≥ s. Moreover, from Hypothesis 1.5.1(ii) implies that φ(t, [75, Proof of Theorem 4.2] we know that ¯ t − s)ψ ≤ φ(t, r
1+
β α
for all t ∈ R, s ≥ r +
ln ρ α
(1.5b)
¯ρ (0, Cr ), where we choose a radius ρ > 1. and ψ ∈ B Now, suppose that I is unbounded below. Given a real sequence (tk )k∈I as in (1.5a) we define the discretized 2-parameter semiflow induced by (FDE) via ¯ k , tκ )ψ ϕ(k, κ)ψ := φ(t
for all κ ≤ k.
(1.5c)
Evidently, ϕ is a 2-parameter semigroup on S = I × Cr and we arrive at Proposition 1.5.2. If Hypothesis 1.5.1 holds, then the 2-parameter semigroup ϕ from (1.5c) has the global attractor ωA ⊆ I × Cr with A = I × B√1+ β (0, Cr ). α
Proof. We check the assumptions of Theorem 1.3.9. First of all, [198, p. 43, Theorem 2.2] guarantees that ϕ is continuous. Let Bˆ be the family of all uniformly bounded nonautonomous sets in S := I × Cr . Then (1.5b) implies that ˆ ˆ ˆ As demonstrated in [75, ϕ is B-dissipative with B-uniformly absorbing set A ∈ B. Proof of Theorem 4.1] it is an easy consequence of the Arzel`a-Ascoli theorem (see ˆ [295, p. 5.7, Theorem 3.1]) that ϕ is B-eventually compact with compactification ˆ time N ≥ r/τ . Then Corollary 1.2.22 implies that ϕ is B-asymptotically compact ˆ ˆ and ωA is a B-attractor. Because B consists of uniformly bounded sets, ωA is the global attractor of ϕ. In certain applications Hypothesis 1.5.1(ii) is too strong and, e.g., for scalar delay differential equation one can deduce alternative conditions: Example 1.5.3 (delay differential equations). Suppose δ > 0, r ≥ 0 are reals and that g : R2 → R is a continuous function. We consider the scalar nonautonomous retarded delay differential equation (DDE, for short) u(t) ˙ = −δu(t) − g(t, u(t − r))
(DDE)
26
1 Nonautonomous Dynamical Systems
under the following assumptions: (i) For every compact K ⊆ R2 there exists a L ≥ 0 such that |g(t, u) − g(t, u¯)| ≤ L |u − u ¯|
for all (t, u), (t, u ¯) ∈ K.
(ii) There exists a γ ≥ 0 such that |g(t, u)| ≤ γ for all t, u ∈ R. In this context, (DDE) is sometimes called Krisztin–Walther equation. For given δ > 0, initial time t0 ∈ R and an initial function ψ ∈ Cr , let us assume the function φ : [t0 − r,∞) → R isa solution of (DDE) with φt0 = ψ. From the elementary d relation dt eδ(t−t0 ) φ(t) = −eδ(t−t0 ) g(t, φ(t − r)) we obtain the estimate e−δ(t−t0 ) φ(t0 ) −
γ γ ≤ φ(t) ≤ e−δ(t−t0 ) φ(t0 ) + δ δ
for all t ≥ t0
¯ s) on Cr , generated by (DDE), satisfies and thus the 2-parameter semiflow φ(t, ¯ s)ψ ≤ e−δ(t−s) |ψ| + γ for all t ≥ s. Arguing as above in the proof of φ(t, r δ r Proposition 1.5.2, we deduce that the discretized 2-parameter semiflow ϕ(k, κ) from (1.5c) has the global attractor ωA ⊆ I × Cr with A = I × Bρ (0, Cr ) and ρ > γδ .
1.5.2 Abstract Evolution Equations In this subsection, we merely prepare assumptions and results for later use. Assume that X, Y are Banach spaces with the embedding X ⊆ Y . We consider abstract nonautonomous evolutionary equations (AE)
ut + B(t)u = f (t, u) in the space Y and make the following standing Hypothesis 1.5.4. Let r ∈ [0, 1) and b, c ≥ 0:
(i) Suppose B(t) : D(B(t)) ⊆ Y → Y , t ∈ R, and that the solution operators for the linear part ut +B(t)u = 0 generate an evolution family (U (t, s))s≤t on X: • • • •
U (t, s) = IX and U (t, s)U (s, τ ) = U (t, τ ) for all τ ≤ s ≤ t. U (·)x : (t, s) ∈ R2 : s < t → L(X), x ∈ X, are continuous. U (t, s)Y ⊆ X for all s < t. There exist reals ω ∈ R, K ≥ 0 such that U (t, s)L(X) ≤ Keω(t−s)
for all s ≤ t,
−r ω(t−s)
U (t, s)L(Y,X) ≤ K(t − s)
e
for all s < t.
1.5 Applications: Discretized Semiflows
27
(ii) Suppose that f : R × X → Y is continuous, satisfies the local Lipschitz condition that for every ρ > 0 there exists a real (ρ) ≥ 0 such that (ρ) := lip2 f |R×B¯ ρ (0,X) < ∞ and f (t, x)Y ≤ b + c xX for all t ∈ R, x ∈ X. First of all, given an evolution family (U (t, s))s≤t on X as above, we obtain a linear continuous and bounded 2-parameter semigroup on I × X via Φ(k, l) := U (tk , tl ) for all l ≤ k. Concerning the nonlinear problem (AE), as in [432, p. 221ff] or [201, p. 52ff] one shows that for every pair (t0 , u0 ) ∈ R × X there exists a unique mild solution u(·; t0 , u0 ) : [t0 , ∞) → X of (AE), i.e., a unique solution u to the integral equation
t
u(t) = U (t, t0 )u0 +
U (t, s)f (s, u(s)) ds
for all t0 ≤ t;
(1.5d)
t0
furthermore, the function u : (t, s, x) ∈ R2 × X : s ≤ t → X is continuous and defines a 2-parameter semiflow on R × X. As above, this solution operator induces a continuous 2-parameter semigroup on I × X, if we define ϕ(k, l)u0 := u(tk ; tl , u0 ) for all l ≤ k.
(1.5e)
This simple idea is important, since it yields a nonautonomous counterpart to the time-h-map of flows generated by autonomous evolutionary equations. For latter use we need to investigate the growth of mild solutions. For this, we have to define a strictly increasing generalization of the exponential function, known as generalized Mittag-Leffler function (cf., e.g., [371, p. 16ff]) Eβ : [0, ∞) → [0, ∞) ,
Eβ (x) :=
∞
xβn , Γ (1 + βn) n=0
where β > 0 and Γ denotes the Gamma function; note that E1 (x) = ex . Next we establish boundedness and Lipschitz continuity of ϕ. Lemma 1.5.5. Under Hypothesis 1.5.4 the following holds true: 1−r 0) (a) It is u(t; t0 , u0 )X ≤ K u0 X eω(t−t0 ) + b (t−t1−r E1−r (μ(t − t0 )) for 1/(1−r)
. all t0 ≤ t, u0 ∈ X, where μ := (KcΓ (1 − r)) (b) For every real ρ > 0 one has the local Lipschitz estimate lip u(t; t0 , ·)|B¯ρ (0,X) ≤ KE1−r (¯ μ(ρ)(t − t0 ))eω(t−t0 )
for all t0 ≤ t
√ 1−r 0) satisfying the estimate max eω(t−t0 ) , E1−r (μ(t − t0 )), b(t−t ≤ 2, 1−r where we have abbreviated μ ¯ (ρ) := [K(2K(ρ + 1))Γ (1 − r)]1/(1−r) .
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1 Nonautonomous Dynamical Systems
Proof. Let t0 ∈ R and u0 , u ¯0 ∈ X. (a) Our assumptions readily imply the inequality t (1.5d) u(t; t0 , u0 ) ≤ Keω(t−t0 ) u0 + Kb (t − s)−r eω(t−s) ds t0 t
+ Kc
(t − s)−r eω(t−s) u(s; t0 , u0 ) ds
t0
for all t0 ≤ t. We abbreviate v(t) := u(t; t0 , u0 ) eω(t0 −t) and deduce t t −r ω(t0 −s) v(t) ≤ K u0 + Kb (t − s) e ds + Kc (t − s)−r v(s) ds t0
(t − t0 )1−r ω(t0 −t) e ≤ K u0 + Kb + Kc 1−r
t0 t
(t − s)−r v(s) ds
t0
for all t0 ≤ t. This estimate enables us to apply the Gronwall–Henry lemma (see [432, p. 625, Lemma D.4]) in order to arrive at 1−r 0) v(t) ≤ K u0 + b (t−t1−r eω(t0 −t) E1−r (μ(t − t0 )) for all t0 ≤ t, which implies assertion (a). (b) Given ρ > 0, thanks to our smallness conditions for the difference t − t0 ≥ 0 one obtains from the inequality shown in (a) that u(t; t0 , u0 ) ≤ 2K(ρ + 1) for ¯ρ (0, X). We abbreviate v(t) := u(t; t0 , u0 ) − u(t; t0 , u all u0 ∈ B ¯0 ) eω(t0 −t) and deduce similarly to (a) from (1.5d) and Hypothesis 1.5.4 that t v(t) ≤ K u0 − u ¯0 + K(2K(ρ + 1)) (t − s)−r v(s) ds. t0
Therefore, again the Gronwall–Henry lemma (see [432, p. 625, Lemma D.4]) implies that v(t) ≤ K u0 − u ¯0 E1−r (¯ μ(ρ)(t − t0 )) and this yields the claim. Abstract Sectorial Equations A relevant special case of abstract equations (AE) are those of sectorial type ut + Bu = f (t, x).
(SE)
As opposed to (AE) the operators B do not depend on t ∈ R and are sectorial, i.e.: • •
B : D(B) ⊆ Y → Y is closed and densely defined. There exist a ∈ R, ϑ ∈ (0, π/2), C ≥ 0 such that
Σa,ϑ := {z ∈ C \ {a} : ϑ ≤ |arg z − a| ≤ π} ⊆ ρ(B) C and the resolvent estimate [zIY − B]−1 L(Y ) ≤ |z−a| holds, t ∈ R, z ∈ Σa,ϑ . • There exists a δ0 > 0 such that σ(B) > δ0 .
1.5 Applications: Discretized Semiflows
29
To formulate our requirements on the nonlinearity in (SE), we recall the concept of fractional power spaces (cf. [201, 432]). For reals α > 0 they are given by Y α := im(aIY − B)−α ,
uα := (aIY − B)uY ,
where the bounded linear operators (aIY − B)−α are defined via 1 (aIY − B)−α := (a − z)−α (zIY − B)−1 dz; 2πi Γ here, (a − z)−α is the principal branch of the reciprocal root function (which is analytic on C \ [a, ∞)). Sectorial operators B generate analytical semigroups (e−Bt )t≥0 and the corresponding evolution operator reads as U (t, s) = e−B(t−s) ; in particular, U (t, s) fulfills Hypothesis 1.5.4(i) with X = Y r , r ∈ [0, 1). If B has a compact resolvent, then also U (t, s), s < t, is compact (cf. [432, p. 97, Theorem 37.5]). Concerning the nonlinear sectorial problem (SE) under Hypothesis 1.5.4, the mild solutions u are continuous in the initial value u0 due to [432, p. 236, Theorem 47.5] and exist globally by [432, p. 239, Theorem 47.7].
1.5.3 Reaction-Diffusion Equations Let t0 ∈ R, d, N ∈ N and Ω ⊆ Rd be a bounded domain with Lipschitz boundary. We consider a nonautonomous reaction-diffusion equation (RDE, for short) ut − DΔu = f (t, u) + g(t, x),
(RDE)
subject to homogeneous Dirichlet initial-boundary conditions u(t, x) = 0 for all t > t0 , x ∈ bd Ω,
u(t0 , x) = u0 (x)
for all x ∈ Ω
(or with periodic or Neumann boundary conditions, cf. [86]) under the somewhat general assumptions: Hypothesis 1.5.6. The diffusion coefficient D ∈ RN ×N possesses a positive symmetric part 12 (D + DT ) ≥ δIRN with δ > 0 and we suppose: (i) g : R × Ω → RN with g ∈ L2loc (R, L2 (Ω)N ) and γ := sup t∈R
t
t+1
g(s, ·)L2 (Ω)N ds < ∞.
(ii) f : R × RN → RN is continuous and there exist real numbers γ1 , . . . , γN > 0, and p1 ≥ . . . ≥ pN ≥ 2, C1 , C2 , C3 > 0 so that for t ∈ R, v, v¯ ∈ RN one has
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1 Nonautonomous Dynamical Systems
f (t, v), v ≤ C1 −
N i=1
pi
γi |vi | ,
N
|fi (t, v)|
pi pi −1
≤ C2
1+
i=1
N
pi
|vi |
,
i=1 2
f (t, v) − f (t, v¯), v − v¯ ≤ C3 |v − v¯| . For the sake of a convenient notation we abbreviate H := L2 (Ω)N . Referring to [86, pp. 114–115, Proposition 2.1], we can employ the Galerkin method to show that for every initial function u0 ∈ H there exists a unique continuous weak solution u(·; t0 , u0 ) : (t0 , ∞) → H such that u(t; t0 , u0 ) ∈ H01 (Ω)N , t0 < t, and λ+1 2C1 + γ for all s > 0 (1.5f) u(t; t − s, u0 )H ≤ u0 2H e−λs + λ λ2 and t ∈ R, where λ = δλ1 and λ1 > 0 is the first eigenvalue of −Δ equipped with zero boundary conditions. If I is unbounded below and (tk )k∈I is as in (1.5a), we define the discretized 2-parameter semiflow of (RDE) by ϕ(k, κ)u0 := u(tk ; tκ , u0 ) for all κ ≤ k; it is a 2-parameter semigroup on S = I × H satisfying the smoothing property with S0 = I × H01 (Ω)N . Proposition 1.5.7. If Hypothesis 1.5.6 holds, then the 2-parameter semigroup ϕ has
the global attractor ωA ⊆ I × H with A = I × Br (0, H) and r >
2C1 λ
+
λ+1 λ2 γ.
Proof. We verify the assumptions of Theorem 1.3.9. By [86, p. 114, Proposition 2.1] we see that u(tk ; tκ , ·), κ ≤ k, and also ϕ(k, κ) : H → H are continuous. Let Bˆ be the family of uniformly bounded nonautonomous sets in I × H and (1.5f) ˆ ˆ ˆ We define the implies that ϕ is B-dissipative with B-uniformly absorbing set A ∈ B. nonautonomous set A1 fiber-wise by A1 (k) := ϕ(k, k − 1)A(k − 1) ⊆ H for k ∈ I and realize that A1 (k) is bounded in H01 (Ω)N , hence compact in the Hilbert space ˆ H (cf. [86, pp. 29–30, Theorem 1.1]). Thus, ϕ is B-compact with compactification ˆ time 1 and Corollary 1.2.22 guarantees that ϕ is B-asymptotically compact. So, ωA ˆ is a B-attractor and because Bˆ consists of uniformly bounded sets, ωA is the global attractor of ϕ. To conclude this subsection we discuss two special cases of (RDE). Example 1.5.8 (scalar reaction-diffusion equation). Let Ω = (−a, a), a > 0 and δ > 0. Consider a scalar nonautonomous 1d reaction-diffusion equation ut − δuxx = f (t, u),
(1.5g)
subject to an initial function u(t0 , x) = u0 (x) and homogenous Dirichlet boundary conditions u(t, −a) = u(t, a) = 0. We suppose f : R × R → R is a continuous function such that there exist reals p ≥ 2, γ1 , C1 , C2 , C3 > 0 with vf (t, v) ≤ C1 − γ1 v p ,
p
p
|f (t, v)| p−1 ≤ C2 (1 + |v| ),
D2 f (t, v) ≤ C3
1.5 Applications: Discretized Semiflows
31
2 for all t, v ∈ R. The eigenvalues of the 1d Laplacian are ( nπ 2a ) , n ∈ N (for this, see Example 3.7.7). An absorbing set for the generated 2-parameter semigroup on I × L2 (−a, a) can be given explicitly as A = I × Br (0, H), where r > 2a 2C /δ. 1 π An interesting special case of (1.5g) is the nonautonomous Chafee–Infante equation
ut − δuxx = u(α1 (t) − α2 (t)u2 ) with continuous bounded reaction functions α1 , α2 : R → (0, ∞), where α2 is uniformly bounded away from 0 (cf. [86, p. 118]). It satisfies the above assumptions 2 with p = 4, C1 = 12 supt∈R αα12(t) , γ1 = 12 inf t∈R α2 (t), C3 = supt∈R α1 (t), and (t) α1 (t)2 1 any r > 2a π δ supt∈R α2 (t) is a feasible radius for an absorbing set I × Br (0, H). Example 1.5.9 (scalar Ginzburg–Landau equation). Let l > 0 be a period. Consider the nonautonomous complex Ginzburg–Landau equation with cubic nonlinearity satisfying l-periodic boundary conditions 2
ut − μ1 (t)u − (1 + iν)uxx + (1 + iμ2 (t)) |u| u = 0 u|t=t0 = u0 ,
in (t0 , ∞) × R,
u(t, x) = u(t, x + l) on (t0 , ∞) × R
(GL)
(cf. [86, p. 118]). Here, ν ∈ R, the instability parameter μ1 : R → R and the dispersion parameter μ2 : R → R are assumed to be continuous functions satisfying μ1 (t) ∈ (R0 , R1 ] ,
μ2 (t) ∈ [−R2 , R2 ] for all t ∈ R
with R0 , R1 , R2 > 0. Separating real and imaginary part in (GL), yields 1 −μ2 (t) 1 −ν 2 u uxx = μ1 (t)u − |u| 1 μ2 (t) ν 1
ut −
and our assumptions hold with p = 4, C1 = R12 , provided R2 ≤ for the absorbing set I × Br (0, H) is given by r > R1 2/λ1 .
√
3. Thus, a radius
1.5.4 Doubly Nonlinear Equations Let t0 ∈ R, d ∈ N and Ω ⊆ Rd be a bounded domain with Lipschitzian boundary. We consider a nonautonomous doubly nonlinear parabolic equation β(u)t − Δu = g(t, x, u),
(nRDE)
subject to homogeneous Dirichlet initial-boundary conditions u(t, x) = 0
for all t > t0 , x ∈ bd Ω,
β(u(t0 , x)) = β(u0 (x))
for all x ∈ Ω.
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1 Nonautonomous Dynamical Systems
Special cases of (nRDE) include reaction-diffusion equations (β(u) = u) as studied in Sect. 1.5.3, or the porous medium equation (β(u) = m |u| sgn u with m ∈ N) and we refer to [125, 126] for a survey of further applications. A theory to describe the initial-boundary value problem (nRDE) within the framework of dynamical systems has been developed in [126]. The following assumptions guarantee that these preparations can be applied. Hypothesis 1.5.10. We suppose the following assumptions: (i) β : R → R is an increasing continuous function, β(0) = 0, and there exist reals c1 , c2 > 0 such that |β(u)| ≤ c1 |u| + c2 for all u ∈ R. (ii) The mapping g : R × Ω × R → R is continuous and D1 g exists such that D1 g(t, x, ·) maps bounded sets into bounded sets. Furthermore, we assume that there exist c3 , c4 , c5 > 0 and p > 2 such that for all t ∈ R, x ∈ Ω and ξ ∈ R, g(t, x, ξ) sgn ξ ≤ c4 − c3 |ξ|p−1 ,
|g(t, x, ξ)| ≤ c5 (|ξ|p−1 + 1).
(iii) There exists a c6 > 0 such that c6 β − g(t, x, ·) is increasing for (t, x) ∈ R× Ω. Now assume that H = L2 (Ω) or H = H01 (Ω). Quoting from [126, Theorem 1] we know that the problem (nRDE) admits for arbitrary initial times t0 ∈ R and states u0 ∈ L2 (Ω) a unique (weak) solution u(·; t0 , u0 ) : (t0 , ∞) → H01 (Ω). We thus can define a 2-parameter semigroup on S = I × H by ϕ(k, l)u0 := u(tk ; tl , u0 ) for all l ≤ k, which satisfies a smoothing property with S0 = I × H01 (Ω). Having this at hand, we readily establish Proposition 1.5.11. Let H ∈ L2 (Ω), H01 (Ω) . If Hypothesis 1.5.10 holds, then the 2-parameter semigroup ϕ has the global attractor ωA ⊆ I × H with A = I × Bρ (0, H), where ρ > 0 depends on c1 , c6 > 0 and d ∈ N only. Proof. Suppose that Bˆ is the family of all uniformly bounded nonautonomous sets in I × H. We again make use of Theorem 1.3.9. Referring to the existence and uniqueness result [126, Theorem 1] we obtain that the 2-parameter semiflow u(t; t0 , ·) : H → H generated by (nRDE) is continuous, which carries over to ϕ. ˆ ˆ The estimate [126, Lemma 12] yields that ϕ is B-dissipative with B-absorbing set A ∈ Bˆ depending only on the data from Hypothesis 1.5.10. As previously in [126, Lemma 11] one sees that for each bounded B ⊆ H there exists a T = T (B) > 0 ˆ such that u(t; s, B) ⊆ H is compact for t − s ≥ T (B). Thus, ϕ is B-eventually ˆ compact and Corollary 1.2.22 yields that ϕ is B-asymptotically compact. Conseˆ quently, ωA is a B-attractor and since Bˆ consists of uniformly bounded sets, ωA is the global attractor of ϕ.
1.6 Remarks
33
1.6 Remarks For many results presented here, the essential assumption on the set I was that it is ordered; thus, the theory can be generalized to the setting where I is for instance a time scale (which is an arbitrary closed subset of the reals, cf. [204]). In particular, and as it is not surprising, there are continuous counterparts to the notions introduced in this chapter, like e.g., of 2-parameter semigroups or attractors, where the discrete interval I is replaced by some real interval. Since we are largely concerned with the corresponding discrete objects, we do not explicitly emphasize the notion of a discrete (semi-) group or a discrete (semi-) dynamical system. Nevertheless, in order to distinguish between the corresponding objects, we speak of (semi-) flows in the continuous time case. Nonautonomous sets and 2-parameter semigroups: The archetypical examples of 2-parameter groups are temporal discretizations of continuous dynamical systems, like flows generated by ordinary differential equations (cf. [9,113,270,435,447]) or wave equations (cf. [432, p. 115ff]) – this includes discretizations of the induced flows, as well as numerical discretizations, provided the stepsize is sufficiently small. A continuous mapping satisfying the properties from Definition 1.1.1 is also called (evolutionary) process – a terminology of [107] in the real case. Our approach via 2-parameter semigroups gives rise to the more general concept of a nonautonomous dynamical system (cf., e.g., [81, 258, 392]), which has its origin as an abstraction of both skew product flows (cf. [418, 429], [192, pp. 43–48]), as well as metric and random dynamical systems (cf. [12, p. 5, Definition 1.1.1]). Definition 1.6.1. Let P be a nonempty set and I = Z. A discrete nonautonomous dynamical system on a metric space X with base space P is a pair of mappings θ : I × P → P , Φ : I × P × X → X with the following properties: (i) The base flow θ satisfies the flow properties θ(0, p) = 0,
θ(k + l, p) = θ(k, θ(l, p)) for all p ∈ P, k, l ∈ I.
(ii) Φ is a cocycle over θ, i.e., for all k, l ∈ I and p ∈ P , x ∈ X we have Φ(0, p, x) = x,
Φ(k + l, p, x) = Φ(k, θ(l, p), Φ(l, p, x)).
(iii) Φ is continuous in the third argument. Provided P is a topological space, a frequent assumption is its compactness. Nevertheless, as seen from Definition 1.1.1, a continuous 2-parameter semigroup ϕ provides such a nonautonomous dynamical system on the base space P = Z via the base flow θ(k, l) := k + l and the cocycle Φ(k, l, ξ) := ϕ(k + l, l)ξ for all ξ ∈ X. Invariant and limit sets: The notion of a nonautonomous set as sequence of (timedependent) sets or spaces allows objects like, e.g., motions to be invariant. Our
34
1 Nonautonomous Dynamical Systems
approach to limit sets is based on the concept of pullback convergence, whose origins can be traced back to the work of [279, 362]; however, in a modern perspective it is due to [256, 268]. Differing from the seemingly more natural “forward convergence”, this notion has advantages, among them the invariance of limit sets. In this context, our Theorem 1.2.8 is taken from [265]. At first glance, the connectedness Proposition 1.2.19 might seem surprising. Indeed, in metric spaces (rather then linear spaces) the situation is more subtle: For autonomous continuous semigroups in connected metric spaces it is shown in [176] that if a global attractor is not connected, then it has infinitely many components, yet it is invariantly connected (meaning that it is not the disjoint union of two nonempty, compact, forward invariant sets). Moreover, in connected metric spaces which are also locally connected, the global attractor is connected. This reference also provides an example of a continuous map defined on a connected subset of R2 that generates an autonomous semigroup whose global attractor is disconnected. Our key “smoothing” property to derive nontrivial results on limit sets was asymptotic compactness. We generalized the corresponding notion for autonomous problems from [432, p. 28]. A related notion, which is typically used in the work of Hale (cf., e.g., [192, 194]), is that of an asymptotically smooth semigroup. The corresponding nonautonomous generalization reads as follows: ˆ Definition 1.6.2. A 2-parameter semigroup is B-asymptotically smooth, if for all ˆ forward invariant B ∈ B there exists a compact {B}-attracting nonautonomous set. In the autonomous case (and if the family Bˆ consists of all bounded sets) the following equivalent characterization of asymptotically compact semigroups is shown in [39, Proposition 2.3(ii)]: A semigroup on a Banach space is asymptotically compact, if and only if it is eventually bounded and asymptotically smooth. Our notion of a contracting semigroup is inspired from [432, p. 24] and has exactly the degree to generality to derive Proposition 1.2.30. Yet it differs from the synonymous autonomous notion in, for instance [192, 194], where the Kuratowski measures of noncompactness is used and the contraction property is formulated in terms of a Darbo condition. The equivalence of different smoothing properties, i.e., being asymptotically compact, contractive or so-called flattening, was shown in [261]. Attractors and global attractors: For autonomous dynamical systems the concept of an attractor, even in finite dimensions, has evolved over the last 40–50 years and we refer to [194] or [432, p. 55ff] for a survey. For autonomous dynamical systems, a comprehensive analysis of many concepts from this chapter can be found in [192, pp. 8–34, Chap. 2]. One of the possible attempts to extend corresponding notions to the nonautonomous case lead to so-called uniform attractors (see [85, 86]), which are subsets of a single state space. A disadvantage of uniform attractors is that they need not to be “invariant” like their autonomous counterparts (in the discrete periodic case, see also [168, 427]). An approach to tackle this problem is to define nonautonomous attraction using skew product dynamics, i.e., extending the state space (see [418, 429], [192, pp. 43–48] or [86, p. 71ff]). Here, the assumption of a
1.6 Remarks
35
compact base space is crucial, which bears the problem that one has to find a suitable topology for the time-translates of the system generating a skew product flow. We avoid such topologically subtle investigations. Indeed, our attractor notion (or of an attracting or absorbing set) is usually denoted as pullback attractor (or pullback attracting, resp., pullback absorbing set). This concept of attraction has been introduced in [256, 268] or [259]. It is possible to present the corresponding theory within the more general framework of nonautonomous dynamical systems from Definition 1.6.1. We do not pursue this approach here, but note that it turned out to be extremely fruitful, particularly in the case of random dynamical systems (cf. [101]). Therefore, pullback attractors are often referred to as cocycle attractors. For a variety of examples using this general setting, see [268]. A comparison of various (nonautonomous) attractor notions is given in [81] and extensions to setvalued dynamical systems are due to [264]. A thorough discussion of attractor and repeller notions in order to describe nonautonomous transitions and bifurcations, is due to [392, pp. 7–50, Chap. 2]. Morse decompositions provide a deeper insight into the “dynamically relevant” objects (i.e., Morse sets and their connecting orbits) of dynamical systems on compact state spaces, and in particular on attractors – for a general approach we refer to [393, 394]. We emphasize that for nonautonomous dynamical systems, whose attractors possess time-varying fibers, such investigations require time-dependent state spaces in a natural way. Referring to Theorem 1.3.12, attractors depend only upper-semicontinuously on parameters, which means that they can collapse under perturbations, but not blowup instantly. Such perturbations include time discretizations and the first results on the behavior of attractors under time discretization are due to [262] (see also [263]), while in the nonautonomous situation we refer to [78, 115, 173, 267]. Also in the finite-dimensional case, various robustness and perturbation results for attractors have been obtained in [182]. In infinite dimensions, [195] study the behavior of attractors for semiflows under discretizations. Under structural assumptions on the attractor, like e.g., exponential convergence, it is possible to derive results on lowersemicontinuous dependence; we refer to [447, pp. 555–562] for a survey and to [173] for corresponding results in varying-stepsize discretizations. Finally, in the autonomous case, a further attractor concept has been introduced in [124, pp. 9–24, Chap. 2], where the attraction rate is always assumed to be exponential. It circumvents certain drawbacks of (pullback) attractors, as possibly slow convergence of motions towards the attractor or lacking robustness properties. This concept can be generalized to our setting as follows: ˆ Definition 1.6.3. A compact nonautonomous set A∗ ⊆ S is called B-exponential attractor, if it is forward invariant, has fibers with finite fractal dimension and if for each k ∈ I, B ∈ Bˆ there exist C ≥ 0, α ∈ (0, 1) so that hS(k) (ϕ(k, k − n)B(k − n), A∗ (k)) ≤ Cαn
for all n ∈ Z+ 0.
36
1 Nonautonomous Dynamical Systems
An introduction to fractal dimensions can be found, for instance, in [453] or [86, p. 51ff]. In fact we totally neglect the relevant question about the dimensionality of attractors in these notes. Corresponding results for continuous semiflows can be found in [86, Chap. IX] or [127, 128]. Applications: The 2-parameter semigroups discussed in Sect. 1.5 are restrictions of 2-parameter semiflows generated by evolutionary differential equations – a survey on generators for nonlinear semigroups is [423], while we refer to [144, p. 478] for the linear (nonautonomous) situation needed in Hypothesis 1.5.4(i). References for the corresponding theories of attractors, etc., are vast in the autonomous setting and include [192, 194, 198, 201, 395, 432, 453]; nonautonomous theory is considered in [86, 337] and the upcoming monograph [79] contains a comprehensive approach to nonautonomous attractors and their structure. Clearly, the discretized 2-parameter semiflows inherit compactness properties from their continuous counterparts; examples read as follows: •
Compact semiflows are frequently generated by sectorial evolutionary equations (see [192, p. 73, Theorem 4.2.2]) and in particular reaction-diffusion equations over bounded domains (see [86, p. 116, Proposition 2.3] for the nonautonomous case). Examples include the Allen–Cahn equation (cf. [432, p. 272, Theorem 51.1]), equations of convection (see [432, p. 318, Theorem 53.4]), the Kuramoto–Sivashinsky equation (see [432, p. 328, Theorem 54.3]) or the Cahn–Hilliard equation (see [432, p. 334, Theorem 55.7]). Moreover, also the Navier–Stokes equations in 2d admit a compact semiflow (see [192, p. 111, Theorem 4.4.5] or [86, p. 109, Proposition 1.2] for a nonautonomous forced problem). • A typical example of an eventually compact semiflow is the solution operator of retarded functional differential equations, where the delay is the compactification time (see [192, p. 61, Theorem 4.1.1]). • Retarded functional differential equations with infinite delay give rise to contracting semiflows (see [192, p. 145ff]). Moreover, abstract nonlinear wave equations induce eventually bounded semiflows (see [432, p. 288, Theorem 52.3]), which are contracting (see [432, p. 291, Theorem 52.4]). • Asymptotically compact semiflows are generated by reaction-diffusion equations on Rn (see [333]) or 2d Navier–Stokes equation (see [76] for a nonautonomous equation). Let us point out that the class of functional differential equations (FDE) is sufficiently general to include ordinary differential equations (r = 0), retarded delay differential equations with bounded time-dependent delays, as well as certain integro-differential equations (cf. [198, p. 39]). For such more specific problems the dissipativity condition Hypothesis 1.5.1(ii) can be relaxed (see [75]). Moreover, differential equations with random delays are discussed in [74]. Finally, we like to mention differential equations with piecewise constant argument as a further area, where proofs are essentially based on discrete techniques (see [359, 360]).
Chapter 2
Nonautonomous Difference Equations
Difference equations are at the cusp between finite- and infinite-dimensional problems. George R. Sell, 2004
The long-time behavior of (autonomous) difference equations or discrete semidynamical systems exhibits various features, which occur in their continuous counterpart of (evolutionary) differential equations only for higher or even infinite-dimensional state spaces. This has various reasons, like absence of monotone solutions, lack of existence and uniqueness of backward solutions, or topological deficits caused by solution sequences rather than connected curves. As a consequence, besides further intricacies there are one-dimensional chaotic maps, there is no Poincar´e–Bendixson theory (cf., e.g., [9, p. 333, Theorem (24.6)]) for planar maps, no limit set dichotomy for monotone maps (see [212]) or a comparatively subtle linearization theory for discrete systems, which are not homeomorphisms (see also Chap. 5). This situation gets even amplified, when considering nonautonomous problems and explicitly time-dependent equations canonically appear as discrete models under external or controlling influences, or in form of varying time-step discretizations. Yet, as opposed to evolutionary differential equations, in many cases the existence of forward solutions for difference equations is trivially given. Apparently, explicit difference equations generate 2-parameter semigroups, where the right-hand side serves as generator. When it comes to applications, however, difference equations are frequently implicit. In particular, in numerical analysis only implicit schemes possess desired stability properties guaranteeing reliable approximations. Therefore, since we aim to apply our results to various discretizations, the central aspects of this chapter are as follows: •
We introduce a fairly general notion of a difference equation or an initial value problem, which is sufficiently flexible to include various problems like explicit or implicit one-step methods, Runge–Kutta schemes, higher order, delay and Volterra, as well as certain partial difference equations. Moreover, we illustrate how external driving canonically leads to nonautonomous problems. • Our tools to solve implicit difference equations are based on fixed point theorems, global inverse function theorems or variational methods in terms of monotone C. P¨otzsche, Geometric Theory of Discrete Nonautonomous Dynamical Systems, Lecture Notes in Mathematics 2002, DOI 10.1007/978-3-642-14258-1 2, c Springer-Verlag Berlin Heidelberg 2010
37
38
•
•
•
•
2 Nonautonomous Difference Equations
operators. Thus, in the latter case, they are modelled after corresponding results for nonlinear elliptic PDEs under monotonicity assumptions (cf., e.g., [87, p. 120ff, Chap. 4]). As a byproduct, under dissipativity conditions we get boundedness for forward solutions of implicit one-step methods. In order to apply the results from Chap. 1 it is of crucial importance to have criteria at hand, which ensure existence and uniqueness of forward solutions inducing 2-parameter semigroups in form of the so-called general (forward) solution. Beyond that we investigate how continuity and compactness properties carry over to the general solution. Having this at hand, we can infer conditions for difference equations to possess absorbing sets and attractors. Stability theory in a nonautonomous framework is more involved than in the autonomous case. On the one hand, we present (uniform) attraction and stability notions, but differing from the classical set-up, they are also applicable to solutions defined on the whole integer line. On the other hand, in order to be consistent with Chap. 1, we also investigate the concepts of pullback stability and attraction. Finally, the relationship between the different notions is addressed. Here, it turns out that stability notions based on pullback convergence are equivalent to classical uniform stability concepts. We will come back to these concepts in future chapters – the same holds for further central terminology introduced in Sect. 2.5, like characteristic exponents or the equation of perturbed motion. We discuss the above concepts in the special case of periodic and autonomous problems and show their equivalence simplifying the stability theory in the autonomous situation. Our applications continue the corresponding preparations from Sect. 1.5. We investigate full discretizations of retarded FDEs using θ-methods. Two discretization strategies are presented for abstract evolution equations: One is based on a 2-stage θ-method, while the other relies on mild solutions and the variation of constants formula – note that the latter is of eminent importance for theoretical investigations; a remark illuminates the special case of sectorial equations. For RDEs we discuss a wide class of full discretizations which include spectral Galerkin, finite element, or finite difference methods in space and θ-methods in time. The resulting finite-dimensional difference equations are shown to be uniformly bounded dissipative. For the complex Ginzburg–Landau equation under finite difference spatial and implicit Euler discretization we show the exis1 tence of absorbing sets in the discrete Sobolev spaces L2N and HN . Finally, a fully implicit Euler scheme is applied to doubly nonlinear RDEs, and it is shown that the corresponding semi-discretization has a uniformly bounded global attractor.
Throughout the chapter, let us suppose that I is a discrete interval, (Xk )k∈I , (Yk )k∈I are sequences of nonempty sets and S ⊆ Xdenotes a nonempty nonautonomous set; we again make use of the union X := k∈I Xk .
2.1 Basics and Examples
39
2.1 Basics and Examples We begin with a general notion for difference equations (recursions, iterations) using a convenient “variable-free” notation. Since we are interested in large time behavior, our concept of a difference equation is biased on the forward time. Definition 2.1.1. Suppose Hk : S(k) → Yk , k ∈ I, and Fk : S(k)×S (k) → Yk+1 , k ∈ I , are mappings. An equation of the form Hk+1 (x ) = Fk (x, x )
(D)
is called an (implicit nonautonomous) difference equation in S with left-hand side Hk and right-hand side Fk . A difference equation (D) is called semi-implicit, if Fk does not depend on x and a semi-implicit difference equation (D) is said to be explicit, if Yk = Xk for k ∈ I and Hk+1 (x ) = x . A sequence φ : J → X defined on a discrete interval J ⊆ I with φ(k) ∈ S(k) for k ∈ J is a solution of (D), if one has the solution identity Hk+1 (φ (k)) ≡ Fk (φ(k), φ (k))
on J .
In case J = I we speak of a complete solution. Next we turn to initial value problems. Definition 2.1.2. Let (κ, ξ) ∈ S and J ⊆ I be a discrete interval with κ ∈ J. A sequence φ : J → X with φ(k) ∈ S(k) for every k ∈ J is called a solution of the initial value problem Hk+1 (x ) = Fk (x, x ),
x(κ) = ξ,
(2.1a)
if φ is a solution of (D) satisfying φ(κ) = ξ. We say κ is the initial time, ξ the initial − state and x(κ) = ξ the initial condition. In case J ⊆ Z+ κ or J ⊆ Zκ we speak of a forward, resp., backward solution of (2.1a). Remark 2.1.3. One refers to (D) as first-order difference equation, since under ambient assumptions on Fk , Hk+1 , only the knowledge one previous value x ∈ S(k) is needed in order to compute the subsequent value x ∈ S (k). Compared to the terminology commonly used in the literature, like e.g., Hk+1 (x(k + 1)) = Fk (x(k), x(k + 1)),
Hk+1 (xk+1 ) = Fk (xk , xk+1 ),
our – at first glance somehow unusual – notation for difference equations dates back at least to [298]. It has the advantage of being compact, convenient and provides further benefits, as we will see later. Nonetheless, the following examples intend to show how general the framework of Definition 2.1.1 actually is. In order to embed certain types of equations into our setting, we deliberately notate them in a traditional fashion. In doing so we neglect the crucial problem to find a feasible topology on their state spaces.
40
2 Nonautonomous Difference Equations
Example 2.1.4 (one-step methods). Let X consist of linear spaces so that the embedding Xk ⊆ Xk+1 , k ∈ I , holds. With a mapping Gk : S(k) × S (k) → Xk+1 , a one-step method is a difference equation of the form (O)
xk+1 = xk + Gk (xk , xk+1 );
it fits into the general set-up of (D) with Hk+1 (x ) = x , Fk (x, x ) = x + Gk (x, x ) and Xk = Yk for all k ∈ I . Here, Gk is called step function. Suppose S is convex and fk : S (k) → Xk+1 . For given θ ∈ [0, 1], the θ-method is a particular one-step method with Gk (x, x ) = fk ((1 − θ)x + θx ). The special case for θ = 1/2 is called midpoint rule. The specific relevance of one-step methods has two reasons: First, numerical schemes for temporal discretizations of differential equations typically lead to onestep methods. They also can be written in the form Δxk = Gk (xk , xk+1 ), where Δxk := xk+1 − xk is the forward difference operator. Second, for such equations many methods originally developed for differential equations, where Δxk substitutes the time derivative x(t), ˙ carry over to our discrete case, as demonstrated systematically using the “time scales calculus” (cf. [204]). Remark 2.1.5. Even in case a one-step method (O) is explicit, its forward solutions might not exist on I+ κ , since IXk + Gk needs not to have images in S (k) for k ∈ I . Example 2.1.6 (Runge–Kutta methods). An important special case of general onestep methods (O) are Runge–Kutta methods. In order to define them, we prescribe mappings fk1 , . . . , fks : S (k) → Xk+1 , k ∈ I , where s ∈ N is the so-called stage, as well as reals aij , bi for i, j ∈ {1, . . . , s}. Then a Runge–Kutta method is a onestep method with step function Gk (x, x ) :=
s
bi fki (gi ).
i=1
Given x ∈ S(k), the vectors g1 , . . . , gs ∈ S (k) are solutions of the equations gi = x +
s
aij fkj (gj )
for all i ∈ {1, . . . , s} .
j=1
These solutions are uniquely determined for explicit Runge–Kutta methods, where aij = 0 for i ≤ j. As a result, forward solutions to one-step methods (O) generated by explicit Runge–Kutta methods exist uniquely if gi ∈ S (k). It became customary and is convenient to symbolize Runge–Kutta methods using the Butcher tableau fk1 .. .
a11 . . . a1s .. .. . .
fks
as1 . . . ass b1 . . . bs
or shorter
f A b.
(2.1b)
2.1 Basics and Examples
41
The explicit (or forward) Euler method is a 1-stage explicit Runge–Kutta method given by the Butcher tableau fk1 0 1
xk+1 = xk + fk1 (xk ),
(2.1c)
where the one-step method (O) is an explicit difference equation in the sense of Definition 2.1.1 with Hk+1 (x ) = x and Fk (x, x ) = x + f 1 (x). The implicit (or backward) Euler method allows the representation fk1 1 1
xk+1 = xk + fk1 (xk+1 )
(2.1d)
and (O) becomes a semi-implicit difference equation with Hk+1 (x ) = x − f 1 (x ) and Fk (x, x ) = x. Obviously, its backward solutions exist and are uniquely determined. A 2-stage θ-method is a Runge–Kutta method depending on a parameter θ ∈ [0, 1] with Butcher tableau fk1 0 0 fk2 1 − θ θ 1−θ θ
xk+1 = xk + (1 − θ)fk1 (xk ) + θfk2 (xk+1 ).
(2.1e)
As for the θ-method from Example 2.1.4, one can switch between the explicit (θ = 0) and implicit (θ = 1) Euler method. However, (2.1e) can be written as semi-implicit equation (D) with Hk+1 (x ) = x − θfk2 (x ) and Fk (x, x ) = x + (1 − θ)fk1 (x). For θ = 12 one speaks of the trapezoidal rule, which in the context of finite difference spatial discretizations of heat equations is also called Crank–Nicholson method. Remark 2.1.7. Whereas we deliberately abandoned a connection to differential equation in Example 2.1.6, commonly Runge–Kutta methods are attached to differential equations x˙ = f (t, x). Here, the functions fki are given in terms of the right-hand side f via fii = hk f (tk + ci hk , ·) with a given discretization mesh (tk )k∈I , stepsizes hk := tk+1 − tk , and prescribed reals ci ∈ [0, 1], 1 ≤ i ≤ s. Example 2.1.8 (higher order difference equations). Given d ∈ Z+ 2 with #I > d, let us suppose (Zk )k∈I is a sequence of nonautonomous sets generating a nonautonomous set Z with the embedding Zk ⊆ Zk+1 for all k ∈ I . With mappings
×Z d
fk :
k+i
→ Zk+d
for all k ∈ I such that k + d ∈ I,
i=0
a so-called d-th order difference equation reads as zk+d = fk (zk , zk+1 , . . . , zk+d ).
(2.1f)
42
2 Nonautonomous Difference Equations
In order to determine the value zk+d ∈ Zk+d , one needs to know d previous values zk , . . . , zk+d−1 . Problems of the form (2.1f) can be embedded into first order sysd−1 tems (D) as follows: We define the spaces Xk = Yk := i=0 Zk+i and mappings
×
Hk+1 (x ) := (x0 , . . . , xd−1 ),
Fk (x, x ) := (x1 , . . . , xd−1 , fk (x0 , . . . , xd−1 , xd−1 )).
Having fixed these objects, assume that J ⊆ I is a discrete interval. A sequence (zk )k∈J in Z is a solution of the d-th order equation (2.1f) satisfying the initial condition zκ+i = ζi ∈ Zκ+i for i = 0, . . . , d − 1, if and only if the sequence φ : J → X, φ(k) := (zk , . . . , zk+d−1 ) solves the first-order initial value problem (2.1a) with the initial condition x(κ) = (ζ0 , . . . , ζd−1 ). Note that (D) is explicit, provided fk does not depend on its last argument. Example 2.1.9 (linear multistep methods). Let d ∈ N with #I > d and suppose Z is a linear space. A linear multistep method is a d-th order difference equation with fk (z0 , z1 , . . . , zd ) =
d i=0
where αi , βi are reals, i ∈ {1, . . . , d} and
αi zi −
d
βi gki (zi ),
i=0
gk1 , . . . , gkd
: Z → Z.
Example 2.1.10 (delay difference equations). Let (Zk )k∈I be a sequence of nonempty sets giving rise to a nonautonomous set Z. By virtue of an index shift, the same procedure as in Example 2.1.8 is also valid for delay difference equations of the form zk+1 = fk (zk+1 , zk , zk−1 , . . . , zk−d ) with constant delay d ∈ N. More general, applying this technique to problems zk+1 = fk (zk+1 , zk , zk−1 , . . . , zk−d(k) ),
(2.1g)
with right-hand side fk : Zk+1 × Zk × . . .× Zk−d(k) → Zk+1 and also time-varying delays d : I → N satisfying d (k) ≤ 1 + d(k)
for all k ∈ I ,
(2.1h)
canonically leads to state spaces changing in time. In order to compute zk+1 one has to know zk , . . . , zk−d(k) and (2.1h) guarantees that this condition is also fulfilled when moving on from k to k+1. Moreover, note that (2.1h) enforces at most linearly growing delays, i.e., it holds d(k) ≤ k − κ + d(κ) for all k ≥ κ. Define the product spaces Xk := Zk ×. . .×Zk−d(k) , which obviously do depend on k, even if the sets Zk are constant. Choose κ ∈ I such that κ − d(κ) ∈ I. Then a sequence (zk )k∈I+ is a solution of (2.1g) satisfying the initial condition κ−d(κ)
zκ = ζ0 , . . . , zd(κ) = ζκ−d(κ) , if and only if φ : I+ κ → X, φ(k) = (zk , . . . , zk−d(k) ) solves the initial value problem (2.1a) with Hk+1 (x ) := (x0 , . . . , xd (k) ),
Fk (x, x ) := (fk (x0 , x), x0 , . . . , xd (k)−1 )
2.1 Basics and Examples
43
equipped with the initial value condition x(κ) = (ζκ , . . . , ζκ−d(κ) ). The resulting first-order equation (D) is explicit, if fk does not depend on its first argument. Example 2.1.11 (Volterra difference equations). Let n0 ∈ Z and I = Z+ n0 . A particular class of delay difference equations are Volterra difference equations xk+1 =
k
fn (k, xn )
n=n0
with given functions fn : I × X → X and a linear space X. The value xk+1 can be computed, provided xn0 , . . . , xk are known. Such Volterra equations can be written as explicit nonautonomous difference equations of the form (D) with constant Yk ≡ X, but time-variable Xk = X k−n0 +1 , a left-hand side Hk given by the embedding operator Hk+1 (xk+1 , xk , . . . , xn0 ) = xk+1 and right-hand side Fk (xk , . . . , xn0 ) =
k
fn (k, xn ).
n=n0
For n0 = −∞ one speaks of a Volterra difference equation with infinite delay, and as initial value to the initial time κ ∈ I a whole sequence (ξk )k≤κ is required. Example 2.1.12 (partial difference equations). Let d ∈ N be the spatial dimension, I be a discrete interval, U be a set and G ⊆ Zd be nonempty – one speaks of a grid or a lattice G. Then an (evolutionary) partial difference equation is an equation uk+1,n = fk (n, (uk,n )n∈G , (uk+1,n )n∈G ) for all n ∈ G
×
(2.1i)
×
with fk : G × n∈G U × n∈G U → U for every k ∈ I , where we interpret k as temporal and n ∈ G as spatial variable. In order to make initial value problems for this recursion well-posed, one has to equip (2.1i) with initial and boundary conditions uκ,n = ξn ∈ U uk,n = ηn ∈ U
for all n ∈ G, for all k ∈ I, n ∈ G0 ,
G G0
respectively, where the subset G0 ⊆ G is the set of “boundary points” of G. Indeed, also the partial difference equation (2.1i) fits in the framework of (D), if we define the constant state space X := {(un )n∈G : un = ηn
for all n ∈ G0 }
and suppose (fk (n, X, X))n∈G ∈ X for every k ∈ I . Then (2.1i) can be written as implicit first order difference equation xk+1 = Fk (xk , xk+1 ) with right-hand side Fk : X × X → X given by Fk (x, y) = (fk (n, x, y))n∈G .
44
2 Nonautonomous Difference Equations
Our final example motivates the study of nonautonomous equations: Example 2.1.13 (driven difference equations). Let I be a discrete interval, P, X be nonempty sets and consider the difference equation x = f (p, x) depending on a parameter p ∈ P . When the fixed value of p is replaced by a sequence (pk )k∈I we speak of a parametrically perturbed difference equation x = f (pk , x).
(2.1j)
For less arbitrary perturbations one prescribes a mapping θ : P → P , which in case of one-sided time I = Z+ 0 needs not to be invertible, and considers x = f (θk p, x);
(2.1k)
the iterates θk are called driving system. No assumption is made about the domain P , nor properties of the map θ. In applications P is typically a metric space (deterministic systems) or a measure space (random systems), with θ : P → P being continuous in the former and measurable in the latter. Both the fruitful concepts (2.1j) and (2.1k) fit in the framework of our explicit difference equations with right-hand side Fk = f (pk , ·) resp. Fk = f (θk p, ·).
2.2 Existence and Boundedness of Solutions We are interested in the existence of solutions for an equation Hk+1 (x ) = Fk (x, x )
(D)
as in Definition 2.1.1, and of corresponding initial value problems Hk+1 (x ) = Fk (x, x ),
x(κ) = ξ,
(2.2a)
with given initial value pair (κ, ξ) ∈ S. The general structure of (D) makes it difficult to obtain universally applicable criteria for the existence of solutions. Therefore, we retreat to a series of special cases: For explicit equations x = Fk (x) the existence of forward solutions is trivially satisfied, provided Fk has values in S (k), whereas backward solutions exist under surjectivity conditions on Fk . • For semi-implicit equations Hk+1 (x ) = Fk (x) forward or backward solutions exist under surjectivity properties of Hk+1 or Fk , respectively. • Finally, for a general equation (2.2a) with invertible left-hand side Hk+1 the −1 existence of forward solutions follows from fixed point results applied to Hk+1 ◦ Fk (x, ·). •
2.2 Existence and Boundedness of Solutions
45
The subsequent results will make these remarks more specific. The situation of backward solutions is essentially captured in the following simple Proposition 2.2.1. Let k ∈ I, inf I < κ and suppose Fk (·, x ) : S(k) → Hk+1 (S (k))
(2.2b)
is onto for all k < κ − 1, x ∈ S (k): (a) Under the assumption Hκ (ξ) ∈ Fκ−1 (S(κ − 1), ξ), there exists a backward solution φ : I− κ → X of (2.2a) in S. (b) If there exists a unique x ∈ S(κ − 1) with Hκ (ξ) = Fκ−1 (x, ξ) and (2.2b) is also one-to-one for all k < κ − 1, x ∈ S (k), then there exists a unique backward solution φ : I− κ → X of (2.2a). It is recursively given by φ(k) =
ξ −1 Fk+1
for k = κ, Hk+1 (φ (k)), φ (k) for k < κ,
where Fk−1 (·, x ) denotes the inverse function of Fk (·, x ) : S(k) → Yk . Remark 2.2.2. When S and Y consist of metric spaces, a typical assumption guaranteeing that (2.2b) is one-to-one, is that Fk (·, x ), x ∈ S (k), is expansive, i.e., for each k ∈ I there exist γk > 0 such that γk dS(k) (x, x¯) ≤ dYk+1 (Fk (x, x ), Fk (¯ x, x )) for all k < κ − 1, x, x ¯ ∈ S(k). Proof. (a) Our assumptions guarantee that there exists a point xκ−1 ∈ S(κ − 1) with Hκ (ξ) = Fκ−1 (x, ξ) and we set φ(κ − 1) := xκ−1 . Since the mapping given in (2.2b) is onto, we inductively construct a sequence (xk )k≤κ in S so that Hk+1 (xk+1 ) = Fk (xk , xk+1 ) for k < κ and get the backward solution φ(k) := xk . (b) Here, the point xκ−1 ∈ S(κ − 1) constructed in (a) is unique. Moreover, the mapping Fk (·, x ) : S(k) → Hk+1 (S (k)) is bijective. Thus, the sequence (xk )k≤κ from (a) is uniquely determined; we set φ(k) := xk .
Now we consider forward solutions starting at time κ ∈ I . They exist and are uniquely determined for explicit initial value problems. A more general situation, which often occurs in applications is that Hk+1 : S (k) → Yk+1
is one-to-one for all k ∈ (I+ κ) ,
Fk (S(k), S (k)) ⊆ Hk+1 (S (k))
for all k ∈ (I+ κ)
(2.2c)
(cf. Example 2.1.8, Example 2.1.10 with constant delay or Example 2.1.12). Moreover, it holds for instance in linearly implicit discretizations, but further criteria for (2.2c) to be true can be found in Sect. B.3. Then the existence of forward solutions
46
2 Nonautonomous Difference Equations
−1 reduces to a fixed point problem. In addition, the inverse Hk+1 is frequently a completely continuous mapping. We can deduce the next result, which trivially holds for semi-implicit equations:
Proposition 2.2.3. Let X consist of Banach spaces and suppose that S consists of bounded, closed and convex sets. If (2.2c) holds and the compositions −1 Hκ+1 ◦ Fκ (ξ, ·) : S (κ) → S (κ), −1 Hk+1 ◦ Fk (x, ·) : S (k) → S (k) for all k > κ, x ∈ S(k)
are χ-set contractions, then (2.2a) has a forward solution φ : I+ κ → X. Proof. Given ξ ∈ S(κ) we can apply Darbo’s Theorem B.2.1 to the fixed point −1 problem xκ+1 = Hκ+1 ◦ Fκ (ξ, xκ+1 ). This yields a point xκ+1 ∈ S (κ) and of fixed point for xk = Hk−1 ◦ successively one constructs a sequence (xk )k∈I+ κ Fk−1 (xk−1 , xk ) yielding the desired forward solution via φ(k) := xk .
An even more simple situation is met for semi-implicit equations. Proposition 2.2.4. If (D) is semi-implicit and (2.2c) holds, then (2.2a) has a unique forward solution φ : I+ κ → X; it is recursively given by φ(k) =
ξ Hk−1 (Fk−1 (φ(k
for k = κ, − 1)) for k > κ.
Proof. The justification is obvious.
Fixed point theorems have been an appropriate tool to prove existence of solutions for initial value problems in general form (2.2a). Now we focus on one-step methods from Example 2.1.4 and the corresponding initial value problems x = x + Gk (x, x ),
x(κ) = ξ
(2.2d)
for given (κ, ξ) ∈ S. In particular, each Xk is a linear space. In this setting, one needs different tools from nonlinear analysis, namely results to solve nonlinear equations. Such problems occur when dealing with fully-implicit discretizations. We suppose throughout that S ⊆ X consists of inner product spaces with the embedding S(k) ⊆ S (k), k ∈ I . Theorem 2.2.5. Provided the step functions Gk : S(k) × S (k) → S (k) satisfy for all k ∈ I that (i) IS (k) − Gk (x0 , ·) is completely continuous for all x0 ∈ S(k), (ii) with reals αk > −1, βk ≥ 0, γk ≤ 1 one has the dissipativity condition 2 Gk (x0 , x), xXk+1 ≤ βk − αk x2Xk+1 − γk x0 2Xk for all x0 ∈ S(k), x ∈ S (k),
(2.2e)
2.2 Existence and Boundedness of Solutions
47
+ then (2.2d) has a forward solution φ : I+ κ → X and for all k ∈ Iκ it satisfies
φ(k)Xk
k−1 2 ≤ e 1−γ (k, κ) ξXκ + 1+α
l=κ
βl e 1−γ (k, l + 1). 1 + αl 1+α
(2.2f)
Proof. We proceed in two steps: (I) Let k ∈ I and x0 ∈ S(k), x ∈ S (k) be fixed. It is our goal to apply Proposition B.2.3 to the mapping Tk : S (k) → S (k), Tk (x) := 2Gk (x0 , x) − 2(x−x0 ). By assumption, the mapping Tk is well-defined, continuous and for every r > 0 the image Tk (Br (0)) ⊆ S (k) relatively compact. Moreover, one has − Tk (x), x = 2 x2 − 2 x0 , x − 2 Gk (x0 , x), x 2
2
2
and thanks to the identity x − x0 = x + x0 − 2 x, x0 , taking the real part in this relation leads to 2
2
2
2
− Tk (x), x = 2 x + x − x0 − x − x0 − 2 Gk (x0 , x), x ≥ x2 − x0 2 − 2 Gk (x0 , x), x (2.2e)
≥ x2 − x0 2 + αk x2 + γk x0 2 − βk
2 (1 − γk ) x0 + βk 2 . = (1 + αk ) x − 1 + αk
2 0 +βk +ε For any ε > 0 choose r(ε) := (1−γk )x and thus for x ∈ bd Br(ε) (0) 1+αk one gets − Tk (x), x > 0. By Proposition B.2.3 this implies that the nonlinear equations Tk (x) = 0 and in turn x−Gk (x0 , x) = x0 admit a solution x ∈ Br(ε) (0). (II) Given an initial value pair (κ, ξ) ∈ S, by step (I) there exists a φ (κ) ∈ S (κ) such that φ (κ) − Gk (ξ, φ (κ)) = ξ holds and the existence of a forward solution φ : I+ κ → X follows by induction. Moreover, in each induction step we can choose r(ε) =
(1−γk )φ(k)2 +βk +ε 1+αk
and due to φ (k) ≤ r(ε) for all ε > 0 we can pass 2
to the limit ε 0 and obtain φ (k) ≤ (1−γk )φ(k) 1+αk the estimate (2.2f) follows from Proposition A.2.3(a).
2
+βk
for all k ∈ (I+ κ ) . Then
While the above theorem is tailor-made for implicit discretizations, the following result is appropriate in the explicit case. Yet, it requires a linearly bounded step function Gk and a uniform dissipativity assumption. Proposition 2.2.6. Suppose J ⊆ I denotes a discrete interval. If the step functions Gk : S(k) × S (k) → S (k) satisfy for all k ∈ J that (i) with reals αk , βk ≥ 0 one has the dissipativity condition 2
2 Gk (x, x ), xXk+1 ≤ βk − αk xXk
for all x ∈ S(k), x ∈ S (k),
48
2 Nonautonomous Difference Equations
(ii) there exist reals γk , δk ≥ 0 with Gk (x, x )Xk+1 ≤
2 δk + γk xXk
for all x ∈ S(k), x ∈ S (k),
(iii) 0 ≤ αk ≤ 1 + γk , then every solution φ : J → X of (O) fulfills for κ ≤ k that φ(k)Xk
k−1 2 ≤ e1−α+γ (k, κ) φ(κ)Xκ + e1−α+γ (k, l + 1)(βl + δl ). l=κ
Proof. For a given solution φ : J → X to the initial value problem (2.2d) we define the real-valued sequence ρ(k) := φ(k)2 and obtain the identity ρ (k) − ρ(k) ≡ φ (k), φ (k) − φ(k) + φ (k) − φ(k), φ(k) (2.2d)
≡ φ(k), Gk (φ(k), φ (k)) + Gk (φ(k), φ (k)), φ(k)
+ Gk (φ(k), φ (k))
2
on J . Passing over to the real parts, our assumptions imply ρ (k)2 ≤ (1 − αk + γk )ρ(k) + βk + δk
for all k ∈ J .
Thus, Proposition A.2.3(a) is applicable to this inequality yields our assertion.
Now suppose the difference equation in (2.2d) is a θ-method as introduced in Example 2.1.4. Thus, S consists of convex sets and we deal with one-step methods x = x + fk ((1 − θ)x + θx )
(2.2g)
with a function fk : S (k) → Xk+1 and a parameter θ ∈ [0, 1]. The next results address the behavior of forward solutions for (2.2g), provided they exist: Proposition
2.2.7. Suppose J ⊆ I denotes a discrete interval and let α ≥ 0, β > 0, θ ∈ 12 , 1 . If the functions fk : S (k) → S (k) fulfill the dissipativity condition 2
for all x ∈ S (k), k ∈ J , α , sup fk (x)Xk+1 < ∞ with ρ := R := sup β ¯ ρ (0,S (k)) k∈J x∈B
fk (x), xXk+1 ≤ α − β xXk+1
then every solution φ : J → X of (2.2g) satisfies for all k ∈ J that ¯ρ+(1−θ)R (0, S(k)) φ(k) ∈ B
⇒
¯ρ+(1−θ)R (0, S (k)). φ (k) ∈ B
(2.2h)
2.2 Existence and Boundedness of Solutions
49
1
Proof. Let k ∈ J and θ ∈ 2 , 1 . For a given solution φ of (2.2g) we define a convex sum φθ (k) := (1 − θ)φ(k) + θφ (k). Taking the inner product of 1 1 (φ (k) − φ(k)) , φ (k) = (φ (k) + φ(k)) + θ − 2 2 θ
fk (φθ (k)) = φ (k) − φ(k) yields the identity 1 1 1 2 2 2 fk (φθ (k)), φθ (k) = φ (k) − φ(k) + (θ − ) φ (k) − φ(k) 2 2 2 and due to θ ≥
1 2
using (2.2h) one gets 2 2 2 φ (k) − φ(k) ≤ 2α − 2β φθ (k) .
(2.2i)
For each k ∈ J one of the following two cases holds true: 2 • α − β φθ (k) ≥ 0: This is equivalent to φθ (k) ≤ ρ and we obtain φ (k) ≤ φθ (k) + (1 − θ) φ (k) − φ(k) ≤ ρ + (1 − θ)R, •
2 α − β φθ (k) ≤ 0: For φ(k) ≤ ρ + (1 − θ)R this implies φ (k)
≤ 2α + φ(k)2 − 2β φθ (k) ≤ (ρ + (1 − θ)R)2 ,
2 (2.2i)
and this gives us the assertion.
Proposition
2.2.8. Suppose J ⊆ I denotes a discrete interval and let αk ≥ 0, β > 0, θ ∈ 12 , 1 . If the functions fk : S (k) → S (k) fulfill the dissipativity condition fk (x), xXk+1 ≤ αk − β x2Xk+1
for all x ∈ S (k), k ∈ J ,
then every solution φ : J → X of (2.2g) satisfies φ(k)Xk where γ :=
2αk 2 ≤ γ k−κ φ(κ)Xκ + 1−γ
1+β(1−θ)|2−3θ| 1+βθ(3θ−1)
for all k ≥ κ,
(2.2j)
∈ (0, 1).
Remark 2.2.9. As the proof below shows, we can replace 2αk better (i.e., smaller) constant (1−γ)[1+βθ(3θ−1)] .
2αk 1−γ
in (2.2j) with the
50
2 Nonautonomous Difference Equations
1
Proof. Let κ, k ∈ J with κ ≤ k, θ ∈ 2 , 1 , φ : J → X be a solution of (2.2g) and we use the notation from the proof of Proposition 2.2.7. The elementary estimate (0.0e) helps us to infer (2.2i) 2 2 2 φ (k) − φ(k) ≤ 2αk − 2β φθ (k) 2 2 ≤ 2αk − 2β (1 − θ)2 φ(k) + θ(1 − θ) φ(k), φ (k) + θ2 φ (k) 2
≤ 2αk + β(1 − θ)(3θ − 2) φ(k)2 + βθ(1 − 3θ) φ (k) , which, in turn, leads to (note θ ∈ 2
φ (k) ≤
1
2, 1
)
1 + β(1 − θ) |2 − 3θ| 2αk 2 φ(k) + 1 + βθ(3θ − 1) 1 + βθ(3θ − 1)
for all κ ≤ k.
Some elementary considerations show 1+β(1−θ)|2−3θ| < 1 and the Gronwall in1+βθ(3θ−1) equality in Proposition A.2.3(a) yields the assertion (2.2j).
Now suppose the difference equation in (2.2d) is a Runge–Kutta method. Proposition 2.2.10. Suppose J ⊆ I denotes a discrete interval and let αk ≥ 0, β > 0. If the equation in (2.2d) is a s-stage Runge–Kutta method with Butcher tableau (2.1b), the functions fk1 , . . . , fks : S (k) → S (k) satisfy the dissipativity condition 2
fi (k, x), xXk ≤ αk − β xXk
for all x ∈ S(k), k ∈ J , 1 ≤ i ≤ s
and there exists a γ ∈ (0, 1) such that the block matrix M (γ) :=
2βeT BA γ + 2β − 1 2βAT Be 2βAT BA + BA + AT B − bbT
∈ R(s+1)×(s+1)
is positive semi-definite, then every solution φ : J → X of (2.2d) satisfies (2.2j), where we define e := (1, . . . , 1)T ∈ Rs and B := diag(b1 , . . . , bs ). Proof. Let φ be a solution of (2.2d), choose κ, k ∈ I with κ ≤ k and by assumption we can fix γ ∈ (0, 1) such that M (γ) is positive semi-definite. Taking the inner product of (2.2d) with itself leads to the estimate 2
φ (k) − γ φ(k)2 − 2αk ≤ −m00 φ(k)2 −
s
mi0 fi (k, gi ), φ(k)
i=1
−
s i=1
m0i φ(k), fi (k, gi ) −
s s i=1 j=1
mij fi (k, gi ), fj (k, gj ) ,
2.3 Difference Equations and 2-Parameter Semigroups
51
where M (γ) = (mij )si,j=0 . The positive semi-definiteness of M (γ) implies the 2 2 relation φ (k) ≤ γ φ(k) + 2αk for all k ∈ J and the Gronwall inequality from Proposition A.2.3(a) yields the assertion.
Example 2.2.11. If the Runge–Kutta method in Proposition 2.2.10 is a 2-stage θ-method, then the matrix M (γ) is of the form ⎛ ⎞ γ + 2β − 1 2β(1 − θ)θ 2βθ2 M (γ) = ⎝2β(1 − θ)θ (1 − θ)2 (2βθ − 1) 2β(1 − θ)θ2 ⎠ ∈ R3×3 , 2βθ2 2β(1 − θ)θ2 (2βθ + 1)θ2 whose principle minors read as γ +2β −1, (θ−1)2 [−4β 2 θ2 +(2βθ−1)(γ +2β −1)] and −θ2 (1 − θ)2 (γ + 2β − 1). Therefore, the only 2-stage θ-method satisfying the assumptions of Proposition 2.2.10 is the implicit Euler method, where we have θ = 1.
2.3 Difference Equations and 2-Parameter Semigroups We return to general difference equations Hk+1 (x ) = Fk (x, x )
(D)
as in Definition 2.1.1. Extending the approach from Sect. 2.2 we are now interested in the problem of well-posedness for (D), meaning the situation where (forward) solutions to initial value problems for (D) exist uniquely and have further convenient properties. Particularly, in order to study the dependence of their solutions on initial conditions, the concept of a general solution becomes convenient. Definition 2.3.1. If for every initial value pair (κ, ξ) ∈ S there exists a unique forward solution φ : I+ κ → X of (D) in S satisfying the initial condition x(κ) = ξ, we set ϕ(·; κ, ξ) := φ and denote ϕ : {(k, κ, ξ) ∈ I × S : κ ≤ k, ξ ∈ S(κ)} → X as general (forward) solution of (D) on S. In case each solution φ exists uniquely on the whole interval I, we denote ϕ : I × S → X as general solution of (D). Proposition 2.3.2. Let (κ, ξ) ∈ S and k, l ∈ I. (a) The general forward solution ϕ of (D) satisfies the semigroup property ϕ(k; κ, ξ) = ϕ(k; l, ϕ(l; κ, ξ)) for all κ ≤ l ≤ k.
(2.3a)
(b) The general solution ϕ of (D) satisfies the group property ϕ(k; κ, ξ) = ϕ(k; l, ϕ(l; κ, ξ)) for all κ, k, l ∈ I.
(2.3b)
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2 Nonautonomous Difference Equations
Remark 2.3.3. (1) The notion of a general backward solution is defined analogously to Definition 2.3.1, provided backward solutions φ : I− κ → X exist uniquely – we refer to Proposition 2.2.1 for corresponding sufficient conditions. The general backward solution defines a backward 2-parameter semigroup. (2) If ϕ is the general (forward, backward) solution to (D), then the (forward, backward) solutions of the explicit equation x = ϕ(k+1; k, x) and of (D) coincide. Moreover, a solution to (D) is a motion for the 2-parameter semigroup ϕ. (3) Provided the general solution ϕ to (D) exists, then the two mappings ϕ(k; κ, ·) : S(κ) → S(k), κ, k ∈ I, and ϕ(κ; k, ·) are inverse to each other. Proof. Let (κ, ξ) ∈ S be given. (a) For any l ∈ I+ κ the sequences φ1 := ϕ(·; κ, ξ) and φ2 := ϕ(·; l, ϕ(l; κ, ξ)) solve (D) and satisfy the initial condition x(l) = φi (l), i = 1, 2; in addition, they are uniquely determined forward solutions to (D), which yields (2.3a). (b) This can be shown analogously.
Thanks to Proposition 2.2.4, the general forward solution of a semi-implicit equation Hk+1 (x ) = Fk (x) (D ) with invertible left-hand side Hk+1 is a 2-parameter semigroup having the generator −1 ϕˆk = Hk+1 ◦Fk , k ∈ I . More general, in case the general forward solution ϕ of (D) exists, it is an immediate consequence of Proposition 2.3.2 that it is a 2-parameter semigroup as discussed in Chap. 1. Analogously, the general backward solution is a backward 2-parameter semigroup. Having this close relationship between difference equations and 2-parameter semigroups at our disposal, we apply terminology from Chap. 1 to nonautonomous equations (D). If I is unbounded below, one can speak of a compact, contracting, dissipative, etc., (D). Here, a somewhat necessary assumption is the invertibility condition Hk+1 : S (k) → Yk+1
is one-to-one for all k ∈ (I+ κ) ,
Fk (S(k), S (k)) ⊆ Hk+1 (S (k))
for all k ∈ (I+ κ)
(2.3c)
which, for instance, is met for linearly implicit discretizations. Corollary 2.3.4. Suppose that (2.3c) holds. A sequence (ϕˆk )k∈I is the generator −1 of ϕ, if and only if ϕˆk (x) ≡ Hk+1 ◦ Fk (x, ϕˆk (x)) holds on S(k), k ∈ I . Proof. This is clear.
Proposition 2.3.5. Suppose that (2.3c) holds and (ϕˆk )k∈I is the generator of ϕ. With respect to (D), a nonautonomous set A ⊆ S is (a) forward invariant, if and only if Fk (A(k), ϕˆk (A(k)) ⊆ Hk+1 (A (k)), (b) backward invariant, if and only if Hk+1 (A (k)) ⊆ Fk (A(k), ϕˆk (A(k)), (c) invariant, if and only if Hk+1 (A (k)) = Fk (A(k), ϕˆk (A(k)) is satisfied for all k ∈ I .
2.3 Difference Equations and 2-Parameter Semigroups
53
Proof. Let k ∈ I and we have ϕˆk = ϕ(k + 1; k, ·). (a) For a forward invariant A one has ϕˆk (A(k)) ⊆ A (k) and Corollary 2.3.4 −1 implies Hk+1 ◦ Fk (A(k), ϕˆk (A(k)) ⊆ A (k). These implications are reversible. (b) and (c) can be shown analogously.
Provided (2.3c) holds true, unique existence and further properties of the general forward solution for (D) can be deduced from fixed point theorems. Theorem 2.3.6. Let X consist of metric spaces and S consist of complete subsets. −1 If (2.3c) holds and Hk+1 ◦ Fk (x, ·) : S (k) → S (k) fulfills a Lipschitz condition −1 ◦ Fk (x, ·) < 1 for all k ∈ I , x ∈ S(k), lip Hk+1
(2.3d)
then the following holds: (a) The general forward solution ϕ of (D) exists on S. −1 (b) If Hk+1 ◦ Fk (·, x ) : S(k) → S (k) is continuous for all k ∈ I , x ∈ S(k) and (2.3d) holds uniformly in x ∈ S (k), then also ϕ is continuous. −1 (c) If lip1 Hk+1 ◦ Fk < ∞ and (2.3d) holds uniformly in x ∈ S (k), then lip ϕ(k; κ, ·) ≤ ec (k, κ) for all κ ≤ k with c(k) :=
−1 lip1 Hk+1 ◦Fk
−1 1−lip2 Hk+1 ◦Fk
.
Remark 2.3.7. (1) Trivially, assertion (a) applies to semi-implicit equations (D ), −1 where S need not to be complete. In any case, the assumption that Hk+1 ◦ Fk has values in S (k) guarantees that solutions do not leave the nonautonomous set S and exist in forward time, accordingly. A similar remark also applies to Proposition 2.2.3. (2) Frequently, difference equations (D) (or the mappings Hk+1 , Fk , k ∈ I ) depend on parameters p from a set P . Examples for p are stepsizes in numerical schemes or certain characteristics in explicit models. Then the general solution is a parameter-dependent 2-parameter semigroup ϕ(·; p). If the assumptions of Theorem 2.3.6 hold uniformly in p ∈ P and P is a first countable topological space, then the mapping (x, p) → ϕ(k; κ, x; p) is continuous. Proof. (a) For each pair (k, x) ∈ S with k ∈ I consider the fixed point equation −1 (Fk (x, x )) x = Hk+1
(2.3e)
in the complete metric space S (k). Its right-hand side is a contraction in x . Thus, from the contraction mapping principle (see Theorem B.1.1) we obtain a unique fixed point ϕˆk (x) ∈ S (k). The corresponding sequence ϕˆk : S(k) → S (k) is the generator for the general forward solution of (D). (b) Now we can apply the uniform C 0 -contraction principle from Theorem B.1.1(a) to the fixed point problem (2.3e) and each ϕˆk , k ∈ I , is continuous. We conclude that ϕ(k; κ, ·) = ϕˆk−1 ◦ . . . ◦ ϕˆκ is a composition of continuous mappings. (c) The assertion follows from Theorem B.1.1(b).
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2 Nonautonomous Difference Equations
−1 Corollary 2.3.8. Let I be unbounded below. If Hk+1 ◦ Fk (·, x ) : S(k) → S (k) is −1 bounded for all x ∈ S (k) and Hk+1 ◦ Fk fulfills a Darbo condition with −1 ◦ Fk ∈ [0, 1] for all k ∈ I , q(k) := dar Hk+1
then the following holds true: ˆ (a) ϕ is bounded and B-contracting for every family Bˆ ⊆ B ⊆ S : lim eq (k, k − n)χk−n (B(k − n)) = 0 n→∞
for all k ∈ I .
ˆ (b) If limn→∞ supk∈I eq (k, k − n) = 0, then ϕ is B-uniformly contracting. Proof. Let k ∈ I . Using the notation from the proof of Theorem 2.3.6 we get from Corollary B.1.2(b) that every generator ϕˆk : S(k) → S (k) is bounded and fulfills a Darbo condition with dar ϕˆk ≤ q(k). Consequently, also ϕ is bounded and the claim follows from Corollary 1.2.29.
Theorem 2.3.9. Let m ∈ Z+ 0 , suppose X consists of Banach spaces and S consists −1 of open sets. If (2.3c) holds and Hk+1 ◦ Fk : S(k) × clXk+1 S (k) → clXk+1 S (k) satisfies −1 (i) lip2 Hk+1 ◦ Fk < 1, −1 (ii) Hk+1 ◦ Fk : S(k) × S (k) → S (k) is of class C m for all k ∈ I ,
then the general forward solution ϕ of (D) exists on S and is of class C m . Remark 2.3.10. If the general solution ϕ of (D) exists as a C 1 -mapping, then D3 ϕ(k; κ, ξ) ∈ GL(Xκ , Xk ) for all k, κ ∈ I and for the generator one has the inclusion Dϕˆκ (ξ) ∈ GL(Xκ , Xκ+1 ) for all κ ∈ I and ξ ∈ Xκ . Proof. Let κ, k ∈ I with κ ≤ k. Thanks to Theorem 2.3.6 the general forward solution ϕ(k; κ, ·) exists as a continuous mapping. Moreover, due to assumption (ii) we can apply the uniform C m -contraction principle from Theorem B.1.5 to the fixed point problem (2.3e). Thus, the generators ϕˆk : S(k) → S (k), k ∈ I are of class C m . Inductively, the general forward solution ϕ(k; κ, ·) is a composition of C m -maps and the chain rule (cf. Theorem C.1.3) implies the assertion.
Due to the implicit structure of our difference equations (D) it becomes notationally involved to describe the higher order derivatives of ϕ(k; κ, ·). We therefore introduce an important abbreviation we are about to use frequently in order to avoid a rather cumbersome row or column notation. Given nonempty sets P, Q and a mapping φ : I × P → Q, for all k ∈ I and p ∈ P we write φ(k, p) φ(k, p) := . (2.3f) φ(k, p) := (φ(k, p), φ(k + 1, p)), φ(k + 1, p)) Note that this notation also serves as parenthesis in sums and differences.
2.3 Difference Equations and 2-Parameter Semigroups
55
Having this at hand, in our next result we see that the derivatives D3n ϕ(·; κ, ξ) solve linear inhomogeneous difference equations (as to be considered in Chap. 3). Corollary 2.3.11 (variational equation). For every k ∈ I we define the function −1 gk := Hk+1 ◦ Fk : S(k) × S (k) → S (k). Given a pair (κ, ξ) ∈ S the partial derivative D3 ϕ(·; κ, ξ) with values D3 ϕ(k; κ, ξ) ∈ L(Xκ , Xk ), k ∈ I+ κ , is a forward solution of the (first order) variational equation IXk+1 − D2 gk (ϕ(k; κ, ξ)) X = D1 gk (ϕ(k; κ, ξ))X
(2.3g)
and satisfies the initial condition D3 ϕ(κ; κ, ξ) = IXκ . Moreover, the partial derivatives D3n ϕ(·; κ, ξ) with values D3n ϕ(k; κ, ξ) ∈ Ln (Xκ ; Xk ), k ∈ I+ κ , 2 ≤ n ≤ m, are forward solutions to the (n-th order) variational equation (n) IXk+1 − D2 gk (ϕ(k; κ, ξ)) X = D1 gk (ϕ(k; κ, ξ))X + gk
(2.3h)
and satisfy the initial condition D3n ϕ(κ; κ, ξ) = 0 ∈ Ln (Xκ ), where the sequence (n) of inhomogeneities gk ∈ Ln (Xκ ; Xk+1 ) is given by (n)
gk x1 · · · xn :=
n
j=2
(N1 ,...,Nj )∈Pj< (n)
Dj gk (ϕ(k; κ, ξ))· #Nj
· D3#N1 ϕ(k; κ, ξ)xN1 · · · D3
ϕ(k; κ, ξ)xNj
for all x1 , . . . , xn ∈ Xκ . Remark 2.3.12. (1) For semi-implicit equations (D ) both the variational equations (2.3g) and (2.3h) are explicit and take the respective form X = D1 gk (ϕ(k; κ, ξ))X,
(n)
X = D1 gk (ϕ(k; κ, ξ))X + gk .
(2) If the nonautonomous set Y consists of Banach spaces and Hk+1 , Fk , k ∈ I , are of class C m , then the first order variational equation of (D) can be represented DHk+1 (ϕ(k + 1; κ, ξ)) − D2 Fk (ϕ(k; κ, ξ)) X = D1 Fk (ϕ(k; κ, ξ))X. (3) Let J ⊆ I be a discrete interval. For a given reference solution φ∗ : J → X of (D) in S, we denote the variational equation DHk+1 (φ∗ (k)) − D2 Fk (φ∗ (k)) X = D1 Fk (φ∗ (k))X on J (or its counterpart above) as linearization of (D) along φ∗ .
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2 Nonautonomous Difference Equations
−1 (4) Suppose the mappings Hk+1 , Fk , k ∈ I , depend m-times continuously differentiable on a parameter p from an open subset P of a Banach space. If the assumptions of Theorem 2.3.9 hold uniformly in p ∈ P , then Theorem B.1.5 shows that also the parameter-dependent 2-parameter semigroup ϕ(k; κ, ·) : S(κ) × P → S(k), κ ≤ k, is of class C m . Its partial derivative D4 ϕ(·; κ, ξ, p) is a forward solution of IXk+1 − D2 gk (ϕ(k; κ, ξ, p)) X = D1 gk (ϕ(k; κ, ξ, p))X.
Proof. From Theorem 2.3.9 we obtain that ϕ(k; κ, ·) : S(κ) → Xk , κ ≤ k, is map−1 ping of class C m . Now we differentiate ϕ(k + 1; κ, ξ) ≡ Hk+1 ◦ Fk (ϕ(k; κ, ξ)) + on (Iκ ) w.r.t. the variable ξ ∈ S(κ) using the higher order chain rule from Theorem C.1.3. The representation as totally unfolded derivative tree (C.1b) yields the claim.
In order to construct a general forward solution for semi-implicit equations Hk+1 (x ) = Fk (x),
(D )
it is sufficient to provide invertibility criteria for Hk+1 , as well as S ⊆ Y and Fk (S(k)) ⊆ S (k),
Hk+1 (S (k)) ⊆ S (k)
for all k ∈ I .
(2.3i)
Under dual assumptions on the right-hand side Fk one can derive criteria for the existence of backward solutions or of the general solution. The indicated invertibility conditions on Hk+1 are formulated in terms of monotonicity assumptions. Corollary 2.3.13. Suppose that X consists of Hilbert spaces and that S consists of closed linear subspaces satisfying (2.3i). Provided for every k ∈ I the left-hand side Hk+1 : S (k) → S (k) is globally Lipschitz and if there exist γk > 0 with 2
γk x − x ¯Xk+1 ≤ Hk+1 (x) − Hk+1 (¯ x), x − x ¯Xk+1
(2.3j)
for all x, x ¯ ∈ S (k), then the following holds: (a) The general forward solution ϕ of (D ) exists on S. (b) ϕ is continuous, if Fk : S(k) → S (k) is continuous for all k ∈ I . Proof. By Theorem 2.3.6 it remains to show that Hk+1 : S (k) → S (k), k ∈ I , is a homeomorphism. This, however, follows from Theorem B.3.3.
Corollary 2.3.14. Suppose that m ∈ N, X consists of Hilbert spaces and S consists of closed linear subspaces satisfying (2.3i). Provided for every k ∈ I the left-hand side Hk+1 : S (k) → S (k) is of class C m and if there exist continuous functions ∞ ωk : [0, ∞) → (0, ∞) with 0 ωkds(s) = ∞ and 2
ωk (xXk+1 ) yXk+1 ≤ DHk+1 (x)y, yXk+1
for all x, y ∈ S (k),
2.3 Difference Equations and 2-Parameter Semigroups
57
then the following holds: (a) The general forward solution ϕ of (D ) exists on S. (b) ϕ is of class C m , if Fk ∈ C m (S(k), Xk+1 ) for all k ∈ I . Remark 2.3.15. In applications the mappings Hk+1 , Fk , k ∈ I , in (D ) depend smoothly, say of class C m , on parameters p from a subset P of a Banach space. The general solution of (D ) is a parameter-dependent 2-parameter semigroup ϕ(·; p). Provided the assumptions of Corollary 2.3.14 hold uniformly in p ∈ P , then also the mapping (x, p) → ϕ(k; κ, x; p) is of class C m (cf. Theorem B.3.5 with Z = P ).
Proof. Use Theorem B.3.5 in the proof of Corollary 2.3.13. Of particular importance for our applications are one-step methods x = x + Gk (x, x ),
(O)
with a mapping Gk : S(k) × S (k) → S (k), k ∈ I , as in Example 2.1.4. Theorem 2.3.16. Let c : I → (0, ∞) be a sequence, suppose that X consists of normed spaces and S ⊆ X consists of linear subspaces. If the step functions Gk (ξ, ·) : S (k) → S (k) are completely continuous and satisfy ¯Xk+1 ≤ x − Gk (ξ, x) − x ¯ + Gk (ξ, x ¯)Xk+1 c(k)−1 x − x
for all k ∈ I
and ξ ∈ S(k), x, x ¯ ∈ S (k), then the general forward solution ϕ of the one-step method (O) exists on S with lip ϕ(k; κ, ·) ≤ ec (k, κ) for all κ ≤ k. Proof. For given k ∈ I and ξ ∈ S(k) solve the equation x − Gk (ξ, x) = ξ for x ∈ S (k). For this we apply Theorem B.3.2 to the mapping T := Gk (ξ, ·) : S (k) → S (k) and obtain a unique solution ϕˆk (ξ) = [I − Gk (ξ, ·)]−1 (ξ) ∈ S (k) with lip ϕˆk ≤ c(k). Then (ϕˆk )k∈I generates the general forward solution to (O).
Theorem 2.3.17. Suppose that m ∈ N, X consists of Hilbert spaces and S consists of closed linear subspaces. If the step functions Gk : S(k) × S (k) → S (k) satisfy for all k ∈ I , ξ ∈ S(k), that (i) Gk : S(k) × S (k) → Xk+1 is of class C m , ∞ (ii) there exist continuous ωk,ξ : [0, ∞) → (0, ∞) with 0
ds ωk (s)
= ∞ and
y2Xk+1 − D2 Gk (ξ, x)y, yXk+1 ≥ ωk,ξ (xXk+1 ) y2Xk+1 for all x, y ∈ S (k), then the general forward solution ϕ of (O) exists on S and is of class C m .
(2.3k)
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2 Nonautonomous Difference Equations
Remark 2.3.18. For explicit one-step methods the assumption (ii) becomes trivial. In case the one-step method (O) is a θ-method as in Example 2.1.4, or a 2-step θ-method as in (2.1e), then the monotonicity condition (2.3k) reduces to 2
2
yXk+1 − θ Dfk (x)y, yXk+1 ≥ ωk,ξ (xXk+1 ) yXk+1 . Proof. Let k ∈ I and ξ ∈ S(k) be arbitrary. Following Theorem B.3.5 the mapping Tk,ξ : S (k) → S (k), Tk,ξ (x) := x − Gk (ξ, x) is a global C m -diffeomorphism, since (2.3k) implies (B.3d). Consequently, the equation ξ = x − Gk (ξ, x) has a unique solution x = ϕˆk (ξ) and ϕˆk : S(k) → Xk+1 is of class C m . We obtain that (ϕˆk )k∈I generates the general forward solution to (O).
Conditions for Dissipativity Now we provide criteria for dissipativity, which has been one of the crucial notions to construct attractors in Chap. 1. In fact, if we can additionally establish continuity ˆ and, e.g., B-asymptotic compactness of the general solution, then Theorem 1.3.9 ˆ guarantees the existence of a B-attractor. Corollary 2.3.19. Suppose ε > 0 is fixed, I is unbounded below and that the assumptions of Theorem 2.2.5 hold with the summability condition ρk :=
k−1 l=−∞
βl e 1−γ (k, l + 1) < ∞ 1 + αl 1+α
for all k ∈ I.
If the general forward! solution of a one-step method "(O) exists, then the nonau√ ˆ for every tonomous set A := (k, x) ∈ S : xXk ≤ ε + ρk is B-absorbing absorption universe ⎪ ⎪ ⎪ ⎪2 Bˆ ⊆ B ⊆ S : lim e 1−γ (k, k − n) ⎪ ⎪B(k − n)⎪ ⎪ = 0 for all k ∈ I , n→∞
1+α
ˆ and B-uniformly absorbing for every absorption universe % $ ⎪ ⎪ ⎪ ⎪2 Bˆ ⊆ B ⊆ S : lim sup e 1−γ (k, k − n) ⎪ ⎪B(k − n)⎪ ⎪ =0 . n→∞ k∈I
1+α
Proof. Since the general forward solution to (O) exists, it coincides with the solutions constructed in Theorem 2.2.5. Let k ∈ I, B be a nonautonomous set from the family Bˆ and choose a sequence ξn ∈ B(k − n) for n ≥ 0. From the estimate (2.2f) we immediately get
ϕ(k, k − n, ξn )Xk
k−1 ⎪ ⎪ ⎪ ⎪2 ≤ e 1−γ (k, k − n) ⎪ ⎪B(k − n)⎪ ⎪ + 1+α
l=k−n
βl e 1−γ (k,l+1) 1+α
1+αl
2.3 Difference Equations and 2-Parameter Semigroups
59
for all n ≥ 0 and by construction of the family Bˆ there exists an N = Nk (B) ≥ 0 such that the inclusion ϕ(k, k − n, ξn ) ∈ A(k) holds for all n ≥ N . We therefore derive ϕ(k; k − n, B(k − n)) ⊆ A(k) for such n ≥ N and B ∈ Bˆ as above. Finally, if the limit relation in the definition of Bˆ is uniform, N does not depend on k ∈ I.
Corollary 2.3.20. Suppose ε > 0 is fixed, I is unbounded below and that the assumptions of Proposition 2.2.6 hold with the summability condition ρk :=
k−1
e1−α+γ (k, l + 1)(βl + δl ) < ∞
for all k ∈ I.
l=−∞
If the general forward! solution of a one-step method "(O) exists, then the nonau√ ˆ tonomous set A := (k, x) ∈ S : xXk ≤ ε + ρk is B-absorbing for every absorption universe ⎪ ⎪ ⎪ ⎪2 Bˆ ⊆ B ⊆ S : lim e1−α+γ (k, k − n) ⎪ ⎪B(k − n)⎪ ⎪ = 0 for all k ∈ I , n→∞
ˆ and B-uniformly absorbing for every absorption universe Bˆ ⊆
$ % ⎪ ⎪ ⎪ ⎪2 B ⊆ S : lim sup e1−α+γ (k, k − n) ⎪ ⎪B(k − n)⎪ ⎪ =0 . n→∞ k∈I
Proof. Using Proposition 2.2.6 this is shown literally as in Corollary 2.3.19.
In order to demonstrate the above dissipativity results, we need the following notion: A real nonnegative sequence (xk )k∈I is called backward tempered, if lim k xk = 0
k→−∞
for all ∈ (1, ∞) .
Backward tempered sequences include bounded sequences, and more general, are allowed to grow polynomially, but not exponentially for k → −∞. Example 2.3.21. Suppose Bˆ is the family of all uniformly bounded subsets of S. We provide criteria for uniformly bounded dissipativity. (1) If the sequences (αk )k∈I and (γk )k∈I in the dissipativity assumption (ii) of 1−γk Theorem 2.2.5 satisfy 0 α + γ, then it is easy to see that 1+α ≤ η for all k ∈ I k holds with some η ∈ (0, 1). This guarantees ⎪ ⎪ ⎪ ⎪ lim sup e 1−γ (k, k − n) ⎪ ⎪B(k − n)⎪ ⎪ = 0 for all B ∈ Bˆ
n→∞ k∈I
1+α
ˆ and (O) is uniformly B-absorbing. The summability condition in Corollary2.3.19 β holds in particular, if 1+α is backward tempered. To justify this, choose ∈ 1, η1 and observe that there exists a L ∈ I such that Thus,
βl 1+αl
≤ −l for all l ≤ L.
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2 Nonautonomous Difference Equations
e 1−γ (k, l + 1) 1+α
βl η k−1 ≤ 1 + αl (η)l
for all k ∈ I, l ≤ L
βl we obtain an yields a summable majorant. In case of a bounded sequence 1+α l l∈I & ' β 1 absorbing set with constant radius ρk ≡ ε + 1−η 1+α . ˆ (2) Similarly one shows that (O) is uniformly B-absorbing, if the sequences (αk )k∈I and (γk )k∈I in Proposition 2.2.6 fulfill γ α; this ensures that 1 − αk + γk ≤ η for all k ∈ I holds with some η ∈ (0, 1). If the sum β + δ is backward tempered, then the summability condition in Corollary 2.3.20 is satisfied. Finally, if (βl + δl )l∈I is bounded, one gets an absorbing set with constant radius ρk ≡ ε + β+δ 1−η . After all we investigate dissipativity properties of θ- or Runge–Kutta methods: Corollary 2.3.22. Suppose ε > 0 is fixed, I unbounded below and that the assumptions of Proposition 2.2.8 or 2.2.10 hold. If the general forward solution of a θ-method (2.2g) or a Runge–Kutta method it is uniformly (O), resp., exists, then 2αk bounded dissipative; in particular, A := (k, x) ∈ S : xXk ≤ ε + 1−γ is ˆ B-absorbing for every absorption universe Bˆ ⊆
$ % ⎪ ⎪ 2 ⎪ n⎪ ⎪ ⎪ B ⊆ S : lim sup γ ⎪B(k − n)⎪ = 0 , n→∞ k∈I
ˆ and B-uniformly absorbing for every absorption universe Bˆ ⊆
$ % ⎪ ⎪ ⎪ ⎪2 B ⊆ S : lim sup γ n ⎪ ⎪B(k − n)⎪ ⎪ =0 . n→∞ k∈I
ˆ As in the proof of Corollary 2.3.19 we infer Proof. Let k ∈ I and suppose B ∈ B. from relation (2.2j) that the inclusion ϕ(k, k − n, B(k − n)) ⊆ A(k) holds for large n. In particular, provided Bˆ consists of uniformly bounded sets, one has the ⎪ ⎪ ⎪ ⎪2 uniform limit relation limn→∞ supk∈I γ n ⎪ ⎪B(k − n)⎪ ⎪ = 0 and a θ-method (2.2g) resp. a Runge–Kutta method (O) is uniformly bounded dissipative.
2.4 Stability This section is by no means meant to be a comprehensive approach to stability theory for solutions or (nonautonomous) sets; see Sect. 2.7 for a survey. In fact, for later reference we merely introduce some basic notions and facts from classical Lyapunov stability theory for nonautonomous difference equations, which is surely based on forward convergence. Beyond that, these notions are supplemented by a corresponding terminology based on the concept of pullback convergence, as used
2.4 Stability
61
throughout Chap. 1. Differing from the well-established forward convergence situation, we do not restrict to solutions defined on intervals bounded below. Indeed, pullback and forward convergence concepts are only comparable for solutions existing on the whole integer axis. Above all, the following word of caution is due: The use of time-variant state spaces and associated norms, enables us to construct examples with seemingly “pathological” or counter-intuitive asymptotic behavior: Example 2.4.1. Let γ > 0 be fixed. Consider the nonautonomous set S = Z × R and equip its k-fibers with the norm |x|k := γ k |x| for all k ∈ Z. The asymptotic behavior of the simple difference equation x = αx, α ∈ R \ {0}, in S crucially depends on the product αγ. Precisely, for κ ∈ Z, ξ = 0 its general solution ϕ fulfills
k
−κ
|ϕ(k; κ, ξ)|k = |αγ| α
ξ −−−− → k→∞
⎧ ⎪ ⎪ ⎨0
for |αγ| < 1, −κ
|α ⎪ ⎪ ⎩∞
ξ|
for |αγ| = 1, for |αγ| > 1.
Yet, having realized and accepted such phenomena, our theory is consistent. Suppose J ⊆ I are discrete intervals, ((Xk , dk ))k∈I denotes a sequence of metric spaces and S ⊆ X is nonempty. We again consider a difference equation Hk+1 (x ) = Fk (x, x )
(D)
in the nonautonomous set S as in Definition 2.1.2. Additionally, we suppose there exists a fixed reference solution φ : J → X of (D) in S and that the general (forward or backward) solution ϕ of (D) exists uniformly in a φ-neighborhood, i.e., for some given ρ0 > 0, the solution ϕ(·; κ, ξ) exists on I± κ for all κ ∈ J and (κ, ξ) ∈ Bρ0 (φ). We introduce measures for the exponential growth of sequences and solutions to difference equations. Given a sequence ψ : J → [0, ∞) we define the upper resp. lower forward characteristic exponents (if J is unbounded above) + + k k χ+ ψ(k), χ+ ψ(k), (2.4a) u (ψ) := lim sup l (ψ) := lim inf k→∞
k→∞
and the upper resp. lower backward characteristic exponents χ− u (ψ) := lim sup
+ ψ(k),
−k
k→−∞
χ− l (ψ) := lim inf
k→−∞
+ ψ(k),
−k
(2.4b)
if J is unbounded below. Furthermore, for ξ ∈ Xκ and a solution φ : J → X of (D) we set ψ(k) := dk (ϕ(k; κ, ξ), φ(k)) and define ± λ± u (φ, ξ) := χu (ψ),
± λ± l (φ, ξ) := χl (ψ).
The real number λ+ u (φ, ξ) indicates the exponential rate at which solutions to (D) starting near φ converge to (or diverge from) the reference solution φ.
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2 Nonautonomous Difference Equations
In our first definition we use the generalized exponential function ea (k, κ) from Sect. A.1. It yields a generalized uniform stability notion sufficiently flexible to capture also non-exponential decay and growth rates occurring in a nonautonomous setting, as well as growth on semiaxes. Definition 2.4.2. A solution φ : J → X to (D) is said to be (Sa+ ) forward a-stable, if there exists a real sequence a : J → [0, ∞) and K ≥ 0, δ > 0, so that for all κ ∈ J, ξ ∈ Bδ (φ(κ), Xκ ) one has dk (ϕ(k; κ, ξ), φ(k)) ≤ Kea (k, κ)dκ (ξ, φ(κ))
for all k ≥ κ,
and if the general backward solution exists uniformly in a φ-neighborhood, (Sa− ) backward a-stable, if there exists a real sequence a : J → [0, ∞) and K ≥ 0, δ > 0, so that for all κ ∈ J, ξ ∈ Bδ (φ(κ), Xκ ) one has dk (ϕ(k; κ, ξ), φ(k)) ≤ Kea (k, κ)dκ (ξ, φ(κ))
for all k ≤ κ.
Remark 2.4.3. (1) A solution φ : J → X to (D) on an interval J unbounded above is called exponentially stable, if it is forward a-stable with a(k) ≡ α ∈ (0, 1). (2) Let κ ∈ J. A forward a-stable solution φ : J → X satisfies + λ+ u (φ, ξ) ≤ χu (ea (·, κ)),
+ λ+ l (φ, ξ) ≤ χl (ea (·, κ))
for all ξ ∈ Bδ (φ(κ))
and dual implications hold for backward a-stable solutions. Yet, as demonstrated in Example 2.4.9, given α > 0 a solution φ : J → X fulfilling λ+ u (φ, ξ) ≤ α for every ξ ∈ S(κ) \ {φ(κ)} needs not to be forward α-stable. This indicates that characteristic exponents are not able to capture uniform stability properties. (3) Under the assumptions of Theorem 2.3.6(c) all solutions of (D) are forward c-stable with c given in Theorem 2.3.6(c). However, often a global bound for c and the exponential growth of (D) is crucial. Furthermore, we need certain attraction notions: Definition 2.4.4. A solution φ : J → X to (D), which is defined on an interval J unbounded above, is said to be (A) attractive, if for every κ ∈ J there exists a δ = δ(κ) > 0, and for every ε > 0 a corresponding N = Nκ (ε) ≥ 0 so that one has ϕ(k; κ, Bδ (φ(κ), Xκ )) ⊂ Bε (φ(k), Xk ) for all k ∈ J, k − κ ≥ N,
(2.4c)
(UA) uniformly attractive, if there exists a δ > 0, and for every ε > 0 a corresponding N = N (ε) ≥ 0 so that for every κ ∈ J one has (2.4c), (GA) globally attractive, if one has the limit relation limk→∞ dk (ϕ(k; κ, ξ), φ(k)) = 0 for all κ ∈ J, ξ ∈ S(κ) and φ is attractive, and on an interval J unbounded below, φ is called
2.4 Stability
63
(PA) pullback attractive, if there exists a δ > 0, and for every ε > 0, k ∈ J a corresponding N = N (ε, k) ≥ 0 so that ϕ(k; k − n, Bδ (φ(k − n), Xk−n )) ⊂ Bε (φ(k), Xk ) for all n ≥ N,
(2.4d)
(UPA) uniformly pullback attractive, if there exists a δ > 0, and for every ε > 0 an N = N (ε) ≥ 0 so that (2.4d) holds for all k ∈ J and n ≥ N . Remark 2.4.5. Let the discrete interval J be bounded below. (1) Rather than given the usual pointwise definitions, we are working with set inclusions in (2.4c) and (2.4d). However, the attraction condition (A) is equivalent to the limit relation ∀κ ∈ J : ∃δ > 0 : lim dk (ϕ(k; κ, ξ), φ(k)) = 0 uniformly for ξ ∈ Bδ (φ(κ)), k→∞
whereas the uniform attraction condition (UA) holds if and only if ∃δ > 0 : lim dk (ϕ(k; κ, ξ), φ(k)) = 0 k→∞
uniformly for (κ, ξ) ∈ Bδ (φ).
(2) If a general forward solution ϕ is continuous on Bρ0 (φ) and ϕ(k; κ, ·) : S(κ) → S(k)
is an open mapping for all κ ≤ k
(2.4e)
holds, then it is sufficient to check the attraction condition (A) (or the equivalent limit relation above) for one fixed κ ∈ J in order to show that φ is attractive. This follows from our continuous dependence on the initial conditions. Criteria for (2.4e) to hold can be deduced for instance using [295, p. 397, Theorem 3.5]. Proposition 2.4.6. For any solution φ : J → X of equation (D) the attraction notions from Definition 2.4.4 are related by: GA ⇒ A ⇐ UA ⇔ ˙ UPA ⇒ PA where the equivalence ⇔ ˙ only holds for J = Z. Proof. Only the equivalence ⇔ ˙ is not evident from Definition 2.4.4. In order to show it, we suppose J = Z. If we set κ = k − n in (2.4c) one immediately obtains (2.4d). Conversely, setting n = k − κ in (2.4d) yields (2.4c).
Proposition 2.4.7. Provided for all κ ∈ J there exists a δ > 0 such that a solution φ : J → X satisfies supξ∈Bδ (φ(κ),Xκ ) λ+ u (φ, ξ) < 1, then φ is attractive. Proof. Let κ ∈ J. We define α := supξ∈Bδ (φ(κ)) λ+ u (φ, ξ) ∈ (0, 1) and infer that + k there exists a K ≥ 0 with dk (ϕ(k; κ, ξ), φ(k)) < 1+α 2 for all k ≥ κ + K and ξ ∈ Bδ (φ(κ)). This yields the uniform limit relation dk (ϕ(k; κ, ξ), φ(k)) ≤
1+α k 2
and therefore Remark 2.4.5(1) implies our claim.
−−−− →0 k→∞
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2 Nonautonomous Difference Equations
In the following examples we fix the nonautonomous set S = Z × R, where each fiber S(k), k ∈ Z, is normed using the absolute value |·|. Example 2.4.8. The zero solution of x = αx satisfies + − − λ+ u (0, ξ) = λl (0, ξ) = λu (0, ξ) = λl (0, ξ) = α for all ξ ∈ R \ {0} ;
further it is forward α-stable, since we have ϕ(k; κ, ξ) = αk−κ ξ. Moreover, in case α ∈ (−1, 1) \ {0}, it is uniformly attractive on every interval J being unbounded above, and uniformly pullback attractive on every interval J unbounded below. One should not confuse a (globally) attractive solution φ : J → X of (D) with the attraction of the nonautonomous set φ = {(κ, φ(κ)) ∈ S : κ ∈ J} as introduced in Definition 1.3.1. This, and more, is illustrated by Example 2.4.9. Consider the scalar difference equation x = ak x,
ak :=
α+
for k ≥ 0,
α−
for k < 0,
whose general forward solution ϕ is the 2-parameter semigroup in Example 1.2.17. + − − One has λ+ u (0, ξ) = λl (0, ξ) = α+ and λu (0, ξ) = λl (0, ξ) = α− for all nonzero ξ ∈ R. The attraction properties of the trivial solution crucially depend on the values of the parameters α− , α+ ∈ R \ {0} (see Fig. 2.1): |α+ | , |α− | < 1: It is uniformly attractive, globally attractive and beyond that forward max {|α+ | , |α− |}-stable. Moreover, it is uniformly pullback attractive. • |α+ | < 1 < |α− |: It is (globally) attractive. If J is additionally bounded below, the zero solution is forward |α+ |-stable and uniformly attractive, whereas both is not true for J = Z. Likewise, the zero solution is not pullback attractive. • |α− | < 1 < |α+ |: It is not attractive, but pullback attractive. On intervals bounded above, it is uniformly pullback attractive, which does not hold for J = Z. Nonetheless, as seen in Example 1.3.5, the set Z × {0} is a global attractor. • 1 < |α+ | , |α− |: It is neither attractive nor pullback attractive. •
Proposition 2.4.10. If J is unbounded below, then a pullback attractive solution φ : J → X of (D) is a local attractor. Proof. Since φ : J → X is a solution of (D), the set φ ⊆ S is invariant. Due to its pullback attraction, there exists a δ > 0, and for each ε > 0, k ∈ J, a corresponding N ≥ 0 in such a way that ξ ∈ Bδ (φ(k − n)) implies d(ϕ(k; k − n, ξ), φ(k)) < 2ε for all n ≥ N . For the Hausdorff separation this, in turn, implies h ϕ k; k − n, Bδ (φ(k − n)) , {φ(k)} =
sup
dist(x, {φ(k)}) < ε
x∈ϕ(k;k−n,Bδ (φ(k−n)))
2.4 Stability
65
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Z
Fig. 2.1 Solutions of the difference equation x = ak x from Example 2.4.9, complete motion φγ and uniformly bounded global attractor A0 from Example 1.3.5
for all n ≥ N . Hence, for each subset B ⊆ Bδ (φ) one deduces the limit relation limn→∞ h(ϕ(k; k−n, B(k−n)), {φ(k)}) = 0 and φ ⊆ S is also a local attractor.
The above attraction notions are based on the concept of convergence. Now we formalize to what extent solutions stay near given ones. Definition 2.4.11. A solution φ : J → X to (D), which is defined on an interval J unbounded above, is said to be (S) stable, if for every ε > 0 and κ ∈ J there is a δ > 0 such that ϕ(k; κ, Bδ (φ(κ), Xκ )) ⊂ Bε (φ(k), Xk ) for all k ≥ κ,
(2.4f)
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2 Nonautonomous Difference Equations
(US) uniformly stable, if for every ε > 0 there is a δ > 0 such that for all κ ∈ J one has the inclusion (2.4f), (AS) asymptotically stable, if it is attractive and stable, (UAS) uniformly asymptotically stable, if it is uniformly attractive and uniformly stable, (GAS) globally asymptotically stable, if it is globally attractive and stable, (S) unstable, if it is not stable, and on an interval J unbounded below, φ is called (PS) pullback stable, if for every ε > 0 and k ∈ J there exists a δ > 0 with ϕ(k; k − n, Bδ (φ(k − n), Xk−n )) ⊂ Bε (φ(k), Xk ) for all n ≥ 0, (2.4g) (UPS) uniformly pullback stable, if for every ε > 0 there exists a δ > 0 such that (2.4g) holds for all k ∈ J and n ≥ 0, (APS) asymptotically pullback stable, if it is pullback stable and pullback attractive, (UAPS) uniformly asymptotically pullback stable, if it is uniformly pullback stable and uniformly pullback attractive. Remark 2.4.12. (1) If J is bounded below, the general forward solution ϕ is continuous on Bρ0 (φ) and moreover (2.4e) holds, then it is sufficient to check the stability condition (S) for one κ ∈ J in order to guarantee that φ is stable. (2) An exponentially stable solution is uniformly asymptotically stable and uniformly asymptotically pullback stable. Proposition 2.4.13. For any solution φ : J → X of (D) the attraction and stability notions from Definitions 2.4.4 and 2.4.11, resp., are related by: S ⇐ ⇑ GAS ⇒ AS ⇐ ⇓ ⇓ GA ⇒ A ⇐
US ⇔ ˙ UPS ⇒ PS ⇑ ⇑ ⇑ UAS ⇔ ˙ UAPS ⇒ APS ⇓ ⇓ ⇓ UA ⇔ ˙ UPA ⇒ PA
where the equivalences ⇔ ˙ only hold for J = Z. Proof. The lower row is justified by Proposition 2.4.6. We only establish implications, which do not immediately follow from Definitions 2.4.4 and 2.4.11. Here, suppose J = Z. (U S ⇔U ˙ P S) Replace κ by k − n in (2.4f) in order to obtain (2.4g). Conversely, setting n = k − κ in (2.4g) gives us (2.4f).
For the remaining section, let ((Xk , dk ))k∈I be a sequence of metric linear spaces such that each metric dk is translation invariant.
2.4 Stability
67
Definition 2.4.14. If φ : J → X is a solution of (D), then Hk+1 (x + φ (k)) = Fk (x + φ(k), x + φ (k))
(Dφ )
is called equation of φ-perturbed (or simply perturbed) motion. It has the extended state space Sφ := {(k, x) ∈ X : k ∈ J, x + φ(k) ∈ S(k)}. Definition 2.4.14 yields a transformation of (D) into a new difference equation (Dφ ) whose behavior near the zero solution is exactly the behavior of the original equation (D) around φ. Theorem 2.4.15. If φ : J → X is a solution of (D), then for any further solution ψ : J → X of (D) the difference ψ − φ : J → X solves the equation of perturbed motion (Dφ ). Moreover, (Dφ ) has the zero solution on J and (a) φ is forward a-stable, if and only if the zero solution of (Dφ ) satisfies the corresponding property w.r.t. (Dφ ), (b) φ satisfies an attraction property from Definition 2.4.4 if and only if the zero solution of (Dφ ) satisfies the corresponding property w.r.t. (Dφ ), (c) φ satisfies a stability property from Definition 2.4.11 if and only if the zero solution of (Dφ ) satisfies the corresponding property w.r.t. (Dφ ). Proof. It is straight forward to check that ψ − φ solves (D) on J. Now we denote the respective general forward solutions of (D) and (Dφ ) by ϕ and ϕφ . For all k, κ ∈ J, κ ≤ k one easily derives the identities ϕφ (k; κ, η) = ϕ(k; κ, η + φ(κ)) − φ(k),
(2.4h)
ϕ(k; κ, ξ) = ϕφ (k; κ, ξ − φ(κ)) + φ(k).
(2.4i)
(a) Let φ : J → X be a forward a-stable solution to (D). Thus, there exist K ≥ 0, δ > 0 such that d(ϕ(k; κ, ξ), φ(k)) ≤ Kea (k, κ)d(ξ, φ(κ)) for all κ ≤ k, where κ ∈ J, ξ ∈ Bδ (φ(κ)). By (2.4h) and the translation invariance of d this yields d(ϕφ (k; κ, η), 0) = d ϕ(k; κ, η + φ(κ)), φ(k) ≤ Kea (k, κ)d(η, 0)
for all κ ≤ k
and initial values η ∈ Bδ (0). This, however, means that the zero solution of (Dφ ) is forward a-stable. The converse follows analogously by relation (2.4i). (b) For the attraction notions based on forward convergence, one can proceed as in the proof of assertion (a). We thus restrict on pullback attraction and suppose J is unbounded below. If φ is pullback attractive, then there exists a δ > 0 such that for each ε > 0 one finds an N ≥ 0 fulfilling (2.4d). This yields the relation (2.4h)
ϕφ (k; k − n, Bδ (0)) = ϕ(k; k − n, φ(k − n) + Bδ (0)) − φ(k) = ϕ k; k − n, Bδ (φ(k − n)) − φ(k) (2.4d)
⊆ Bε (φ(k)) − φ(k) = Bε (0) for all n ≥ N
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and consequently 0 is a pullback attractive solution to (Dφ ). For the converse direction one argues using (2.4i). Furthermore, the assertion for uniform pullback attraction follows along the same lines. (c) The claim for the various stability concepts can be shown using the translation relations (2.4h) and (2.4i) as in the above steps.
2.5 Periodic and Autonomous Difference Equations In our subsequent considerations on periodic equations we aim to embed the classical theory of autonomous dynamical systems into our wider approach. We focus on stability, while various further aspects will be tackled later on. As in Sect. 1.4 we can replace the discrete interval I by the integers Z. Definition 2.5.1. A difference equation Hk+1 (x ) = Fk (x, x )
(D)
as in Definition 2.1.1 is said to be p-periodic (or simply periodic), if there exists a p ∈ N such that for all (k, x, x ) ∈ S × S one has S(k + p) = S(k), Hk+p (x ) = Hk (x ),
Yk+p = Yk , Fk+p (x, x ) = Fk (x, x ).
In case p = 1 we say (D) is autonomous. Remark 2.5.2. (1) For autonomous difference equations the state spaces S(k) are constant and Hk+1 , Fk do not depend on k; we can write H(x ) = F (x, x ). (2) Suppose φ : J → X is a solution of a difference equation (D). Note that even if (D) is autonomous, the associated equation of φ-perturbed motion (Dφ ) is nonautonomous, as soon as the solution φ is not constant. This underlines the significance of a nonautonomous theory. The next result shows that all our preparations from Sect. 1.4 apply to periodic difference equations. Proposition 2.5.3. Let p ∈ N. The general (forward) solution of a p-periodic difference equation (D) is a p-periodic 2-parameter (semi)group. Proof. For arbitrary pairs (κ, ξ) ∈ S both sequences ϕ(·; κ, ξ) and ϕ(· + p; κ + p, ξ) satisfy the initial condition x(κ) = ξ and solve (D). Thus, they coincide and we have ϕ(k; κ, ξ) = ϕ(k + p; κ + p, ξ) for all k ≥ κ.
The period map Πκ is a tool to prove existence of periodic solutions. Corresponding compactness conditions on Πκ are given in Propositions 1.4.6 and 1.4.7.
2.5 Periodic and Autonomous Difference Equations
69
Corollary 2.5.4. Let q, r ∈ N, κ ∈ Z and suppose (D) is r-periodic. (a) If (D) possesses a q-periodic solution φ : Z+ κ → X, then φ(κ) ∈ S(κ) is a fixed lcm(q,r)/r point of Πκ . (b) If an iterate Πκq possesses a fixed point ξ ∈ S(κ), then there exists a complete qr-periodic solution φ : Z → X with φ(κ) = ξ for (D). Proof. Let κ ∈ Z be given. (a) Above all, we define p := lcm(q, r). If φ : Z+ κ → X is a q-periodic solution p/r of (D), we obtain φ(κ) = φ(κ+p) = ϕ(κ+p; κ, φ(κ)) = Πκ (φ(κ)) and therefore p/r φ(κ) ∈ S(κ) is a fixed point of Πκ . (b) Conversely, if ξ ∈ S(κ) is a fixed point of Πκq , then Proposition 1.4.4(c) implies that the 2-parameter semigroup has a qr-periodic complete motion φ, which is a complete solution of (D).
Proposition 2.5.5. Let q, r ∈ N. Given a q-periodic solution φ : J → X of an r-periodic equation (D), the attraction and stability notions from Definitions 2.4.4 and 2.4.11, resp., are related by: S ⇔ US ⇔ ˙ UPS ⇔ PS ⇑ ⇑ ⇑ ⇑ GAS ⇒ AS ⇔ UAS ⇔ ˙ UAPS ⇔ APS ⇓ ⇓ ⇓ ⇓ ⇓ GA ⇒ A ⇔ UA ⇔ ˙ UPA ⇔ PA where the equivalences ⇔ ˙ only hold for J = Z. Proof. Let φ : J → X be a q-periodic solution of the r-periodic equation (D) on a discrete interval J. Setting p := lcm(q, r) we obtain that φ and (D) are p-periodic. Let ϕ denote the general forward solution of (D) and by Proposition 2.5.3 we have ϕ(k + np; κ + np, ξ) = ϕ(k; κ, ξ) for all κ ≤ k
(2.5a)
and n ∈ Z. By Proposition 2.4.13 we only have to show four assertions: (S ⇒ U S) Let the solution φ be stable on an interval J unbounded above. Hence, for every ε > 0, k0 ∈ J there exists a δ(ε, k0 ) > 0 such that ξ ∈ Bδ (φ(k0 ))
(2.4f)
⇒
d(ϕ(k; k0 , ξ), φ(k)) < ε
for all k ≥ k0 .
0 +p−1 We choose Δ := minkl=k δ(ε, l) > 0 and ξ ∈ BΔ (φ(κ)). For κ ∈ J we pick 0 n ∈ Z such that k0 ≤ κ + np < k0 + p. By our choice of Δ and (2.5a) this yields
d(ϕ(k; κ, ξ), φ(k)) = d(ϕ(k + np; κ + np, ξ), φ(k)) < ε i.e., the desired uniform stability of φ.
for all k ≥ κ,
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(A ⇒ U A) Let the solution φ : J → X be attractive. This means that for every k0 ∈ J there exists an η(k0 ) > 0 such that for all ξ ∈ Bη (φ(k0 )) and ε > 0 there is a K = K(ε, k0 , ξ) ≥ 0 with d(ϕ(k; k0 , ξ), φ(k)) < ε for all k − k0 ≥ K (cf. (2.4c)). 0 +p−1 η(l). For given κ ∈ J pick an integer n ∈ Z Now we choose N := minkl=k 0 such that k0 ≤ κ + np < k0 + p and we obtain for initial values ξ ∈ BN (φ(κ)) that (cf. (2.5a)) d(ϕ(k; κ, ξ), φ(k)) = d(ϕ(k + np; κ + np, ξ), φ(k)) < ε for all k − κ ≥ K(ε, k0 , ξ). For a fixed such ξ, it remains to verify that K can be chosen to be independent of the initial time k0 . For this, we define the real valued sequence σ(r) := sup
sup
r≤u k0 ≤κ
d(ϕ(κ + u; κ, ξ), φ(κ + u)),
since d(ϕ(· + u; ·, ξ), φ(· + u)) is p-periodic one gets σ(r) = sup sup d(ϕ(κ + u; κ, ξ), φ(κ + u)) r≤u κ∈J
and σ : Z+ 0 → [0, ∞) is a positive sequence decreasing to zero. Thus, for each ε > 0 we can choose an integer K(ε) ≥ 0 such that σ(K(ε)) < ε holds. By definition of the sequence σ, this is the desired constant. (P S ⇒ U P S) Let the solution φ satisfy (2.4g) on an interval J unbounded below. Using the crucial translation invariance (2.5a), and by passing over to the minimum over a whole period, one verifies as above that the constant δ in (2.4g) can actually be chosen independently from κ ∈ J. This is our claim. (P A ⇒ U P A) Again, the uniformity is guaranteed by (2.5a) and one deduces that pullback attractive solutions φ are uniformly pullback attractive.
2.6 Applications Don’t worry, the continuous heritage is not a total waste. Doron Zeilberger (cf. [466])
In this section, we illustrate the results from Chaps. 1 and 2 by applying them to various discretizations of evolutionary equations. Concerning a temporal discretization, suppose that we have a discretization mesh (tk )k∈I satisfying hk := tk+1 − tk ∈ [T, T ]
for all k ∈ I
(2.6a)
for some given bound T > 0 and ∈ (0, 1] balancing between maximal and minimal possible stepsizes.
2.6 Applications
71
2.6.1 Fully Discretized Functional Differential Equations Using previous notation and terminology introduced in Sect. 1.5.1, we consider a nonautonomous retarded FDE u(t) ˙ = f (t, ut )
(FDE)
with right-hand side f : R × Cr → Rd . First of all, we introduce a finite-dimensional approximation of the natural state space Cr = C([−r, 0], Rd ), r > 0. For this, we choose N ∈ N, set h := Nr and define the subspace of piecewise linear functions ! " Cr,N := ψ ∈ Cr : ψ|[ih−r,(i+1)h−r] is affine linear for i = 0, . . . , N − 1 ⊆ Cr . Evidently, the d(N + 1)-dimensional spaces Cr,N and Rd(N +1) are isometrically isomorphic by virtue of the linear mapping Γr,N : Cr,N → Rd(N +1) explicitly given by Γr,N ψ := (ψ(0), . . . , ψ(−N h)). The extension of Γr,N to Cr also serves as a projection onto Cr,N required to transform initial conditions for (FDE) to its spatial discretization. In order to avoid cumbersome notations we will frequently identify Cr,N with Rd(N +1) henceforth. We use h = Nr as temporal stepsize yielding a discretization mesh tk = kh, k ∈ I, as in (2.6a). We obtain a full discretization of (FDE) using the θ-methods1 uk+1 = uk + hf (tθk , (1 − θ)uk + θuk+1 , . . . , (1 − θ)uk−N + θuk−N +1 ), uk+1 = uk + h(1 − θ)f (tk , uk , . . . , uk−N ) + hθf (tk+1 , uk+1 , . . . , uk−N +1 ) (2.6b) with tθk = (1 − θ)tk + θtk+1 and θ ∈ [0, 1]. Both equations in (2.6b) are delay difference equations in the sense of Example 2.1.10. Therefore, the first equation in (2.6b) is equivalent to the difference equation x = Fk (x, x ), (2.6c) ⎛ ⎞ x0 + hf (tθk , (1 − θ)x0 + θy0 , . . . , (1 − θ)xN + θyN ) ⎜ ⎟ x0 ⎜ ⎟ Fk (x, y) := ⎜ ⎟, .. ⎝ ⎠ . xN −1
1
The notational inconsistency between ut in (FDE) and uk , uk+1 , etc., causes no confusion here.
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whereas the second equation in (2.6b) is equivalent to ⎛ ⎞ x0 + h(1 − θ)f (tk , x0 , . . . , xN ) ⎜ ⎟ x0 ⎜ ⎟ x = fk1 (x) + fk2 (x ), fk1 (x) := ⎜ ⎟ , (2.6d) .. ⎝ ⎠ . xN −1 fk2 (x) := (hθf (tk+1 , x0 , . . . , xN ), 0, . . . , 0)T and we address existence and uniqueness questions for solutions to (2.6c), (2.6d): Proposition 2.6.1. Suppose that f (t, ·) : Cr → Rd , t ∈ R, is of class C m , m ≥ 0. If lip2 f < ∞ with θh lip2 f < 1 holds, then the general forward solutions of (2.6c) and (2.6d) exist as C m -functions on Cr,N . Proof. Apply Theorems 2.3.6 and 2.3.9 to the respective equations (2.6c) and (2.6d)
with left-hand side Hk (x) = x. Similar results can be obtained using Theorem B.3.5 under monotonicity assumptions on f (t, ·). Furthermore, the dissipativity conditions of Theorem 2.2.5 or 2.2.7 are not suitable for (2.6b). However, we will provide corresponding criteria for FDEs in Sect. 4.9.1. For discretizations of (DDE) we deduce Example 2.6.2 (delay differential equations). Let δ > 0, r ≥ 0, Bˆ is the family of uniformly bounded subsets of I × Cr,N and moreover suppose that g : R2 → R is a continuous function. As full discretization of the delay differential equation (DDE) from Example 1.5.3 we consider the delay difference equation uk+1 = (1 − δh)uk − hg(tk , uk−N )
(2.6e)
under the standing conditions δh < 1, i.e., δr < N , and γ := sup(t,u)∈R2 |g(t, u)| . Note that (2.6e) fits into the setting of (2.6b) with d = 1, θ = 0, although our assumptions are weaker than the ones imposed in Example 1.5.3. Clearly, (2.6e) gives rise to a continuous 2-parameter semigroup ϕ on the extended state space S = I × Cr,N (cf. Proposition 2.6.1). Given an initial time κ ∈ I one deduces using [376, Lemma 3.1] that the estimate −(1 − δh)k−κ − γδ uκ ≤ uk ≤ (1 − δh)k−κ + γδ uκ
for all k ∈ I+ κ
holds. Thus, the 2-parameter semigroup ϕ(k, l) : Cr,N → Cr,N , l ≤ k, has a
N +1 ˆ uniformly bounded B-absorbing set containing the closed box − γδ , γδ – hence, the radius of the absorbing set coincides with the differential equation case from ˆ Example 1.5.3. Due to dim Cr,N = N + 1 we know that ϕ is B-compact and admits a global attractor by Theorem 1.3.9.
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2.6.2 Time-Discretized Abstract Evolution Equations Let X, Y be Banach spaces with X ⊆ Y . We are interested in two different approaches to obtain temporal discretizations of abstract evolutionary equations (AE)
ut + B(t)u = f (t, u)
in the space Y . In the first approach it is not necessary to impose Hypothesis 1.5.4. It goes as follows: In order to obtain a temporal discretization for (AE) we formally apply the 2-stage θ-method from Example 2.1.6 to (AE) yielding an implicit difference equation u = Thk θ (tk+1 )[IY − hk (1 − θ)B(tk )]u + hk (1 − θ)Thk θ (tk+1 )f (tk , u) + hk θThk θ (tk+1 )f (tk+1 , u ) with the resolvent operator Th (t) := (IY + hB(t)) following smoothing property:
(2.6f) −1
∈ L(Y ). We impose the
Hypothesis 2.6.3. Suppose B(t) : D(B(t)) ⊆ Y → Y are closed linear operators and that there exists a τ0 ≥ 0 with Th (t) ∈ L(Y, X) for all t ∈ R, h > τ0 . Proposition 2.6.4. Let f : R × X → Y and suppose that for given θ ∈ (0, 1] the discretization mesh (2.6a) satisfies τ0 < θhk ,
− θh1 k ∈ σ(B(tk+1 )) for all k ∈ I .
If Hypothesis 2.6.3 holds with lip f (tk+1 , ·) < ∞ for all k ∈ I , then the general forward solution ϕ of (2.6f) exists on I × X and is continuous, provided one of the following conditions holds: (a) hk θ lip Thk θ (tk+1 ) ◦ f (tk+1 , ·) < 1 for all k ∈ I , (b) X is a Hilbert space and for each k ∈ I there exists a γk > 0 such that for all u, v ∈ X, 2
hk θ Thk θ (tk+1 ) (f (tk+1 , u)−f (tk+1 , v)) , u − vX ≤ (1 − γk ) u − vX . Proof. (a) We can apply Theorem 2.3.6 with Hk+1 = Thk θ (tk+1 )−1 and a nonlinearity Fk (u, u ) = (IY − hk (1 − θ)B(tk ))u + hk (1 − θ)f (tk , u) + hk θf (tk+1 , u ). Indeed, Thk θ (tk+1 )−1 satisfies (2.3c) and due to lip2 Fk ≤ rθhk lip2 f , as well as the continuity of Thk θ (tk+1 )(IY − hk (1 − θ)B(tk )) we can infer the claim.
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(b) We can also interpret (2.6f) as semi-implicit equation (D ) by defining Hk+1 (u ) := u − hk θThk θ (tk+1 )f (tk+1 , u ), Fk (u) := Thk θ (tk+1 )(IY − hk (1 − θ)B(tk ))u+hk (1 − θ)Thk θ (tk+1 )f (tk , u); our assumptions yield that Hk+1 : X → X is globally Lipschitz and strongly monotone, i.e., (2.3j) holds. Thus, Corollary 2.3.13 implies our claim.
Corollary 2.6.5. Suppose that Thk θ (tk+1 ) ∈ L(Y, X), k ∈ I , is compact and the family Bˆ consists of all bounded subsets B ⊆ I × X. If the assumptions of ˆ Proposition 2.6.4(a) hold, then (2.6f) is B-uniformly contracting. Proof. We use the notation from the proof of Proposition 2.6.4(a). Due to the global −1 Lipschitz continuity required for f (tk+1 , ·), the mapping Hk+1 ◦ Fk is bounded. Moreover, by compactness of Thk θ (tk+1 ) it satisfies a trivial Darbo condition with −1 dar Hk+1 ◦ Fk = 0. Then Corollary 2.3.8(b) implies the assertion.
As explained below, our second approach is essential in analytical, as well as discretization theory and its importance is hard to overestimate. It falls in the setting of Sect. 1.5.2 and is based on the fact that mild solutions exist for (AE). Hence, we work under Hypothesis 1.5.4, which guarantees the existence of an evolution family (U (t, s))s≤t on X for the linear part and of a continuous 2-parameter semiflow u on R × X. As a temporal discretization of the evolution equation (AE) we investigate the semiflow u evaluated at the discrete points tk , k ∈ I. The resulting sequence in X satisfies an explicit nonautonomous difference equation x = Ak x + fk (x)
(AΔE)
in X = I × X with the functions (cf. (1.5d)) . tk+1 Ak := U (tk+1 , tk ), fk (x) := U (tk+1 , s)f (s, u(s; tk , x)) ds. tk
This means that the general forward solution ϕ of (AΔE) and the mild solutions u to (AE) are related by (cf. (1.5e)) ϕ(k; l, u0 ) = u(tk ; tl , u0 ) for all l ≤ k. This implies that various properties from the 2-parameter semiflow carry over to the difference equation (AΔE), like for instance compactness or absorbing sets. Hence, to any evolutionary differential equation (AE) allowing a variation of constants formula (1.5d), we attach an explicit difference equation (AΔE) interpolating its solution values restricted to a given discretization mesh. The benefit from this procedure is as follows: •
Asymptotic properties of (AΔE) extend to (AE) (see the arguments and references [83, 169, 200, 285, 285, 343] from the Preface, or [246]). • Stability properties are transferred from (AΔE) to the differential equation (AE). Particularly in an infinite-dimensional setting the corresponding discrete theory for (AΔE) is more amenable than (AE), due to smoothing properties of the integral in the definition of fk or the fact that Ak is a bounded operator.
2.6 Applications •
75
In temporal discretizations of (AE), e.g., using (2.6f) or a general Runge–Kutta method, the global discretization error typically grows exponentially in time (cf., for instance, [55, 447]). For this reason, it is easier to compare solution and discretization values on basis of the right-hand side of (AE) and, e.g., (2.6f), instead of working with the associate solution semigroups.
Time-Discretized Abstract Sectorial Equations For evolution equations ut + Bu = f (t, u), where the linear part ut + Bu = 0 is autonomous with a sectorial operator B (see Sect. 1.5.2), the smoothing property required in Hypothesis 2.6.3 is satisfied with X = Y r , r ∈ [0, 1) (see [438, Lemma 1]). Moreover, Corollary 2.6.5 can be applied, if θ = 1 and B has a compact resolvent.
2.6.3 Fully Discretized Reaction-Diffusion Equations Let d ∈ N. In order to motivate our hypothesis made below, consider the following scalar semilinear nonautonomous parabolic initial-boundary value problem ut − δ(t)Δu = f (t, x, u)
for t > t0 , x ∈ Ω, for t ≥ t0 , x ∈ bd Ω, for x ∈ Ω
u(t, x) = 0 u(t0 , x) = u0 (x)
(RDE)
equipped with homogeneous Dirichlet boundary conditions. Here, Ω ⊆ Rd is a bounded domain with smooth boundary bd Ω (depending on the context, smooth means Lipschitz or polygonal) and a time-variable diffusion rate in form of a function δ : R → [0, ∞). Conditions guaranteeing the existence of (e.g., weak) solutions for (RDE) can be found, for instance, in Sect. 1.5.3 or in [432, p. 269ff]. The same holds for dissipativity properties (see Proposition 1.5.7). We, however, focus on spatial discretizations of (RDE) leading to initial value problems for ordinary differential equations in RN of the form v(t0 ) = v0 ,
M v˙ + δ(t)Av = M F (t, v),
(2.6g)
where A, M ∈ RN ×N are positive definite and the mapping F : R × RN → RN will be specified later. One denotes A as stiffness matrix and M as mass matrix. Instead of the Euclidean state space RN for (2.6g), or more general FN , we use the discrete Lebesgue space L2N := FN equipped with the inner product
x, yL2 := N
N 1 1
x, y = xl y¯l Nd Nd l=1
and the induced norm xL2 := N
1 Nd
/N l=1
2
|xl | .
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2 Nonautonomous Difference Equations
Hypothesis 2.6.6. Let m ∈ N, λ ∈ R, b > 0 and α, β : R → R are functions with: 2 (i) M ∈ GL(RN ) and M −1 Av, v L2 ≥ λ vL2 for all v ∈ RN , N
N
(ii) F (t, ·) ∈ C m (RN , RN ) and for all t ∈ R, u, v ∈ RN one has 2
F (t, v), vL2 ≤ α(t) − b vL2 , N
N
2
D2 F (t, u)v, vL2 ≤ β(t) vL2 . N
N
We clarify that these assumptions are realistic by the following examples: Example 2.6.7 (finite differences). For simplicity we suppose a rectangular domain d Ω = i=1 (−ai , ai ) with a1 , . . . , ad > 0 and introduce the grids
×
G=
×[−N , N ] d
j
j Z
G0 =
,
j=1
×(−N , N ) d
j
j Z
j=1
with integers N1 , . . . , Nd > 1. Moreover, we have to introduce the mesh size vector h = (h1 , . . . , hd ) with hi = ai /Ni . For fixed instants t, the solution of (RDE) is represented as vector u = (ui )i∈G0 , where i = (i1 , . . . , id ) is a multiindex to specify the grid point xi = (h1 ii , . . . , hd id ). Having this at hand, we use the finite difference Laplacian Δh from Sect. 3.7.4. This reduces the initialboundary value problem (RDE) to an N -dimensional ordinary differential equation (2.6g) with A = Δh ,
M = IRN ,
F (t, v) = (f (t, ih, vi ))i∈G
0 and initial values v0 = (u0 (ih))i∈G0 . More precisely, one has N = dj=1 (2Nj −1). Thus, spatial finite difference discretizations lead to the identity as mass matrix M and a symmetric, positive definite matrix A (cf. Lemma 3.7.9). It turns out stiffness /d 2 π 2 −2 that λ = j=1 4Nj aj sin 4Nj . The remaining spatial discretization schemes discussed in this subsection are Galerkin methods (cf., e.g., [148, p. 81ff]), which are based on a variational formulation of the reaction-diffusion equation in (RDE). Indeed, Galerkin methods are variation problems
ut , vV + δ(t)a(u, v) = f (t, ·, u), vV
for all v ∈ V,
where the test space V ⊆ H01 (Ω) is a suitable subspace of finite dimension N and a bounded bilinear form a : V × V → R. We choose a basis φ1 , . . . , φN of V and suppose a solution of the above variational problem is of the form u(t, x) =
N l=1
vl (t)φl (x),
2.6 Applications
77
which leads to the relation N
φl , φi V v˙ l + δ(t)
l=1
N
1 2 a(φl , φi )vl =
f
t, ·,
l=1
N
3
4 , φi
vl φl
l=1
V
for all i ∈ {1, . . . , N }. This problem, however, can be written in the form (2.6g) N with A = (a(φl , φi ))N i,l=1 , M = ( φl , φi V )i,l=1 and the nonlinearity 21 2 F (t, v) = M
−1
f
t, ·,
N
3 vl φl
l=1
4 3N , φi
. V
(2.6h)
i=1
Under consistency assumptions (for details, see [148, p. 91, Remark 2.20]) the stiffness matrix A is positive-definite. Moreover, provided φ1 , . . . , φN is an orthogonal basis of V one knows that M is diagonal with positive elements. Now we briefly discuss special cases of general Galerkin methods: Example 2.6.8 (spectral Galerkin method). For spectral Galerkin methods it is ! " V = span φl ∈ L2 (Ω) : l = 1, . . . , N , where φ1 , . . . , φN are the normalized eigenvectors corresponding to the first N eigenvalues of the Laplace operator Δ on L2 (Ω) equipped with zero boundary conditions, which form a basis of H01 (Ω). Canonically, the inner product on V is the H01 (Ω)-inner product and the symmetric bilinear form a is . ∇u∇v.
a(u, v) := Ω
Obviously, here M is the identity matrix on RN and A is symmetric. Example 2.6.9 (finite elements). Let T denote a triangulation of Ω into simplices S ⊆ Rd . For (piece-wise linear) finite element methods the subspace V consists of continuous functions vanishing on bd Ω, such that for each v ∈ V and every simplex S ∈ T the restriction v|S is affine linear. One chooses a basis of V according to [148, p. 46ff] and the inner product, as well as the bilinear form a on V are as for the spectral Galerkin method in Example 2.6.8. Then A and M are symmetric positive definite (cf. [148, p. 145, Proposition 3-64]). Example 2.6.10 (collocation methods). We also briefly mention the class of collocation methods (cf. [388, p. 380ff]). Here, one chooses V to be the space of (piece-wise) polynomials in d variables. As significant difference to the above schemes, for collocation methods the integrals occurring in the H01 (Ω)-inner product and in the bilinear form a are replaced by their Gauß–Lobatto numerical discretization. While the mass matrix M becomes diagonal, the stiffness matrix is not necessarily symmetric for collocation methods.
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2 Nonautonomous Difference Equations
In order to obtain a temporal discretization of (2.6g), and consequently a full discretization of (RDE), we employ two related implicit temporal discretizations for (2.6g), namely the 2-stage θ-method M
v − v + A [(1 − θ)δ(tk )v + θδ(tk+1 )v ] hk = (1 − θ)M F (tk , v) + θM F (tk+1 , v ),
(2.6i)
and the classical θ-method M
v − v + δ(tθk )A [(1 − θ)v + θv ] = M F (tθk , (1 − θ)v + θv ) hk
(2.6j)
with tθk := (1 − θ)tk + θtk+1 , k ∈ I . Both schemes depend on θ ∈ [0, 1] and we are interested in the question, whether they induce 2-parameter semigroups. Proposition 2.6.11. Suppose that Hypothesis 2.6.6 holds true and θ ∈ [0, 1]. (a) If θhk (β(tk ) − δ(tk+1 )λ) < 1 for all k ∈ I , then the general forward solution ϕ(·; θ) of (2.6i) exists and ϕ(k; κ, ·; θ) ∈ C m (RN , RN ), κ ≤ k. (b) If the stepsizes in (2.6a) fulfill θhk β(tθk ) − δ(tθk )λ < 1 for all k ∈ I ,
(2.6k)
then the general forward solution ϕ(·; θ) of (2.6j) exists and additionally one has ϕ(k; κ, ·; θ) ∈ C m (RN , RN ), κ ≤ k. Proof. Since the proofs of assertions (a) and (b) are similar, we focus on part (a). For this, we write (2.6i) as semi-implicit equation (D ) with Hk+1 (v ) = v + θhk δ(tk+1 )M −1 Av − θhk F (tk+1 , v ), Fk (v) = v − (1 − θ)hk δ(tk )M −1 Av + (1 − θ)hk F (tk , v) and spaces Xk = Yk = L2N . We obtain from Hypothesis 2.6.6 the inequality
DHk+1 (u)v, v = v2 + θhk δ(tk+1 ) M −1 Av, v − θhk D2 F (tk+1 , u)v, v 2
≥ [1 + θhk δ(tk+1 )λ − θhk β(tk+1 )] v
for all k ∈ I, u, v ∈ RN . In conclusion, Corollary 2.3.14 yields our claim.
Next we study dissipativity properties of the schemes (2.6i) and (2.6j). Referring to Example 2.2.11 we know that the adequate Proposition 2.2.10 is applicable to (2.6i), only if θ = 1, i.e., only for the implicit Euler scheme. This method, however, is also contained in schemes of the form (2.6j), and we restrict to them. Proposition 2.6.12. Let ε > 0 and 1 I be unbounded below. Suppose Hypothesis 2.6.6 holds with θ ∈ and (2.6k). For stepsizes with 2,1
2.6 Applications
79
β∗ := inf k∈I hk b + δ(tθk )λ > 0, the θ-method (2.6j) is uniformly bounded dissipative with absorbing set Aθ :=
(k, x) ∈ I × R
N
: xL2 ≤ N
ε+
5 2hk α(tθk )(1+β∗ θ(3θ−1)) β∗ θ(3θ−1)−β∗ (1−θ)|2−3θ|
.
Remark 2.6.13. For discretization meshes (2.6a) with small stepsizes hk also the constant β∗ becomes small. Yet, the size of the absorbing set Aθ effectively depends only on the ratio 1/ of the stepsize bounds in (2.6a), since we have 2 sup
α(tθ )
k∈I k + θ(3θ − 1) 2hk α(tθk )(1 + β∗ θ(3θ − 1)) inf k∈I (b+δ(tθk )λ) ≤ =: ρ0 β∗ θ(3θ − 1) − β∗ (1 − θ) |2 − 3θ| θ(3θ − 1) − (1 − θ) |2 − 3θ|
for all k ∈ I . Note that the absorbing set Aθ blows up as θ 12 . On the other hand, the radius ρ0 > 0 achieves its minimal value
supk∈I α(tθk ) inf k∈I (b+δ(tθk )λ)
+ 1 for θ = 1.
Proof. The general forward solution of (2.6j) exists due to Proposition 2.6.11(b). For this, we write (2.6j) in the form (2.2g) with a step function fk : RN → RN , fk (v) = −hk δ(tθk )M −1 Av + hk F (tθk , v) and harvest from Hypothesis 2.6.6 the estimate
fk (v), v = −hk δ(tθk ) M −1 Av, v + hk F (tθk , v), v 2 ≤ hk α(tθk ) − hk b + δ(tθk )λ v for all (k, v) ∈ I × RN . Thus, Corollary 2.3.22 and Proposition 2.2.10 yield our claim.
Proposition 2.6.14. Under the assumptions of Proposition 2.6.12 the θ-method (2.6j) has a uniformly bounded global attractor ωAθ ⊆ I × L2N and the mapping θ → ωAθ (k), k ∈ I, is upper-semicontinuous in θ ∈ 12 , 1 . Remark 2.6.15. An analysis of the explicit constant stepsize Euler method applied to an autonomous ODE (2.6g) can be found in [142, Sect. 4]. Here, absorbing sets and attractors exist, if the temporal stepsize T is restricted in terms of the initial data. Without this assumption, the scheme may blow up and a global attractor cannot exist for the explicit Euler method as dynamical system on RN . 2 Proof. Let Bˆ be the family of uniformly bounded nonautonomous sets in I × LN 1 and θ ∈ 2 , 1 . We check the assumptions of Theorem 1.3.9. First, the general forward solution of (2.6j) exists by Proposition 2.6.11(b) as a continuous mapping. ˆ ˆ Proposition 2.6.12 implies that ϕ(·; θ) is B-dissipative with a B-uniformly absorbˆ Due to dim L2 < ∞ the 2-parameter semiflow ϕ(·; θ) is clearly ing set Aθ ∈ B. N ˆ ˆ and beB-compact with compactification time 1. In conclusion, ωAθ is a B-attractor ∗ ˆ cause B consists of uniformly bounded sets, Aθ := ωAθ must be the global attractor of ϕ(·; θ).
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2 Nonautonomous Difference Equations
Using Remark 2.3.15 we see that ϕ(·; θ) is a continuous 2-parameter semigroup. By the explicit form of the absorbing sets Aθ in Proposition 2.6.12 it is easy to construct an absorbing set independent of θ ∈ int U , where U ⊆ 12 , 1 is compact. Therefore, the upper-semicontinuity assertion follows from Theorem 1.3.12.
Example 2.6.16. We continue the above Example 2.6.7 by a finite difference discretization to the Chafee–Infante equation ut − δ(t)uxx = u(α1 (t) − α2 (t)u2 ) as discussed in Example 1.5.8, but here with a continuous time-dependent diffusion coefficient δ : R → [0, ∞). The resulting ODE is of the form (2.6g) with M = IRN , N F (t, v) = vi (α1 (t) − α2 (t)vi2 ) i=1 ;
A = Δh = h−2 diag(−1, 2, −1),
consequently, we obtain the relations (cf. Lemma 3.7.10) −1 M Δh v, v L2 ≥ N
D2 F (t, u)v, vL2
N
1 2
N +1 2 1 − cos Nπ+1 v2L2 , a N
N N 2 1 1 2 α1 (t) − 3α2 (t)ui vi ≤ = α1 (t)vi2 N i=1 N i=1 2
= α1 (t) vL2
N
and thanks to supx∈R (2α1 (t)x2 − α2 (t)x4 ) =
F (t, v), vL2
N
=
α1 (t)2 α2 (t)
also
N 1 α1 (t)vi2 − α2 (t)vi4 = N i=1
N N 1 1 α1 (t)2 2 − α1 (t) vL2 2α1 (t)vi2 − α2 (t)vi4 − α1 (t)vi2 ≤ N N i=1 N i=1 α2 (t)
for all t ∈ R, u, v ∈ RN . We conclude that Hypothesis 2.6.6 holds with α(t) = α1 (t)2 1 N +1 2 1 − cos Nπ+1 > 0 and β(t) = α1 (t). α2 (t) , b = inf t∈R α1 (t), λ = 2 a
2.6.4 Fully Discretized Finite Difference Ginzburg–Landau Equation Returning to the previous Example 1.5.9, we consider the nonautonomous complex Ginzburg–Landau equation with cubic nonlinearity equipped with 1-periodic initialboundary conditions 2
ut − μ1 (t)u − (1 + iν)uxx + (1 + iμ2 (t)) |u| u = 0 in (t0 , ∞) × R, u|t=t0 = u0 ,
u(t, x) = u(t, x + 1) on (t0 , ∞) × R,
(GL)
2.6 Applications
81
and a given initial time t0 ∈ R. The instability parameter μ1 : R → R and the dispersion parameter μ2 : R → R are assumed to be continuous functions with μ1 (t) ∈ (R0 , R1 ] ,
μ2 (t) ∈ [−R2 , R2 ] for all t ∈ R,
bounds R0 , R1 , R2 > 0 and ν ∈ R. It is shown in [86, p. 118] that (GL) is wellposed in the space L2 ,√ has regular complete solutions and a uniform attractor (under the assumption R2 ≤ 3, see also Example 1.5.9). Let us turn to a finite difference version of the initial-boundary value problem (GL). For the sake of a spatial discretization, we subdivide the periodicity interval [0, 1] into N ≥ 3 uniform subintervals of length h = 1/N and the state space for a finite difference approximation of (GL) respecting periodic boundary conditions is the set of N -periodic sequences in C, which will be identified with CN . For the discrete Laplacian Δh and the corresponding discrete Lebesgue and Sobolev spaces we refer the declined reader to our explanations in Sect. 3.7.4. Since the nonlinear term in (GL) does not contain spatial derivatives, let us define FN : CN → CN by T FN (x) := |x1 |2 x1 , . . . , |xN |2 xN . Our full discretization of (GL) consists of such a finite difference approximation in space (represented by the matrix Δh ), leading to the ODE in CN , x˙ + Δ˜h x = F (t, x)
(2.6l)
with Δ˜h := (1+iν) [ICN + Δh ], F (t, x) := (μ1 (t)+1+iν)x−(1+iμ2(t))FN (x) and a fully implicit variable stepsize Euler method for (2.6l), i.e., we arrive at the implicit recursion x − x + Δ˜h x = F (tk+1 , x ), (ΔGL) hk which will be denoted as fully discretized Ginzburg–Landau equation. First of all, in order to study the behavior of (ΔGL) we need technical tools: Lemma 2.6.17. The map FN : L2N → L2N satisfies a one-sided Lipschitz condition (1 + iμ2 (tk+1 )) FN (x) − FN (¯ x), x − x ¯L2 N + 2 2 2 ≤ 1 + μ2 (tk+1 ) xL∞ + ¯ ¯2L2 xL∞ x − x N
N
N
for k ∈ I , x, x ¯ ∈ L2N .
Proof. The proof follows along the lines of the autonomous situation considered in [310, Lemma 3.11].
Next we need to investigate whether (ΔGL) is well-posed in the sense that forward solutions exist. Obviously, (ΔGL) is a one-step method as in (O).
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2 Nonautonomous Difference Equations
Lemma 2.6.18. If the temporal stepsizes (2.6a) satisfy 2hk μ1 (tk+1 ) < 1
for all k ∈ I ,
(2.6m)
N then for every pair (κ, ξ) ∈ I × CN there exists a forward solution φ : I+ κ → C of (ΔGL) satisfying φ(κ) = ξ.
Proof. We consider (ΔGL) as one-step scheme of the form (O) in the extended state N space S = I × C and the step function Gk (x ) := −hk Δ˜h x − F (tk+1 , x ) . From Lemma 3.7.11 one obtains for all x ∈ CN the relation (ΔGL) 2 Gk (x), x = −hk x + Δh x, x − (μ1 (tk+1 ) + 1) x + FN (x), x ≤ −hk ( Δh x, x − μ1 (tk+1 ) x, x) ≤ hk μ1 (tk+1 ) x
2
and therefore Theorem 2.2.5 is applicable with αk = 2hk μ1 (tk+1 ) and βk = γk = 0. Hence, our stepsize assumption ensures αk > −1 and the claim follows.
2 2R1 Lemma 2.6.19. Let J ⊆ I be a discrete interval, define ρ0 := R0 and finally choose ρ > ρ0 . If φ : J → L2N is a solution of (ΔGL), then the following holds: ¯ρ (0, L2 ) for k ∈ J , ¯ ρ (0, L2 ) ⇒ φ (k) ∈ B (a) One has the implication φ(k) ∈ B N N 2 (b) for every uniformly bounded B ⊆ I× LN there exists an N0 = N0 (B, ρ) ≥ 0 so ¯ρ (0, L2 ) for k ∈ J, n ≥ N0 , k − n ∈ J. that φ(k − n) ∈ B(k − n) ⇒ φ(k) ∈ B N 2
Proof. Let φ : J → L2N be a solution to (ΔGL) and abbreviate u(k) := φ(k)L2 . N Taking the L2N -inner product of (ΔGL) with φ (k) ∈ L2N , passing over to the real part and using (0.0e), we arrive at 4
u (k) ≤ u(k) + 2μ1 (tk+1 )hk u (k) − 2hk u (k) − 2hk φ (k)L4
N
(2.6n)
for all k ∈ J . Thanks to the relation minx∈R x4 − 4μ1 (tk+1 )x2 = −4μ1 (tk+1 ) and xL4 ≥ xL2 (cf. (3.7i)) we deduce N
N
4
4
φ (k)L4 ≥ −4μ1 (tk+1 )2 + 4μ1 (tk+1 ) φ (k)L4 N
N
≥ −4μ1 (tk+1 )2 + 4μ1 (tk+1 )u (k) and therefore we obtain from (2.6n) that 4
(1 − 2μ1 (tk+1 )hk ) u (k) ≤ u(k) − 2hk u (k) − hk φ (k)L4
N
+ 4μ1 (tk+1 )2 hk − 4μ1 (tk+1 )hk u (k),
(2.6o)
2.6 Applications
83
which, in turn, yields the estimate u (k) ≤
1 4μ1 (tk+1 )2 hk u(k) + 1 + 2μ1 (tk+1 )hk 1 + 2μ1 (tk+1 )hk
for all k ∈ J .
¯ρ (0, L2 ), we consequently obtain (a) Let k ∈ J . Given φ(k) ∈ B N 4R12 T ρ2 + 4μ1 (tk+1 )2 hk (2.6a) ρ2 + ≤ ρ2 , ≤ 1 + 2μ1 (tk+1 )hk 1 + 2R0 T 1 + 2R0 T
u (k) ≤
since obviously ρ > ρ0 holds true and assertion (a) is verified. (b) In addition, in order to show (b), this enables us to apply the Gronwall inequality from Proposition A.2.3(a), which guarantees k−1
2
u(k) ≤ ea (k, k − n) φ(k − n)L2 + 4 N
ea (k, l + 1)
l=k−n
μ1 (tl+1 )2 hl 1 + 2μ1 (tl+1 )hl
for all n ≥ 0 with the sequence a(k) := (1 + 2μ1 (tk+1 )hk )−1 ∈ (0, 1). Thus, our assumptions on the function μ1 and the stepsizes (2.6a) imply 2
u(k) ≤ ea (k, k − n) φ(k − n)L2 + ρ20 N
and for sufficiently large N ≥ 0, depending on ρ > ρ0 , we have deduced the ¯ ρ (0, L2 ). This yields our claim. inclusion φ(k − n) ∈ B
N In the following result a stepsize restriction is formulated in terms of the Lambert W-function W : [−e−1 , ∞) → R (see [99]). Lemma 2.6.20. Let J ⊆ I be a discrete interval, choose δ > 1, let ρ0 > 0 be the radius of the absorbing L2N -ball from Lemma 2.6.19 and define 6
4R12 2ρ20 √ + , (2.6p) +e ρ1 :=
qk := 2μ1 (tk+1 ) + 54(1 + μ2 (tk+1 )2 ) 1 + 54(1 + μ2 (tk+1 )2 )ρ40 ρ40 . ρ20
√δ
supk∈I qk
Moreover, suppose that the temporal stepsizes (2.6a) fulfill hk qk < 1 +
W (−δe−δ ) δ
for all k ∈ I
(2.6q)
1 solves (ΔGL), then the following holds: and choose ρ > ρ1 . If φ : J → HN
¯ρ (0, H 1 ) for k ∈ J . ¯ ρ (0, H 1 ) ⇒ φ (k) ∈ B (a) One has the implication φ(k) ∈ B N N 1 (b) For every uniformly bounded B ⊆ I × HN there exists an N1 = N1 (B, ρ) ≥ 0 ¯ρ (0, H 1 ) for k ∈ J, n ≥ N1 , k − n ∈ J. with φ(k − n) ∈ B(k − n) ⇒ φ(k) ∈ B N
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2 Nonautonomous Difference Equations
Proof. The positive invariance assertion (a) can be proven as in Lemma 2.6.19 and 1 we only show the claimed dissipativity of (ΔGL) w.r.t. the HN -norm. Suppose that 1 ∗ φ : J → HN is a solution of (ΔGL). First note that φ(k ) = 0 for some k ∗ ∈ J immediately implies φ(k) = 0 for all k ≥ k ∗ and therefore our assertion. Thus, w.l.o.g. we can suppose φ (k)L2 = 0 from now on. Referring to our preparations N in Lemma 2.6.19 and (3.7j) this foremost requires an estimate for φ(k) ∈ CN in 2 the |·|Δh -seminorm. For this, we abbreviate u(k) := |φ(k)|Δh , take the ·, ·Δh inner product of (ΔGL) with φ (k) and pass over to the real part in order to obtain u (k) ≤
1 2
2
(u (k) + u(k)) + μ1 (tk+1 )hk u (k) − hk Δh φ (k)L2
+
N
3 2
N + hk + δh φj (k) φj+1 (k) 1 + μ2 (tk+1 )2 N j=1
2
+ φj (k)
2
;
in the latter sum one has to respect the periodicity condition φN +1 = φ1 and the Cauchy-Schwarz inequality for finite sums yields N 1 + δh φj (k) φj+1 (k) N j=1
2
+ φj (k)
2
2 2 ≤ 2 Dh+ φ (k)L4 φ (k)L4 N
N
and therefore u (k) − u(k) 2 ≤ 2μ1 (tk+1 )u (k) − 2 Δh φ (k)L2 N hk + + 2 2 + 6 1 + μ2 (tk+1 )2 Dh φ (k)L4 φ (k)L4 . N
N
Next we make use of Lemma 3.7.13(c) in order to arrive at u (k) − u(k) (3.7q) 4 ≤ 2μ1 (tk+1 ) + 54(1 + μ2 (tk+1 )2 ) φ (k)L4 u (k) N hk 2
2
+ φ (k)L2 − Δh φ (k)L2 , N
(2.6r)
N
4
but further estimates are due in order to control the term φ (k)L4 . Thanks to the N interpolation inequality from Lemma 3.7.13(a) we have 2
− Δh φ (k)L2
N
(3.7m)
≤ −
4
|φ (k)|Δh 2
φ (k)L2
N
2.6 Applications
85
and the discrete Gagliardo–Nirenberg inequality from Lemma 3.7.13(c) with parameters p = 4, q = 2 implies 4
φ (k)L4
N
2 2 φ (k)L2 + 2 φ (k)L2 |φ (k)|Δh φ (k)L2 N N N 1 2 2 ≤ φ (k)L2 + φ (k)L2 + k |φ (k)|Δh φ (k)L2 , N N N k
(3.7o)
≤
2
where in the last step we made use of the elementary inequality 2xy ≤ x + y 2 for all > 0 (see [432, p. 621]). We insert the last two relations into (2.6r) and obtain
u (k) − u(k) ≤ 2μ1 (tk+1 ) + 54(1 + μ2 (tk+1 )2 ) hk 1 2 2 φ (k)L2 + k |φ (k)|Δh φ (k)L2 × φ (k)L2 + N N N k 4 |φ (k)|Δh 2 u (k) + φ (k)L2 . − 2 N φ (k)L2 N
−1
and obtain from Lemma 2.6.19(b) that Now we set k := 54(1 + μ2 (tk+1 )2 )ρ40 for a sufficiently large period of time given by N0 ≥ 0 one arrives at the inclusion ¯ρ (0, L2 ) with ρ > ρ0 . This finally assures u (k)−u(k) ≤ qk u (k) + ρ2 , φ (k) ∈ B N hk which is equivalent to u (k) ≤ a(k)u(k) + b(k) with coefficient sequences a(k) := hk ρ2 1 1−qk hk , b(k) := 1−qk hk ; note here that (2.6q) implies qk hk < 1 due to W (δ) < 0 for δ > 1. Having this at hand, we are in the position to apply the uniform Gronwall inequality from Proposition A.2.3(b). Relying on its notation, we check the corresponding assumptions: •
1 First of all, one has a(k) ≥ 1 and due to the elementary inequality 1−x ≤ eδx −δ for all x ∈ [0, 1 + W (−δe )/δ], our stepsize restriction (2.6q) implies
α1 = sup
¯ −1 k+7 N
κ≤k
n=k
¯ −1 k+7 N 1 ¯ ≤ sup eδqn hn ≤ eδ(N +1)T supk∈I qk . 1 − qn hn κ≤k n=k
/k+N¯ b(n) /k+N¯ ¯ + 1)T. One has α2 = supκ≤k n=k a(n) = ρ2 supκ≤k n=k hk ≤ ρ2 (N • Recalling the proof of Lemma 2.6.19 we obtain from (2.6o) that •
2
2
2
φ (n)L2 − φ(n)L2 + 2μ1 (tn+1 )hn φ (n)L2 + 2hn |φ (n)|Δh N
N
4
+ hn φ (n)L4 ≤ 4μ1 (tn+1 )2 hn , N
N
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2 Nonautonomous Difference Equations
¯ to deduce we can sum this estimate over n from k to k + N ¯ k+ N
2
2
2
φ (n)L2 − φ(n)L2 + 2μ1 (tn+1 )hn φ (n)L2 + 2hn |φ (n)|Δh N
N
n=k 4
+ hn φ (n)L4
N
N
¯ + 1) ≤ 4R12 T (N
which yields for k ≥ N0 so large that we are inside the absorbing L2N -ball that ¯ k+ N
2
2μ1 (tn+1 )hn φ (n)L2 + 2hn |φ (n)|Δh N
n=k
4
+ hn φ (n)L4
N
¯ + 1) + ρ2 ≤ 4R12 T (N
and from these preparations we can draw the conclusion
T α3 = T
¯ k+ N
¯ + 1) + ρ2 . u(k) ≤ 4R12 T (N
n=k
Since all the above bounds for α1 , α2 , α3 are uniform in k, so large that the solutions ¯ρ (0, L2 ), our Proposition A.2.3(b) guarantees with r = (N ¯ + 1)T that remain in B N ¯) φ(k + N
2 Δh
α3 4R12 r + ρ2 ≤ eδr supk∈I qk ρ2 r + ≤ α1 α2 + ¯ ¯ + 1) T (N N +1 8 9 4 1 + R12 . = eδr supk∈I qk ρ2 r + r
1 has the global minimum Keeping in mind that the function r → r + r 1 √ interval (0, ∞) for r = , this readily implies
¯) φ(k + N
2 Δh
≤ 2e
√δ
supk∈I qk
2R12 ρ2 √ +
√2
in the
for all k ≥ N0 .
1 With this we have deduced a uniform bound on the HN -seminorm, independent of the spatial mesh size h, the temporal stepsize H and the initial state of φ. This gives ¯ ρ (0, H 1 ), with radius ρ > ρ1 (cf. (3.7j)). us the desired absorbing ball B
N
Proposition 2.6.21. Let I be unbounded below. Choose δ > 1, let ρ0 , ρ1 > 0 be the radii of the absorbing balls from the respective Lemmata 2.6.19 and 2.6.20, and ρ > ρ1 . If the temporal stepsizes fulfill (2.6m), (2.6q) and + hk 2μ1 (tk+1 ) + 12 1 + μ2 (tk+1 )2 ρ2 < 1
for all k ∈ I ,
(2.6s)
2.6 Applications
87
then the fully discretized Ginzburg–Landau equation (ΔGL) ¯ ρ (0, H 1 ), (a) generates a Lipschitzian general forward solution ϕ on I × B N 1 1 (b) has the global attractor ωA ⊆ I × HN with A = I × Bρ (0, HN ). Proof. (a) From Lemma 2.6.18 we know that forward solutions to (ΔGL) exist and before showing their continuous dependence on the initial conditions, we have to 1 establish uniqueness. Given κ ∈ I we pick two solutions φ, φ¯ : I+ κ → HN of (ΔGL) and obtain the solution identities φ (k) ≡ φ(k) − hk [μ1 (tk+1 )φ (k) − ν˜Δh φ (k) − μ ˜ (tk+1 )FN (φ (k))] ,
¯ − hk μ1 (tk+1 )φ¯ (k) − ν˜Δh φ¯ (k) − μ ˜(tk+1 )FN (φ¯ (k)) φ¯ (k) ≡ φ(k) with ν˜ := 1 + iν and μ ˜(t) := 1 + iμ2 (t). Subtracting these relations, taking the L2N -inner product with φ (k) − φ¯ (k) and passing to the real part yields u (k) − u(k) ≤ 2μ1 (tk+1 )hk u (k) − 2hk (1 + iμ2 (tk+1 )) FN (φ (k)) − FN (φ¯ (k)), φ (k) − φ¯ (k) L2 , N
¯ 2 2 . Thanks to Lemma 2.6.17 where we have abbreviated u(k) := φ(k) − φ(k) LN and the continuous embedding (3.7o) from Lemma 3.7.13(b) this means (3.7o)
u (k) − u(k) ≤ 2μ1 (tk+1 )hk u (k) + 2 2 + 2hk 1 + μ2 (tk+1 )2 3 φ (k)H 1 + 3 φ¯ (k)H 1 u (k) N
N
¯ ρ (0, H 1 ) (cf. Lemma 2.6.20(a)) one conseand inside the positively invariantset B N + quently has u (k) − u(k) ≤ hk 2μ1 (tk+1 ) + 12 1 + μ2 (tk+1 )2 ρ2 u (k). This ¯ estimate helps us in two aspects: First of all, for identical initial values φ(k) = φ(k) 1 ¯ one has u(k) = 0 and due to (2.6s) one derives u (k) = 0, i.e., inside Bρ (0, HN ) 1 ¯ forward solutions are uniquely determined. Thus, inside the ball Bρ (0, HN ) the general forward solution ϕ of (ΔGL) exists. Second, the above estimate implies ¯ 2 ϕ(k + 1; k, ξ) − ϕ(k + 1; k, ξ) L
N
≤
1
1−hk 2μ1 (tk+1 )+12
√
1+μ2 (tk+1
)2 ρ2
ξ − ξ¯ 2 L
N
for all k ∈ I
¯ρ (0, H 1 ). Inductively the general forward solution is Lipschitzian. and ξ, ξ¯ ∈ B N ˆ ¯ρ (0, H 1 ). Due to dim H 1 < ∞ (b) Let B denote the family of subsets of I × B N N ˆ we know that the 2-parameter semigroup ϕ is B-compact (see Remark 1.2.21(1)) and the claim follows from Theorem 1.3.9.
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2.6.5 Time-Discretized Doubly Nonlinear Equations In this subsection, we return to the doubly nonlinear equation briefly introduced in Sect. 1.5.4. More precisely, we again study β(ut ) − Δu = g(t, x, u) u=0
for t > t0 , x ∈ Ω, for t > t0 , x ∈ bd Ω,
β(u(t0 , x)) = β(u0 (x))
for x ∈ Ω,
(nRDE)
where Ω ⊆ Rd is a bounded domain with Lipschitzian boundary bd Ω. The precise assumptions are given in Hypothesis 1.5.10. For simplicity, we restrict to constant stepsize discretizations, i.e., the situation = 1 and tk := kT for k ∈ I in (2.6a). Let us consider fully-implicit Euler discretization of (nRDE) given as β(u ) − β(u) = Δu + g(tk+1 , x, u ) T β(u(tκ , 0)) = β(u0 ),
for x ∈ Ω,
(nRΔE)
where the recursion (nRΔE) is understood in a weak sense (cf. [125]). Lemma 2.6.22. Suppose that Hypothesis 1.5.10 holds. If the stepsize fulfills T < 1/c6 , then for all initial value pairs (κ, u0 ) ∈ I × L2 (Ω) there exists a unique 2 1 ∞ forward solution φ : I+ κ → L (Ω) of (nRΔE) with φ(k) ∈ H0 (Ω) ∩ L (Ω) for all k > κ. Proof. See [125, Theorems 4.2 and 4.3].
From this lemma we can conclude that the general forward solution ϕ of the implicit recursion (nRΔE) is well-defined on L2 (Ω) and H01 (Ω). Lemma 2.6.23. Suppose that Hypothesis 1.5.10 holds and I is unbounded below. If the stepsize fulfills T < 1/c6 , 1/T ∈ N, then (nRΔE) is uniformly bounded dissipative and the absorbing set has T -independent fibers A(k) ⊆ H01 (Ω) ∩ L∞ (Ω), k ∈ I. Proof. (I) For the necessary estimate yielding an absorbing set in H01 (Ω) ∩ L∞ (Ω), we refer to [125, Lemma 7.6] and provide only a sketch. Let κ ∈ I be arbitrary 1 and φ : I+ of the implicit recursion (nRΔE). κ → H0 (Ω) denote a forward solution u We define the function G : R → R, G(u) := − 0 g(tk+1 , x, s) ds and derive the estimate . . 1 1 2 2 φ (k)H 1 + G(φ (k, x)) dx ≤ φ(k)H 1 + G(φ(k, x)) dx 0 0 2 2 Ω Ω 2 1 for all k ∈ (I+ κ ) . Let us abbreviate u(k) := 2 φ(k)H01 + Ω G(φ(k, x)) dx and then the above inequality simplifies to u (k) ≤ u(k). We aim to apply the uniform
2.7 Remarks
89
Gronwall lemma from Proposition A.2.3(b) with a(k) ≡ 1, b(k) ≡ 0 and N = 1/T ∈ N, by assumption. Thanks to [125, Theorem 5.2] we deduce the existence of a real α3 ≥ 0, independent of T , and of an integer κT > 0 such that T
k+N
u(n) ≤ α3
for all k ≥ κT with k + N ∈ I.
n=k
Therefore, Proposition A.2.3(b) implies the relation u(k) ≤ α3 for all k ≥ κT . Since the sequence φ(k) is bounded in L∞ (Ω) for k ≥ κT (see Lemma 2.6.22 and [125, Lemma 7.4ff]), by definition of u(k) we can deduce that there exists a R > 0 such that max φ(k)L∞ , φ(k)H 1 ≤ R for all k ≥ κT . We conclude that 0
after a sufficiently large time, depending on the H01 -norm of φ(κ), the solution of ¯R (0, H01 (Ω)) ∩ B ¯ R (0, L∞ (Ω)). (nRΔE) satisfies the inclusion φ(k) ∈ B ˆ (II) Let B denote the family of nonautonomous sets uniformly bounded both in the H01 - and the L∞ -norm. Given ε > 0 we deduce from step (I) that the nonauton ˆ omous set A := I × BR+ε (0, H01 (Ω)) ∩ BR+ε (0, L∞ (Ω)) is B-absorbing.
Proposition 2.6.24. Suppose that Hypothesis 1.5.10 holds and I is unbounded below. If the stepsize fulfills T < 1/c6 and 1/T ∈ N, then (nRΔE) has a uniformly bounded global attractor ωA in both the sets I × H01 (Ω) and I × L∞ (Ω).
Proof. Let us suppose that Bˆ is the family of nonautonomous sets which are uniformly bounded both in the H01 - and the L∞ -norm. With Lemma 2.6.22 and [125] we can deduce that the general forward solution of (nRΔE) exists as continuous 2-parameter semigroup ϕ(k; κ, ·) : L2 (Ω) → L2 (Ω) fulfilling the smoothing property ϕ(k + 1; k, L2 (Ω)) ⊆ H01 (Ω) for all k ∈ I ; by the compact embedˆ ding H01 (Ω) L2 (Ω) (see [432, p. 607, Theorem B.1(2)]) it is also B-compact. Referring to Lemma 2.6.23, we also have a B-absorbing set A and therefore Theorem 1.3.9 implies our claim.
2.7 Remarks Basics and examples: Thorough introductions to explicit difference equations following didactical principles are, e.g., the textbooks [133, 248, 334], which include a variety of examples from various applications. On a more ambitious level, a rich source of results provide the monographs [3, 294] and more advanced topics are presented in [4]. The calculus of the forward difference operator is summarized in [3, p. 26ff, Sect. 1.8]. Finally, we gladly mention the book [425], which has a focus on applications to economical models. The particularities of our approach are as follows: First, the foundation of all our considerations have been first order difference equations – a legitimate restriction, since higher order (incl. finite delay) equations can always be brought into this from (cf. Examples 2.1.8 and 2.1.10). Despite this
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2 Nonautonomous Difference Equations
philosophy, we do not want to conceal that the special structure of higher order equations allows to employ a series of specific tools which are not applicable in our general setting. Hence, we survey some related references: •
•
•
•
•
For a start, such qualitative properties and special methods applying to higher order difference equations are discussed in monographs [276, 289] or, for instance, the paper [282]. As excellent source for results on difference equations appearing as discretizations of (ordinary) differential equations, like for example Runge–Kutta or multistep methods, we refer to the monographs [70] or [188, 189]. The examined methods are of central importance in the numerical analysis of evolutionary differential equations. A remark concerning discretization methods is incomplete without mentioning “nonstandard schemes” (cf. the monograph [335] or [336]) or “mimetic methods”. Here, as opposed to classical numerical analysis, the focus is not to minimize the discretization error, rather than to capture the qualitative dynamical behavior (e.g., boundedness, monotonicity, convergence) – a philosophy which led to the concept of dynamical consistency. Delay difference equations occur in a large number of applications ranging from population dynamics [130,131] over economic models [426] to discretizations of delay differential equations [47, 318]. Nevertheless, the author is not aware of a monograph dealing exclusively with delay difference equations, apart from a single chapter in [184, Chap. 7]. The right-hand side of a delay difference equation written as first order system is also called delay endomorphism (cf. [376, 403]). The theory of Volterra difference equations is getting increasingly voluminous and [133, p. 232ff, Sects. 5.3–5.6] serves as introduction. We only refer to [38, 442], [52] dealing with stability questions of linear equations or [329] concerned with stability problems in nonlinear systems. Discretization methods for Volterra differential equations are treated in [67]. Partial difference equations canonically appear as finite difference discretizations of partial differential equations and have a rich background literature (cf., for example, [92, 100, 440]). A comprehensive approach to their theory (in case of two independent discrete variables) can be found in [84].
Second, our equations live in infinite-dimensional state spaces. Besides as temporal discretizations, difference equations in Banach spaces occur in biological applications in order to describe the dispersal of univoltine distributed populations (cf., e.g., [278]); the global dynamics of the model in the latter reference has been investigated in [109]. Since an integral operator is involved, one speaks of integrodifference equations. Third, a further important aspect of our approach is to consider nonautonomous problems. Here, the concept of driven difference equations x = f (θk p, x)
(2.7a)
as nonautonomous equations with right-hand sides fk = f (θk p, ·) : X → X, is very fruitful from an applied point of view (see [80, 226, 259]). On the one hand,
2.7 Remarks
91
the iterates Φ(k, p, x) := f (θk p, ·) ◦ . . . ◦ f (p, ·) define a discrete nonautonomous dynamical system as in Definition 1.6.1. On the other hand, one has the examples: •
A sequence of maps fk on X is chosen periodically or, perhaps less regularly, from a finite family of maps {g1 , . . . , gr }. Then a difference equation x = gik (x), ik ∈ {1, . . . , r}, can be written as (2.7a) with P being the set of sequences from I into {1, . . . , r} and θ is the shift operator on P defined by θ(ik ) = ik+1 . The space P can be given the structure of a compact metric space with metric d(p, p¯) := (1 + r)−|k| |pk − p¯k | . k∈I
For parametrically perturbed difference equations x = g(x, q) and parameters q from a compact space Q, we define P to be the space of sequences p : I → Q. Equipped with the metric d(p, p¯) :=
2−|k| |pk − p¯k | ,
k∈I
it is a compact metric space and x = g(x, qk ) fits in the setting of (2.7a). • In order to incorporate random or stochastic influences, one considers metric dynamical systems (Ω, F, P, θ), i.e., a probability space together with a measurable map θ : Ω → Ω such that θP = P. A random difference equation (see [12]) is of the form x = f (θk ω, x), where f (ω, ·) : X → X is assumed to be measurable with some space X. For fixed ω, i.e., for a path-wise consideration, this is a nonautonomous difference equation (cf. [12, p. 50ff, Sect. 2.1] or [459]). In the end, a further feature of our approach is that state spaces are allowed to vary in time (see also [117] for a simple example). The motivation for this is at least threefold: First, in principle we want to include full discretizations of PDEs with time-dependent spatial refinements. In applications, such adaptive schemes yield a reasonable strategy, since the smoothness of solutions to parabolic problems in general varies considerably in space and time. The numerical analysis of these problems is due to [145–147]. Second, discrete time random dynamical systems provide another important class of such examples (see [226]). Third, one can consider difference equations restricted to invariant (nonautonomous) sets, like e.g., (global) attractors, invariant vector bundles given as ranges of dichotomy projectors (see Chap. 3) or invariant fiber bundles (see Chaps. 4–5). Existence and boundedness of solutions: Even in low dimensions, the lacking existence or uniqueness of backward solutions makes the theory of difference equations more complex and richer than their continuous counterpart. Yet, we suppressed corresponding considerations and focussed on forward solutions. Implicit difference equations typically occur as discretizations of evolutionary differential equations, where a particular numerical stability property is required,
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like e.g., A- or A(θ)-stability (see [189, p. 42ff]). In order to show existence of forward solutions, one has the full array of nonlinear analysis at hand. This includes fixed point methods (see Theorems 2.2.3 and 2.3.6) and global inverse functions (see Corollaries 2.3.13 and 2.3.14). For instance, the paper [10] applies global inverse function theorems in order to prove the existence of solutions to implicit difference equations. Keeping in mind that temporal discretizations of semilinear parabolic differential equations yield a sequence of nonlinear elliptic problems, also methods involving monotone operators are appropriate (cf. Theorem B.3.3). The dissipativity condition for the θ-method from Proposition 2.2.8 is an adaption of [222, Theorem 3.1] to our nonautonomous situation. Similarly, our approach to dissipative Runge–Kutta methods in Proposition 2.2.10 is essentially taken from [207, Theorem 2.4] or [223, Theorem 3.8]. Similar results for multistep methods are due to [208]. Indeed, it has been shown in [207] that consistent, DJ-irreducible, algebraically stable Runge–Kutta methods with a stability function R satisfying |R(∞)| < 1 fulfill the positive semi-definiteness assumption of Proposition 2.2.10. In [74] it is shown that a split implicit Euler scheme associated to a random delay differential system generates a discrete time random dynamical system, which also possesses a stochastic stationary solution with the same attracting property, and which converges to the stationary solution of the random DDE pathwise as the stepsize goes to zero. Difference equations and 2-parameter semigroups: Conditions for a set to be (forward) invariant w.r.t. a given difference equation depend strongly on the structure of the equation. For instance, invariant rectangles for finite difference discretizations of RDEs have been constructed in [213]. An easy to verify criterion to detect invariant intervals for delay difference equations is due to [231]. The analysis of attractors under numerical discretization dates back to the papers [262, 263] with extensions in [181, 182] (autonomous ODEs under one- and multistep schemes), [367] for a general perturbation approach or [195] for abstract semigroups and applications to hyperbolic and parabolic PDEs. For the nonautonomous case of (pullback) attractors we refer to [80, 260, 268] or [257]. In the latter reference, the existence and upper-semicontinuous convergence of a (pullback) attractor is established for difference equations under spatial discretization, including effects due to finiteness of computer arithmetics. It is well-known that attractors can be characterized using Lyapunov functions. This remains valid for time-dependent problems and a so-called converse theorem into this direction is due to [255]. One approach to get a Morse decomposition for the global attractor of difference equations are integer-valued Lyapunov functions as analyzed in[318]. Finally, as opposed to our approach, the paper [250] defines and investigates limit sets for nonautonomous difference equations as single sets. Stability: Without question, the stability theory of (nonautonomous) difference equations is a fairly classical topic. Basic introductions can be found in [133, p. 154ff, Chap. 4], [294, p. 87ff, Chap. 4] or [298]. For more advanced and different techniques we refer to [3, p. 234ff, Chap. 5] and [4, pp. 84–111, Sects. 8–11]. An approach related to Lyapunov functions is given, in particular, by [322,
2.7 Remarks
93
pp. 139–182]. Furthermore, [3, p. 251ff] contains a number of examples showing that the implications from Proposition 2.4.13 are not reversible. Stability issues for difference equations in normed spaces are investigated in [175]. Moreover, a quantitative version of the classical attraction and stability notions given in Definitions 2.4.4 and 2.4.11 can be found in [187] in terms of K- and KL-functions. The related literature in form of journal papers and proceedings contributions is abundant. Without claiming completeness, early trend-setting contributions include [448, 449] (stability results via comparison technique), [450] (discrete Lyapunov functions), [424] (asymptotic stability, instability and conditional stability for semilinear equations in Banach spaces), [297] or [227, p. 1ff] (spectral conditions for autonomous equations). For further local stability results on equilibria of nonautonomous equations, we refer to, e.g., [14]. The attraction and stability notions based on pullback convergence are quite recent and rarely studied in the literature so far. However, [391] or [392, p. 126, Theorem 5.9] provides corresponding local criteria for attraction and repulsion of solutions based on linearization. Thanks to Proposition 2.4.13 these pullback notions are closely related to classical uniform stability concepts. We also mention results from [326, 328] or [84, p. 147ff, Chap. 6], which apply to what they call “discrete reaction-diffusion equations”, i.e., time-explicit finite difference discretizations of 1d RDEs ut = uxx + f (t, u). Using Gronwall- or Bihari inequalities, conditions for boundedness or asymptotic stability of solutions are derived. In addition, [327] deals with linear equations. We refer to [74] for the stability of solutions to random DDEs under discretization. A quite different approach to infer attraction properties of difference equations has been developed in [150]. It is based on the idea to formulate a given difference equation as operator equation in an ambient space of sequences decaying to zero. While convergence is generically exponential in the autonomous situation, time-dependent problems feature a wider class of possible decays. Accordingly, by choosing an appropriate norm on the sequence space, one tries to capture the desired kind of decay. Various fixed point results have been employed in [150] in order to solve the operator equation. Other corresponding tools from nonlinear analysis involving measures of noncompactness have been used in [377] leading to various convergence results. Periodic and autonomous difference equations: A survey on results concerning periodic equations and solutions is given in [108]. Many of these results can be traced back to [190]. As interesting newer result, globally asymptotically stable periodic solutions of continuous autonomous difference equations (on connected metric spaces) turned out to be necessarily constant (cf. [137]). In [136] this has been generalized to q-periodic solutions of r-periodic difference equations, in which global asymptotic stability requires q to divide r; for further results we refer to [57]. A remarkable fact about periodic equations is that attraction or stability properties are uniform. Such questions appear to be more subtle for general timedependencies, e.g., in form of almost-periodicity. A corresponding counterexample of a stable, but not uniformly stable, almost periodic (scalar linear) difference
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equation can be constructed: One considers the ODE given in [463, pp. 42–43] and restricts its solutions to the integers. Results on the existence of almost periodic solutions can be found in [428]. Applications: For difference equations occurring as discretizations of delay and FDEs we refer to the survey [467] or the monograph [47]. Our spatial discretization approach to general FDEs is from [155]. A spatial discretization of scalar dissipative DDEs similar to ours is studied in [174]; here, the goal is to understand the behavior of Morse decompositions. The behavior of attracting sets for DDEs under Runge– Kutta discretization has been discussed in [269]. Discretization and approximation schemes for abstract differential equations have been surveyed in [183] and we refer to [273] for an application. As illustrated in [438] smoothing properties of the semiflows persist when passing over to Euler discretizations. Understandably, there is a significant numerical literature on temporal semidiscretizations of parabolic differential equations among which we only quote [55, 315] dealing with Runge–Kutta methods. For the necessary perturbation theory for discretization of parabolic equations we refer to [446]. In [195] the authors provide conditions that full and temporal discretizations of parabolic and hyperbolic PDEs have attractors converging to the ODE of the original PDE. A linearly implicit Euler time discretization of the 1d RDE ut − uxx = f (u) subject to Dirichlet boundary conditions is considered in [347]; it is shown that the corresponding solution flow in H01 (0, 1) has gradient structure, compact attractor, Morse–Smale properties, as well as structural stability w.r.t. the attractor. See [129] for similar results involving full finite difference discretizations. Section 2.6.3 dealing with full discretization of reaction-diffusion equations is inspired by [142]. Concerning the autonomous situation, under appropriate mesh conditions and using discrete Lyapunov functions, they show that the gradient structure of (RDE) persists under full discretization in the sense that numerical orbits converge to equilibria. Moreover, the existence of absorbing sets and attractors is investigated for various implicit schemes, as well as multistep backward differentiation methods. The corresponding analysis requires the uniform Gronwall inequality from Proposition A.2.3(b). Related results involving the θ-method can be found in [222]. Our set-up of a finite difference discretization for the Ginzburg–Landau equation is a nonautonomous extension of [309]. Another grateful source for applications is the Kuramoto–Sivashinsky PDE. For this problem, it has been shown in [163] that dissipativity is preserved under Galerkin discretization; similar results for finite difference discretizations have been obtained in [162, 272]. Using appropriate weighted norms, the existence of attractors for an implicit Euler, finite difference discretization of a system of RDEs on the whole space has been shown in [61]. Time discretizations of doubly nonlinear parabolic equation are addressed in [125] and we filled the missing link involving nonautonomous attractors in Sect. 2.6.5. See [51] for related problems on the p-Laplacian.
Chapter 3
Linear Difference Equations
If the values of A were assumed to be invertible, we should obtain a theory that is a pallid and essentially trivialized copy of the theory for differential equations [. . . ]; our main concern is, therefore, to develop our analysis without any assumption on the invertibility of the “transition operators”.1 C.V. Coffman and J. Sch¨affer (cf. [94])
Already in an analysis of nonlinear systems, linear problems frequently occur in form of variational equations (cf. Corollary 2.3.11) when linearizing along a given reference solution. Provided this solution does lack a specific time-dependence (e.g., (almost) periodicity, or being convergent), then the resulting variational equations are nonautonomous in the general sense. A further reason for the importance of linear equations is that the difference of two solutions to a nonlinear problem always solves a linear homogeneous difference equation, as follows from an easy application of the mean value theorem (cf. [295, p. 341, Theorem 4.2]). Finally, a solid linear theory opens the door to utilize appropriate perturbation results and to generalize a global geometric theory to semilinear equations. In this chapter, we present the corresponding theory. More detailed: •
In the first section, we introduce our particular notion of a linear difference equation and discuss basic concepts like the characteristic superposition principle, the evolution operator and the crucial variation of constants formula; compared to the continuous time case, the latter features an asymmetry between forward and backward time. Moreover, for linear equations stability notions are global in nature and can be characterized using properties of the evolution operator. • As an interlude, we briefly discuss periodic equations and their particular features including stability notions and a Floquet theory for invertible problems. • Technically important concepts are discussed in Sect. 3.3, namely invariant splittings and exponential boundedness. Such an invariant splitting and the resulting invariant vector bundles allow us to extend the idea of (generalized) eigenspaces known from the autonomous theory, to our general time-variant situation. Furthermore, we prepare some results relevant for the analysis in spaces of exponentially growing sequences. • The subsequent Sect. 3.4 discusses possible hyperbolicity notions for nonautonomous linear equations. A classical example shows that hyperbolicity based on asymptotic stability and corresponding Lyapunov exponents is not sufficiently 1
In our terminology, a “transition operator” is called evolution operator (see Definition 3.1.7).
C. P¨otzsche, Geometric Theory of Discrete Nonautonomous Dynamical Systems, Lecture Notes in Mathematics 2002, DOI 10.1007/978-3-642-14258-1 3, c Springer-Verlag Berlin Heidelberg 2010
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robust under perturbations. The same problem occurs with the notion of an exponential forward dichotomy, while exponential dichotomies turn out to be sufficient. Furthermore, they assure a dynamical characterization of invariant vector bundles as equivalence classes of solutions allowing a particular exponential growth bound. Unlike the Lyapunov spectrum, the resulting dichotomy spectrum proves to be appropriate in our context. However, so far its detailed structure is only understood for finite-dimensional equations, and as a substitute we work with so-called exponential splittings, which additionally allow a wider class of possible growth rates. • In Sect. 3.5 we prove various results on the behavior of solutions to linear difference equations under inhomogeneous and semilinear perturbations. In particular, it is shown that spaces of exponentially bounded sequences are admissible, i.e., exponentially bounded perturbations yield corresponding solutions – and essential prerequisite to construct invariant fiber bundles using the Lyapunov–Perron approach. Moreover, for later applications we provide a perturbation result for equations in spaces of multilinear mappings. • The main advantage of exponential dichotomies and also of N -splittings is their robustness under small bounded homogeneous perturbations. We prove a corresponding so-called roughness theorem for general N -splittings. • In the section on applications we essentially collect some tools needed for diverse purposes related to the spectrum of linear problems. This begins with results on companion matrices occurring in discretizations of retarded FDEs. We leave the finite-dimensional realm when dealing with time discretizations of abstract linear evolution and parabolic equations. However, also fully discretized diffusion equations are considered with a focus on the 1d case – including an introduction to the corresponding discrete Lebesgue and Sobolev spaces. Let I denote a discrete interval. Throughout the whole chapter, we suppose that (Xk )k∈I and (Yk )k∈I are sequences of linear spaces over the field F.
3.1 Basics Due to the algebraic structure of each fiber Yk , k ∈ I , we can assume w.l.o.g. that linear equations are semi-implicit. Definition 3.1.1. A difference equation (D) is called linear inhomogeneous (or simply linear), if Fk , Hk+1 are affine-linear mappings, i.e., if (D) is of the form Bk+1 x = Ak x + gk
(Lg )
with mappings Ak ∈ Hom(Xk , Yk+1 ), k ∈ I , Bk ∈ Hom(Xk , Yk ), k ∈ I and a so-called inhomogeneity gk ∈ Yk+1 for k ∈ I . In case gk = 0 for all k ∈ I one denotes (Lg ) as linear homogeneous and obtains Bk+1 x = Ak x.
(L0 )
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97
A specific feature of linear equations is an affine solution space. More detailed, given a discrete interval J ⊆ I, we define the solution space for (Lg ), LJ (g) := {φ : J → X| Bk+1 φ (k) ≡ Ak φ(k) + gk on J } and investigate its algebraic properties: Proposition 3.1.2 (superposition principle). With inhomogeneities gk , g˜k ∈ Yk+1 in (Lg ) and (Lg˜ ), resp., one has the implication g) φ ∈ LJ (g), φ˜ ∈ LJ (˜
⇒
cφ + c˜φ˜ ∈ LJ (cg + c˜g˜) for all c, c˜ ∈ F.
In particular, LJ (0) is a linear space over F.
Proof. The straight forward verification is omitted.
Corollary 3.1.3. (a) If (Lg ) has a unique bounded (periodic) solution, then the zero solution is the unique bounded (complete) solution of (L0 ). (b) Conversely, if 0 is the unique bounded (periodic) solution of (L0 ), then (Lg ) has at most one bounded (periodic) solution. Proof. We only give the proof in the periodic case and I = Z. In case of bounded solutions one replaces the word “periodic” by “bounded” in the argument below. (a) Let φ∗ : Z → X be the unique periodic solution to (Lg ) (with period p ∈ N). If there exists a further periodic solution φ (say with period P ∈ N) of (L0 ) besides the trivial one, then by Proposition 3.1.2 also φ∗ + φ solves (Lg ) with (φ∗ + φ)(k + pP ) ≡ φ∗ (k + pP ) + φ(k + P p) ≡ (φ∗ + φ)(k)
on Z.
Thus, also the sum φ∗ + φ is periodic and by uniqueness we get φ = 0. (b) If there exist two periodic solutions φ1 , φ2 : Z → X of (Lg ) with respective periods p1 , p2 ∈ N, then the difference φ1 − φ2 solves (L0 ) and is p1 p2 -periodic. The uniqueness assumption guarantees φ1 = φ2 . The following result is important and a linear counterpart to Theorem 2.3.6. Proposition 3.1.4. Suppose gk ∈ im Bk+1 , k ∈ I , holds. (a) Under the assumption im Ak ⊆ im Bk+1 ,
ker Bk+1 = {0}
for all k ∈ I
(3.1a)
the general forward solution ϕ of (Lg ) exists. (b) Under the assumption im Bk+1 ⊆ im Ak ,
ker Ak = {0}
the general backward solution ϕ of (Lg ) exists.
for all k ∈ I
(3.1b)
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3 Linear Difference Equations
Remark 3.1.5. Provided (3.1a) and (3.1b) hold simultaneously, one gets −1 Ak ∈ Iso(Xk , Xk+1 ) for all k ∈ I . Bk+1
(3.1c)
Proof. (a) Let (k, x) ∈ X with k ∈ I . By the left inclusion in (3.1a) we obtain the existence of a x ∈ Xk+1 such that Bk+1 x = Ak x + gk and since Bk+1 is one-toone, this x is unique and we define Φˆk (x) := x . This sequence Φˆk : Xk → Xk+1 , k ∈ I , generates the forward solution ϕ of (Lg ). (b) can be shown similarly. The next result shows that Xκ and LI± (0) have the same dimension: κ Corollary 3.1.6. For every κ ∈ I the mapping ξ → ϕ(·; κ, ξ) is linear and bijective, i.e., a vector space isomorphism Xκ → LI± (0). κ Proof. Under assumption (3.1a) the general forward solution ϕ exists. Thus, the above mapping is well-defined and linear, since for c, c˜ ∈ F and ξ, ξ˜ ∈ Xκ , ˜ ≡ cϕ(k; κ, ξ) + c˜ϕ(k; κ, ξ) ˜ ϕ(k; κ, cξ + c˜ξ)
on I+ κ.
(3.1d)
˜ OneIndeed, both sides solve (L0 ) and satisfy the initial condition x(κ) = cξ + c˜ξ. to-oneness immediately follows from the unique solvability of (L0 ) in forward time. Finally, the mapping is onto, since for arbitrary φ ∈ LI± (0) the pre-image in Xκ is κ given by φ(κ). Definition 3.1.7. An evolution operator of (Lg ) (or of (L0 )) is a linear mapping Φ(k, κ) ∈ Hom(Xκ , Xk ) so that for every initial time κ ∈ I the sequence Φ(·, κ) solves the initial value problem Bk+1 X = Ak X,
X(κ) = IXκ .
In particular, a forward evolution operator Φ(k, κ) is defined for κ ≤ k, while a backward evolution operator Φ(k, κ) is defined for k ≤ κ. Example 3.1.8. Let ϕ be the general forward solution of the nonlinear equation (D). Under the assumptions of Corollary 2.3.11, the partial derivative D3 ϕ(·, ξ) ∈ L(Xκ , Xk ), ξ ∈ Xκ , is the forward evolution operator of the variational equation (2.3g). Proposition 3.1.9. Let k, κ ∈ I. (a) Under condition (3.1a) the forward evolution operator Φ(k, κ) of (L0 ) is uniquely determined and given by Φ(k, κ) = ϕ(k; κ, ·) for all κ ≤ k, −1 Φ(k, κ) = Bk−1 Ak−1 . . . Bκ+1 Aκ
for all κ < k.
(3.1e)
(b) Under condition (3.1b) the backward evolution operator Φ(k, κ) of (L0 ) is uniquely determined and given by Φ(k, κ) = ϕ(k; κ, ·) for all k ≤ κ, −1 Φ(k, κ) = A−1 k−1 Bk . . . Aκ−1 Bκ
for all k < κ.
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99
Example 3.1.10. Given a sequence a : I → F, the generalized exponential function ea from Sect. A.1 is the evolution operator of the explicit scalar equation x = a(k)x;
(3.1f)
moreover, in case a(k) = 0 it is also a backward evolution operator. Proof. Let k ∈ I . Assumption (3.1a) implies that for each x ∈ Xk there exists a −1 unique x ∈ Xk+1 with Bk+1 x = Ak ξ and the mapping Bk+1 : im Ak → Xk+1 −1 exists; we define Φ(k + 1, k)x := x = Bk+1 Ak x. Now assertion (a) yields by mathematical induction, and (b) can be shown analogously. Our following observation guarantees that the forward evolution operator Φ is a −1 linear 2-parameter semigroup on X with generator Φˆk := Bk+1 Ak , provided (3.1a) holds. In particular, the notions of a compact, contracting, etc., semigroup from Chap. 1 are applicable to the linear equations (L0 ) or (Lg ). Similarly, under (3.1b) the backward evolution operator is a linear backward 2-parameter semigroup. Under both (3.1a) and (3.1b) one obtains a linear 2-parameter group Φ. Proposition 3.1.11. Let k, l, κ ∈ I. (a) Under assumption (3.1a) one has the linear semigroup property Φ(k, l)Φ(l, κ) = Φ(k, κ) for all κ ≤ l ≤ k.
(3.1g)
(b) Under assumption (3.1b), property (3.1g) holds for all k ≤ l ≤ κ. (c) Under the assumptions (3.1a) and (3.1b), the group property (3.1g) holds for all k, l, κ ∈ I and one has Φ(κ, k) = Φ(k, κ)−1 for all k, κ ∈ I. Remark 3.1.12. Under the assumptions of (b) one sees from (3.1g) that Φ(κ, ·) is a backward solution of the initial value problem X = XA−1 k−1 Bk , X(κ) = I. Proof. The claim follows from Proposition 2.3.2 together with Proposition 3.1.9. Corollary 3.1.13. Let V ⊆ X be a vector bundle, suppose (3.1a) holds and that the inhomogeneity gk fulfills gk ∈ Bk+1 V (k), k ∈ I . Given a linear inhomogeneous equation (Lg ), then the nonautonomous set V is: (a) Forward invariant, if and only if Ak V(k) ⊆ Bk+1 V (k) for all k ∈ I . (b) Backward invariant, if and only if Ak V(k) ⊇ Bk+1 V (k) for all k ∈ I . (c) Invariant, if and only if Ak V(k) = Bk+1 V (k) for all k ∈ I . Proof. We apply Proposition 2.3.5 with the affine-linear mappings Hk+1 (x ) = Bk+1 x and Fk (x, x ) = Ak x+gk . Here, the inclusions (3.1a) imply the invertibility condition (2.3c) where Yk = im Bk . Example 3.1.14. The super-stable vector bundle U0+ := (κ, ξ) ∈ X | ∃n ∈ Z+ κ : Φ(nκ , κ)ξ = 0
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is forward invariant and consists of linear spaces. In the invertible situation (3.1c) one clearly has U0+ = I × {0}, but in general this must not hold. Beyond the properties of general invariant sets stated in Proposition 1.2.6, in our linear setting the following holds for vector bundles: Corollary 3.1.15. With a finite set I, suppose {Vi }i∈I is a family of vector bundles Vi ⊆ X . Provided (3.1a) holds and the inhomogeneity gk fulfills gk ∈ Bk+1 Vi (k) for all k ∈ I , i ∈ I, then the algebraic sum i∈I Vi and the Whitney sum i∈I Vi are forward invariant (backward invariant, invariant) w.r.t. (Lg ), if each Vi , i ∈ I, is forward invariant (backward invariant, invariant, resp.). Proof. Let I be a finite set and suppose each Vi , i ∈ I, is forward invariant. Then the previous Corollary 3.1.13(a) implies Ak
i∈I
Vi (k) =
Ak Vi (k) ⊆
i∈I
Bk+1 Vi (k) = Bk+1
i∈I
Vi (k)
i∈I
for all k ∈ I and the algebraic sum i∈I Vi is forward invariant. The remaining assertions are shown similarly using Corollary 3.1.13(b) and (c). For linear inhomogeneous equations an explicit formula for the general solution can be given. Under our usual assumptions the general solution of (Lg ) is an affinelinear mapping, which is the sum of the general (forward resp. backward) solution of the homogeneous equation (L0 ) and a particular solution of (Lg ). Theorem 3.1.16 (variation of constants). Let k ∈ I and (κ, ξ) ∈ X . (a) Under assumption (3.1a) with gk ∈ im Bk+1 , k ∈ I , the general forward solution of (Lg ) is uniquely determined and given by the formula ϕ(k; κ, ξ) = Φ(k, κ)ξ +
k−1
−1 Φ(k, n + 1)Bn+1 gn
for all κ ≤ k.
(3.1h)
n=κ
(b) Under assumption (3.1b) with gk ∈ im Ak , k ∈ I , the general backward solution of (Lg ) is uniquely determined and given by the formula ϕ(k; κ, ξ) = Φ(k, κ)ξ −
κ−1
Φ(k + 1, n + 1)A−1 n gn
for all k ≤ κ.
(3.1i)
n=k
Remark 3.1.17. (1) The asymmetry between the two variation of constants formulas (3.1h) and (3.1i) can be bypassed: Under both the assumptions (3.1a) and (3.1b) one concludes (3.1c) and therefore ϕ(k; κ, ξ) = Φ(k, κ)ξ −
κ−1 n=k
−1 Φ(k, n + 1)Bn+1 gn
for all k ≤ κ.
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101
(2) With the aid of Corollary 3.1.6 we see that the linear space Xκ and the affinelinear space LI± (g) have the same dimension. κ Proof. For fixed (κ, ξ) ∈ X we abbreviate ϕ(k) := ϕ(k; κ, ξ). (a) Thanks to Proposition 3.1.9(a) the forward evolution operator Φ of (L0 ) exists. We have ϕ(κ) = ξ and
(3.1h)
Φ(k + 1, κ)ξ +
Bk+1 ϕ (k) = Bk+1
Φ(k
−1 + 1, n + 1)Bn+1 gn
n=κ
= Bk+1
k
Φ(k + 1, κ)ξ +
k−1
−1 −1 Φ(k + 1, n + 1)Bn+1 gn + Bk+1 gk
n=κ (3.1e)
= Ak Φ(k, κ)ξ + Ak
k−1
−1 Φ(k, n + 1)Bn+1 gn + gk = Ak ϕ(k) + gk (Lg )
n=κ
for all k ≥ κ, which yields the validity of (3.1h). (b) This is shown analogously to (a) using Proposition 3.1.9(b).
So far our considerations have been purely algebraic. Now assume that the extended state space X consists of topological linear spaces. Corollary 3.1.18. Let κ, k ∈ I and ϕ denotes the general solution of (Lg ): −1 Ak ∈ L(Xk , Xk+1 ) for all κ ≤ k, one has (a) Under assumption (3.1a) and Bk+1 Φ(k, κ) ∈ L(Xκ , Xk ) and ϕ(k; κ, ·) ∈ C(Xκ , Xk ) for κ ≤ k. (b) Under assumption (3.1b) and A−1 k−1 Bk ∈ L(Xk , Xk−1 ) for all k < κ, one has Φ(k, κ) ∈ L(Xκ , Xk ) and ϕ(k; κ, ·) ∈ C(Xκ , Xk ) for k ≤ κ.
Remark 3.1.19. In applications, the above boundedness assumption −1 Ak ∈ L(Xk , Xk+1 ) for all k ∈ I Bk+1
(3.1j)
is frequently established using the closed graph theorem (cf., for example, [244, p. 166, Theorem 5.20]). Often the following examples are encountered, where (Yk )k∈I , (Zk )k∈I denote sequences of Banach spaces. If we suppose Ak ∈ L(Xk , Yk+1 ), k ∈ I , then each of the upcoming conditions guarantees (3.1j): Bk : D(Bk ) ⊆ Zk → Yk is closed with ker Bk = {0}, im Bk = Yk and Xk = D(Bk ) for all k ∈ I (cf. [244, p. 167, Problem 5.21]). • Bk : D(Bk ) ⊆ Xk → Yk is closed, im Bk ⊆ Yk is closed and (3.1a) holds. −1 • Bk : Xk → Yk with Bk : im Bk → Xk is closable and (3.1a) holds (cf. [244, p. 167, Problem 5.22]). •
Moreover, if the conditions (3.1a), (3.1b) and (3.1j) are satisfied, one can conclude −1 Bk+1 Ak ∈ GL(Xk , Xk+1 ) (cf. [96, p. 91, 12.5]). Proof. Clearly, by Proposition 3.1.9, the operator Φ(k, κ) is bounded as composition of bounded operators. Since the map ϕ(k; κ, ·) is (affine-)linear by Theorem 3.1.16, it consequently has to be continuous.
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Corollary 3.1.20. Suppose that X consists of Banach spaces and let κ, k ∈ I with κ < k. If the conditions (3.1a), (3.1j) hold, then dar Φ(k, κ) = dar ϕ(k; κ, ·) ≤
k−1
−1 dar Bl+1 Al .
n=κ −1 Al is compact for an l ∈ [κ, k)Z , then Φ(k, κ) ∈ L(Xκ , Xk ) is In particular, if Bl+1 compact and ϕ(k; κ, ·) ∈ C(Xκ , Xk ) completely continuous.
Remark 3.1.21. Assume that X consists of identical copies of a Banach space X.
−1 n For each l ∈ I the limit limn→∞ dar(Bl+1 Al )n equals the radius of the essential −1 Al ∈ L(X) (cf. [192, p. 14, Lemma 2.3.3]). spectrum of the operator Bl+1
Proof. By Proposition 3.1.9 the mapping Φ(k, κ) is a composition of bounded linear operators. Then the claim follows from [35, p. 39, Proposition 5.3(b)].
Linear Stability Theory We suppose that X consists of metric linear spaces with a translation invariant metric. For linear equations attraction or stability is a property of the equation and not merely of single solutions as before in the nonlinear situation of Sect. 2.4. Proposition 3.1.22. Let φ : I → X be a solution of (Lg ) and suppose (3.1a): (a) φ is forward a-stable, if and only if the zero solution of (L0 ) satisfies the corresponding property w.r.t. (L0 ). (b) φ satisfies an attraction property from Definition 2.4.4, if and only if the zero solution of (L0 ) satisfies the corresponding property w.r.t. (L0 ). (c) φ satisfies a stability property from Definition 2.4.11, if and only if the zero solution of (L0 ) satisfies the corresponding property w.r.t. (L0 ). Proof. Every solution φ of (Lg ) gives rise to the same equation of perturbed motion, namely (L0 ). Thus, the claim follows from Theorem 2.4.15. In case X consists of normed linear spaces, it is possible to characterize the above stability notions for linear equations using their evolution operator. Proposition 3.1.23. Assume that (3.1a) and (3.1j) holds. A linear inhomogeneous equation (Lg ) is: (a) Forward a-stable, if and only if there exists a C > 0 with Φ(k, κ)L(Xκ ,Xk ) ≤ Cea (k, κ)
for all κ ≤ k.
(3.1k)
(b) Stable, if and only if for all κ ∈ I there exists a C > 0 such that Φ(k, κ)L(Xκ ,Xk ) ≤ C
for all κ ≤ k.
(3.1l)
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103
(c) Pullback stable, if and only if for all k ∈ I there exists a C > 0 such that (3.1l) holds. (d) Uniformly stable (resp. uniformly pullback stable), if and only if there exists a C > 0 such that the estimate (3.1l) holds for all k, κ ∈ I, κ ≤ k. (e) Asymptotically stable, if and only if for every ε > 0 and κ ∈ I there exists an N = Nκ (ε) ≥ 0 such that Φ(k, κ)L(Xκ ,Xk ) < ε
for all k − κ ≥ N.
(3.1m)
(f) Asymptotically pullback stable, if and only if for every ε > 0 and k ∈ I there exists an N = N (ε, k) > 0 such that (3.1m) holds. (g) Uniformly asymptotically stable (resp. uniformly asymptotically pullback stable), if and only if there exist C > 0, α ∈ (0, 1) such that (3.1k) holds for all κ ≤ k with the constant growth rate a(k) ≡ α. −1 Remark 3.1.24. (1) Let I be bounded below. If Bk+1 Ak ∈ L(Xk , Xk+1 ), k ∈ I , are onto, then the condition (2.4e) holds by the open mapping theorem (cf., e.g., [295, p. 386, Theorem 1.3]) and it is sufficient to check the stability condition in assertion (b) resp. the asymptotic stability in (e) for only one instant κ ∈ I, in order to guarantee that (Lg ) is stable resp. asymptotically stable. (2) A linear difference equation (L0 ) satisfying (3.1a) and (3.1j), also fulfills the −1 estimate (3.1k) with C = 1 and a(k) = Bk+1 Ak . Conversely, (3.1k) guarantees −1 the estimate Bk+1 Ak ≤ Ca(k) for all k ∈ I (cf. (3.1e)). In case (3.1k) hold with a sequence a bounded above, we say (L0 ) has bounded forward growth. (3) Provided its backward evolution operator exists, a linear equation (Lg ) is backward b-stable, if and only if there exists a C > 0 with
Φ(k, κ)L(Xκ ,Xk ) ≤ Ceb (k, κ) for all k ≤ κ.
(3.1n)
We say (L0 ) has bounded backward growth, if b is bounded away from 0 and in case of bounded forward and backward growth we simply speak of bounded growth. Suppose the forward and backward evolution operators Φ for (L0 ) exist. If (Lg ) is forward a-stable in the sense that (3.1k) holds, then 1/Cea (k, κ) ≤ Φ(k, κ)L(Xκ ,Xk )
for all k ≤ κ
and if (L0 ) is backward b-stable in the sense that (3.1n) holds, then 1/Ceb (k, κ) ≤ Φ(k, κ)L(Xκ ,Xk )
for all κ ≤ k.
−1 Ak )k∈I is norm-wise bounded by C. (4) If (Lg ) is uniformly stable, then (Bk+1 (5) A linear equation (Lg ) is asymptotically stable or asymptotically pullback stable, if and only if one has the respective limit relations
lim Φ(k, κ) = 0
k→∞
for all κ ∈ I,
lim Φ(k, κ) = 0
κ→−∞
for all k ∈ I.
(3.1o)
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(6) The notions of uniform asymptotic (pullback) stability and exponential stability coincide for linear difference equations (cf. Remark 2.4.12(2)). Moreover, forward 1-stability coincides with uniform (pullback) stability. (7) The above characterization of various stability notions for linear equations also has consequences in a nonlinear set-up. Keeping in mind that ea is the evolution operator of (3.1f), depending on a : J → (0, ∞), it helps us to illustrate how flexible the notion of forward a-stability is. Indeed, a forward a-stable solution of the general nonlinear equation (D) is: Stable, if supκ≤k ea (k, κ) < ∞ for all κ ∈ J Pullback stable, if supκ≤k ea (k, κ) < ∞ for all k ∈ J Uniformly (pullback) stable, if supκ∈J supκ≤k ea (k, κ) < ∞ • Asymptotically stable (globally attractive), if limk→∞ ea (k, κ) = 0 for all κ ∈ J • Asymptotically pullback stable, if limκ→−∞ ea (k, κ) = 0 for all k ∈ J • Uniformly asymptotically (pullback) stable, if there exists a α ∈ (0, 1) such that a(k) ≤ α for all k ∈ J • • •
Proof. Thanks to Proposition 3.1.22 all solutions of (Lg ) share the same attraction resp. stability behavior, and w.l.o.g. we can restrict to the zero solution of the linear homogeneous equation (L0 ). Referring to Proposition 3.1.9(a), its general solution reads as ϕ(k; κ, ξ) = Φ(k, κ)ξ for all κ ≤ k, ξ ∈ Xκ : (a) The assertion is evident from condition (Sa+ ) in Definition 2.4.2. (b) The proof is similar to the less classical situation shown in (c). (c) If (L0 ) is pullback stable, for each k ∈ I there exists a δ > 0 such that Φ(k, k − n)Bδ (0) ⊆ B1 (0) for all n ≥ 0 (cf. (2.4g)). By linearity, this implies the inclusion Φ(k, k − n)B1 (0) ⊆ B1/δ (0) and thus Φ(k, k − n) ≤ 1δ for all n ≥ 0. Therefore, (3.1l) holds with C = 1/δ depending on k ∈ I. ε Conversely, let ε > 0, k ∈ I and suppose (3.1l) holds. We set δ := 2C and Φ(k, k − n)Bδ (0) ⊆ CBδ (0) ⊂ B (0) for all n ≥ 0 simply means that (Lg ) is pullback stable. (d) Here, δ in independent of κ ∈ I and the characterization of uniform stability is shown analogously to (b). Concerning uniform pullback stability, we assume I is unbounded below. If the zero solution to (L0 ) is uniformly pullback stable, then there exists a δ > 0 such that Φ(k, k−n)Bδ (0) ⊆ B1 (0) for all k ∈ I and n ≥ 0. Linearity implies Φ(k, k − n)B1 (0) ⊆ B1/δ (0) and thus the estimate Φ(k, k − n) ≤ 1/δ for all k ∈ I and n ≥ 0, which obviously yields (3.1l). For the converse implication proceed as in (c), where C > 0 and also δ > 0 do not dependent on κ ∈ I. (e) See the similar proof for asymptotic pullback stability considered next. (f) Let the zero solution to (L0 ) be asymptotically pullback stable. In particular, it is pullback attractive, i.e., there exists a ρ > 0, and for each ε > 0, k ∈ I a corresponding N ≥ 0 such that Φ(k, k − n)Bρ (0) ⊆ Bρε/2 (0) for all n ≥ N (cf. (2.4d)). Linearity guarantees the inclusion Φ(k, k − n)B1 (0) ⊆ Bε/2 (0) and this implies the estimate Φ(k, k − n) < ε for all n ≥ N , an inequality yielding (3.1m).
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105
For the converse it is clear that (3.1m) implies the pullback attractivity of the trivial solution. Moreover, (3.1m) guarantees that each sequence (Φ(k, k − n))n≥0 is bounded, say Φ(k, k − n) ≤ Ck for all n ≥ 0 with some real Ck ≥ 0. By the characterization shown in (c) this implies that (L0 ) is also pullback stable. (g) Since the classical characterization of uniform asymptotic stability can be shown similarly, we restrict to the pullback case. Suppose the trivial solution of (L0 ) is uniformly asymptotically pullback stable. Then it is uniformly pullback attractive, i.e., there exists an integer K ≥ 0 such that Φ(l, l − K) ≤
1 2
for all l ∈ I.
(3.1p)
Since (L0 ) is also uniformly pullback stable, our assertion (d) guarantees the existence of a C > 0 such that Φ(k, l) ≤ C2 for all l ≤ k. For given κ ≤ k we choose m ∈ Z+ 0 maximal with κ + mK ≤ k. Then Proposition 3.1.11(a) yields (3.1g)
Φ(k, κ) ≤ Φ(k, κ + mK) Φ(κ + mK, κ) ≤
C 2
m−1
Φ(κ + (n + 1)K, κ + nK)
n=0 (3.1p)
≤
with α :=
K
1 2
C 2
1 m 2
≤C
1 K m+1 2
K
≤ Cαk−κ
∈ (0, 1). The converse direction is clear.
for all κ ≤ k
A typical feature of the linear theory is that the various attractivity notions imply the corresponding stability concepts. Therefore, for linear equations the lower two rows in the diagram from Proposition 2.4.13 are equivalent: Proposition 3.1.25. For linear equations (Lg ) on normed spaces, the attraction and stability notions from Definitions 2.4.4 and 2.4.11, resp., are related by: S ⇐ ⇑ GAS ⇔ AS ⇐ GA ⇔ A ⇐
US ⇔ ˙ UPS ⇒ PS ⇑ ⇑ ⇑ UAS ⇔ ˙ UAPS ⇒ APS UA ⇔ ˙ UPA ⇒ PA
where the equivalences ⇔ ˙ only hold for I = Z. Proof. Due to Proposition 2.4.13 it remains to show the following implications: (A ⇒ AS) Since (Lg ) is attractive, we derive from Proposition 3.1.23(e) that for each κ ∈ I there exists a constant Cκ > 0 such that Φ(k, κ) ≤ Cκ for all κ ≤ k. Using Proposition 3.1.23(b) this means that (Lg ) is stable, and in turn, asymptotically stable.
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(A ⇒ GA) This results from the proof of Proposition 3.1.23(e): Indeed, if (L0 ) is attractive, then the left limit relation in (3.1o) holds and we obtain global attractivity. (AS ⇒ GAS) Since (L0 ) is asymptotically stable, we have the left limit relation (3.1o) and therefore global asymptotic stability. (GA ⇒ GAS) is clear from the above implications. (U A ⇒ U AS) can be shown as in the above implication (A ⇒ AS), where C ≥ 0 does not depend on κ ∈ I now. (P A ⇒ AP S) For a pullback attractive equation (Lg ) one shows as in the proof of Proposition 3.1.23(f) that the right limit relation in (3.1o) holds. In order to show that (L0 ) is also pullback stable, one argues as in the above implication (A ⇒ AS), but now with the aid of Proposition 3.1.23(c). (U P A ⇒ U AP S) We have to show that (L0 ) is uniformly pullback stable. Since it is uniformly pullback attractive, there exist ρ, N > 0 such that Φ(k, k − n)Bρ (0) ⊆ B1 (0) and consequently Φ(k, k − n) ≤ 1/ρ for all k ∈ I, n ≥ N . Since N > 0 does not depend on k ∈ I, the transition operator fulfills (3.1l) and the claim follows from Proposition 3.1.23(c). The next proposition states that uniformly asymptotically pullback stable linear equations have global attractors, whose fibers consist of singletons. As opposed to the nonlinear problem, this requires no compactness assumption on the semigroup. Proposition 3.1.26. Suppose that I is unbounded below. If a linear equation (Lg ) −1 is uniformly asymptotically pullback stable and supk∈I Bk+1 gk X < ∞ holds, k then the global attractor of (Lg ) is the nonautonomous set φ∗ ⊆ X given by the k−1 −1 Lyapunov–Perron sum φ∗ : I → X, φ∗ (k) := n=−∞ Φ(k, n + 1)Bn+1 gn . Proof. By assumption we know from Theorem 3.1.23(g) that the evolution operator Φ(k, κ) of (L0 ) satisfies the relation (3.1k) with a(k) ≡ α ∈ (0, 1). One can show as in Theorem 3.5.3 below that φ∗ : I → X is a bounded solution of (Lg ) and thus the nonautonomous set φ∗ is uniformly bounded and invariant. Now suppose B ⊆ X denotes a uniformly bounded nonautonomous set with B ⊆ BR for some R > 0. For arbitrary sequences ξn ∈ B(k − n) we obtain using Theorem 3.1.16(a) that k−n−1 −1 Φ(k, l + 1)Bl+1 gl dist(ϕ(k; k − n, ξn ), {φ∗ (k)}) = Φ(k, k − n)ξn + (3.1h)
l=−∞
(3.1k)
≤ CRαn +
k−n−1
Φ(k, l + 1)B −1 gl l+1
l=−∞
and consequently 0 ≤ h(ϕ(k; k − n, B(k − n)), {φ∗ (k)}) ≤ CRαn +
k−n−1
Φ(k, l + 1)B −1 gl l+1
l=−∞
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107
for all k ∈ I and n ≥ 0. It is easy to see that the right-hand side in this estimate converges to 0 as n → ∞. Thus, φ∗ is an invariant set attracting uniformly bounded nonautonomous sets, i.e., the global attractor for (Lg ). Provided the discrete interval I is unbounded above, we can introduce a first spectral notion for linear difference equations. For this, the upper resp. lower forward Lyapunov exponents of (L0 ) are given by
+ k k λ+ (ξ) := lim sup Φ(k, κ)ξ , λ (ξ) := lim inf Φ(k, κ)ξXk , u Xk l k→∞
k→∞
+ for all ξ ∈ Xκ . Clearly, one has λ+ l (ξ) ≤ λu (ξ) and it is not difficult to show that + + λu (ξ), λl (ξ) are independent of κ ∈ I. In fact, they are characteristic exponents associated to the trivial solution. Similarly to our approach in Sect. 2.4 one can also define backward Lyapunov exponents for equations defined in backward time.
Definition 3.1.27. Suppose that I is unbounded above and (3.1a), (3.1j) hold true. The forward Lyapunov spectrum of (L0 ) is given by + λ+ ΣL+ (A, B) := l (ξ), λu (ξ) . ξ∈Xκ \{0}
As it will turn out, the Lyapunov spectrum does not exhibit robustness properties needed for our purposes (see Example 3.4.1). We, therefore, only collect some elementary observations rather than providing a comprehensive approach. Corollary 3.1.28. If a linear equation (L0 ) is: (a) Forward a-stable, then ΣL+ (A, B) ⊆ [0, χ+ u (ea (·, κ))]. (b) Stable, then ΣL+ (A, B) ⊆ [0, 1]. (c) Uniformly asymptotically stable, then ΣL+ (A, B) ⊆ [0, 1). Proof. (a) We fix a pair (κ, ξ) ∈ X , where ξ is nonzero. From Proposition 3.1.23(a) we know that there exists a C > 0 such that Φ(k, κ) ≤ Cea (k, κ) for all k ≥ κ and thus the claim is a consequence of lim sup k Φ(k, κ)ξ ≤ lim sup k Cea (k, κ) ξ = lim sup k ea (k, κ). k→∞
k→∞
k→∞
The remaining assertions (b) and (c) follow similarly from Proposition 3.1.23.
Corollary 3.1.29. If ΣL+ (A, B) ⊆ [0, 1), then (L0 ) is asymptotically stable. Proof. We proceed indirectly and suppose (L0 ) is not asymptotically stable. Then there exists a pair (κ, ξ) ∈ X , a δ > 0 and a sequence (kn )n∈N of positive integers with limn→∞ kn = ∞ such that Φ(kn , κ)ξ ≥ δ. This implies √ k kn Φ(kn , κ)ξ ≥ lim n δ = 1, λ+ u (ξ) ≥ lim sup n→∞
n→∞
which contradicts our assumption λ+ u (ξ) < 1 for all nonzero ξ ∈ Xκ .
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For later reference, we close this section by introducing a type of linear transformation for linear equations, which preserves attraction and stability properties. ˜ k )k∈I is a sequence of normed linear spaces and (Y˜k )k∈I is a For this, suppose (X ˜ k , Y˜k+1 ) and B ˜ k ∈ Hom(X ˜ k , Y˜k ), sequence of linear spaces. Given A˜k ∈ Hom(X we consider a linear homogeneous equation ˜k+1 x = A˜k x B
(3.1q)
in the nonautonomous set X˜ and introduce Definition 3.1.30. Let J ⊆ I be a discrete interval and suppose both (L0 ) and (3.1q) satisfy (3.1a). The linear homogenous equation (L0 ) and (3.1q) are called linearly conjugated on J, if there exists a sequence (Λ(k))k∈J with values Λ(k) ∈ ˜ k ), k ∈ J, solving the identity X B ˜ −1 A˜k ≡ B −1 Ak X on J and also GL(Xk , X k+1 k+1 admitting the boundedness relations sup Λ(k)L(Xk ,X˜ k ) < ∞, sup Λ(k)−1 L(X˜ ,X ) < ∞. k∈J
k∈J
k
k
The sequence (Λ(k))k∈J is called Lyapunov transformation or linear conjugation. Remark 3.1.31. (1) Linearly conjugated equations are also called kinematically similar. This means that x → Λ(k)x transforms (L0 ) into (3.1q). Such a Lyapunov transformation can be applied to linear inhomogeneous equation (Lg ) yielding ˜k+1 Λ(k)B −1 gk ˜k+1 x = A˜k x + B B k+1 −1 and preserving boundedness properties of the inhomogeneity sequence Bk+1 gk for k ∈ I . In addition, also (global) Lipschitz continuity of nonlinear equations (D) is preserved under linear conjugacy. (2) In case X = X˜ , linear conjugacy defines an equivalence relation on the set of all linear homogenous equations (L0 ) satisfying (3.1a). (3) The forward evolution operators Φ of (L0 ) and Φ˜ of (3.1q) are related by
˜ l) for all l ≤ k. Φ(k, l)Λ(l) = Λ(k)Φ(k,
(3.1r)
As a consequence, Proposition 3.1.23 yields that attraction and stability properties of (Lg ) and (L0 ) persist under linear conjugacy. Moreover, the forward Lyapunov ˜ B). ˜ spectrum is invariant under linear conjugacy, i.e., ΣL+ (A, B) = ΣL+ (A,
3.2 Periodic Linear Equations Periodic = Autonomous mod Floquet Bernd Aulbach, 1998 Let a period p ∈ N be given. In the beginning of this section we focus on linear homogeneous difference equations
3.2 Periodic Linear Equations
109
Bk+1 x = Ak x
(L0 )
as introduced in Definition 3.1.1, which satisfy (3.1a) and are p-periodic. Referring to Proposition 3.1.9(a) the evolution operator Φ(k, l) ∈ Hom(Xl , Xk ) is defined in forward time, by Proposition 2.5.3 one obtains Φ(k + p, l + p) = Φ(k, l) for all l ≤ k
(3.2a)
and the period map of (L0 ) reads as Φκ := Φ(κ + p, κ) ∈ Hom(Xκ ), κ ∈ Z. Definition 3.2.1. Let κ ∈ Z. Then σp (Φκ ) ⊆ C is denoted as Floquet spectrum of (L0 ) and each element of σp (Φκ ) is a Floquet multiplier. For (L0 ) with invertible evolution operators the Floquet spectrum does not depend on κ ∈ Z; more precisely, we have Proposition 3.2.2. Let κ ∈ Z. In case ker Φ(k, l) = {0} holds for all k, l ∈ Z satisfying 0 ≤ k − l < p, then σp (Φk ) = σp (Φl ) for all k, l ∈ Z. Proof. Thanks to Φk = Φk+p (cf. Proposition 1.4.4(a)) it suffices to show the assertion for k, l ∈ Z with 0 ≤ k − l < p. Given λ ∈ σp (Φl ) we pick a ξ ∈ Xl with Φl ξ = λξ and obtain the relation (3.1g)
(3.1g)
Φk Φ(k, l)ξ = Φ(k + p, l)ξ = Φ(k + p, l + p)Φl ξ = λΦ(k, l)ξ. By assumption, Φ(k, l)ξ ∈ Xk is nonzero and consequently λ ∈ σp (Φk ). This establishes σp (Φl ) ⊆ σp (Φk ) and in order to show the converse inclusion we suppose λ ∈ σp (Φk ). If η ∈ Xk is a corresponding eigenvector, i.e., Φk η = λη, we set ξ := Φ(l + p, k)η = 0 and obtain Φl ξ = Φ(l + p, k − p)η = λΦ(l + p, k)η = λξ from (3.1g); therefore, λ ∈ σp (Φl ) and the claim follows. Proposition 3.2.3. Let κ ∈ Z and λ ∈ F. Then λ ∈ σp (Φκ ) holds, if and only if (L0 ) admits a nontrivial solution φ : Z+ κ → X satisfying φ(k + p) = λφ(k)
for all k ∈ Z+ κ.
(3.2b)
Proof. (⇒) Given λ ∈ σp (Φκ ), there exists a ξ = 0 such that Φκ ξ = λξ. We define φ(k) := Φ(k, κ)ξ to obtain the assertion from Proposition 2.5.3 and for all k ∈ Z+ κ, φ(k + p) = Φ(k + p, κ)ξ = Φ(k + p, κ + p)Φκ ξ = λΦ(k, κ)ξ = λφ(k). (⇐) Conversely, if a nontrivial solution φ : Z+ κ → X satisfies the relation (3.2b), then Proposition 3.1.9(a) yields Φκ φ(κ) = φ(κ + p) = λφ(κ). Since φ is nontrivial, this finally shows λ ∈ σp (Φκ ).
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3 Linear Difference Equations
Proposition 3.2.4. Let m ∈ N. Then (L0 ) has a nontrivial mp-periodic solution, if and only if a Floquet multiplier of (L0 ) is an m-th root of unity. Proof. Let κ ∈ Z. (⇒) Corollary 2.5.4(a) shows that Φm κ has a nontrivial fixed point and so 1 ∈ σp (Φm ). κ (⇐) Let λ ∈ σp (Φκ ) with λm = 1. Then Proposition 3.2.3 implies that (L0 ) has a nontrivial solution with φ(k + mp) = λm φ(k) = φ(k) for all k ∈ Z+ κ. Consequently, φ is mp-periodic. Proposition 3.2.5. Let m ∈ N. If a Floquet multiplier of (L0 ) is an m-th root of unity, then the vector bundle P := {(k, x) ∈ X : x ∈ ker(Φm κ − IXk )} is invariant w.r.t. (L0 ) and consists of complete mp-periodic solutions. Proof. Let κ ∈ Z. The periodicity of the sequence Φ· , i.e., Φk+mp = Φk , carries over to the vector bundle P. In order to establish the desired invariance property Φ(k, l)P(l) = P(k) for all l ≤ k it is therefore sufficient to show two inclusions: (I) Claim: Φ(k, l)P(l) ⊆ P(k) for 0 ≤ k − l < mp: We choose η ∈ Φ(k, l)P(l) and thus there exists a ξ ∈ P(l) such that η = Φ(k, l)ξ. Consequently, Proposition 2.5.3 implies η ∈ P(k), since we have (3.1g)
m η = Φ(k, l)ξ = Φ(k, l)Φm l ξ = Φ(k + mp, l + mp)Φl ξ = Φ(k + mp, l)ξ (3.1g)
(3.1g)
= Φ(k + mp, k)Φ(k, l)ξ = Φm k η.
(II) Claim: Φ(k, l)P(l) ⊇ P(k) for 0 ≤ k − l < mp: For a given ξ ∈ P(k) we define η := Φ(l + mp, k)ξ and Proposition 2.5.3 yields (3.1g)
(3.1g)
m m Φm l η = Φl Φ(l + mp, k)ξ = Φ(l + 2mp, k)ξ = Φ(l + 2mp, k + mp)Φk ξ = Φ(l + mp, k)ξ = η,
thus η ∈ P(l). Having this at hand, the claim follows from (3.1g)
Φ(k, l)η = Φ(k, l)Φ(l, k − mp)ξ = Φ(k, k − mp)ξ = ξ and we deduce that P is invariant. (III) It remains to show that P consists of complete mp-periodic solutions. For this, choose l ≥ κ, ξ ∈ P(l) and since Φ(l, κ) is onto between P(κ) and P(l), there exists a ξ0 ∈ ker(Φm κ − IXκ ) such that ξ = Φ(l, κ)ξ0 . Because ξ0 is a fixed point of Φm , we arrive at κ ξ = Φ(l, κ)ξ0 = Φ(l, κ)Φm κ ξ0 = Φ(l + mp, κ + mp)Φ(κ + mp, κ)ξ0 (3.1g)
= Φ(l + mp, l)Φ(l, κ)ξ0 = Φm l ξ
and ξ is a fixed point of Φm l . Then Corollary 2.5.4(b) implies the assertion.
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111
The value of the subsequent theorem is that the qualitative study of periodic systems reduces to the investigation of an autonomous problem: Theorem 3.2.6 (Floquet). Let κ ∈ Z and assume that (3.1c) holds. If there exist m ∈ {1, 2} and an operator R ∈ Iso(Xκ ) with mp , Φm κ =R
(3.2c)
then there exists an mp-periodic sequence Λ(k) ∈ Iso(Xκ , Xk ), k ∈ Z, such that the Floquet representation Φ(k, κ) = Λ(k)Rk holds for all k, κ ∈ Z. Remark 3.2.7. Let p ∈ Z+ 2 and m ∈ {1, 2}. An operator R ∈ Hom(Xκ ) satisfying Rp = Φm is called a p-th root of Φm κ κ . Existence and uniqueness results for operator roots are strongly related to the spectrum of Φm κ ∈ L(Xκ ). Therefore, it is sometimes important to work in a complex Banach space Xκ and passing over to the complexification of Φκ (see points (1) and (2) below): (0) Let Xκ be linear space. In order to have a p-th root, invertibility of Φm κ is m 2m essential in the following sense: If 0 < dim ker Φm < p and ker Φ = ker Φ κ κ κ , m then Φκ has no p-th root (cf. [97, Proposition 1.5]). (1) Let Xκ be a complex Banach space. The Riesz functional calculus (see, e.g., [96, p. 199ff]) enables us to show the existence of operator roots. Precisely, if there exists a continuous curve C ⊆ C leading from the origin 0 to the point at infinity such that σ(Φm κ ) ∩ C = ∅, then (cf. [346, Theorem 1]): √ • { p z ∈ C : 0 = z ∈ C}∩{0} is a union of p curves with only one common point 0, which divides the plane into n sectorial domains D1 , . . . , Dp ⊆ C. • Each Dk corresponds a unique p-th root Rn of Φm ) ⊆ Dn . κ such that σ(Rn p m • If an operator R is a p-th root of Φm κ , then it is of the form Φκ = n=1 Rn Pn , where the operators P1 , . . . , Pn ∈ L(Xκ ) satisfy Pi Pj = δij Pj for arbitrary indices i, j ∈ {1, . . . , p} and P1 + . . . + Pp = IXκ . (2) Let Xκ be a complex separable Hilbert space. The functional calculus for normal operators (see, e.g., [96, p. 285ff]) guarantees that every normal operator Φm κ has a p-th root. Moreover, the interior of Rp := {T ∈ L(Xκ )| ∃R ∈ L(Xκ ) : T = Rp } can be characterized as follows: Φm κ ∈ int Rp if and only if one of the upcoming conditions holds (see [97, Theorem 5.6]): • •
0 and ∞ do belong to the same component of C∞ \ σ(Φm κ ). 0 is an isolated eigenvalue of Φm that belongs to the unbounded component of κ the complement of σ(Φm ) \ {0} and whose Riesz projection has rank 1. κ
Furthermore, a positive operator Φm κ has a unique positive p-th root (see [66]), which is self-adjoint, provided Φm κ is (see [239]). (3) If Xκ is a linear space over the field F with dim Xκ < ∞, then a p-th root of Φm κ can be explicitly computed using the Jordan normal form (see [170, pp. 231–241]). Moreover, in case m = 2 and F = R one can find a real root. In this finite-dimensional situation, from R ∈ Iso(Xκ ) we infer Φκ ∈ Iso(Xκ ) and conse−1 quently also the inclusion Bk+1 Ak ∈ Iso(Xk , Xk+1 ) follows. Thus, the evolution
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3 Linear Difference Equations
operator of (L0 ) satisfies Φ(k, κ) ∈ Iso(Xκ , Xk ) for all k ∈ Z. Thus, we obtain the −1 invertibility of Bk+1 Ak from the corresponding properties of R. Proof. We simply define Λ(k) := Φ(k, κ)R−k ∈ Iso(Xκ , Xk ) and due to (3.1g)
−k−mp Λ(k + mp) = Φ(k + mp, κ)R−k−mp = Φ(k + mp, κ + mp)Φm κ R (3.2c)
= Φ(k, κ)R−k = Λ(k)
for all k ∈ Z, this sequence is mp-periodic.
Corollary 3.2.8. If (3.1c) holds, then a linear p-periodic equation (L0 ) is linearly conjugated on Z to the explicit autonomous equation y = Ry by virtue of the Lyapunov transformation (Λ(k))k∈Z . Proof. Suppose that κ ∈ Z is arbitrarily given. Since the sequence (Λ(k))k∈Z is mp-periodic, it is trivially bounded together with its inverse. We apply the Lyapunov transformation x = Λ(k)y with Λ(k) = Φ(k, κ)R−k to (L0 ) and obtain the linear −1 equation y = Λ (k)−1 Bk+1 Ak Λ(k)y, where the coefficient operator simplifies to (3.1g)
−1 −1 Λ (k)−1 Bk+1 Ak Λ(k) ≡ Rk+1 Φ(κ, k + 1)Bk+1 Ak Φ(k, κ)R−k ≡ R
on Z and this was our claim.
A further observation is that Floquet multipliers and Lyapunov exponents are related as follows: Corollary 3.2.9. Let κ ∈ Z. If (3.1a) holds and λ ∈ σp (Φκ ) with associated eigen − p |λ|. vector ξ ∈ Xκ , then λ+ u (ξ) = λl (ξ) = Proof. Let κ ∈ Z and suppose that the evolution operator Φ of (L0 ) is given as Φ(k, κ) = Λ(k)Rk . Since λ is a Floquet multiplier, we obtain Φκ ξ = λξ with some eigenvector ξ = 0. For arbitrary κ, k ∈ Z satisfying κ ≤ k we now define integers nk := max {n ∈ Z : κ + np ≤ k} and consequently obtain k − p < κ + nk p ≤ k. nk This yields k−p−κ < nkk ≤ k−κ kp kp and thus limk→∞ k = 1/p. We conclude k Φ(k, κ)ξ
n k Φ(k, κ + nk p)Φnκk ξ = k |λ| k Φ(k, κ + nk p)ξ nk 1 → |λ| p , = |λ| k k Φ(k, κ + nk p)ξ −−−−
(3.1g)
=
k→∞
which is our claim. Now we turn our attention to p-periodic linear inhomogeneous equations Bk+1 x = Ak x + gk
(Lg )
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113
satisfying (3.1a). Due to the variation of constants formula from Theorem 3.1.16(a) we know that the period map Πκ : Xκ → Xκ of (Lg ) is given by Πκ (ξ) = Φκ ξ +
κ+p−1
−1 Φ(κ + p, n + 1)Bn+1 gn .
(3.2d)
n=κ
Proposition 3.2.10. Suppose that X consists of topological linear spaces and that (3.1j) holds. A linear p-periodic equation (Lg ) has a unique complete p-periodic solution, if there exists a κ ∈ Z such that 1 ∈ σ(Φκ ). Proof. By assumption we have IXκ − Φκ ∈ GL(Xκ ) and define ξ ∗ := (IXκ − Φκ )−1
κ+p−1
−1 Φ(κ + p, n + 1)Bn+1 gn .
n=κ
Consequently, ξ ∗ ∈ Xκ is a fixed point of Πκ and by Corollary 2.5.4(b) there exists a complete p-periodic solution φ of (Lg ). Its uniqueness can be seen as follows. ¯ Then the difference φ − φ¯ is Suppose there exists another p-periodic solution φ. a nontrivial p-periodic solution of (L0 ) and with Proposition 3.2.3 we arrive at the contradiction 1 ∈ σp (Φκ ) ⊆ σ(Φκ ). Theorem 3.2.11. Suppose that X consists of Banach spaces, (3.1j) holds and that Φκ ∈ L(Xκ ) is compact for one κ ∈ Z. A linear p-periodic equation (Lg ) has a bounded solution, if and only if it has a p-periodic solution. Proof. The only nontrivial implication is that the existence of bounded solutions implies p-periodic solutions. From Corollary 2.5.4 we know that (Lg ) has a p-periodic solution if and only if there exist a pair (κ, ξ) ∈ X such that (3.2e)
ξ = Πκ (ξ) = Φκ ξ + η
κ+p−1 −1 with η := n=κ Φ(κ + p, n + 1)Bn+1 gn (cf. (3.2d)). Hence, it suffices to show that any solution of (Lg ) is unbounded, if one cannot solve (3.2e). From a theorem of Fredholm (cf., e.g., [465, pp. 372–373, Sect. 8.5]), we obtain that (3.2e) is not solvable, if and only if there is a x ∈ Xκ such that x = Φκ x
and
η, x = 0;
(3.2f)
note here that, by assumption, Φκ is a compact operator. For an arbitrary solution φ of (Lg ) satisfying φ(κ0 ) = ξ we have from Theorem 3.1.16(a) that (3.1h)
φ(k) = Φ(k, κ0 )ξ +
k−1 n=κ0
−1 Φ(k, n + 1)Bn+1 gn
for all κ0 ≤ k
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3 Linear Difference Equations
and therefore φ(κ0 + p) = Φκ ξ + η, as well as (see (Lg )) Bk+np+1 φ (k + np) = Ak+np φ(k + np) + gk+np = Ak φ(k + np) + gk for n ∈ Z+ 0 . By the uniqueness of solutions we get that φn (k) := φ(k + np) is a solution of the initial value problem (Lg ), x(κ0 ) = φ(κ0 + np). Thus, (3.2e) gives us φk (κ0 + p) = φ(κ0 + (n + 1)p) = Φκ φ(κ0 + np) + η and this, in turn, yields n−1 φn (κ0 + p) = Φnκ ξ + j=0 Φjκ η by mathematical induction. We get n
φn (κ0 + p), x = ξ, [Φκ ] x +
k−1
j
η, [Φκ ] x
for all n ∈ Z+ 0
j=0
from (3.2f) and because of η, x = 0 we have φ(κ0 + np) x ≥ φ(κ0 + np), x = φn (κ0 + p), x −−−−→ ∞. n→∞
Thus, the sequence φ is unbounded.
Finally, we summarize how the different attraction and stability notions from Sect. 2.4 simplify for linear periodic equations: Corollary 3.2.12. Suppose p ∈ N, X consists of normed spaces and (L0 ) is p-periodic. Then the attraction and stability notions for (Lg ) from Definitions 2.4.4 and 2.4.11, resp., are related by: S ⇔ US ⇔ UPS ⇔ PS ⇑ ⇑ ⇑ ⇑ GAS ⇔ AS ⇔ UAS ⇔ UAPS ⇔ APS GA ⇔ A ⇔ UA ⇔ UPA ⇔ PA Proof. The assertion is a combination of Propositions 2.5.5 and 3.1.25.
3.3 Invariant Splittings and Exponential Growth In finite-dimensional state spaces X the archetypical examples for invariant subspaces of autonomous linear difference equations x = Ax, A ∈ L(X), with spectrum σ(A) = {λ1 , . . . , λN } are the generalized eigenspaces Eigλn A ⊆ X. In particular, they yield a composition X=
N j=1
Eigλj A
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115
and our aim is to extend this situation to a time-variant setting. For this purpose, in this section’s first part we are interested in nonautonomous sets consisting of linear spaces, which are invariant w.r.t. a homogeneous linear difference equation Bk+1 x = Ak x
(L0 )
as in Definition 3.1.1. Evident examples of such invariant vector bundles are the graph of the trivial solution I × {0} or the extended state space X itself. In these cases we speak of trivial vector bundles. Another example is the super-stable vector bundle U0+ introduced in Example 3.1.14. The notion of an invariant projector allows to describe further invariant vector bundles, where we impose various conditions: The invariance condition ensures that a vector bundle is forward invariant w.r.t. (L0 ), whereas the regularity condition implies the existence of backward solutions. The strong regularity condition yields both above conditions and allows to introduce a Green’s function. Definition 3.3.1. A sequence P = (P (k))k∈I of mappings P (k) ∈ Hom(Xk ) with P (k) = P (k)2 , k ∈ I , is called a projector and we set Q := {(k, x) ∈ X : x ∈ ker P (k)} ,
P := {(k, x) ∈ X : x ∈ im P (k)} .
If the extended state space X consists of topological linear spaces and the inclusion P (k) ∈ L(Xk ), k ∈ I , holds, one denotes P as bounded projector. Remark 3.3.2. (1) Let P, P¯ be projectors. We briefly write ker P = ker P¯ to abbreviate ker P (k) = ker P¯ (k) for all k ∈ I and proceed similarly with im P = im P¯ . (2) The complementary projector I − P is given by (I − P )(k) := IXk − P (k) for k ∈ I ; evidently, I − P is also a projector and we consistently write Q(k) := IXk − P (k) for all k ∈ I. Furthermore, both the nonautonomous sets Q, P are closed for a bounded projector P . In addition, one has the Whitney sum Q ⊕ P = X.
(3.3a)
Definition 3.3.3. A projector P = (P (k))k∈I is called invariant projector for (L0 ), if the invariance condition ker Bk+1 |Q (k) = {0} ,
im Ak Q(k) ⊆ im Bk+1 Q (k) for all k ∈ I
(3.3b)
holds. Moreover, one speaks of a regular projector for (L0 ), provided one has the regularity condition ker Ak |P(k) = {0} ,
im Bk+1 P (k) ⊆ im Ak P (k) for all k ∈ I .
(3.3c)
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3 Linear Difference Equations
Remark 3.3.4. (1) For explicit equations x = Ak x in X the invariance condition (3.3b) simplifies to the inclusion im Ak Q(k) ⊆ Q (k). (2) The identity P (k) = IXk is a trivial example of an invariant projector. In this situation the conditions (3.3c) ensure the backward evolution operator of (L0 ) to exists (cf. Proposition 3.1.4(b)). Thus, (3.3c) is a weakened invertibility condition. (3) On the other hand, the zero projector P (k) = 0 is always regular and in this situation, the invariance condition (3.3b) implies the existence of the forward evolution operator of (L0 ) (cf. Proposition 3.1.4(a)). (4) Referring to Corollary 3.4.25 we make the convention that invariant projectors of p-periodic equations (L0 ) are also p-periodic. Under the invariance condition (3.3b) one can show the existence of a forward invariant vector bundle for (L0 ). Indeed, if P is an invariant projector, one derives that Bk+1 |Q (k) : Q (k) → im Bk+1 Q (k) is bijective and consequently Bk+1 |−1 Q (k) Ak : Q(k) → Q (k) is well-defined. Therefore, Q is a forward invariant vector bundle, i.e., (L0 ) is an equation in Q with forward evolution operator Φ+ P (k, κ) : Q(κ) → Q(k), −1 −1 Φ+ P (k, κ) := Bk |Q(k) Ak−1 · . . . · Bκ+1 |Q (κ) Aκ
for all κ ≤ k
(3.3d)
and one has the commutativity relation (cf. Fig. 3.1) + Q(k)Φ+ P (k, l) = ΦP (k, l)Q(l) for all l ≤ k.
(3.3e)
Dually, the regularity condition (3.3c) yields that (L0 ) has a backward invariant vector bundle. For a regular projector P of (L0 ), the mappings Ak |P(k) : P(k) → im Ak P (k) are bijective and Ak |−1 P(k) Bk+1 : P (k) → P(k) are well-defined. We therefore obtain a backward invariant vector bundle P and introduce a backward 2-parameter semigroup Φ− P (k, κ) : P(κ) → P(k),
Φ− P (l, κ)
P(κ)
P (k)
P (l) Q(κ) Xκ
Q (l) Xl
I Q (k) (k, l) Φ+ P
Fig. 3.1 Complementary invariant vector bundles Q and P
Xk
3.3 Invariant Splittings and Exponential Growth −1 −1 Φ− P (k, κ) := Ak−1 |P(k−1) Bk · . . . · Aκ |P(κ) Bκ+1
117
for all k ≤ κ,
(3.3f)
which additionally fulfills (cf. Fig. 3.1) − P (k)Φ− P (k, l) = ΦP (k, l)P (l) for all k ≤ l.
(3.3g)
By definition, one obtains the Whitney sum (3.3a); hence, an invariant regular projector P induces a splitting (3.3a) into a forward invariant bundle Q and a backward invariant bundle P of (L0 ). If both P and the complementary projector Q are invariant, one gets a splitting of the extended state space X into two forward invariant vector bundles. Corollary 3.3.5. Assume that (L0 ) satisfies ker Bk+1 = {0}, k ∈ I , and P is an invariant projector: (a) If im Ak P (k) ⊆ im Bk+1 , k ∈ I , then the forward evolution operator Φ of (L0 ) exists and satisfies Q(k)Φ(k, l) = Φ(k, l)Q(l) for all l ≤ k.
(3.3h)
(b) If im Ak P (k) ⊆ im Bk+1 P (k), k ∈ I , then also the complementary projector Q is invariant. (c) If also I − P is an invariant projector, then the forward evolution operator of (L0 ) exists and (3.3h) holds. Proof. Let k ∈ I be given. (a) It remains to show the left inclusion in (3.1a) in Proposition 3.1.4(a). For this, for arbitrary x ∈ Xk we write Ak x = Ak Q(k) + Ak P (k)x, where the invariance condition (3.3b) and our assumption guarantee that both summands are in im Bk+1 . Then (3.3h) is a direct consequence of (3.3e). (b) Clearly, ker Bk+1 = {0} yields ker Bk+1 |P (k) = {0}. (c) For each y ∈ im Ak there exists a ξ ∈ Xk such that y = Ak ξ. Since both P and I − P are invariant projectors, we obtain ξ1 , ξ2 ∈ Xk such that (3.3b)
y = Ak P (k)ξ + Ak Q(k)ξ = Bk+1 P (k)ξ1 + Bk+1 Q(k)ξ2 ∈ im Bk+1 , i.e., the inclusion im Ak ⊆ im Bk+1 is satisfied. Hence, assumption (3.1a) holds and the forward evolution operator exists by Proposition 3.1.4(a). For a regular invariant projector P , Green’s function associated to a linear equation (L0 ) is given as GP (k, l) ∈ Hom(Xl , Xk ),
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3 Linear Difference Equations
GP (k, l) :=
Φ+ P (k, l)Q(l)
−Φ− P (k, l)P (l)
for l ≤ k, for k < l
(3.3i)
and we easily deduce the subsequent properties of GP (·, l). Lemma 3.3.6. If a projector P satisfies the strong regularity condition ker Bk+1 = {0} , ker Ak |P(k) = {0} ,
im Ak Q(k) ⊆ im Bk+1 Q (k), im Bk+1 P (k) = im Ak P (k)
(3.3j)
for all k ∈ I , then it is invariant, regular w.r.t. (L0 ) and the following holds: (a) The forward evolution operator Φ of (L0 ) exists and (3.3h) holds. (b) Φ− P (k, l) ∈ Iso(P(l), P(k)) for all l ≤ k and P is invariant. (c) Green’s function GP fulfills Bk+1 GP (k + 1, l) = Ak GP (k, l) + δk,l−1 Bk+1
for all k, l ∈ I.
(3.3k)
(d) If (L0 ) and P are p-periodic, then GP (k + p, l + p) = GP (k, l) for all k, l ∈ Z. Remark 3.3.7. (1) If P (k) ≡ IXk holds, then the strong regularity condition (3.3j) coincides with the assumptions (3.1a) and (3.1b). Therefore, one has the invertibility relation (3.1c) and the general solution to (L0 ) exists. (2) On the other hand, if P (k) ≡ 0, then (3.3j) reduces to (3.1a) guaranteeing the existence of the general forward solution to (L0 ). (3) Since dim P(k), k ∈ I, is constant by assertion (b), we write dim P. Proof. Let k ∈ I . Above all, it is clear that the strong regularity condition (3.3j) implies both the conditions (3.3b) and (3.3c): (a) This follows directly from Corollary 3.3.5(a). (b) Since the projector P satisfies the regularity condition (3.3c), we know that Ak |P(k) : P(k) → im Ak P (k) is bijective. On the other hand, the linear mapping Bk+1 : Xk+1 → Yk+1 is one-to-one and with the inclusion im Bk+1 P (k) ⊆ im Ak P (k) it follows that Bk+1 |P (k) : P (k) → im Bk+1 P (k) is bijective. This guarantees that also the composition Ak |−1 P(k) Bk+1 : P (k) → P(k) is bijective and with (3.3f) this yields our claim. − −1 (c) Thanks to assertion (a) we can define Φ− for l ≤ k and P (k, l) := ΦP (l, k) − extend ΦP to I × I. Therefore, we infer (3.3i)
Bk+1 GP (k + 1, l) = Bk+1 Φ+ P (k + 1, l)Q(l) = Bl Q(l) (3.1e)
− = −Bl Φ− P (l, l − 1)ΦP (l − 1, l)P (l) + Bk+1 IXl
= −Al−1 Φ− P (l − 1, l)P (l) + Bk+1 IXl (3.3i)
= Ak GP (l − 1, l) + Bk+1 IXl
for k = l − 1,
3.3 Invariant Splittings and Exponential Growth
119
(3.3i)
Bk+1 GP (k + 1, l) = Bk+1 Φ+ P (k + 1, l)Q(l) = Ak Φ+ P (k, l)Q(l) = Ak GP (k, l) for all l ≤ k,
(3.3d)
and in the remaining case k ≤ l − 2 we also get the desired relation (3.3i)
− Bk+1 GP (k + 1, l) = −Bk+1 Φ− P (k + 1, l)P (l) = −Ak ΦP (k, l)P (l) (3.3i)
= Ak GP (k, l).
(d) By the p-periodicity of P , the claim follows from (3.2a).
Often the assumptions of Proposition 3.1.4(a) are satisfied and we get Corollary 3.3.8. Suppose (3.1a) holds. Then P is (a) an invariant projector, if one has the commutativity relation −1 −1 P (k)Bk+1 Ak = Bk+1 Ak P (k) for all k ∈ I ,
(3.3l)
(b) a regular projector, if −1 Ak ∈ Iso(P(k), P (k)) for all k ∈ I Bk+1
(3.3m)
and a projector P also fulfills the strong regularity condition (3.3j), provided both (3.3l) and (3.3m) are satisfied. Remark 3.3.9. Under assumption (3.1a) we know that P is an invariant projector for (L0 ), if and only if (3.3h) holds. Proof. Let k ∈ I be given. (a) The first kernel condition in (3.3b) follows by (3.1a). For y ∈ im Ak Q(k) there exists a x ∈ Xk such that y = Ak Q(k)x and since (3.3l) readily im−1 plies Bk+1 Q (k)Bk+1 Ak = Ak Q(k), we get y ∈ im Bk+1 Q (k). Thus, (3.3b) is satisfied. (b) We check the regularity condition (3.3c). Given x ∈ P(k) with Ak x = 0, the −1 assumption (3.3m) guarantees Bk+1 Ak x = 0 and consequently x = 0. Moreover, for an arbitrary y ∈ im Bk+1 P (k) there exists a x ∈ Xk+1 with y = Bk+1 P (k)x. −1 Due to (3.3m) we can represent P (k)x ∈ P (k) as P (k)x = Bk+1 Ak ξ for some −1 ξ ∈ P(k), and infer y = Bk+1 P (k)x = Bk+1 Bk+1 Ak ξ = Ak P (k)ξ which finally yields (3.3c). In order to establish the strong regularity condition (3.3j), it remains to show the inclusion im Ak P (k) ⊆ im Bk+1 P (k). Given y ∈ im Ak P (k) we find (3.3l)
−1 y = Ak P (k)x = Bk+1 P (k)Bk+1 Ak x
for some x ∈ Xk and therefore the claim.
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3 Linear Difference Equations
When interested in a detailed dynamical description, it is useful to split the extended state space of a linear equation (L0 ) into two or more nontrivial invariant vector bundles. In order to quantify this idea, we proceed as follows: For the remainder, N will be a natural number or ∞. In this context we use the convenient notation 1 ≤ n < N + 1, since it simultaneously allows us to indicate that n varies over the finite range {1, . . . , N } (if N ∈ N), as well as over N (if N = ∞). Definition 3.3.10. Let N ∈ N ∪ {∞}. The extended state space X is said to allow an N -splitting (Pn )N n=1 , if there exist projectors Pn (k) ∈ Hom Xk such that one has for all k ∈ I : (i) If N < ∞, then Pn (k)Pm (k) = δmn Pn (k) for all 1 ≤ m, n < N + 1 and N
Pn (k)x = x for all x ∈ Xk ,
n=1
(ii) if N = ∞, then the conditions from (i) hold, where X consists of topological linear spaces and Pn (k) ∈ L(Xk )
for all 1 ≤ n < N.
(3.3n)
An N -splitting with (3.3n) is called bounded. For a minimal N -splitting one additionally assumes Pn = 0 for all 1 ≤ n < N . Remark 3.3.11. (1) A minimal 1-splitting necessarily consists of the identity. (2) If N < ∞ and (3.3n) holds, then PN (k) ∈ L(Xk ), k ∈ I . Example 3.3.12. If a nonautonomous set X consists of separable Hilbert spaces with constant dimension N = dim Xk for all k ∈ I, then X admits a bounded N -splitting. Indeed, keeping k ∈ I fixed, every separable Hilbert space Xk has a countable basis (xn )N n=1 , which by the Gram-Schmidt procedure (cf. [96, p. 14, 4.6]) can assumed to be orthonormal. We define the orthogonal projectors Pn (k)x := xn , xXk xn
for all k ∈ I, 1 ≤ n < N + 1,
which are bounded by the Cauchy-Schwarz inequality and one has the orthogonal sum Xk = n∈N im Pn (k), k ∈ I (see [342, p. 444, Theorem 6.8.2]). The existence of an N -splitting has a variety of useful geometric consequences. n First, also the sequences Pm , Qnm given by the sums n (k) := Pm
Qnm (k)
n j=m
Pj (k)
for n < m,
0
:= IXk −
for k ∈ I, 1 ≤ m ≤ n < N + 1,
n Pm (k)
for all k ∈ I
3.3 Invariant Splittings and Exponential Growth
121 P 12
Q3 Q1 ⊕ Q3 Q12
P2
P1 Q1
P3
Q2
P 23
P1 ∩ P3
Fig. 3.2 Hierarchical inclusions for a 3-splitting
are projectors on Xk , which are bounded provided Pm , . . . , Pn are. Mimicking notation from Definition 3.3.1 we can define further nonautonomous sets Qn := {(k, x) ∈ X : x ∈ ker Pn (k)} , Qnm
:= {(k, x) ∈ X : x ∈
n ker Pm (k)} ,
Pn := {(k, x) ∈ X : x ∈ im Pn (k)} , n n Pm := {(k, x) ∈ X : x ∈ im Pm (k)}
and obtain the intersection, resp., Whitney sum Qnm =
n
Qj ,
j=m
n Pm =
n
Pj
for all m ≤ n
(3.3o)
j=m
n := I × {0} for n < m. For a supplemented by the conventions Qnm := X and Pm minimal splitting we obtain two hierarchical inclusions (see Fig. 3.2 for N = 3)
. . . ⊂ Q31 ⊂ Q21 ⊂ Q1 ∩ ∩ . . . ⊂ Q32 ⊂ Q2 ∩ . . . ⊂ Q3 . ..
P1 ⊂ P12 ⊂ P13 ⊂ . . . ∪ ∪ P2 ⊂ P23 ⊂ . . . ∪ P3 ⊂ . . . .. .
(3.3p)
n Moreover, in case of a bounded splitting, the sets Qnm and Pm are closed. n Lemma 3.3.13. Let P = (Pn )N n=1 be an N -splitting of X . If every projector P1 , 1 ≤ n < N , is bounded, then also P is bounded.
Proof. Given k ∈ I we proceed by induction over n. For n = 1 one clearly has P1 (k) = P1n (k) ∈ L(Xk ) by assumption. In the induction step n − 1 → n we know by our induction hypothesis that P1n−1 (k) is bounded and our assumption yields the boundedness of P1n (k). Thus, we have Pn (k) = P1n (k) − P1n−1 (k) ∈ L(Xk ) yielding our claim.
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3 Linear Difference Equations
Definition 3.3.14. An N -splitting (Pn )N n=1 of X is said to be invariant for (L0 ), if each sequence Pn is an invariant projector for (L0 ). Example 3.3.15 (spectral splitting). Let X, Y be Banach spaces and suppose the operators A ∈ Hom(X, Y ), B ∈ Hom(Y ) are such that ker B = {0} and furthermore B −1 A : D(B −1 A) ⊆ X → X is closed. We say the operator B −1 A admits a spectral splitting, if σ(A, B) =
N
σi
for a finite N ∈ N
i=0
can be separated into nonempty spectral sets σ0 , . . . , σN , where each σi ⊆ C with 1 ≤ i ≤ N is bounded and enclosed in a closed curve γi ⊂ ρ(T ), the γi lying outside one another, whereas σ0 ⊆ C may be unbounded and is excluded by the curve γN . Then [244, p. 178ff] yields that the spectral projectors 1 [IX − ζB −1 A]−1 dζ ∈ L(X) for all 1 ≤ i ≤ N Pi := − 2πi γi N and P0 := IX − i=1 P1 define a bounded N + 1-splitting of X. Moreover, it is invariant w.r.t. the autonomous equation Bx = Ax. In case X is finite-dimensional with d = dim X, then σ(B −1 A) consists of up to d distinct (and isolated) eigenvalues. Hence, every finite-dimensional linear difference equation allows a spectral splitting with N ≤ d. Example 3.3.16. Let X be an infinite-dimensional Hilbert space and suppose the operators A, B ∈ Hom(X, Y ) are such that ker B = {0}, B −1 A ∈ L(X) is compact and normal with σ(B −1 A) = {0} ∪ {λn }n∈N and nonzero eigenvalues λn arranged in decreasing order of magnitude. Consequently, we can apply [244, p. 260, Theorem 2.10] to B −1 A ∈ L(X). Let Pn be the spectral projector associated to the spectral set σn = {λn }, n ∈ N, and P0 be the orthogonal projection onto ker B −1 A. Then (Pn )∞ n=0 is a bounded invariant splitting of X, whose projectors are pairwise orthogonal. Proposition 3.3.17. If a linear equation (L0 ) allows an invariant N -splitting (Pn )N n=1 , then the following holds for 1 ≤ m ≤ n < N + 1: n (a) Pm is an invariant projector and Qnm a forward invariant vector bundle. n (b) If Pm , . . . , Pn satisfy the regularity condition (3.3c), then also Pm and the vecn tor bundles Pm are invariant.
Proof. Let k ∈ I and 1 ≤ m ≤ n < N + 1. n n (a) Obviously, Pm is a projector and it remains to show that Pm is invariant. Since each Pj is an invariant projector, we obtain the implication Bk+1 |Qj (k) ξj = 0
(3.3b)
⇒
ξj = 0
for all ξj ∈ Qj (k), m ≤ j ≤ n.
3.3 Invariant Splittings and Exponential Growth
123
Now suppose ξ ∈ Qnm (k + 1) with Bk+1 ξ = 0. The intersection property (3.3o) yields ξ ∈ Qj (k) for all m ≤ j ≤ n and this guarantees ξ = 0, i.e., the first n inclusion in (3.3b) is satisfied or Pm . Concerning the second inclusion, we suppose n η ∈ im Ak Qm (k), i.e., there exists a ξ ∈ Qnm (k) such that η = Ak ξ. Because of ξ ∈ Qnm (k) we deduce ξ ∈ Qj (k) for all m ≤ j ≤ n. Since each Pj is an invariant projector, we obtain the existence of ξ1 ∈ Qj (k) such that Ak ξ = Bk+1 Qj (k)ξ1 for all m ≤ j ≤ n and consequently the desired inclusion η ∈ im Bk+1 Qnm (k + 1). −1 n n n (k) : P (b) We prove that Bk+1 Ak |Pm m (k) → Pm (k + 1) is bijective. This is n n equivalent to show that for every η ∈ Pm (k + 1) there exists a unique ξ ∈ Pm (k) −1 satisfying Bk+1 Ak ξ = η. For such η and m ≤ j ≤ n we know by assumption that −1 there exist unique ξj ∈ Pj (k) with Bk+1 Ak ξj = Pj (k)η. Thus, η=
n
Pj (k)η =
j=m
n
−1 −1 Bk+1 Ak ξj = Bk+1 Ak
j=m
n
ξj
j=m
n and setting ξ := ξm + . . . + ξn ∈ Pm (k) yields our assertion.
Pij
of (L0 ), also their inHaving identified the invariant vector bundles Qi1 and tersections Qi1 ∩ Pij and Whitney sums Qi1 ⊕ Pij , 1 ≤ i, j < N + 1, form invariant vector bundles. In case N < ∞ this gives rise to 2N − 2 invariant vector bundles for (L0 ). Of particular interest among them are those which persist under nonlinear −1) perturbations (see Sect. 4.2). For this reason, we consider the following (N +2)(N 2 nontrivial invariant vector bundles of (L0 ). Corollary 3.3.18 (hierarchy of invariant vector bundles). In case a linear equation (L0 ) allows a regular invariant minimal N -splitting (Pn )N n=1 , then ∩ P1j Uij := Qi−1 1
for all 1 ≤ i ≤ j < N + 1
are forward invariant vector bundles of (L0 ) such that: j ⊃ Uij ⊂ Uij+1 for all 1 < i ≤ j < N . (a) One has the inclusions Ui−1 (b) In case N < ∞ one has the extended hierarchy
Uij = Pij
and
U11 ⊂ U12 ⊂ . . . ⊂ U1N −1 ⊂ X ∪ ∪ ∪ U22 ⊂ . . . ⊂ U2N −1 ⊂ U2N ∪ ∪ .. .. .. . . . ∪ ∪ N −1 N UN −1 ⊂ UN −1 ∪ N UN
(3.3q)
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3 Linear Difference Equations
Proof. Consider pairs (i, j) with 1 ≤ i ≤ j < N + 1. From Proposition 3.3.17 we derive the forward invariance of U1j = Q01 ∩ P1j = P1j ,
UiN = Qi−1 ∩ P1N = Qi−1 1 1
for N < ∞
and Proposition 1.2.6(a) yields the forward invariance of Uij . (a) By definition, we have the two inclusions Uij
=
Qi−1 1
∩
P1j
⊂
j ∩ P1j = Ui−1 Qi−2 1
Qi−1 1
∩
P1j+1
=
Uij+1
for 1 < i, for j < N.
(b) We already verified the first row and the right column of (3.3q) in the hierarchical inclusions (3.3p). Moreover, one deduces the hierarchy (3.3q) from Uij = Qi−1 ∩ P1j = ker P1i−1 ∩ im P1j = im PiN ∩ im P1j = im Pij = Pij 1
and the corollary is shown.
In the upcoming Sect. 3.4 we will establish the concept of an exponential splitting by describing the dynamics in the invariant vector bundles Pij and Qji . For this purpose, we need to propagate the notion of exponential boundedness which plays a key role for our further analysis and is important for dimension estimates of invariant fiber bundles. Here, we refer to Sect. A.1 for the generalized exponential function ea and suppose X consists of normed spaces. Provided a linear equation (Lg ) or (L0 ) is a-stable, then the growth (or decay) of every solution can be estimated by ea . In a more systematic fashion, one introduces Definition 3.3.19. Let c, d : I → (0, ∞). A sequence φ in X is called (a) c+ -bounded, if there exists a κ ∈ I such that sup ec (κ, k) φ(k)Xk < ∞
κ≤k
and c− -bounded, if there exists a κ ∈ I such that sup ec (κ, k) φ(k)Xk < ∞,
k≤κ
(b) c, d-bounded, if there exists a κ ∈ I such that max sup ec (κ, k) φ(k)Xk , sup ed (κ, k) φ(k)Xk < ∞ κ≤k
k≤κ
± and c-bounded, if it is c, c-bounded. By X± κ,c we denote the set of all c -bounded sequences, by Xc,d the set of c, d-bounded, and by Xc is the set of c-bounded sequences. Given operators Bk+1 ∈ Hom(Xk+1 , Yk+1 ) with ker Bk+1 = {0}, k ∈ I ,
3.3 Invariant Splittings and Exponential Growth
125
−1 ± the set X± κ,c,B consists of sequences ψ with ψ(k) ∈ im Bk+1 and B·+1 ψ ∈ Xκ,c . Similarly one defines the sets Xc,d,B and Xc,B .
Remark 3.3.20. Let φ : I → X be a sequence in X . (1) If a sequence c : I → (0, ∞) satisfies c 1, we call φ ∈ X+ κ,c exponentially decaying in forward time, • 1 c, we call φ ∈ X− κ,c exponentially decaying in backward time •
(cf. Lemma A.1.3). In addition, the relationship between characteristic exponents and c± -boundedness is as follows: • •
+ If I is unbounded above and φ ∈ X+ κ,c , then χu (φ) ≤ c . − if I is unbounded below and φ ∈ Xκ,c , then χ− u (φ) ≤ 1/ !c".
(2) One has φ ∈ Xc,d if and only if φ is c+ - and d− -bounded. − (3) The sets X+ κ,c , Xκ,c and Xc,d are nonempty, since they contain the zero sequence or the space of all sequences which are eventually 0. On bounded intervals I the sets X± κ,c , Xc,d coincide and consist of all sequences in X . Periodic sequences are contained in X1,1 . More general, on arbitrary intervals, conventional boundedness corresponds to the case c = 1 and we can generalize Corollary 3.1.3. Corollary 3.3.21. (a) If (Lg ) has a unique c± -bounded (or c, d-bounded) solution, then the trivial solution is the unique c± -bounded (or c, d-bounded) solution of (L0 ). (b) Conversely, if 0 is the unique c± -bounded (or c, d-bounded) solution of (L0 ), then (Lg ) has at most one c± -bounded (or c, d-bounded) solution. Proof. In the proof of Corollary 3.1.3, simply replace the word “periodic” or “bounded” by c± -bounded (resp. c, d-bounded). Finally we will discuss the consequences of an N -splitting of X to the notion of c± - and c, d-boundedness. Primarily, it helps us to achieve somewhat better constants in our further analysis. More precisely, differing from the usual norm ·Xκ on Xκ , the norm from the following result yields conditions of the form max {x, y} rather than sums x + y. Lemma 3.3.22. Let κ ∈ I. If X has a bounded N -splitting (Pn )N n=1 , then |x|n := max P1n (κ)xXκ , Qn1 (κ)xXκ
for 0 ≤ n < N + 1
defines a family of equivalent norms on Xκ , which are equivalent to ·Xκ . Proof. Let 1 ≤ n < N + 1, since in case n = 0 there is nothing to prove. The reader may show that each |·|n is actually a norm on Xκ . Then the two inequalities |x|n ≤ max {P1n (κ) , Qn1 (κ)} x , x ≤ P1n (κ)x + Qn1 (κ)x ≤ 2 |x|n
for all x ∈ Xκ
imply that ·Xκ and |·|n are equivalent norms on Xκ . By transitivity, also the norms |·|n are pairwise equivalent.
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3 Linear Difference Equations
Proposition 3.3.23. Let κ ∈ I, c : I → (0, ∞) and suppose that X admits a bounded N -splitting (Pn )N n=1 with sup P1n (k)L(Xk ) < ∞
for all 1 ≤ n < N + 1.
(3.3r)
k∈I
For every 0 ≤ n < N + 1 a sequence φ in X is: ec (κ, k) |φ(k)|n < ∞. (a) c± -bounded, if and only if φ± κ,c := supk∈I± κ − (b) c, d-bounded, if and only if φκ,c,d := max φ+ κ,c , φκ,d < ∞. (c) c-bounded, if and only if φκ,c := supk∈I ec (κ, k) |φ(k)|n < ∞. Remark 3.3.24. (1) For arbitrary 0 ≤ n < N + 1 a sequence φ in X satisfies φ ∈ X± κ,c
⇒
φ ∈ Xc,d
⇒
±
|φ(k)|n ≤ ec (k, κ) φκ,c for all k ∈ I± κ, + for all k ∈ I+ ec (k, κ) φκ,c κ, |φ(k)|n ≤ − for all k ∈ I− ed (k, κ) φκ,d κ.
(2) Let c, d : I → (0, ∞) be bounded above and 1 ≤ n < N + 1. If φ ∈ X± κ,c , then also the sequences φ and φ defined in (2.3f) are c± -bounded with (cf. (0.0b)) ± φ(k) = φ(k) ≤ max {1, c } φκ,c ec (k, κ) for all k ∈ (I± κ); n
n
the same holds on I for c-bounded sequences φ. ± (3) Similarly to Proposition 3.3.23, for sequences ψ in Xκ,c,B , Xc,d,B or Xc,B −1 ± we define the respective mappings ψκ,c,B := supk∈I± ec (κ, k) Bk+1 ψ(k) n , κ −1 − −1 + ψ κ,c , B·+1 ψ κ,d , ψκ,c,d,B := max B·+1
ψκ,c,B := ψκ,c,c,B .
Proof. The straight forward proof is left to the reader. In particular, note that the norm ·κ,c,d depends on κ ∈ I, whereas the spaces Xc,d do not. We continue by investigating the fundamental algebraic and functional analytical properties of the sets X± κ,c and Xc,d . Lemma 3.3.25. Let κ ∈ I and c, d : I → (0, ∞). ±
±
± (a) X± κ,c is a linear space with seminorm ·κ,c ; if I = Iκ then ·κ,c is a norm on ± ± Xκ,c and Xκ,c is a Banach space, if each Xk , k ∈ I, is complete. (b) Xc,d is a linear space with norm ·κ,c,d and Xc,d is a Banach space, if each Xk , k ∈ I, is complete.
Proof. We omit the elementary proof.
3.3 Invariant Splittings and Exponential Growth
127
Lemma 3.3.26. If κ ∈ I and c, c¯, d, d¯ : I → (0, ∞) with c ≤ c¯ and d¯ ≤ d, then we have the continuous embeddings − X− κ,d → Xκ,d¯,
+ X+ κ,c → Xκ,¯ c,
Xc,d → Xc¯,d¯
and the corresponding embedding operators are bounded above by 1. Proof. For every sequence φ ∈ X+ κ,c we obtain for all k ≥ κ that +
ec¯(κ, k) φ(k)Xk ≤ ec (κ, k) φ(k)Xk ≤ φκ,c and passing over to the least upper bound for k yields our claim. The assertion for d− -bounded or c, d-bounded sequences follows analogously. For later use in Sect. 4.4 we need some additional preparations for sequences with values in spaces of multilinear mappings. For this, given κ ∈ I and n ∈ Z+ 0 we n define the set Lκ := k∈I Ln (Xκ ; Xk ); note that L0κ = X. Having this at hand, we ± introduce the spaces X0,± κ,c := Xκ,c , n ± Xn,± κ,c := Φ : I → Lκ | Φ(k) ∈ Ln (Xκ ; Xk ) for all k ∈ I and Φ is c -bounded and obtain the following Lemma 3.3.27. Let c : I → (0, ∞) and n ∈ N. If each Xk , k ∈ I, is complete, then n−1,± Xn,± ) are isometrically isomorphic Banach spaces. κ,c and L(Xκ , Xκ,c n−1,± Proof. Given κ ∈ I consider the mapping J : Xn,± ) given by κ,c → L(Xκ , Xκ,c
± (JΦ)x (k) := Φ(t)x for k ∈ Zκ and x ∈ Xκ . In order to prove that J is the desired norm isomorphism, we choose Φ ∈ Xn,± κ,c and obtain
Φ(k)xLn−1 (Xκ ,Xk ) ec (κ, k) ≤ Φ(k)Ln (Xκ ;Xk ) ec (κ, k) x ≤ Φ± κ,c x for all k ∈ Z± κ . Thus, the continuity of the evidently linear mapping J follows from ±
n−1,± = sup (JΦ)x ≤ Φ JΦL(Xκ ,Xκ,c κ,c . )
x =1
Vice versa, the inverse J −1 : L(Xκ , Xn−1,± ) → Xn,± κ,c κ,c of the operator J is given −1 ¯ ¯ by (J Φ)(t)x := (Φx)(t). By the open mapping theorem (see [295, p. 388, Corollary 1.4]), J −1 is continuous and it remains to show that it is non-expanding. Thereto we choose Φ¯ ∈ L(Xκ , Xn−1,± ) and x ∈ Xκ arbitrarily to get κ,c −1 (J Φ)(k)x ¯ ¯ e (κ, k) = (Φx)(t) e (κ, k) Ln−1 (Xκ ,Xk ) c Ln−1 (Xκ ,Xk ) c ± ¯ ≤ Φ¯ x for all k ∈ Z± ≤ Φx κ, κ,c L(Xκ ,Xn−1,± ) κ,c
Φ¯ ¯ and this estimate yields (J −1 Φ)(k) e (κ, k) ≤ , which c L(Xκ ,Xn,± κ,c ) −1 ± L n (Xκ ;Xk ) ¯ ¯ gives us the desired J Φ ≤ Φ n,± . Thus, J is an isometry. κ,c
L(Xκ ,Xκ,c )
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3 Linear Difference Equations
Lemma 3.3.28. Let m, n ∈ Z+ 0 , c : I → (0, ∞), Ω = ∅ be an open subset of a Banach space and suppose that X consists of Banach spaces. If f ∈ C m (Ω, Xn,± κ,c ),
m then one has the inclusion f (·) (k) ∈ C (Ω, Ln (Xκ ; Xk )) for every k ∈ I. Proof. For k ∈ I the evaluation map evk : Xn,± κ,c → Ln (Xκ ; Xk ), evk (φ) := φ(k) is a continuous homomorphism and hence of class C ∞ . It follows by the chain rule
that the composition f (·) (k) = evk ◦ f inherits its smoothness from f .
3.4 Dichotomies and Splittings Ein unerwartetes Beispiel Ta Li (cf. [304]) We suppose that X consists of normed spaces and J ⊆ I is a discrete interval. In the upcoming Chap. 4 we show that invariant vector bundles of linear equations Bk+1 x = Ak x,
(L0 )
as in Definition 3.1.1, persist under nonlinear perturbations – including their dynamical characterization. Therefore one needs sufficiently robust hyperbolicity assumptions on the linear part, which preserve stability properties. In the autonomous case, it yields from Proposition 3.1.23(g) and Corollary 3.2.12 that the notions of asymptotic and exponential stability are equivalent. Thus, asymptotic stability of the linear part will be sufficient to persist under small nonlinear perturbations. This, however, fails for nonautonomous equations, as demonstrated by the following slight modification of a classical example due to Li (cf. [304, Sect. 2]). Example 3.4.1. Suppose X = N × R2 and consider the explicit equation
0 x = Ak x + γ |x1 |
,
Ak :=
α 0 β 0 αγ βk+1 k
,
(3.4a)
where βk := exp ψ(k) with the function ψ : [1, ∞) → R, ψ(t) := t sin ln t and constants α, γ > 0 satisfying ! −π α ∈ e−(1+e )/γ , e−1/γ . (3.4b) We successively verify the following claims: (I) Claim: The sequence Ak is bounded: The mean value theorem (cf. [295, p. 341, Theorem 4.2]) implies the existence of a τ ∈ (k, k + 1) with √ |ψ(k + 1) − ψ(k)| = |cos ln τ + sin ln τ | ≤ 2 for all k ∈ I, therefore
βk+1 βk
∈ [e−
√ 2
,e
√ 2
] and thus Ak is bounded.
3.4 Dichotomies and Splittings
129
(II) Claim: The linear part of (3.4a) has Lyapunov exponents in (0, 1): The evolution operator for the linearization of (3.4a) reads as Φ(k, l) =
αk−l
for all l ≤ k.
(αγ )k−l ββkl
By the inclusion (3.4b) one can choose a δ > 1 with αδ < 1, eαγ δ < 1 and due to βk (αγ δ)k = (ek sin ln k αγ δ)k ≤ (eαγ δ)k for k ∈ I this yields the limit relation limk→∞ δ k Φ(k, l) = 0 for all l ∈ I. Thus, max α, δ −1 < 1 is an upper bound for the upper Lyapunov exponents of the linear part and the linearization of (3.4a) is asymptotically stable by Corollary 3.1.29. (III) Claim: The trivial solution of (3.4a) is unstable: Using Theorem 3.1.16(a) the general forward solution of (3.4a) is ϕ(k; l, ξ) =
(αγ )k−l ββkl ξ2 +
αk−l ! ξ1
|ξ1 | α
γ
(αγ )k βk
k−1 n=l
−1 βn+1
for all l ≤ k.
Due to −γ ln α < 1 + e−π (cf. (3.4b)) it is possible to find r, ρ ∈ (0, 1) with ρ ≤ cos x + γ ln α + e−π+y−x cos y
for all x, y ∈ (0, r).
Given n ∈ I we define integers k, κ via the relations
k = exp 2n + 12 π − 1,
κ + 1 = exp 2n − 12 π − 1,
where [·] : R → Z denotes the integer It is possible reals
function.
to choose
εn , δn > 0 such that k = exp 2n + 12 π + εn , κ + 1 = exp 2n − 12 π + δn and the sequences εn , δn converge to 0 as n → ∞. In particular, for sufficiently large n we have εn , δn ∈ (0, r) and the inequality κ + 1 = ke−π+δn −εn < k is easy to see. Then the relations
ψ(k) = sin 2n + 12 π + εn = cos εn ,
ψ(κ + 1) = sin 2n − 12 π + δn = − cos δn yield (αγ )k βk
k−1
−1 −1 βn+1 ≥ (αγ )k βk βκ+1 = e(cos εn +γ ln α)k+cos δn (κ+1)
n=l −π+δn −εn )k
= e(cos εn +γ ln α+cos δn e
≥ eρk
and from the above we see that the second component of the general solution ϕ decays to zero only if the initial value satisfies ξ1 = 0. Indeed, the trivial solution of (3.4a) is unstable.
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3 Linear Difference Equations
Consequently, we need more robust concepts based on uniform asymptotic stability, or the more general forward a-stability. Under weak invertibility assumptions, a first approach are forward dichotomies. Note that for explicit equations no backward solutions are required and one only needs an invariance inclusion for P. Definition 3.4.2. A linear equation (L0 ) has an exponential forward dichotomy on J, if there exists a bounded projector P such that P , I − P are invariant for (L0 ), satisfy Bk+1 |−1 Q (k) Ak ∈ L(Q(k), Q (k)),
Bk+1 |−1 P (k) Ak ∈ L(P(k), P (k))
for all k ∈ J and that the following holds: (i) M := supk∈J P (k)L(Xk ) < ∞, (ii) there exist constants K ± ≥ 1 and growth rates a, b : J → (0, ∞) with a b, b bounded above, such that for all k, l ∈ J, l ≤ k one has + Φ (k, l)x ≤ K + ea (k, l) x P Xl X k + 1 ΦQ (k, l)x ≥ K − eb (k, l) xXl Xk
for all x ∈ Q(l), for all x ∈ P(l).
(3.4c)
In case a 1 b one denotes (L0 ) as forward hyperbolic. Remark 3.4.3. (1) The second dichotomy estimate in (3.4c) implies that the operator Φ+ Q (k, l) : P(l) → Xk , l ≤ k, is one-to-one. Hence, for unique backward solutions in P, one has to impose corresponding surjectivity conditions. (2) We call the linear homogeneous equation Bk+1 x = c(k)Ak x
(L(c)0 )
a scaled linear equation, where c : I → F is a given sequence with nonzero values. Provided the forward evolution operator Φ of (L0 ) exists, then the forward evolution operator Φc of (L(c)0 ) reads as Φc (k, l) = ec (k, l)Φ(k, l) for all l ≤ k and the forward dichotomy spectrum of (L0 ) is defined by Σf (A, B) := γ > 0 : (L(γ −1 )0 ) is not forward hyperbolic . Example 3.4.4 (forward dichotomy spectrum). In finite dimensions X = J × Rd the forward dichotomy spectrum of a linear equation (L0 ) satisfying (3.1a) has been characterized in [28, Spectral Theorem] as follows: The forward dichotomy spectrum of (L0 ) is either empty or it is the disjoint union of N closed intervals (called spectral intervals) where N does not exceed d + 1. To be more explicit, we either have Σf (A, B) = ∅ or Σf (A, B) = (0, ∞) or one of the four cases ⎧ ⎧ ⎫ ⎫ ⎨[αN , βN ]⎬ ⎨ [α1 , β1 ] ⎬ Σf (A, B) := ∪ [αN −1 , βN −1 ] ∪ . . . ∪ [α2 , β2 ] ∪ or or ⎩ ⎩ ⎭ ⎭ (0, βN ] [α1 , ∞)
3.4 Dichotomies and Splittings
131 Sf (A, B, P )
b3
a2
b2
a1
b1
a0
Fig. 3.3 Ordering of the rates bn+1 ≤ an bn in an exponential (forward) splitting
occurs, where 0 < αN ≤ βN < αN +1 ≤ . . . < α1 ≤ β1 . For one-sided time, i.e., − J = Z+ κ or J = Zκ , κ ∈ I, one has 0 ≤ N ≤ d and the case N = d + 1 implies that (0, βN ] is a spectral interval. Furthermore, systems with bounded forward growth in the sense of relation (3.1k) satisfy Σf (A, B) ⊆ (0, a ], and bounded growth ensures a nonempty forward dichotomy spectrum. Given a (bounded) invariant splitting (Pn )N n=1 , by Proposition 3.3.17(a) also the sum P1n is an invariant projector for (L0 ) and we can introduce (see Fig. 3.3). Definition 3.4.5. A linear equation (L0 ) is said to admit an exponential forward N -splitting on J, in symbols ⎧ (−∞, a0 ) ⎪ ⎪ ⎨ Sf (A, B; P ) = (b1 , ∞) ⎪ ⎪ ⎩ N −1 i=0 (bi+1 , ai )
if N = 0, if N = 1,
(3.4d)
if N > 1,
if there exist growth rates an , bn : J → (0, ∞), a0 ≤ ∞, an bn 0 ≤ bN , bn+1 ≤ an
for all 1 ≤ n < N, for all 0 ≤ n < N,
(3.4e)
such that: • • •
When N = 0, then (L0 ) is forward a0 -stable with a0 ∞. When N = 1, then (L0 ) is backward b1 -stable with 0 b1 . When N > 1, then there exists an N -splitting (Pn )N n=1 of X such that (L0 ) admits an exponential forward dichotomy on J with projector P1n and data an , bn , Kn+ , Kn− for every 1 ≤ n < N .
An exponential forward 3-splitting is called exponential forward trichotomy. Remark 3.4.6. (1) A 0-splitting corresponds to the situation of a forward dichotomy with P = 0, while the case P = I describes a 1-splitting. (2) The vector bundles Qn1 , P1n are forward invariant. (3) In case !c" > 0 an exponential forward N -splitting of (L0 ) carries over to the scaled equation (L(c)0 ), where the growth rates need to be modified to can , cbn , and this yields Sf (γ −1 A, B; P ) = γ −1 Sf (A, B; P ) for every γ > 0.
132
3 Linear Difference Equations Sf (A, B, P ) b2
a1
α3
β3
α2
a0
b1
β2
α1
β1
Σf (A, B)
Fig. 3.4 Exponential forward splitting (dotted lines) and forward dichotomy spectrum (solid lines)
(4) When the forward dichotomy spectrum of (L0 ) consists of N < ∞ spectral intervals, then there exists an invariant N -splitting such that the formal inclusion Σf (A, B) ⊆ Sf (A, B; P ) holds in the sense that bn αn ,
βn an−1
for all 1 ≤ n < N
(cf. Fig. 3.4). In this setting, the forward dichotomy spectrum yields the finest exponential forward splitting. However, in general an interval (bi+1 , ai ) in (3.4d) can contain more one spectral interval. The notion (3.4d) has to be understood that Nthan −1 the union i=0 (bi+1 , ai ) forms a cover of Σf (A, B). Proposition 3.4.7. Let N > 1. Provided (L0 ) admits an exponential forward N -splitting (3.4d) on J, then for all 1 ≤ n < N and c : J → (0, ∞) one has: (a) If J is unbounded above, then there exists a solution φ : Z+ κ →X Qn1 ⊆ (κ, ξ) ∈ X : , of (L0 ) with φ(κ) = ξ and φ ∈ X+ κ,c
if an ≤ c,
and in case ker Bk+1 = {0}, k ∈ J, it holds Qn1 = (κ, ξ) ∈ X : Φ(·, κ)ξ ∈ X+ κ,c ,
if an ≤ c bn .
(b) If J is unbounded below and ker Bk+1 = {0}, k ∈ J , then there exists a solution φ : Z− κ →X (κ, ξ) ∈ X : ⊆ P1n , if an c. of (L0 ) with φ(κ) = ξ and φ ∈ X− κ,c Proof. Let κ ∈ J and 1 ≤ n < N . (a) Let an ≤ c. In order to show the first set inclusion, we choose (κ, ξ) ∈ Qn1 , hence, ξ = Qn1 (κ)ξ and therefore the estimate (3.4c) ec (κ, k) Φ+ (k, κ)ξ ≤ Kn+ e acn (k, κ) ξ ≤ Kn+ ξ n P 1
for all k ∈ Z+ κ
3.4 Dichotomies and Splittings
133
+ which implies that φ := Φ+ P1n (·, κ)ξ is c -bounded. Obviously, φ is a forward solution of (L0 ) yielding the first claim. Concerning the converse inclusion, we remark that the forward evolution operator Φ exists by Corollary 3.3.5(c). Thus, the relation
Φ(·, κ)Qn1 (κ)ξ − Φ(·, κ)P1n (κ)ξ = Φ(·, κ)ξ ∈ X+ κ,c in connection with (3.4c) requires P1n (κ)ξ = 0, i.e., ξ ∈ Qn1 (κ). (b) Suppose that there exists a c− -bounded backward solution φ : Z− κ → X of (L0 ) with φ(κ) = ξ. Due to Corollary 3.3.5(c) the forward evolution operator of (L0 ) exists, we have Φ(κ, k)φ(k) = ξ for all k ≤ κ and consequently Qn1 (κ)ξ = Qn1 (κ)Φ(κ, k)φ(k) (3.4c)
(3.3e)
= Φ(κ, k)Qn1 (k)φ(k) ≤ Kn+ ean (κ, k) φ(k) − ≤ Kn+ e acn (k, κ) φκ,c for all k ∈ Z− κ.
Thanks to Lemma A.1.3(b), passing over to k → −∞ guarantees ξ ∈ P1n (κ).
The above Proposition 3.4.7 guarantees that at least the invariant vector bundles Qn1 allow a dynamical characterization using forward solutions with a specific growth behavior, i.e., independent of the projectors P1n . However, assuming only a forward dichotomy (or splitting), we show in the following example that this is not possible for the vector bundles P1n . Example 3.4.8. For real parameters 0 < α < β consider the explicit equation x = Ak x
(3.4f)
in X = Z × R3 , where the coefficient matrix Ak ∈ L(R3 ) is given by ⎛ ⎞ α 0 0 Ak := ⎝ 0 βδk 0 ⎠ , 0 0 β
δk :=
0
for k ≤ 0,
1
for k > 0.
It is not difficult to infer that the evolution operator of (3.4f) reads as ⎞ ⎛ k−l 0 0 α Φ(k, l) = ⎝ 0 (βδk )k−l 0 ⎠ 0 0 β k−l
for all l ≤ k,
where 00 is understood to be 1. For every b ∈ R the matrix sequence ⎛ 0 bδk
!k−1 ⎞ 0 ⎟ 0⎠ δk
0
0
⎜ P (k) := ⎝0
α β
1
for all k ∈ Z
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3 Linear Difference Equations
defines a projector, whose kernels do not depend on b. It is straight forward to verify + that (3.4f) √ has an exponential forward dichotomy with M1 = 1 + |b|, K1 = 1,3 − K1 = 1 + b2 and growth rates a1 (k) ≡ α, b1 (k) ≡ β on Z, provided R is equipped with the Euclidean norm. Yet, the ranges of P (k) depend on b and therefore it is not possible to obtain a complete dynamical characterization of P1 , depending only on growth properties of solutions.
Exponential Splittings Assuming the following stronger notions, one is in the satisfying position to have a complete dynamical characterization of P1n in terms of exponentially bounded backward solutions (see also Fig. 3.3). Definition 3.4.9. A linear equation (L0 ) has an exponential dichotomy on J, if there exists a bounded invariant regular projector P for (L0 ) satisfying Bk+1 |−1 Q (k) Ak ∈ L(Q(k), Q (k)),
Ak |−1 P(k) Bk+1 ∈ L(P(k), P(k − 1))
for all k ∈ J and that there exist real constants K + , K − ≥ 1 and also growth rates a, b : J → (0, ∞) with a b, b bounded above, such that + Φ (k, l)Q(l) ≤ K + ea (k, l) for all l ≤ k, P L(Xl ,Xk ) − Φ (k, l)P (l) ≤ K − eb (k, l) for all k ≤ l P L(X ,X ) l
(3.4g)
k
and k, l ∈ J. In case a 1 b one denotes (L0 ) as hyperbolic. Remark 3.4.10. (1) In case (3.4g) holds with a = b = 1, one speaks of an ordinary dichotomy. Linear equations (L0 ) admitting an exponential dichotomy with arbitrary splitting a b are also called pseudo-hyperbolic. (2) An exponential dichotomy with P = 0 is equivalent to forward a-stability, whereas backward b-stability corresponds to a dichotomy with P = I. (3) We define the dichotomy spectrum of (L0 ) as Σ(A, B) := γ > 0 : (L(γ −1 )0 ) is not hyperbolic . (4) Frequently, (L0 ) appears as variational equation of a nonlinear equation (D) along a reference solution φ∗ as in Remark 2.3.12(3). In this case, one denotes φ∗ as hyperbolic, if (L0 ) has an exponential dichotomy, or equivalently 1 ∈ Σ(A, B). Example 3.4.11 (dichotomy spectrum). Suppose we have X = J × Rd . Following [28, 29] the dichotomy spectrum of (L0 ) is either empty or the disjoint union of N closed spectral intervals, where N does not exceed d + 1, but N ≤ d holds for systems satisfying (3.1a) and (3.1b). Precisely, we either have Σ(A, B) = ∅ or Σ(A, B) = (0, ∞) or one of the four cases
3.4 Dichotomies and Splittings
135
⎫ ⎫ ⎧ ⎧ ⎪ ⎪ ⎬ ⎬ ⎨[αN , βN ]⎪ ⎨ [α1 , β1 ] ⎪ ∪ [αN −1 , βN −1 ] ∪ . . . ∪ [α2 , β2 ] ∪ Σ(A, B) := or or ⎪ ⎪ ⎪ ⎪ ⎭ ⎭ ⎩ ⎩ (0, βN ] [α1 , ∞) occurs, where 0 < αN ≤ βN < αN +1 ≤ . . . < α1 ≤ β1 . When dealing with one− sided time J = Z+ κ or J = Zκ , κ ∈ I, one has 0 ≤ N ≤ d and the case N = d + 1 implies that (0, βN ] is a spectral interval. It has been shown in [436, Corollary 2.2] that the dichotomy spectrum is invariant under linear conjugacy. Equations with bounded growth have nonempty and compact dichotomy spectrum. Definition 3.4.12. A linear equation (L0 ) is said to admit an exponential N -splitting on J, in symbols ⎧ (−∞, a0 ) ⎪ ⎪ ⎨ S(A, B; P ) = (b1 , ∞) ⎪ ⎪ ⎩ N −1 i=0 (bi+1 , ai )
if N = 0, if N = 1,
(3.4h)
if N > 1,
if there exist growth rates an , bn : J → (0, ∞) satisfying (3.4e) such that: When N = 0, then (L0 ) is forward a0 -stable with a0 ∞. When N = 1, then (L0 ) is backward b1 -stable with 0 b1 . • When N > 1, then there exists an N -splitting (Pn )N n=1 of X and growth rates an , bn : J → (0, ∞) satisfying (3.4e) such that (L0 ) admits an exponential dichotomy on J with projector P1n and data an , bn , Kn+ , Kn− for every 1 ≤ n < N . • •
An exponential 3-splitting is called an exponential trichotomy. Remark 3.4.13. (1) The dimensions dim P1n (k) are constant for all k ∈ I. (2) Exponential splittings can be extended over compact intervals. More detailed, let ¯ J ⊆ I be a discrete interval with J ⊆ ¯ J such that ¯J ∩ J is finite. If (L0 ) has a ¯ bounded invariant splitting on J, then an exponential N -splitting on J carries over to each larger interval ¯ J. (3) Also an exponential N -splitting of (L0 ) carries over to the scaled difference equation (L(c)0 ), where the growth rates need to be modified to can , cbn , provided one has !c" > 0. For reals γ > 0 it is S(γ −1 A, B; P ) = γ −1 S(A, B; P ). (4) Thanks to (3.1r), an exponential N -splitting persists under linear conjugacy as introduced in Definition 3.1.30. (5) A linear system (L0 ) with a dichotomy spectrum consisting of N spectral intervals allows an invariant N -splitting such that Σ(A, B) ⊆ S(A, B; P ), where S(A, B; P ) is the finest possible invariant splitting. Yet, each interval (bi+1 , ai ) can contain more than one spectral interval and S(A, B; P ) forms a cover of Σ(A, B), which for infinite-dimensional state spaces has a possibly complex structure.
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3 Linear Difference Equations
Next we investigate the relationship between exponential forward splittings and exponential splittings: Proposition 3.4.14. Let N > 1 and ker Bk+1 = {0} for k ∈ J . (a) Provided (L0 ) admits an exponential forward N -splitting (3.4d) on J and −1 Ak P1n (k + 1) ⊆ im Bk+1
for all k ∈ J , 1 ≤ n < N
(3.4i)
holds, then (3.4h) is satisfied and P1n is an invariant vector bundle. (b) Conversely, if an exponential N -splitting (3.4h) holds and im Ak P1n (k) ⊆ im Bk+1 P1n (k + 1) for all k ∈ J , 1 ≤ n < N,
(3.4j)
then the forward evolution operator of (L0 ) exists and (3.4d) holds. Remark 3.4.15. Under the above assumptions one has Σ(A, B) = Σf (A, B). Proof. Let l ∈ J and 1 ≤ n < N . (a) Suppose (L0 ) has an exponential forward N -splitting (Pn )N n=1 with Mn := supk∈J P1n (k) < ∞. Every sequence P1n is an invariant projector and we clearly have the estimate (3.4c) + ΦP1n (k, l)Qn1 (l)x ≤ Kn+ ean (k, l) Qn1 (l)x ≤ (1 + Mn )Kn+ ean (k, l) x ¯1 (0) yields an for all x ∈ Xl and l ≤ k. Passing to the supremum over x ∈ B estimate as the first one in (3.4g). Now let l ∈ J , ξ0 ∈ P1n (l) and from (3.4c) (3.3d) − −1 ebn (l + 1, l) ξ0 ≤ Kn− Φ+ Qn (l + 1, l)ξ0 = Kn Bl+1 |P n (l+1) Al ξ0 1
1
we deduce that the map Bl+1 |−1 P n (l+1) Al is one-to-one (cf. Remark 3.4.3(1)). Thanks 1
n n to our assumption (3.4i), the composition Bl+1 |−1 P1n (l+1) Al : P1 (l) → P1 (l + 1) is also onto and therefore bijective. Thus, for l ≤ k the composition −1 n n Bk |−1 P n (k) Ak−1 · . . . · Bl+1 |P n (l+1) Al : P1 (l) → P1 (k) 1
1
is the inverse operator to Φ− defined via (3.3f). The regularity condition (3.3c) Qn 1 n n holds for (L0 ). By assumption, Bk+1 |−1 P n (k+1) Ak ∈ L(P1 (k), P1 (k + 1)) and [295, 1
p. 388, Corollary 1.4] yields that also the inverse Ak |−1 P1n (k) Bk+1 is bounded. In parn ticular, each P1 is an invariant vector bundle. It remains to show the backward estimate in (3.4g). For this, by Corollary 3.3.5(c) the forward evolution operator Φ exists and we obtain
3.4 Dichotomies and Splittings
137
(3.3g) n − n n ebn (l, k) Φ− P1n (k, l)P1 (l)x ≤ ebn (l, k) P1 (k)ΦP1n (k, l)P1 (l)x (3.4c) n ≤ Kn− Φ(l, k)P1n (k)Φ− P1n (k, l)P1 (l)x (3.3g)
= Kn− P1n (l)x
for all k ≤ l.
¯1 (0) we arrive at Passing to the least upper bound over x ∈ B − Φ (k, l)P1n (l) ≤ Kn− ebn (k, l) P1n (l) ≤ Mn Kn− ebn (k, l) Pn for all k ≤ l, i.e., a remaining inequality as in (3.4g). (b) Now assume (L0 ) admits an exponential N -splitting. It is an immediate consequence of (3.4g) that we have P1n (l) ≤ Kn+ for all l ∈ J, which implies the boundedness property on the invariant projectors. Moreover, Corollary 3.3.5(b) and (3.4j) yield that also Qn1 is an invariant projector for (L0 ). In addition, Corollary 3.3.5(c) guarantees that the forward evolution operator Φ(k, l) exists. For arbitrary x ∈ Xl we have the estimates (3.4g)
Φ(k, l)Qn1 (l)x ≤ Φ(k, l)Qn1 (l) Qn1 (l)x ≤ Kn+ ean (k, l) Qn1 (l)x for all l ≤ k, yielding the first relation in (3.4c), and n n P1n (l)x ≤ Φ− P n (l, k)P1 (k) Φ(k, l)P1 (l)x 1
(3.4g)
≤ Kn− ebn (l, k) Φ(k, l)P1n (l)x
for all l ≤ k, implying the second estimate in (3.4c). This was our claim.
We obtain the following dynamical characterization of the vector bundles. Proposition 3.4.16. Let N > 1 and ker Bk+1 = {0}, k ∈ J . Provided (L0 ) has an exponential N -splitting (3.4h), then for all 1 ≤ n < N and c : J → (0, ∞) it holds: (a) If J is unbounded above and (3.4j) holds, then
n (κ, ξ) ∈ X : Φ(·, κ)ξ ∈ X+ κ,c ⊆ Q1 ,
if c bn .
(b) If J is unbounded below, then P1n
there exists a solution φ : Z− κ →X , ⊆ (κ, ξ) ∈ X : of (L0 ) with φ(κ) = ξ and φ ∈ X− κ,c
if c ≤ bn .
Remark 3.4.17 (pseudo-stable hierarchy). Let 1 ≤ n < N . (1) In case of an interval J unbounded above and an ≤ cn bn we obtain with Proposition 3.4.7(a) the dynamical characterization Qn1 = (κ, ξ) ∈ X : Φ(·, κ)ξ ∈ X+ κ,cn
(3.4k)
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3 Linear Difference Equations
and each Qn1 is called a pseudo-stable vector bundle. Furthermore, (L0 ) is an equation in Qn1 and the restriction to Qn1 is forward cn -stable. Given a minimal splitting, directly from (3.4k) and Lemma 3.3.26 we obtain the pseudo-stable hierarchy J × {0} ⊆ U0+ ⊂ . . . ⊂ Q31 ⊂ Q21 ⊂ Q1 ⊂ X of forward invariant vector bundles, where U0+ is the super-stable bundle. If there exists a minimal n∗ such that an∗ 1, then all solutions in Qn1 ∗ decay exponentially to 0 and one denotes Us := Qn1 ∗ as stable vector bundle of (L0 ); in addition, nontrivial Qn1 with n > n∗ are called strongly stable vector bundles and form the stable hierarchy J × {0} ⊆ U0+ ⊂ . . . ⊂ Q1n∗ +1 ⊂ Qn1 ∗ . For 1 ≤ an∗ and thus 1 bn∗ one denotes Ucs := Qn1 ∗ as center-stable vector bundle. (2) For J = I the exponential growth of forward solutions gives rise to the relation
(κ1 , ξ1 )
∼+ n
(κ2 , ξ2 ) :⇔
there exist solutions φi : Z+ κi → X to (L0 ) with φi (κi ) = ξi and φ2 − φ1 ∈ X+ κ,c for an ≤ c bn ,
with κ = max {κ1 , κ2 }, which is easily seen to be an equivalence relation on X . + Thus, the equivalence classes [·]n form a partition of X and we have Qn1 = [0]+ n. Remark 3.4.18 (pseudo-unstable hierarchy). Let 1 ≤ n < N . (1) Dually, for a discrete interval J unbounded below and an c ≤ bn we infer with Proposition 3.4.7(b) the dynamical characterization P1n =
there exists a solution φ : Z− κ →X . (κ, ξ) ∈ X : of (L0 ) with φ(κ) = ξ and φ ∈ X− κ,cn
(3.4l)
Here, every P1n is a pseudo-unstable vector bundle of (L0 ). We deduce that (L0 ) is an equation in P1n , the restriction to P1n is backward cn -stable and borrowing notation from Corollary 3.3.18 it is U1n = P1n . For a minimal splitting, from (3.4l) we get the pseudo-unstable hierarchy J × {0} ⊂ P1 ⊂ P12 ⊂ P13 ⊂ . . . ⊂ X of invariant vector bundles. If there exists a maximal index n∗ with 1 bn∗ , then all solutions in P1n∗ exist and decay exponentially to 0 in backward time. Hence, Uu := P1n∗ is denoted as unstable vector bundle and the nontrivial P1n with n < n∗ are strongly unstable vector bundles forming the unstable hierarchy J × {0} ⊂ . . . ⊂ P1n∗ −1 ⊂ P1n∗ . Finally, for bn∗ ≤ 1 and consequently an∗ 1 one denotes the vector bundle Ucu := P1n∗ of (L0 ) as center-unstable.
3.4 Dichotomies and Splittings
139
(2) For J = I the exponential growth of backward solutions allows us to define (κ1 , ξ1 ) ∼− n (κ2 , ξ2 ) :⇔
there exist solutions φi : Z− κi → X to (L0 ) with φi (κi ) = ξi and φ2 − φ1 ∈ X− κ,c for an c ≤ bn ,
with κ = min {κ1 , κ2 }. Also this is an equivalence relation on X , the equivalence n − classes [·]− n form a partition of X and we have P1 = [0]n . Remark 3.4.19 (pseudo-center hierarchy). Let J = Z. (1) Given an N ∈ Z+ 2 we suppose (3.4j). If we choose cn : Z → (0, ∞) with an cn bn , then the nonautonomous sets Uij from the extended hierarchy (3.3q) in Corollary 3.3.18 allow a dynamical characterization: ⎧ ⎪ ⎨
⎫ Φ(·, κ)ξ ∈ X+ κ,ci−1 and there exists ⎪ ⎬ Uij = (κ, ξ) ∈ X : a solution φ : Z− κ → X of (L0 ) ⎪ ⎪ ⎩ ⎭ with φ(κ) = ξ and φ ∈ X− κ,cj for all 1 < i ≤ j < N – we speak of the pseudo-center hierarchy (3.3q) of (L0 ). (2) For the exponential growth of complete solutions we introduce the equivalence relation ⎧ ⎪ there exist complete solutions φi : Z → X to ⎪ ⎨ j (κ1 , ξ1 ) ∼i (κ2 , ξ2 ) :⇔ (L0 ) with φi (κi ) = ξi and φ2 − φ1 ∈ Xc,d for ⎪ ⎪ ⎩ all ai−1 c ≤ bi−1 and aj d ≤ bj on X , whose equivalence classes [·]ji form a partition of X and Uij = [0]ji . Remark 3.4.20 (classical hierarchy). If (L0 ) has an exponential N -splitting with N > 2 and bn∗ +1 ≤ 1 ≤ an∗ , we then get the classical invariant vector bundles: • •
•
• •
+1 Stable vector bundle Us = Unn∗∗+1 : Because of cn∗ +1 bn∗ +1 and the dynamical characterization, solutions of (L0 ) on Us tend to 0 exponentially for k → ∞. Center-stable vector bundle Ucs = Unn∗∗ +1 : All solutions of (L0 ) which are not growing too fast as k → ∞ (meaning they are c+ 1 -bounded with cn∗ +1 ≤ bn∗ +1 ) are contained in Ucs , like e.g., solutions bounded in forward time. Center-unstable vector bundle Ucu = U1n∗ : All solutions of (L0 ) which exist and are not growing too strong as k → −∞ (in the sense of c− n∗ +1 -boundedness with an∗ +1 ≤ cn∗ +1 ) lie on Ucu , like e.g., solutions bounded in backward time. Unstable vector bundle Uu = U1n∗ −1 : All solutions on the unstable vector bundle exist in backward time and converge exponentially to 0 as k → −∞. Center vector bundle Uc := Unn∗∗ : The center vector bundle consists of those solutions which are contained both in the center-stable and the center-unstable vector bundle. Particularly, all bounded complete solutions lie on Uc .
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3 Linear Difference Equations
Proof of Proposition 3.4.16. Let κ ∈ J and 1 ≤ n < N . (a) Combining Corollary 3.3.5(b) and (c) we see using (3.4j) that the forward evolution operator Φ for (L0 ) exists. Now, pick ξ ∈ Xκ such that Φ(·, κ)ξ is c+ bounded and there exists a C ≥ 0 with Φ(k, κ)η ≤ Cec (k, κ) for all k ∈ Z+ κ. Referring to the second splitting estimate in (3.4g), this yields (3.3g) n P1n (κ)η ≤ Φ− P1n (κ, k)P1 (k) Φ(k, κ)η (3.4g)
≤ Kn− ebn (κ, k) Φ(k, κ)η ≤ CKn− e bcn (k, κ) for all k ∈ Z+ κ
and thanks to Lemma A.1.3(a) the right-hand side of this inequality tends to 0 in the limit k → ∞. Hence, we have η ∈ Qn1 (κ). + (b) For ξ ∈ P1n (κ) we define φ := Φ− P1n (·, κ)ξ and clearly, φ : Zκ → X is a solution of (L0 ) with φ(κ) = ξ. Moreover, we get (3.4g) ec (κ, k) φ(k) = ec (κ, k) Φ− (k, κ)ξ ≤ Kn− e bn (k, κ) ξ ≤ Kn− ξ n P 1
for all k ≤ κ and this implies that φ is c− -bounded.
c
Corollary 3.4.21. Let 1 ≤ n < N and c : J → (0, ∞). (a) If J is unbounded above and c bn , then there exists a unique c+ -bounded forward solution of (L0 ) in P1n , namely the trivial one. (b) If J is unbounded below and an c, then there exists a unique c− -bounded solution of (L0 ) in Qn1 , namely the trivial one. (c) If J = Z and c, d ∈ (an , bn ), then the unique c, d-bounded solution of (L0 ) is the trivial one. Proof. Let κ ∈ J and 1 ≤ n < N . (a) From Proposition 3.4.16(a) and (3.4k) we can infer that an arbitrary c+ n bounded forward solution φ : J+ κ → X of (L0 ) satisfies φ(κ) ∈ Q1 (κ). On the n other hand, we assumed φ(κ) ∈ P1 (κ) and this yields φ(κ) = 0. (b) This follows dually using Proposition 3.4.16(b) and (3.4l). (c) This is an immediate consequence of (a) and (b). In our following corollary we address the question, whether invariant projectors are uniquely determined. For splittings on semiaxes this is only asymptotically true, but kernels resp. images have to agree; we precisely get: Corollary 3.4.22. Suppose that (L0 ) has an exponential N -splitting w.r.t. different ¯ N projectors (Pn )N n=1 and (Pn )n=1 , but the same growth rates an , bn : (a) If J is unbounded above, then ker Pn = ker P¯n . (a) If J is unbounded below, then im Pn = im P¯n . (c) If J = Z, then Pn = P¯n for all 1 ≤ n < N .
3.4 Dichotomies and Splittings
141
Proof. Let k ∈ J and 1 ≤ n < N . (a) Since the dynamical characterization (3.4k) from Remark 3.4.17 is indepen¯ n1 . Using mathematical induction we now dent of the projectors, we obtain Qn1 = Q show ker Pn (k) = ker P¯n (k): This clearly holds for n = 1. As mathematical induction step n → n + 1 we suppose that ker Pj (k) = ker P¯j (k) for 1 ≤ j ≤ n. Now choose η ∈ ker P1n+1 (k) and due to η ∈ ker P1n (k) = ker P¯1n (k) we obtain 0=
n
P¯j (k)η + P¯n+1 (k)η =
j=1
n
Pj (k)η + P¯n+1 (k)η = P¯n+1 (k)η.
j=1
This means η ∈ ker P¯n+1 (k) and consequently ker Pn+1 (k) ⊆ ker P¯n+1 (k). With a symmetric argument one shows ker P¯n+1 (k) ⊆ ker Pn+1 (k). (b) This follows analogously to (a) using (3.4l) in Remark 3.4.18. (c) Due to (a) and (b) the projectors Pn , P¯n have the same kernel and image; thus, they agree. Corollary 3.4.23. Suppose that (L0 ) fulfills (3.1a), (3.1b) and admits an exponential N -splitting (3.4h). Provided also (P¯n )N n=1 is a regular invariant splitting for (L0 ), then the following holds for all 1 ≤ n < N : (a) If J is unbounded above and ker P1n (κ) = ker P¯1n (κ) for one κ ∈ J, then S(A, B; P ) = S(A, B; P¯ ),
lim P¯1n (k) − P1n (k) = 0.
k→∞
(a) If J is unbounded below and im P1n (κ) = im P¯1n (κ) for one κ ∈ J, then S(A, B; P ) = S(A, B; P¯ ),
lim P¯1n (k) − P1n (k) = 0.
k→−∞
Proof. Let k ∈ J and 1 ≤ n < N . By assumption, we can apply Proposition 3.1.4 and obtain that the evolution operator Φ(k, l) of (L0 ) exists for all k, l ∈ J. (a) Our assumption on the kernels implies that we have P¯1n (κ)P1n (κ) = P1n (κ),
P1n (κ)P¯1n (κ) = P¯1n (κ),
(3.4m)
¯ n1 ) which, in turn, implies (and similarly for the complementary projectors Qn1 , Q P¯1n (κ) − P1n (κ) = [I − P1n (κ)] P¯1n (κ) = P¯1n (κ)P1n (κ) − P1n (κ)P¯1n (κ)P1n (κ) = [I − P1n (κ)] P¯1n (κ) − P1n (κ) P1n (κ). We define P1n (k) := Φ(k, κ)P1n (κ)Φ(κ, k), k ∈ J, and from this we can deduce the desired limit relation
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3 Linear Difference Equations
n P¯1 (k) − P1n (k) = Φ(k, κ) P¯1n (κ) − P1n (κ) Φ(κ, k) = Φ(k, κ) I − P1n (κ) P¯1n (κ) − P1n (κ) P1n (κ)Φ(κ, k) ≤ Kn+ Kn− e an (k, κ) P¯1n (κ) − P1n (κ) −−−−→ 0
(3.4g)
bn
k→∞
¯ n are bounded. It remains to (cf. Lemma A.1.3). Moreover, the sequences P¯1n , Q 1 show the dichotomy estimates (3.4g) for the splitting (P¯n )N n=1 . They follow from n (3.4g) Φ(k, l)Q ¯ n1 (l) (3.4m) ¯ 1 (k) ¯ n1 (l) ≤ Kn+ ean (k, l) sup Q = Φ(k, l)Qn1 (l)Q j∈J
for all l ≤ k and a dual estimate for k ≤ l. (b) The assertion (b) can be shown analogously.
Linear 2-parameter semigroups have finite-dimensional unstable vector bundles, provided a compactness property is given. More detailed, given respective measures of noncompactness χk on Xk , k ∈ J, one has Proposition 3.4.24. Suppose (L0 ) has an exponential N -splitting (3.4h) on an interval J unbounded below. ˆ (a) If dim P1i < ∞ for some 1 ≤ i < N , then (L0 ) is B-contracting with Bˆ = B ⊆ X : lim eai (k, k − n)χk−n (B(k − n)) = 0 for all k ∈ J . n→∞
ˆ with (b) Conversely, if (L0 ) is B-contracting Bˆ =
⎪ ⎪ ⎪ ⎪ B ⊆ X : sup ⎪ ⎪B(k)⎪ ⎪<∞
for all κ ∈ J
k≤κ
and 1 ≤ bi for some 1 ≤ i < N , then dim P1j < ∞ for all 1 ≤ j ≤ i. Proof. (a) Let k ∈ J and B ∈ Bˆ be given. Due to dim P1i < ∞ one has the relation χk (Φ(k, k − n)P1i (k − n)B(k − n)) = 0 for all n ≥ 0 and consequently χk (Φ(k, k − n)B(k − n)) (c3 )
≤ χk (Φ(k, k − n)P1i (k − n)B(k − n)) + χk (Φ(k, k − n)Qi1 (k − n)B(k − n)) = χk (Φ(k, k − n)Qi1 (k − n)B(k − n))
(3.4g)
≤ Ki+ eai (k, k − n)χk−n (B(k − n)) −−−−→ 0. n→∞
ˆ Therefore, (L0 ) is B-contracting.
3.4 Dichotomies and Splittings
143
¯ 1 (0, P1i (κ)) be the closed unit ball in P1i (κ). Since (b) Let κ ∈ J and Sκ := B i i : P1 (κ) → P1 (k) is a toplinear isomorphism, we can define the nonau-
Φ− (k, κ) P1i
(k, κ)Sκ for k ≤ κ; thanks to tonomous set B via its fibers B(k) := Φ− Pi 1
⎪ ⎪ (3.4g) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ − ⎪ ⎪ ⎪Sκ⎪ ⎪ (k, κ)S ≤ Ki− ⎪ Φ ⎪ ⎪B(k)⎪ ⎪=⎪ ⎪ κ i ⎪ ⎪ P 1
for all k ≤ κ
ˆ Thus, because (L0 ) is B-contracting, ˆ it satisfies B ∈ B. we deduce (κ − n, κ)Sκ ) = χκ (Φ(κ, κ − n)B(κ − n)) −−−−→ 0 χκ (Sκ ) = χκ (Φ(κ, κ − n)Φ− Pi 1
n→∞
and so χκ (Sκ ) = 0. This yields that Sκ ⊆ P1i (κ) is compact and by [295, p. 39, Corollary 3.15] one has dim P1i < ∞. The claim results from Remark 3.4.18(1). Autonomous and Periodic Equations We close this section with results on autonomous and periodic equations. On basis of the following result we establish the convention that periodic equations admitting an exponential splitting have periodic projectors. Corollary 3.4.25. Let p ∈ N. Suppose that (L0 ) is p-periodic and satisfies (3.1a), (3.1b). If (L0 ) has an exponential N -splitting (3.4h) with An := sup max {ean (k + p, k), ean (k, k + p)} < ∞, k∈Z
Bn := sup max {ebn (k + p, k), ebn (k, k + p)} < ∞, k∈Z
then also Pn are p-periodic for all 1 ≤ n < N . Proof. Let 1 ≤ n < N and since (L0 ) is periodic, we can assume J = Z. Moreover, by Proposition 3.1.4 the evolution operator Φ(k, l) exists for all k, l ∈ Z. Defining the projector P¯n (k) := Pn (k + p) we obtain Φ(k, l) I − P¯1n (l) (3.2a) = Φ(k + p, l + p)Qn1 (l + p) (3.4g)
≤ Kn+ ean (k + p, l + p)≤Kn+ A2n ean (k, l) for all l ≤ k
and analogously Φ(k, l)P¯1n (l) ≤ Kn− Bn2 ebn (k, l) for all k ≤ l. This implies an exponential splitting w.r.t. the projectors P¯n and Corollary 3.4.22(c) yields the claim.
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3 Linear Difference Equations
Fig. 3.5 Spectral decomposition required to define the adapted norms in Lemma 3.4.26
σ− σ− σ+
α
β
In Fig. 3.5 we see the spectral decomposition required in the next lemma. Lemma 3.4.26 (adapted norms). If T ∈ L(X) is an operator on a Banach space X allowing a spectral splitting into sets σ− , σ+ ⊆ C with ˙ −, σ(T ) = σ+ ∪σ
α := max |λ| < min |λ| =: β, λ∈σ+
λ∈σ−
then the following holds: (a) X = X+ ⊕ X− with closed subspaces X+ , X− ⊆ X satisfying T X+ ⊆ X+ = im P+ , σ(T |X+ ) = σ+ ,
T X− ⊆ X− = im P− , σ(T |X− ) = σ− ,
where P− , P+ are the spectral projectors associated with σ− , σ+ , resp.. (b) For every ε > 0 there exists a norm |·|ε , equivalent to the given one, with |T P+ x|ε ≤ (α + ε) |P+ x|ε , (β − ε) |P− x|ε ≤ |T P− x|ε
for all x ∈ X
and provided both spectral sets σ− , σ+ are nonempty, |P+ |ε = |P− |ε = 1,
|x|ε = |P+ x|ε + |P− x|ε
for all x ∈ X.
Proof. See [227, p. 6, Technical Lemma 1]. Lemma 3.4.27. If T ∈ L(X) is an operator on a Banach space X and α := max |λ| , λ∈σ(T )
β := min |λ| , λ∈σ(T )
then the following holds: (a) If the spectral points of T with |λ| = α are semisimple isolated eigenvalues, then there exists a K+ ≥ 1 such that T k ≤ K+ αk for all k ∈ Z+ 0.
3.4 Dichotomies and Splittings
145
(b) If the spectral points of T with |λ| = β > 0 are semisimple isolated eigenvalues, then there exists a K− ≥ 1 such that T k ≤ K− β k for all k ∈ Z− 0. Proof. (a) By assumption the operator T ∈ L(X) admits a spectral decomposi˙ + , where σα consists of the semisimple eigenvalues of T with tion σ(T ) = σα ∪σ modulus α, and Pα , P+ are the corresponding spectral projectors. Here, im Pα is finite-dimensional and invariant under T . Therefore, [458, p. 131, Satz A.3.5] can be applied to T Pα ∈ L(im Pα ) yielding the existence of a constant Kα ≥ 1 such that T k Pα ≤ Kα αk for all k ≥ 0. On the other hand, ρ(T P+ ) < α and we can make use of Lemma 3.4.26 in order to infer T k P+ ≤ K+ αk . Combining these two estimates, we arrive at k k T = T (Pα + P+ ) ≤ (Kα + K+ )αk for all k ∈ Z+ . 0 (b) By assumption the operator T is invertible and the claim follows, if one applies (a) to the inverse T −1 . Theorem 3.4.28 (splitting for autonomous equations). Let X, Y be Banach spaces and suppose that A, B ∈ Hom(X, Y ) fulfill ker B = {0}, im A ⊆ im B with B −1 A ∈ L(X). If there exists a finite N > 1 and reals α0 ≤ ∞, 0 ≤ βN ,
αn βn βn+1 ≤ αn
for all 1 ≤ n < N, for all 0 ≤ n < N,
(3.4n)
N such that σ(A, B) = ˙ i=1 σi with compact spectral sets σi = ∅ and max |λ| < αi < βi < min |λ|
λ∈σi+1
λ∈σi
for all 1 ≤ i < N,
then the following holds: (a) The autonomous equation
Bx = Ax,
N −1
(3.4o)
has an exponential N -splitting S(A, B; P ) = i=1 (βi+1 , αi ) on Z. (b) P consists of the (constant) spectral projectors associated to the sets σi . (c) If the spectral points μ ∈ σi+1 with |μ| = maxλ∈σi+1 |λ| are isolated semisimple eigenvalues, one can choose αi = maxλ∈σi+1 |λ|. Dually, if the spectral points μ ∈ σi with |μ| = minλ∈σi |λ| > 0 are isolated semisimple eigenvalues, one can choose βi = minλ∈σi |λ|. Remark 3.4.29. (1) If ρ(A, B) < α0 , then autonomous equations (3.4o) are forward α0 -stable, i.e., admit a 0-splitting on Z. Moreover, if the spectral points μ with modulus |μ| = ρ(A, B) are semisimple eigenvalues, one can choose α0 = ρ(A, B). Similarly, if there exists a β1 > 0 with β1 < minλ∈σ(A,B) |λ|, then (3.4o) is backward β1 -stable, i.e., has an exponential 1-splitting on Z. In fact, one can choose β1 = minλ∈σ(A,B) |λ|, provided the spectral points μ with modulus μ are semisimple eigenvalues.
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3 Linear Difference Equations
(2) An autonomous equation (3.4o) is hyperbolic, if and only if the spectrum σ(A, B) is disjoint from the unit circle S1 in C. Particularly, a hyperbolic autonomous equation (3.4o) with σ(A, B) ⊆ B1 (0) is exponentially stable. (3) In case the above assumptions hold with N = 2, then the following proof shows that there exists an equivalent renorming of X with −1 k−l (B A) Q1 ≤ eα1 (k, l), (B −1 A)l−k P1 ≤ eβ1 (l, k) for all l ≤ k, i.e., the constants K1+ , K1− can be chosen to be 1. N Proof. We have σ(A, B) = σ(B −1 A) = ˙ i=1 σi and define the spectral projectors 0 Pn ∈ L(X) n as in Example 3.3.15. We furthermore introduce the projectors P1 := 0, P1n := i=1 Pi and complementary projectors Qn1 := IX − P1n . Introducing the n n sets σm := ˙ i=m σi yields that each disjoint union N ˙ n+1 σ(A, B) = σ1n ∪σ
for all 1 ≤ n < N
is a spectral decomposition for T = B −1 A as required in Lemma 3.4.26, where Qn1 , P1n are the respective spectral projectors. Since the compact disjoint sets |σ(B, A)| N −1 and i=1 [αi , βi ] have positive distance, we obtain norms |·|n , equivalent to the given one on X, such that |T Qn1 x|n ≤ αn |Qn1 x|n ,
βn |P1n x|n ≤ |T P1n x|n
for all x ∈ X, 1 ≤ i < N.
Due to 0 ∈ σ1n = σ(T P1n ), the operator T P1n is invertible and mathematical induction implies k n T Q1 x ≤ αkn |Qn1 x| , T −k P1n x ≤ βn−k |P1n x| n n n n
for all x ∈ X, 0 ≤ k.
Since the norms |·|n are equivalent on X, this guarantees that (3.4o) has the claimed exponential N -splitting. We have established the assertions (a) and (b). Concerning (c), we apply Lemma 3.4.27 to the projectors P1n or Qn1 . Theorem 3.4.30 (splitting for compact equations). Let X be an infinite-dimensional Hilbert space and suppose A, B ∈ Hom(X, Y ) with ker B = {0}, im A ⊆ im B. If B −1 A ∈ L(X) is compact and normal with σ(B −1 A) = {0} ∪ {νn }n∈N and nonzero eigenvalues νn arranged in decreasing order of magnitude, then: (a) For positive real sequences (αn−1 )n∈N , (βn )n∈N with αn < βn ≤ αn−1 ,
|νn | ∈ (βn , αn−1 ) for all n ∈ N, ∞ one has an exponential N -splitting S(A, B; P ) = n=1 (βn+1 , αn ) on Z for (3.4o) with constants Kn± = 1.
3.4 Dichotomies and Splittings
147
(b) The splitting P consists of the (constant) orthogonal spectral projectors Pn associated to the singletons {νn }. Proof. By assumptions, we are in the framework of Example 3.3.16 and can abbreviate T := B −1 A. Suppose Pn is the spectral projector associated to the spectral set {νn }, n ∈ N, and P0 be the orthogonal projection onto ker T . We obtain a bounded invariant splitting (Pn )∞ n=1 of X, whose projectors are pairwise orthogonal. Thus, / T P1n x, T P1n x
=
n j=1
νj Pj x,
/
= νn ν¯n
n
0
j=1 n i=1
Pi x
≥ νn ν¯n
νj Pj x
n
n
Pj x, Pj x
j=1
0
for all n ∈ N
, Pj x
j=1
and consequently it is T P1n x ≥ |νn | P1n x. On the other hand, with the spectral radius ρ it is T Qn1 = ρ(T Qn1 ) = νn+1 for all n ∈ N and we arrive at the estimates k n T Q1 ≤ |νn+1 |k ,
[T P1n ]−k ≤ |νn |−k
for all k ∈ Z+ 0
and this was our claim.
Finally, Floquet theory shows that periodic equations are linearly conjugated to autonomous problems as discussed above. However, we avoid the corresponding assumption (cf. Theorem 3.2.6) that operator roots exist and follow a direct approach: Theorem 3.4.31 (splitting for periodic equations). Let p ∈ N, X , Y consist of Banach spaces and suppose that Ak ∈ Hom(Xk , Yk+1 ), Bk ∈ Hom(Xk , Yk ) are −1 p-periodic, ker Bk+1 = {0}, im Ak ⊆ im Bk+1 with Bk+1 Ak ∈ GL(Xk , Xk+1 ). If there exists a finite N > 0 and reals as in (3.4n) satisfying σ(Φκ ) =
˙ N n=1
σn ,
max |λ| < αn < βn < min |λ|
λ∈σn+1
λ∈σn
for all 1 ≤ n < N
with compact spectral sets σn = ∅, then the following holds: (a) The p-periodic equation
Bk+1 x = Ak x (3.4p) N −1 √ has an exponential N -splitting S(A, B; P ) = n=0 ( p βn+1 , p αn ). (b) The splitting P consists of the p-periodic projectors. (c) If the spectral points μ ∈ σn+1 with |μ| = maxλ∈σn+1 |λ| are isolated semisimple eigenvalues, one can choose αi = maxλ∈σn+1 |λ|. Dually, if the spectral points μ ∈ σn with |μ| = minλ∈σn |λ| > 0 are isolated semisimple eigenvalues, one can choose βn = minλ∈σn |λ|. Remark 3.4.32. (1) If the spectral radius satisfies ρ(Φκ ) < α0 , then periodic equa√ tions (3.4p) are forward p α0 -stable, i.e., admit a 0-splitting on Z. Furthermore, if
148
3 Linear Difference Equations
the spectral points μ with modulus |μ| = ρ(Φκ ) are semisimple eigenvalues, one can choose α0 = ρ(Φκ ). Similarly, if there exists a β1 > 0 with β1 < minλ∈σ(Φκ ) |λ|, √ then (3.4p) is backward p β1 -stable, i.e., has an exponential 1-splitting on Z. In fact, one can choose β1 = minλ∈σ(Φκ ) |λ|, provided the spectral points μ with modulus μ are semisimple eigenvalues. (2) A p-periodic equation (3.4p) is hyperbolic, if and only if the spectrum σ(Φκ ) is disjoint from the unit circle S1 in C. In particular, it is exponentially stable, provided σ(Φκ ) ⊆ B1 (0). Proof. Let κ ∈ Z and Φκ ∈ GL(Xκ ) be the associated period map for (3.4p). Initially, we remark that one can apply Theorem 3.4.28 to the explicit autonomous equation x = Φκ x in the Banach space Xκ , which yields the estimates k−l n Φκ Q1 (κ) ≤ Kn+ αk−l n ,
l−k n Φκ P1 (κ) ≤ Kn− βnl−k
for all l ≤ k
and 1 ≤ n < N . Here, P1n (κ) and Qn1 (κ) are the complementary spectral projectors corresponding to the spectral decomposition via the annulus with radii αn , βn . Moreover, since our assumptions guarantee that the evolution operator Φ(k, l) for (3.4p) exists for all k, l ∈ Z, we define P1n (k) := Φ(k, κ)P1n (κ)Φ(κ, k) for all κ ≤ k < κ + p and extend P1n periodically on Z. Now let l ≤ k be given. Since Φκ , κ ∈ Z, is p-periodic (see Proposition 1.4.4(a)), w.l.o.g., we can assume that l ≤ κ < l + p. We define ˆ n± := Kn± K
max
0≤|k−l|
{Φ(k, l) , Φ(k, l)Qn1 (l) , Φ(k, l)P1n (l)}
and remark that these constants exist by the periodicity of (3.4p) and P1n . With this we easily deduce the desired estimates Φ(k, l)Qn1 (l) ≤ Φ(k, κ + np) Φ(κ + np, κ)Qn1 (κ) Φ(κ, l)Qn1 (l) ≤ Φ(k, κ + np) Φnκ Qn1 (κ) Φ(κ, l)Qn1 (l) p−1 −l √ k−l ˆ + αn ≤ K ˆ + max ≤K αn p αn for all l ≤ k n n n l=0
and Φ(l, k)P1n (k) ≤ Φ(l, κ + np) Φ(κ + np, κ)P1n (κ) Φ(κ, k)P1n (k) ≤ Φ(l, κ + np) Φnκ P1n (κ) Φ(κ, k)P1n (k) l−k p−1 −l ˆ n− βnn ≤ K ˆ n− max ≤K βn p βn for all l ≤ k, l=0
which yields the assertion (b). Finally, our claim (c) follows via an application of Lemma 3.4.27 to x = Φκ x. Theorem 3.4.33. Let J = Z, κ ∈ Z and suppose (L0 ) is a p-periodic equation satisfying (3.1a) and (3.1b). With given statements
3.4 Dichotomies and Splittings
149
(a) (L0 ) has an exponential dichotomy with constant growth rates α, β > 0, (b) The unique γ-bounded solution of (L0 ) is the trivial one, where γ ∈ (α, β), (c) |σp (Φκ )| ∩ (αp , β p ) = ∅ one has the implication (a) ⇒ (b) ⇒ (c) and in case dim X < ∞ the above statements are equivalent. Proof. Above all, we infer from Theorem 3.1.9 that the evolution operator Φ(k, κ) of (L0 ) exists for all k, κ ∈ Z. We consequently obtain the implications: (a) ⇒ (b) follows immediately from Corollary 3.4.21(c). (b)⇒ (c) We proceed indirectly and assume there exists a point λ ∈ σp (Φκ ) with p |λ| ∈ (α, β). For the corresponding eigenvector ξ ∈ Xκ one obtains n
Φ(κ + np, κ)ξ = Φnκ ξ = |λ| ξ
for all n ∈ Z
and thus as in the proof of Theorem 3.4.31 there exist K1 , K2 ≥ 0 such that n
Φ(k, κ)ξ ≤ Φ(k, κ + np) Φ(κ + np, κ)ξ ≤ K1 |λ| ξ ≤ K2
k−κ p |λ| ξ
for all κ, k ∈ Z. Therefore, Φ(·, κ)ξ is a nontrivial complete and p |λ|-bounded solution to (L0 ). This, however, contradicts (b). For the claimed equivalence it remains to show (c) ⇒ (a). By dim X < ∞ we have σ(Φκ ) = σp (Φκ ) and so the claim results from Theorem 3.4.31. We conclude this section with the following word of caution. In many ways, an exponential dichotomy is the appropriate hyperbolicity notion for nonautonomous equations. This is particularly exemplified by the fact that such equations −1 form an open set in the space B of (L0 ) with bounded coefficient operators Bk+1 Ak ∞ equipped with the usual -topology. This will be a consequence of the main result of Sect. 3.6. Yet, differing from the autonomous or periodic situation, in general an exponential dichotomy is not a generic property, i.e., dichotomic equations form an open but not a dense subset of the above space B. The lacking denseness can be seen as follows: Example 3.4.34. Suppose X = N × R and consider a scalar equation x = a(k)
βk+1 x, βk
(3.4q) √
where a : N → R denotes a sequence satisfying e− 2 ≤ |a(k)| ≤ ρ for all k ∈ N, ln k ρ < e−1 and βk = ek sin is given as in Example 3.4.1. In particular, one has the −√2 √2 βk+1 for all k ∈ N. The evolution operator for (3.4q) is ,e inclusion βk ∈ e given by Φ(k, l) = ea (k, l) ββkl , l ≤ k, this yields
150
3 Linear Difference Equations
|Φ(k, l)| ≤
ρ k βk (ρesin ln k )k (ρe)k = ≤ − −−−→ 0 for all l ∈ N ρ l βl ρ l βl ρl βl k→∞
and accordingly (3.4q) is asymptotically stable by Proposition 3.1.23(e). In order to show that (3.4q) is not uniformly asymptotically stable, assume the contrary. Then, by Proposition 3.1.23(g) there exist α ∈ (0, 1), C ≥ 1 such that |Φ(k, l)| αl−k ≤ C
for all l ≤ k.
√ √ First of all, we choose γ ∈ (1, 2) such that eγ− 2 > α holds true. From elementary trigonometry one easily deduces the implication
t ∈ [ξn− , ξn+ ] for one n ∈ N0
⇒
sin ln t + cos ln t ≥ γ,
γ √ + 2πn . where ξn− := exp arcsin √γ2 − π4 + 2πn , ξn+ := exp 3π 4 − arcsin 2 Note that these interval bounds satisfy both the limit relations limn→∞ ξn− = ∞, as well as limn→∞ (ξn+ − ξn− ) = ∞. Hence,√ for sufficiently large n ∈ N0 one finds γ− 2 k−l > C. For such integers we obtain positive integers l, k ∈ [ξn− , ξn+ ] with e α
βk = ek sin ln k−l sin ln l = exp βl
k
sin ln s + cos ln s ds
≥ eγ(k−l)
l
and finally arrive at the contradiction C ≥ |Φ(k, l)| α
l−k
βk = e αa (k, l) ≥ βl
√
e− α
2
√ 2 k−l
!k−l β eγ− k ≥ βl α
> C.
Therefore, no equation in the class (3.4q) is uniformly asymptotically stable. For instance, the equation !β √ − 2 −1 k+1 x = e 2+e x (3.4r) βk clearly is in the class (3.4q) with a(k) ≡ perturbation of (3.4r) we consider x = with a(k) =
e−
√
2
e−
+e−1 2
√
!
2
+e−1 2
!β
k+1
βk
e−
√
2
+e−1 2
on N. As linear homogeneous
x + b(k)x = a(k)
βk+1 x βk
k + ββk+1 b(k) and for sufficiently small supremum norm of
√
b : N → R, we can enforce e− 2 ≤ |a(k)| ≤ ρ for all k ∈ N. Thus, no system in a whole neighborhood of (3.4r) is uniformly asymptotically stable.
3.5 Admissibility
151
3.5 Admissibility In this section, we are interested in inhomogeneous perturbations Bk+1 x = Ak x + gk ,
(Lg )
of a linear homogenous difference equation (L0 ), as well as in semilinear perturbations Bk+1 x = Ak x + fk (x, x ). Thereby, let the extended state space X consist of normed linear spaces, J ⊆ I be a discrete interval and suppose gk ∈ im Bk+1 , k ∈ J , throughout. We will provide criteria for the existence of exponentially bounded solutions. Let us begin to investigate the linear problem (Lg ) and we suppose Hypothesis 3.5.1. Let N > 0, suppose (L0 ) fulfills (3.1a) and admits a strongly regular exponential N -splitting on J in form of S(A, B; P ) =
N −1
(bi+1 , ai );
i=0
the associated Green’s functions are abbreviated by Gi := GP1i , 1 ≤ i < N + 1. Remark 3.5.2. Thanks to the strongly regular exponential splitting for (L0 ) and Lemma 3.3.6, one has the following relations, where 1 ≤ i < N , (3.3g)
Qi1 (k)Φ(k, κ) = Φ(k, κ)Qi1 (κ) for all κ ≤ k, (3.3g)
P1i (k)Φ− (k, κ) = Φ− (k, κ)P1i (κ) for all k ≤ κ. Pi Pi 1
(3.5a)
1
Theorem 3.5.3. Let κ ∈ J and suppose Hypothesis 3.5.1 is fulfilled. If c : J → (0, ∞) satisfies c ∈ (ai , bi ) for one 1 ≤ i < N , then the following holds true: (a) Provided J is unbounded above and g ∈ X+ κ,c,B , then for every ξ ∈ Xκ there + exists a unique forward solution φ : Zκ → X of (Lg ), satisfying φ ∈ X+ κ,c and Qi1 (κ)φ(κ) = Qi1 (κ)ξ. It is given by the Lyapunov–Perron sums φ(k) := Φ(k, κ)Qi1 (κ)ξ +
∞
−1 Gi (k, n + 1)Bn+1 gn
for all k ∈ Z+ κ
n=κ
and fulfills the estimate + + φκ,c ≤ Ki+ Qi1 (κ)ξ Xκ + Ci (c) gκ,c,B .
(3.5b)
(b) Provided J is unbounded below and g ∈ X− κ,c,B , then for every ξ ∈ Xκ there − exists a unique backward solution φ : Zκ → X of (Lg ), satisfying φ ∈ X+ κ,c and P1n (κ)φ(κ) = P1n (κ)ξ. It is given by the Lyapunov–Perron sums
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3 Linear Difference Equations κ−1
φ(k) := Φ− (k, κ)P1i (κ)ξ + Pi 1
−1 Gi (k, n + 1)Bn+1 gn
for all k ∈ Z− κ
n=−∞
and fulfills the backward estimate − − φκ,c ≤ Ki− P1i (κ)ξ Xκ + Ci (c) gκ,c,B with Ci (c) := max
Ki+ Ki− c−ai , bi −c
(3.5c)
.
The estimates on the Lyapunov–Perron sums in the following proof have prototypical character for our subsequent considerations in Chaps. 4 and 5. Proof. Let κ ∈ J. Since both assertions of Lemma 3.5.3 can be shown in a very similar fashion, we present only the proof of (b). Concerning (a) we remark that Lemma 3.3.6 ensures the existence of the forward evolution operator Φ for (L0 ). For given ξ ∈ Xκ it is not difficult to verify that φ : Z− κ → X is a solution of the linear inhomogeneous equation (Lg ). Indeed, we have the identity (3.1e)
−1 Ak Φ− (k, κ)P1i (κ)ξ + φ (k) ≡ Bk+1 Pi 1
κ−1
−1 Gi (k + 1, n + 1)Bn+1 gn
n=−∞
κ−1
−1 −1 −1 i Bk+1 ≡ Bk+1 Ak Φ− (k, κ)P (κ)ξ+ Ak Gi (k, n + 1) + δk,n Bn+1 gn i 1 P
(3.3k)
1
≡
−1 Bk+1 Ak φ(k)
n=−∞
+
−1 Bk+1 gk
on Z− κ−1 .
Now we show that φ is c− -bounded. Using Lemma A.1.5(b) we have i (3.5a) + P (k)φ(k) ≤ ΦP i (k, κ)P1i (k) P1i (κ)ξ 1 1
+
κ−1
+ −1 gn ΦP i (k, n + 1)P1i (n + 1) P1i (n + 1)Bn+1 1
n=k
≤ Ki− ebi (k, κ) P1i (κ)ξ
(3.4g)
+Ki−
κ−1
ebi (k, n + 1)ec (n, κ) g− κ,c,B
n=k
ec (k, κ) − gκ,c,B ≤ Ki− ebi (k, κ) P1i (κ)ξ + Ki− !bi − c"
(A.1e)
and accordingly using Lemma A.1.5(a) one gets
3.5 Admissibility
153
i (3.5a) k−1 + −1 Q1 (k)φ(k) ≤ gn ΦP i (k, n + 1)Qi1 (n + 1) Qi1 (n + 1)Bn+1 1
n=−∞ (3.4g)
≤ Ki+
k−1
eai (k, n + 1)ec (n, κ) g− κ,c,B
n=−∞ (A.1d)
≤ Ki+
ec (k, κ) g− κ,c,B , !c − ai "
which finally leads to ec (k, κ) φ(k)Xk ≤
Ki− P1i (κ)ξ Xκ
+ max
Ki− Ki+ , !c − ai " !bi − c"
−
gκ,c,B
− for all k ∈ Z− κ . This implies φ ∈ Xκ,d , as well as the desired estimate (3.5c). To infer uniqueness of φ, let φ¯ ∈ X− κ,c denote another solution of (Lg ) satisi i ¯ fying P1 (κ)φ(κ) = P1 (κ)ξ. Due to Lemma 3.3.25(a) the difference φ − φ¯ is ¯ c− -bounded and a solution of (L0 ) with P1i (κ) φ(κ) − φ(κ) = 0. However, by Corollary 3.4.21(b) the trivial solution is the only c− -bounded solution of (L0 ) in Qi1 , i.e., φ¯ = φ.
Theorem 3.5.4. Suppose Hypothesis 3.5.1 is fulfilled with J = Z. If c, d : Z → (0, ∞) satisfy c, d ∈ (ai , bi ) for one 1 ≤ i < N , then for each inhomogeneity g ∈ Xc,d,B there exists a unique complete solution φ∗ ∈ Xc,d of (Lg ). It is given by the Lyapunov–Perron sum −1 φ∗ (k) := Gi (k, n + 1)Bn+1 gn for all k ∈ Z n∈Z
and fulfills the estimate φ∗ κ,c,d ≤ Di (c, d) gκ,c,d,B with Di (c, d) := max
Ki+ c−ai
+
Ki+ Ki− d−ai , bi −d
+
for all κ ∈ Z Ki− bi −c
(3.5d)
.
Remark 3.5.5. For a p-periodic equation (L0 ) also the solution ϕ : Z → X is p-periodic. This easily follows from Lemma 3.3.6(d). Proof. First of all, we know that φ∗ : Z → X is a solution to (Lg ), because from Lemma 3.3.6 we obtain the identity −1 φ∗ (k) ≡ Gi (k + 1, n + 1)Bn+1 gn n∈Z (3.3k)
≡
−1 −1 Bk+1 Ak Gi (k, n + 1) + δk,n Bn+1 gn
n∈Z −1 −1 ≡ Bk+1 Ak φ∗ (k) + Bk+1 gk
on Z.
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3 Linear Difference Equations
Given κ ∈ Z, we proceed as in the proof of the above Proposition 3.5.1. From the dichotomy estimate (3.4g) and Lemma A.1.5(a) we deduce i Q1 (k)φ∗ (k) ec (κ, k) ≤ i Q1 (k)φ∗ (k) ed (κ, k) ≤
Ki+ Ki+ + !d − ai " !c − ai "
Ki+ gκ,c,d,B !d − ai "
gκ,c,d,B
for all k ∈ Z+ κ,
for all k ∈ Z− κ
and dually, (3.4g) and Lemma A.1.5(b) imply i P1 (k)φ∗ (k) ec (κ, k) ≤
Ki− gκ,c,d,B for all k ∈ Z+ κ, !bi − c" i Ki− Ki− P1 (k)φ∗ (k) ed (κ, k) ≤ + gκ,c,d,B for all k ∈ Z− κ. !bi − d" !bi − c" If we combine the four above estimates, then the sequence φ∗ : Z → X is welldefined, c, d-bounded and fulfills the claimed estimate (3.5d). Moreover, φ∗ is the unique c, d-bounded solution of (Lg ). To prove this, let φ¯∗ ∈ Xc,d denote another solution of (Lg ). We see from Lemma 3.3.25(b) that also the difference φ∗ − φ¯∗ is c, d-bounded and a solution of (L0 ). Yet, referring to Corollary 3.4.21(c) the trivial solution is the only c, d-bounded solution of (Lg ), i.e., φ¯∗ = φ∗ . The following example shows the importance of a (strongly) regular splitting for (L0 ) in the above admissibility results: Example 3.5.6. We return to the explicit equation (3.4f) from Example 3.4.8, which admits an exponential forward dichotomy on Z. Nevertheless, we have im Ak =
R × {0} × R
for k ≤ 0,
3
for k > 0, R ⎧ ⎪ {0} × {0} × R ⎪ ⎪ ⎛ !k−1 ⎞ ⎛ ⎞ ⎪ ⎪ ⎨ α 0 b ⎜ β ⎜ ⎟ ⎟ im P (k) = ⎜ ⎜ ⎟ ⎟ ⎪ R + R ⎪ 1 ⎝ ⎝0⎠ ⎠ ⎪ ⎪ ⎪ ⎩ 1 0
for k ≤ 0, for k > 0,
consequently, the condition (3.4i) is violated for k ≥ 0 and we cannot infer an exponential dichotomy from Proposition 3.4.14(a). Now we consider a linear inhomogeneous perturbation x = Ak x + gk of (L0 ), where ⎛
0
⎞
gk := ⎝δk,0 ⎠ 0
for all k ∈ Z
3.5 Admissibility
155
and compute its general forward solution ⎛ k ⎞ α ξ1 3 ϕ(k; 0, ξ) = ⎝β k−1 ⎠ for all k ∈ Z+ 1, ξ ∈R . k β ξ3 So we observe that, independently from the initial vector ξ ∈ R3 , no solution of (3.4f) is γ + - or γ-bounded for γ ∈ (α, β). Thus, the perturbation results from Theorem 3.5.3 and Theorem 3.5.4 cannot hold for merely forward dichotomous equations. We continue our investigations with semilinear equations Bk+1 x = Ak x + fk (x, x ) + gk
(Sg )
under the following assumptions: Hypothesis 3.5.7. Let X consist of Banach spaces. Suppose fk : Xk × Xk+1 → Yk+1 with fk (Xk , Xk+1 ) ⊆ im Bk+1 for all k ∈ I and (i) fk (0, 0) ≡ 0 on I, −1 (ii) One has the Lipschitz estimates Lj := lipj supk∈I Bk+1 fk < ∞ for j = 1, 2. It is our upcoming goal to show that the assertions of Theorems 3.5.3 and 3.5.4 carry over to (Sg ) for sufficiently small nonlinear perturbations fk . Theorem 3.5.8. Let κ ∈ J and suppose Hypotheses 3.5.1 and 3.5.7 are fulfilled. If a sequence c : J → (0, ∞) satisfies c ∈ (ai , bi ),
(L1 + c L2 )Ci (c) < 1
for one 1 ≤ i < N,
(3.5e)
then the following holds true: (a) Provided J is unbounded above and g ∈ X+ κ,c,B , then for every ξ ∈ Xκ there → X of (Sg ), satisfying φ ∈ X+ exists a unique forward solution φ : Z+ κ κ,c and i i Q1 (κ)φ(κ) = Q1 (κ)ξ. It fulfills the estimate +
φκ,c ≤
1 1−(L1 + cL2 )Ci (c)
! + Ki+ Qi1 (κ)ξ Xκ + Ci (c) gκ,c,B .
(3.5f)
(b) Provided J is unbounded below and g ∈ X− κ,c,B , then for every ξ ∈ Xκ there + exists a unique backward solution φ : Z− κ → X of (Sg ), satisfying φ ∈ Xκ,c n n and P1 (κ)φ(κ) = P1 (κ)ξ. It fulfills the backward estimate −
φκ,c ≤
1 1−(L1 + cL2 )Ci (c)
! − Ki− P1i (κ)ξ Xκ + Ci (c) gκ,c,B ,
where the constant Ci (c) is defined in Theorem 3.5.3.
(3.5g)
156
3 Linear Difference Equations
Remark 3.5.9. (1) Equation (S0 ) has the trivial solution, and if (L0 ) has a • •
0-splitting, then the trivial solution is forward c-stable with a0 c, 1-splitting, then the trivial solution is backward c-stable with c b1 .
In particular, for a 0-splitting where one can choose c 1 so that a0 + K1+ (L1 + c L2 ) c, the trivial solution of (S0 ) is globally exponentially stable. (2) As a consequence of Theorem 3.5.8(a) one obtains the principle of linearized asymptotic stability: If (L0 ) is uniformly asymptotically stable and lim sup lip fk |Bρ (0,Xk )×Bρ (0,Xk+1 ) = 0,
ρ0 k∈J
then the trivial solution of (S0 ) is exponentially stable. For the proof, one applies Theorem 3.5.8(a) to Bk+1 x = Ak x + fkρ (x, x ), where for sufficiently small ρ > 0 the nonlinearity fkρ is constructed as in Proposition C.2.5. (3) It can be seen from the proof of Theorem 3.5.3 that the norm Qi1 (κ)ξ oc curring in (3.5b), (3.5f) and the norm P1i (κ)ξ from (3.5c), (3.5g) can be replaced by the expression min {|ξ|i , ξ}. Proof. Let κ ∈ J and ξ ∈ Xκ . (a) Suppose the sequences ψ, ψ¯ ∈ X+ κ,c are given. Then also the shifted sequence ψ is c+ -bounded, since we have +
ec (κ, k) ψ (k) = c(k)ec (κ, k + 1) ψ (k) ≤ c ψκ,c
for all k ∈ Z+ κ.
We define an inhomogeneity rψ : J → X by rψ (k) := fk (ψ(k)) + gk and together with Hypothesis 3.5.7 the triangle inequality yields ! −1 −1 B −1 k+1 rψ (k) ≤ Bk+1 fk (ψ(k)) − fk (0, 0) + Bk+1 (fk (0, 0) + gk ) −1 ≤ L1 ψ(k) + L2 ψ (k) + Bk+1 gk for all k ∈ Z+ κ. Therefore, rψ ∈ X+ κ,c,B , we have the estimate +
+
+
rψ κ,c,B ≤ (L1 + c L2 ) ψκ,c + gκ,c,B
(3.5h)
and obtain from Theorem 3.5.3(a) that the inhomogeneous system Bk+1 x = Ak x + rψ (k)
(3.5i)
possesses a unique c+ -bounded solution; it is denoted by Tξ ψ : Z+ κ → X and fulfills Qn1 (κ)φ(κ) = Qn1 (κ)ξ. In addition, we can represent this solution as
3.5 Admissibility
157
(Tξ ψ)(k) := Φ(k, κ)Qi1 (κ)ξ +
∞
−1 Gi (k, n + 1)Bn+1 rψ (n) for all k ∈ Z+ κ.
n=κ + In particular, the operator Tξ : X+ κ,c → Xκ,c is well-defined and the difference Tξ ψ − Tξ ψ¯ ∈ X+ κ,c (see Lemma 3.3.25(a)) solves the linear equation
¯ Bk+1 x = Ak x + fk (ψ(k)) − fk (ψ(k)) ¯ = 0. Its inhomogeneity is c+ -bounded, since with Qi1 (κ) (Tξ ψ)(κ) − (Tξ ψ)(κ) ! + −1 ¯ Bk+1 fk (ψ(k)) − fk (ψ(k)) ≤ (L1 + c L2 ) ψ − ψ¯ κ,c ec (k, κ) for all k ∈ Z+ κ . We use Theorem 3.5.3(a) in order to infer (set ξ := 0 in (3.5b)) Tξ ψ − Tξ ψ¯ + ≤ (L1 + c L2 )Ci (c) ψ − ψ¯ + ; κ,c κ,c hence, Tξ is a contraction on X+ κ,c (cf. (3.5e)). By Lemma 3.3.25(a) the normed space X+ is complete and the contraction mapping principle (see, for example, κ,c [295, p. 361, Lemma 1.1]) guarantees a unique fixed point φ ∈ X+ κ,c of Tξ with Qi1 (κ) (φ(κ) − ξ) = 0. Moreover, φ is a solution of the inhomogeneous system (3.5i) (where ψ = φ) and, in turn, also of the nonlinear equation (Sg ). By means of Theorem 3.5.3 (a) applied to (3.5i) one deduces +
(3.5b)
+
φκ,c ≤ Ki+ ξ + Ci (c) rφ κ,c 2 1 (3.5h) + + ≤ Ki+ ξ + Ci (c) (L1 + c L2 ) φκ,c + gκ,c,B and (3.5e) implies the claimed estimate (3.5f). (b) One proceeds analogously to (a) and uses Theorem 3.5.3(b).
Theorem 3.5.10. Suppose Hypotheses 3.5.1 and 3.5.7 are fulfilled with J = Z. If sequences c, d : Z → (0, ∞) satisfy c, d ∈ (ai , bi ),
(L1 + L2 max { c , d })Di (c, d) < 1
(3.5j)
for one 1 ≤ i < N , then for each inhomogeneity g ∈ Xc,d,B there exists a unique complete solution φ∗ ∈ Xc,d of (Sg ). It satisfies the estimate φ∗ κ,c,d ≤
Di (c, d) gκ,c,d,B 1 − (L1 + L2 max { c , d })Di (c, d)
for all κ ∈ Z, where the constant Di (c, d) is defined in Theorem 3.5.4.
(3.5k)
158
3 Linear Difference Equations
Remark 3.5.11. Let p ∈ N. Note that p-periodicity of (Sg ) carries over to the solution φ∗ : Z → X. Proof. Let κ ∈ Z be given, as well as ψ, ψ¯ ∈ Xc,d . Consider the linear equation Bk+1 x = Ak x + rψ (k),
(3.5l)
whose inhomogeneity rψ : Z → X, rψ (k) := fk (ψ(k)) + gk is an element of the space Xc,d,B , since we have ! −1 −1 −1 B rψ (k) ≤ f (ψ(k)) − f (0, 0) B + Bk+1 gk k k k+1 k+1 −1 gk for all k ∈ Z ≤ L1 ψ(k) + L2 ψ (k) + Bk+1 and consequently (cf. the above proof of Theorem 3.5.3(a)) rψ κ,c,d,B ≤ (L1 + L2 max { c , d }) ψκ,c,d + gκ,c,d,B .
(3.5m)
Therefore, the operator T : Xc,d → Xc,d , (T ψ)(k) :=
Gi (k, n + 1)rψ (n)
n∈Z
is well-defined as demonstrated in Theorem 3.5.4 and T ψ : Z → X represents the unique complete solution of (3.5l) in Xc,d. On the one hand, the difference T ψ − T ψ¯ is c, d-bounded by Lemma 3.3.25(b), and on the other hand also a so¯ lution of the equation Bk+1 x = Ak x + fk (ψ(k)) − fk (ψ(k)). Its inhomogeneity is easily seen to be in Xc,d,B and satisfies the estimates ! −1 ¯ ec (κ, k) Bk+1 fk (ψ(k)) − fk (ψ(k)) ≤ (L1 + c L2 ) ψ − ψ¯ κ,c,d,B , ! −1 ¯ fk (ψ(k)) − fk (ψ(k)) ed (κ, k) Bk+1 ≤ (L1 + d L2 ) ψ − ψ¯ κ,c,d,B − for k ∈ Z+ κ resp. k ∈ Zκ . Following the proof of Theorem 3.5.4 this implies
T ψ − T ψ¯
κ,c,d
≤ (L1 + L2 max { c , d })Di (c, d) ψ − ψ¯ κ,c,d
(3.5d)
and thanks to (3.5j), the operator T : Xc,d → Xc,d is a contraction on the Banach space Xc,d (cf. Lemma 3.3.25(b)). The contraction mapping principle (see [295, p. 360, Lemma 1.1]) ensures a unique fixed point φ∗ ∈ Xc,d. Besides the solution identity for (3.5l) (with ψ = φ∗ ) and consequently for (Lg ), the sequence φ∗ additionally satisfies the inequality
3.5 Admissibility
159
(3.5d)
φ∗ κ,c,d ≤ Di (c, d) rψ κ,c,d,B
! ≤ Di (c, d) (L1 + L2 max { c , d }) φ∗ κ,c,d + gκ,c,d,B .
(3.5m)
Keeping (3.5j) in mind, this implies the remaining estimate (3.5k).
In particular to meet the requirements of Sect. 4.6 we are interested in linear equations, where each k-fiber of the extended state space is the space of n-linear mappings Ln (Xk ); we set Ln := k∈I Ln (Xk ). Referring to the notation introduced in (0.0c), these difference equations are of the form Bk+1 XB −1
= Ak XP+ (k) ,
Bk+1 XB −1 k+1 Ak P− (k)
= Ak XP− (k) ,
k+1 Ak P+ (k)
(3.5n)
where P− (k) ∈ L(Xk ) is a regular invariant projector for (L0 ) and P+ (k) the associated complementary projector. Consequently, the backward evolution operator Φ− P− (k, κ) : im P− (κ) → im P− (k) exists for k ≤ κ. It is easy to see that, given initial time κ ∈ I and initial state Ξ ∈ Ln (Xκ ; im P− (κ)) with ΞP+ (κ) = Ξ, Λ+ (k, κ)Ξ := Φ− P− (k, κ)ΞΦ(κ,k)P+ (k)
for all k ∈ Z− κ
(3.5o)
defines the uniquely determined backward solution Λ+ (·, κ)Ξ of the first equation in (3.5n) satisfying (Λ+ (k, κ)Ξ)P+ (k) = Λ+ (k, κ)Ξ for all k ∈ Z− κ. In the same way, given κ ∈ I and Ξ ∈ Ln (Xκ ; im P+ (κ) with ΞP− (κ) = Ξ, Λ− (k, κ)Ξ := Φ(k, κ)ΞΦ−
P− (κ,k)P− (k)
for all k ∈ Z+ κ
(3.5p)
defines the uniquely determined forward solution Λ− (·, κ)Ξ of the second equation in (3.5n) which fulfills (Λ− (k, κ)Ξ)P− (k) = Λ− (k, κ)Ξ for all k ∈ Z+ κ. The following result can be considered as a counterpart to Theorem 3.5.3 for equations of the form (3.5n). Here, it suffices to restrict ourselves to exponential dichotomies instead of splittings. Lemma 3.5.12. Let n ∈ N, κ ∈ J, c : J → (0, ∞) and assume that (L0 ) has an exponential dichotomy with data a, b, K + , K − and projector P− on J. If
−1 a sequence H ± satisfies Bk+1 H ± (k) ∈ Ln Xk ; im P∓ (k) for all k ∈ J and H ± ∈ (Ln )± κ,c,B , then the linear inhomogeneous equation Bk+1 XB −1
k+1 Ak P± (k)
= Ak X(k)P± (k) + H ± (k)P± (k)
(3.5q)
has the following properties: (a) In case J is unbounded above and c solution Υ+ : J → Ln of (3.5q) with
b an ,
there exists a unique c-bounded
160
3 Linear Difference Equations
Υ+ (k) = Υ+ (k)P+ (k) ∈ Ln (Xk ; im P− (k))
for all k ∈ J,
(3.5r)
given by the Lyapunov–Perron sum Υ+ (k) := −
∞
−1 + Φ− P− (k, l + 1)Bl+1 H (l)Φ+
P+ (l,k)P+ (k)
l=k +
K Kn
(3.5s)
+
− + + and satisfying the estimate Υ+ κ,c ≤ b−ca n H κ,c,B . a (b) In case J is unbounded below and bn c, there exists a unique c-bounded solution Υ− : J → Ln of (3.5q) with
Υ− (k) = Υ− (k)P− (k) ∈ Ln (Xk ; im P+ (k))
for all k ∈ J,
given by the Lyapunov–Perron sum Υ− (k) :=
k−1
−1 − Φ+ P+ (k, j + 1)Bl+1 H (l)Φ−
P− (l,k)P− (k)
l=−∞ −
and satisfying the estimate Υ− κ,c ≤
n K+ K− cbn −a
−
H − κ,c,B .
Proof. (a) We subdivide the proof into two steps: (I) We first consider H + (k) ≡ 0 on J. Then (3.5q) coincides with the first equation in (3.5n). Let Υ+ : J → Ln be a c-bounded solution of (3.5q) satisfying (3.5r). Passing over to the limit k → ∞ in the inequality P+ (k,κ)P+ (κ) (3.5r) − ≤ ΦP− (κ, k)P− (k) Υ+ (k)Φ+ (k,κ)P+ (κ) P
− Υ+ (κ) = ΦP− (κ, k)Υ+ (k)Φ+ (3.5o)
+
n (0.0d) + ≤ Φ− (κ, k)P (k) Υ (k) (k, κ)P (κ) Φ − + + P− P+ (3.4g)
+
n ≤ K− K+ e can (k, κ) Υ+ κ,c b
for all k ∈ Z+ κ
yields Υ+ (κ) = 0 (see Remark A.1.4(1)). Since κ ∈ J was arbitrary, the zero solution of (3.5q) is the only c-bounded solution satisfying (3.5r). (II) We now omit the restriction on H + and note that the sequence Υ+ from (3.5s) is well-defined, since the estimate ∞ − Φ (k, l + 1)P− (l)B −1 H + (l) + Υ+ (k) ≤ l+1 Φ P− (3.5s)
l=k
(l,k)P (k) + P+
3.6 Roughness
161
∞ (0.0d)
n − −1 H + (l) Φ+ (l, k)P (k) ΦP− (k, l + 1)P− (l) Bl+1 + P+
≤
l=k (3.4g)
n ≤ K− K+ ec (k, κ)
∞
+ eb (k, j + 1)ecan (l, k) H + κ,c,B
l=k
≤
n K− K+ H + + e (k, κ) κ,c,B c n !b − ca "
for all k ∈ J
holds with the aid of Lemma A.1.5(a), which in turn yields the claimed estimate for the c+ -norm of Υ+ . Moreover, it is easy to see from (3.5a) that Υ+ satisfies the relation (3.5r). In addition, Υ+ is a solution of (3.5q) since Υ+ (k)B −1
k+1 Ak P+ (k)
(3.5s)
∞
(3.5a)
l=k+1 ∞
≡ − ≡ −
−1 + Φ− P− (k + 1, l + 1)Bl+1 H (l)Φ+
−1 P+ (l,k+1)Q(k+1)Bk+1 Ak P+ (k)
−1 −1 + Bk+1 Ak Φ− P− (k, l+1)Bk+1 H (l)Φ+
P+ (l,k)P+ (k)
l=k (3.5s)
≡
−1 Bk+1 Ak Υ+ (k)P+ (k)
−1 + Bk+1 H + (k)P+ (k)
−1 +Bk+1 H + (k)P+ (k)
on J.
Finally, the uniqueness statement results from step (I), because the difference of any two c-bounded solutions of (3.5q) is a c-bounded solution of (3.5n) and therefore identically vanishing. (b) This can be shown similarly using Lemma A.1.5(b).
3.6 Roughness The primary advantage of exponential dichotomies or more general exponential splittings above other nonautonomous hyperbolicity notions is their persistence under a wide class of perturbations. Accordingly, there are various methods in order to establish corresponding so-called roughness theorems. Among them, we use a fixed point argument, which beyond conceptional clarity has the benefit that its proof generalizes to parameter-dependent equations. Consequently, it is possible to extend this approach for instance to show smooth dependence of the invariant projectors by means of the uniform contraction principle (cf. Theorems B.1.1 and B.1.5). Moreover, it generalizes to different types of dichotomies (see, e.g., [217]). More detailed, given a linear difference equation Bk+1 x = Ak x,
(L0 )
162
3 Linear Difference Equations
as in Definition 3.1.1, which admits an exponential splitting as in Hypothesis 3.5.1. Thus, there exist (pseudo-) hyperbolic splittings X = P1i ⊕ Qi1 , 1 ≤ i < N , of the extended state space X into invariant vector bundles consisting of solutions with a particular growth behavior in forward (see Qi1 ) resp. backward time (see P1i ). In the present section we investigate homogeneous perturbations of the form
¯k+1 x = Ak + A¯k x Bk+1 + B
(P)
¯k ∈ Hom(Xk , Yk ), and provide sufficient with operators A¯k ∈ Hom(Xk , Yk+1 ), B criteria such that also (P) is dichotomous or has an exponential splitting. In order ¯ i1 , to tackle this problem, we construct a pseudo-hyperbolic splitting X = P¯1i ⊕ Q i ¯ , as well 1 ≤ i < N , for (P). For this, the forward evolution operator of (P) in Q 1 as the backward evolution operator in P¯1i are characterized as fixed points of an appropriate linear equation. For this we need the following Hypothesis 3.6.1. Let J ⊆ I be a discrete interval. Suppose X consists of Banach ¯ k fulfill spaces, the coefficient operators Ak , A¯k , Bk , B ¯k+1 ⊆ im Bk+1 , im Ak , im A¯k , im B −1 ¯ Bk+1 Ak ∈ L(Xk , Xk+1 ),
ker Bk+1 = {0} , −1 ¯ Bk+1 Bk+1 ∈ L(Xk+1 )
¯k+1 satisfy for all k ∈ J , and the perturbations A¯k , B −1 δ1 := sup Bk+1 A¯k L(X
k ,Xk+1 )
k∈J
,
−1 ¯k+1 δ2 := sup Bk+1 B L(X k∈J
k+1 )
.
Moreover, for sequences c, d : J → (0, ∞) we define constants δ(c) := δ1 + c δ2 , qi (c, d) := max
Ki− Ki+ , !c − ai " !bi − d"
for all 1 ≤ i < N.
Before proceeding, we define the lattices J2+ := (k, l) ∈ J2 : l ≤ k ,
J2− := (k, l) ∈ J2 : k ≤ l
and establish our functional analytical set-up as follows. Given κ ∈ J and a sequence c : J → (0, ∞), we introduce the union L := k,l∈J L(Xl , Xk ), as well as the linear spaces of operators L0 := X : J → L| X(k) ∈ L(Xk ) and sup X(k) < ∞ , k∈J
2 := X : J → L| X(k, l) ∈ L(X , X ) and sup X(k, l) e (l, k) < ∞ , L+ l k c c + l≤k
+ − Lc := X : J− → L| X(k, l) ∈ L(Xl , Xk ) and sup X(k, l) ec (l, k) < ∞ . k≤l
3.6 Roughness
163
First, let us canonically equip L0 with the sup-norm X1 := supk∈J |X(k)|i , where |·|i denotes the operator norm induced by the corresponding norm from Lemma 3.3.22. All three sets become Banach spaces, where the latter two are equipped with the respective norms +
Xc := sup |X(k, l)|i ec (l, k), l≤k
−
Xc := sup |X(k, l)|i ec (l, k). k≤l
For later reference we abbreviate the expression ˆ ± (n, l) := B −1 A¯n Ξ ± (n, l) − B ¯n+1 Ξ ± (n + 1, l) , Ξ n+1 which is obviously linear in Ξ ± ∈ L± c , and easily deduce the exponential estimates + ˆ+ Ξ (n, l) ≤ δ(c) Ξ + c ec (n, l) for all l ≤ n, − ˆ− Ξ (n, l) ≤ δ(c) Ξ − c ec (n, l) for all n ≤ l
(3.6a)
from Hypothesis 3.6.1. On the above spaces we introduce three operators, whose properties are the purpose of our subsequent lemmas. Convention: As in the situation below, we often encounter sequence-valued mappings like, e.g., T : Z → L0 or φ : Z → X± κ,c , defined on some given set Z. For a less awkward notation we use the sometimes imprecise, but convenient abbrevia tion φ(k, z) := φ(z) (k) ∈ Xk . We proceed accordingly with nonautonomous sets S(z) ⊆ X depending on parameters z, i.e., we write S(k, z) := S(z)(k). Lemma 3.6.2. Let J = Z+ κ , κ ∈ I, and c, d : J → (0, ∞). If Hypotheses 3.5.1 and 3.6.1 are satisfied with ai d and c bi for one 1 ≤ i < N , then the − operator T0 : L+ c × Ld → L0 , T 0 (k, Ξ + , Ξ − ) := Qi1 (k) − −
k−1 n=κ ∞
ˆ − (n, k) Φ(k, n + 1)Qi1 (n + 1)Ξ i ˆ+ Φ− Pi (k, n + 1)P1 (n + 1)Ξ (n, k)
n=k
for all k ∈ Z+ κ is affine-linear, well-defined with 0 + − T (Ξ , Ξ ) ≤ K + + qi (d, c)δ(d) max Ξ + + , Ξ − − i c d 1
(3.6b)
and one has the Lipschitz estimates lip1 T 0 ≤
Ki− δ(c) , !bi − c"
lip2 T 0 ≤
Ki+ δ(d) . !d − ai "
(3.6c)
164
3 Linear Difference Equations
− − Proof. Let Ξ + ∈ L+ c , Ξ , ∈ Ld be given and above all one has the inclusion 0 + − T (k, Ξ , Ξ ) ∈ L(Xk ), k ∈ Z+ κ . Thanks to the exponential splitting of (L0 ), from Lemma A.1.5 we obtain the estimates k−1 i (3.4g) ˆ− Q1 (k)T 0 (k, Ξ + , Ξ − ) ≤ K + + K + e (k, n + 1) Ξ (n, k) a i i i n=κ (3.6a)
≤ Ki+ + Ki+ δ(d)
k−1
− eai (k, n + 1)ed (n, k) Ξ − d
n=κ
K + δ(d) Ξ − − , ≤ Ki+ + i d !d − ai " ∞ i (3.4g) ˆ+ P1 (k)T 0 (k, Ξ + , Ξ − ) ≤ K − e (k, n + 1) Ξ (n, k) bi i (A.1d)
(3.6a)
≤ Ki−
n=k ∞
+ ebi (k, n + 1)ec (n, k) Ξ + c
n=k
Ki− δ(c) Ξ + + ≤ c !bi − c"
(A.1e)
for all k ∈ Z+ κ.
0 + − Passing over to the least upper bounds for k ∈ Z+ κ implies T (Ξ , Ξ ) ∈ L0 and − + therefore T0 : Lc × Ld → L0 is well-defined satisfying (3.6b). Along the same lines one deduces the Lipschitz estimates (3.6c).
Lemma 3.6.3. Let J = Z+ κ , κ ∈ I, and c : J → (0, ∞). If Hypotheses 3.5.1 and 3.6.1 hold with c ∈ (ai , bi ) for one 1 ≤ i < N , then the operator T + : + L+ c × L0 → Lc , T + (k, l, Ξ + , Ξ) := Φ(k, l)Qi1 (l)Ξ(l) +
∞
ˆ + (n, l) Gi (k, n + 1)Ξ
(3.6d)
n=κ
for all l ≤ k, is affine-linear, well-defined with + + T (Ξ , Ξ) + ≤ K + Ξ + qi (c, c)δ(d) Ξ + + i 1 c c
(3.6e)
and one has the Lipschitz estimates lip1 T + ≤ qi (c, c)δ(c),
lip2 T + ≤ Ki+ .
(3.6f)
Proof. As in Lemma 3.6.2 the proof follows from straight forward estimates of the Lyapunov–Perron sums using the estimates (3.4g) and Lemma A.1.5. Lemma 3.6.4. Let J = Z+ κ , κ ∈ I, and d : J → (0, ∞). If Hypotheses 3.5.1 and 3.6.1 hold with d ∈ (ai , bi ) for one 1 ≤ i < N , then the operator T − : − L− d × L0 → Ld ,
3.6 Roughness
165
i T − (k, l, Ξ − , Ξ) := Φ− Pi (k, l)P1 (l)[IXl − Ξ(l)] +
∞
ˆ − (n, l) Gi (k, n + 1)Ξ
n=κ
(3.6g) for all k ≤ l, is affine-linear, well-defined with − − T (Ξ , Ξ) − ≤ K − Ξ + qi (d, d)δ(d) Ξ − − i 1 d d
(3.6h)
and one has the Lipschitz estimates lip2 T − ≤ Ki− .
lip1 T − ≤ qi (d, d)δ(d),
(3.6i)
Proof. The proof is omitted, since it is dual to the one of Lemma 3.6.3.
This brings us to our central robustness result. For the sake of brevity we formulate it for exponential dichotomies only, but a generalization to general exponential splittings is possible by means of mathematical induction. It shows that the growth rates for the perturbed system (P) can be chosen arbitrarily close to the original ones of (L0 ), provided the bounds δ1 , δ2 > 0 are small. Theorem 3.6.5 (roughness theorem). Let θ ∈ (0, 1) and J be an unbounded interval. Suppose that Hypotheses 3.5.1 and 3.6.1 hold. If the sequences a ¯, ¯b : J → ¯ (0, ∞) fulfill ai a ¯ b bi and
¯) + qi (¯ a, ¯b) δ(¯b) ≤ θ, 2 max Ki− , Ki+ qi (¯b, a
(3.6j)
then the perturbed linear homogeneous equation (P) has an exponential dichotomy − + 2 ¯ ± := max{Ki ,Ki } and a projector P¯ i satisfying on J with rates a ¯, ¯b, constants K i
i P1 (k) − P¯1i (k) L(X
k)
≤
1
1−θ
K+ Ki− + 3¯ i 4 !bi − a ¯" b − ai
δ(d)
for all k ∈ J. (3.6k)
Remark 3.6.6. Provided one can choose δ1 , δ2 > 0 so small that
K+ Ki− + 3¯ i 4 !bi − a ¯" b − ai
δ(d) < 1
holds true, then the projectors P1i and P¯1i are conjugated, i.e., there exist linear operators Si (k) ∈ GL(Xk ) such that the relations P¯1i (k) = Si (k)−1 P1i (k)Si (k) and P¯1i (k) = Si (k)P1i (k) (cf. [244, pp. 32–34]) hold. Proof. We give a proof only for the case J = Z+ κ with some κ ∈ I. The case of a dichotomy in backward time or on the whole integer axis can be done similarly.
166
3 Linear Difference Equations
Suppose that the sequences a ¯, ¯b : J → (0, ∞) fulfill a ¯, ¯b ∈ (ai , bi ) for a given 1 ≤ i < N and a ¯ ¯b. In applications of (3.6j) we often make use of the relation − max qi (¯ a, a ¯), qi (¯b, ¯b) = qi (¯ a, ¯b). On the product L+ × L we use a norm topola ¯ ¯ b + − ogy as in (0.0b). We introduce the affine-linear operator T : La¯ × L¯− → L+ , a ¯ × L¯ b b
T (Ξ + , Ξ − ) := T + (Ξ + , T 0 (Ξ + , Ξ − )), T − (Ξ − , T 0 (Ξ + , Ξ − )) , which is clearly well-defined due to the above Lemmata 3.6.2, 3.6.3 and 3.6.4. The following proof is split into three steps. − − (I) Claim: T : L+ → L+ is a contraction with lip T ≤ θ. a ¯ × L¯ a ¯ × L¯ b b + ¯ + ∈ La¯ , Ξ − , Ξ ¯ − ∈ L¯− and using the triangle inWith given sequences Ξ + , Ξ b equality, from Lemma 3.6.3 we have the Lipschitz estimates + + 0 + − ¯ + , Ξ − )) + T (Ξ , T (Ξ , Ξ )) − T + (Ξ¯ + , T 0 (Ξ a ¯ ¯ + + + K + T 0 (Ξ + , Ξ − ) − T 0 (Ξ ¯ + , Ξ − ) ≤ qi (¯ a, a ¯)δ(¯ a) Ξ + − Ξ i a ¯ 1 − + (3.6c) a) K K δ(¯ Ξ + − Ξ¯ + + , ≤ qi (¯ a, a ¯)δ(¯ a) + i i a ¯ !bi − a ¯" + + 0 + − T (Ξ , T (Ξ , Ξ )) − T + (Ξ + , T 0 (Ξ + , Ξ ¯ − )) + (3.6f)
a ¯
¯ − ) ≤ Ki+ T 0 (Ξ + , Ξ − ) − T 0 (Ξ + , Ξ 1
(3.6f)
(3.6c)
≤
Ki+ Ki+ δ(¯b) ¯ − ¯− 3 4 Ξ − − Ξ b ¯b − ai
and analogously using Lemma 3.6.4 we have − − 0 + − T (Ξ , T (Ξ , Ξ )) − T − (Ξ ¯ − , T 0 (Ξ + , Ξ ¯ − )) ¯− b (3.6i) K − K + δ(¯b) Ξ − − Ξ¯ − ¯− , ≤ qi (¯b, ¯b)δ(¯b) + 3i¯ i 4 b b − ai − − 0 + − ¯ + , Ξ − )) ¯− T (Ξ , T (Ξ , Ξ )) − T − (Ξ − , T 0 (Ξ b (3.6c)
≤
Ki− Ki+ δ(¯ a) Ξ + − Ξ ¯ + + . a ¯ !bi − a ¯"
Together with our assumption (3.6j) this yields lip T ≤ θ. − (II) Claim: The unique fixed point (Ξκ+ , Ξκ− ) ∈ L+ of T satisfies a ¯ × L¯ b − + + + − K max Ki , Ki i (Ξκ , Ξκ ) + − ≤ La ×L ¯ ¯ 1−θ b
(3.6l)
3.6 Roughness
167
+ By step (I) the mapping T is a contraction on the Banach space L+ and has a a ¯ × L¯ b + − unique fixed point Ξκ = (Ξκ , Ξκ ). Using well-known arguments one has
Ξκ = T (Ξκ ) ≤
1 T (0, 0) 1−θ
and the claimed estimate (3.6l) for the norm of Ξκ follows from the relations + (3.6e) (3.6b) T (0, T 0 (0, 0)) + ≤ K + T 0 (0, 0) ≤ K + K + , i i i c (3.6h) (3.6b) − − T (0, T 0 (0, 0)) ≤ K − T 0 (0, 0) ≤ K − K + . i i i d (III) Claim: The sequences ¯ i (k) := T 0 (k, Ξ + , Ξ − ), Q i κ κ
P¯1i (k) := I − Qi1 (k) for all k ∈ Z+ κ
(3.6m)
define invariant projectors for the perturbed equation (P). We insert (3.6d) and (3.6g) defining the operators T + resp. T − into the defining first relation (3.6m), which implies Ξκ+ (k, l) = Φ(k, l)Qi1 (l) − −
∞
l−1
ˆ − (n, l) Φ(k, n + 1)Qi1 (n + 1)Ξ κ
n=κ i ˆ+ Φ− Pi (k, n + 1)P1 (n + 1)Ξκ (n, l)
n=k
+
k−1
ˆ + (n, l) Φ(k, n + 1)Qi1 (n + 1)Ξ κ
(3.6n)
n=l
for all l ≤ k and i Ξκ− (k, l) = Φ− Pi (k, l)P1 (l) +
∞
i ˆ+ Φ− Pi (k, n + 1)P1 (n + 1)Ξκ (n, l)
n=l
+
k−1
ˆ − (n, l) Φ(k, n + 1)Qi1 (n + 1)Ξ κ
n=κ
−
l−1
i ˆ− Φ− Pi (k, n + 1)P1 (n + 1)Ξκ (n, l)
n=k
for all k ≤ l, and in particular for k = l we obtain Ξκ+ (k, k) = Qi1 (k) −
∞
i ˆ+ Φ− Pi (k, n + 1)P1 (n + 1)Ξκ (n, k)
n=k
−
k−1 n=κ
ˆκ− (n, k), Φ(k, n + 1)Qi1 (n + 1)Ξ
(3.6o)
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3 Linear Difference Equations
Ξκ− (k, k) = P1i (k) + ∞
+
k−1
ˆκ− (n, l) Φ(k, n + 1)Qi1 (n + 1)Ξ
n=κ i ˆ+ Φ− Pi (k, n + 1)P1 (n + 1)Ξκ (n, l).
(3.6p)
n=k
This immediately shows the identities Ξκ+ (k, k) + Ξκ− (k, k) ≡ I
(3.6q)
¯ i (k) on J. In the following we show that Q ¯ i (k) ∈ L(Xk ), k ∈ J, and Ξκ+ (k, k) = Q 1 1 are projectors. For this, we multiply (3.6n) from the right with Ξ + (l, l) ∈ L(Xl ) and correspondingly (3.6o) with Ξ − (l, l) ∈ L(Xl ). Afterwards, we insert the result into the two relations (3.6p) and make use of (3.6q), which yields Ξκ+ (k, l)Ξ + (l, l) = Φ(k, l)Qi1 (l) − 2
l−1
ˆ − (n, l) Φ(k, n + 1)Qi1 (n + 1)Ξ κ
n=κ
+
l−1
ˆκ− (n, l)Ξ − (l, l) Φ(k, n + 1)Qi1 (n + 1)Ξ
n=κ
−
∞
i + ˆ+ Φ− Pi (k, n + 1)P1 (n + 1)Ξκ (n, l)Ξ (l, l)
n=k
+
k−1
ˆκ+ (n, l)Ξ + (l, l) Φ(k, n + 1)Qi1 (n + 1)Ξ
n=l
for all l ≤ k and Ξκ− (k, l)Ξ − (l, l) = Φ(k, l)P1i (l) + 2
∞
i ˆ+ Φ− Pi (k, n + 1)P1 (n + 1)Ξκ (n, l)
n=l
−
∞
i + ˆ+ Φ− Pi (k, n + 1)P1 (n + 1)Ξκ (n, l)Ξ (l, l)
n=l
+
k−1
ˆ − (n, l)Ξ − (l, l) Φ(k, n + 1)Qi1 (n + 1)Ξ κ
n=κ
−
l−1
i − ˆ− Φ− Pi (k, n + 1)P1 (n + 1)Ξκ (n, l)Ξ (l, l)
n=k
for all k ≤ l. In these two equations we substitute the operator-valued variables Φ¯+ (k, l) := Ξκ+ (k, l)Ξ + (l, l) for l ≤ k and Φ¯− (k, l) := Ξκ− (k, l)Ξ − (l, l) for − k ≤ l. This yields a new fixed point equation for the pair (Φ¯+ , Φ¯− ) ∈ L+ and a ¯ × L¯ b
3.6 Roughness
169
thanks to our assumption (3.6j) there exists a unique fixed point. This fixed point however, is given by (Ξκ+ , Ξκ− ), and one has Ξκ+ (k, l) = Ξκ+ (k, l)Ξ + (l, l) for all l ≤ k, Ξκ− (k, l) = Ξκ− (k, l)Ξ − (l, l) for all k ≤ l. ¯ i (k) and Ξ − (k, k) are projections. Along the same lines Particularly, Ξ + (k, k) = Q 1 + ¯ i1 , while Ξ − is a one shows that Ξ represents a linear 2-parameter semigroup in Q i ¯ backward 2-parameter semigroup in the complementary set P1 . (III) It is an immediate consequence of (3.6l) that we have − + + + Ξκ (k, l)Q ¯ i1 (l) ≤ Ki max Ki , Ki ea¯ (k, l) for all l ≤ k 1−θ and similarly one obtains the corresponding dichotomy estimate in backward time for the operator Ξκ+ (k, l)P¯1i (l), k ≤ l. It remains to establish the estimate (3.6k). For this, from Lemma 3.6.2 we obtain i (3.6m) Q ¯ (k) − Qi (k) ≤ T 0 (Ξ + , Ξ − ) − T 0 (0, 0) 1 1 κ κ (3.6c) Ki+ δ(d) Ki− δ(c) 4 + 3¯ for all k ∈ J. ≤ !bi − a ¯" b − ai
This finishes the proof of Theorem 3.6.5.
We close this somewhat technical section with an example. It illustrates that, as opposed to dichotomies and splittings, merely forward dichotomies and splittings as introduced in Definitions 3.4.2 and 3.4.5, resp., lack a ∞ -roughness property. This necessitates an unperturbed system with noninvertible right-hand side. Example 3.6.7. Consider (3.4f) from Example 3.4.8, which admits an exponential forward dichotomy on Z. We perturb it homogeneously to x = (Ak + A¯k )x
(3.6r)
with the matrix ⎛
0 0 A¯k := ⎝0 (1 − δk )ε 0 0
⎞ 0 0⎠ , 0
δk :=
0
for k ≤ 0,
1
for k > 0,
where ε ∈ (0, α] can be arbitrarily small. Yet, the perturbed equation (3.6r) cannot have an exponential forward dichotomy with growth rates γ < δ and γ, δ ∈ (α, β). In order to see this, we suppose the contrary, i.e., there exists a projector P¯ : Z → L(R3 ) yielding an exponential forward dichotomy for (3.6r) on Z. We would
170
3 Linear Difference Equations
+ trivially also obtain forward dichotomies on the two semiaxes Z− 0 and Z1 with the same data. If Φ¯ denotes the evolution operator for (3.6r), then Proposition 3.4.7(a) guarantees
¯ 1) ∈ X+ dim Q1 (1) = dim ξ ∈ R3 : Φ(·, 1,γ = dim ξ ∈ R3 : ξ2 = ξ3 = 0 = 1. On the other hand, by construction, the right-hand side of (3.6r) is invertible and the regularity condition holds. Thus, Propositions 3.4.7(b) and 3.4.16(b) imply ¯ 0) ∈ X− dim Q1 (0) = 3 − dim P1 (0) = 3 − dim ξ ∈ R3 : Φ(·, 0,δ 3 = 3 − dim ξ ∈ R : ξ1 = ξ2 = 0 = 2, in this yields the contradiction 1 = dim Q1 (1) = dim(A0 + A¯0 )Q1 (0) = 2, since it is A0 + A¯0 ∈ GL(R3 ). In conclusion, (3.6r) cannot admit an exponential forward dichotomy on Z with rates γ, δ.
3.7 Applications We again suppose (tk )k∈I is a discretization mesh satisfying hk := tk+1 − tk ∈ [τ, T ]
for all k ∈ I
with given stepsize bounds 0 < τ ≤ T . Many of the subsequent results are stated for autonomous equations only, in particular when it comes to quantitative questions. Yet, using the robustness Theorem 3.6.5 or Remark 3.5.9(1) it is possible to obtain information for nonautonomous equations, provided the time-varying perturbations are sufficiently small. We exemplify this in Example 3.7.3.
3.7.1 Discretized Linear Functional Differential Equations In the notation of Sect. 1.5.1 we consider linear retarded FDEs (LDE)
u(t) ˙ = L(t)ut , where the continuous right-hand side L : R → L(Cr , Rd ) is given by L(t)ψ =
0
d[η(t, ϑ)]ψ(ϑ) −r
for all t ∈ R.
3.7 Applications
171
Here, η : R × R → Rd×d is supposed to be continuous in the first argument. In the second argument, it is normalized (η(t, ϑ) = 0 for ϑ ≥ 0, η(t, ϑ) = η(t, −r) for ϑ ≤ −r), continuous from the left in ϑ on (−r, 0) and has bounded variation on [−r, 0] (cf. [198, p. 166ff]). In order to avoid a singularity in our discretization below, we suppose that η is atomic, i.e., limϑ−r η(t, ϑ) = η(t, −r) for all t ∈ R. We apply the spatial discretization from Sect. 2.6.1 to (LDE) and the θ-method for temporal discretization, which leads us to the recursion uk+1 = uk + (1 − θ)
N
h [η(tk , (1 − i)h) − η(tk , ih)] uk−i
n=1
+θ
N
(3.7a)
h [η(tk+1 , (1 − i)h) − η(tk+1 , ih)] uk+1−i
n=1
for θ ∈ [0, 1]. So, we can write (LDE) as linear difference equation x = Ak (θ, h)x
(LΔE)
in Rd(N +1) ∼ = Cr,N with the (N + 1) × (N + 1)-block matrix ⎛ Ak (θ, h) :=
⎞
IRd + hθ ηˆk1 h(1 − θ)ˆ ηk1 + hθ ηˆk2 . . . h(1 − θ)ˆ ηkN−1 + hθ ηˆkN hθ ηˆkN ⎜ I Rd ⎟ ⎜ ⎟ .. ⎜ ⎟, . ⎜ ⎟ .. ⎝ ⎠ . I Rd 0Rd
where ηˆki := η(tk , (−i + 1)h) − η(tk , −ih) ∈ Rd×d . In the scalar case d = 1 and an autonomous equation (LDE) such a companion matrix Ak (θ, h) ≡ A(θ, h) does not depend on k ∈ I and has the following properties: Lemma 3.7.1. Let d = 1, θ ∈ [0, 1] and h > 0. (a) σ(A(θ, h)) = {λ1 , . . . , λn }, n ≤ d, where λ1 , . . . , λn ∈ C are the roots of the characteristic equation η 1 )tN − h tN +1 − (1 + hθˆ
N −1
(1 − θ)ˆ η l + θˆ η l+1 tN −l − hθˆ η N = 0.
l=1 N −1 (b) Each (λN , . . . , 1) ∈ Cr,N , l = 1, . . . , n, is an eigenvector of A(θ, h). l , λl (c) The Euclidean norm of A(θ, h) is
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3 Linear Difference Equations
5
1 2 4 4 N 4 A(θ, h)2 := 2 α + α − 4h θ (ˆ η ) 2 (1 − θ)ˆ η l + θˆ η l+1 + (hθˆ η N )2 . √ Remark 3.7.2. (1) In the limit one has limh0 A(θ, h)2 = 2 > 1 and therefore Euclidean norm bounds are inadequate for estimates of A(θ, h)n 2 , n ∈ N, in order to achieve asymptotic stability. One rather has to work with adapted norms and information on σ(A(θ, h)) as in Lemma 3.4.26. (2)A survey of methods to detect gaps, i.e., annuli disjoint from σ(A(θ, h)) in the complex plane C, is given in [405]. with α := 1 + (1 + hθˆ η 1 )2 + h2
N −1 l=1
Proof. The assertions (a) and (b) are straight forward to prove. For assertion (c), we remark that the Euclidean norm and the largest singular value of A(θ, h) coincide. However, the singular values of companion matrices are explicitly known (cf. [251, Theorem 3.1]) yielding the above formula. Example 3.7.3 (delay differential equations). Let δ > 0, r > 0 and ϑ : R → R. We consider the delay differential equation (DDE) from Example 1.5.3 with a linear function g(t, u) = ϑ(t)u. A corresponding full discretization is the delay difference equation uk+1 = (1 − δh)uk − hϑ(tk )uk−N
(3.7b)
under the standing conditions δh < 1, i.e., δr < N . We write (3.7b) as an explicit system x = Ax + A¯k x in Cr,N with operators ⎞
⎛
1 − δh ⎜ 1 0 ⎜ A := ⎜ . .. ... ⎝ 1
⎛ ⎞ 0 . . . 0 −hϑ(tk ) ⎜0 . . . 0 ⎟ 0 ⎜ ⎟ A¯k := ⎜ . ⎟. . .. ⎝ .. ⎠ 0 0 ... 0 0
⎟ ⎟ ⎟, ⎠ 0
Thanks to σ(A) = {0, 1 − δh} we deduce from Proposition 3.4.28 (see [376, Example 4.3] for details) that x = Ax has an exponential dichotomy on Z with constant rates α ∈ (0, 1 − δh) and 1 − δh, K ± = 2α−K 1 + (1 − δh)−K and a one-dimensional pseudo-unstable subspace. Choose 6reals α < α ¯ <77β¯ < 6 !N 7 6 !N −N 4 N 1 + 1 + and 1 − δh =: β, define δ˜ := αN 1 + N N N −rδ N −rδ α under the condition δ˜ max
2r 2r , β− α−α ¯ β¯
sup |ϑ(t)| < 1, t∈R
which is sufficient for (3.6j), also x = Ax + A¯k x admits an exponential dichotomy on I by Theorem 3.6.5. On the other hand, in case ϑ is constant, [290, Theorem 3] provides necessary and sufficient conditions for (3.7b) to be uniformly asymptotically stable in terms of the constants hδ and hϑ.
3.7 Applications
173
3.7.2 Time-Discretized Linear Abstract Evolution Equations For Banach spaces X, Y satisfying an embedding X ⊆ Y we consider an abstract linear nonautonomous evolutionary equation ut + B(t)u = 0
(LAE)
in Y and suppose that Hypothesis 1.5.4(i) holds true. Since (LAE) generates an evolution family (U (t, s))s≤t on X, we obtain a linear continuous bounded 2-parameter semigroup on X := I × X via Φ(k, l) := U (tk , tl ) for all l ≤ k. If we define Ak := U (tk+1 , tk )
for all k ∈ I ,
then Φ(k, l) ∈ L(X), l ≤ k, is the forward evolution operator of x = Ak x. Hypothesis 3.7.4. Let K − , K + ≥ 1 and α < β be reals. Suppose that (LAE) possesses an exponential dichotomy on R, i.e., there exists a projection-valued mapping P¯ : R → L(X) satisfying for all s ≤ t that P¯ (t)U (t, s) = U (t, s)P¯ (s), U (t, s)|im P¯ (s) ∈ GL(im P¯ (s), im P¯ (t)), U (t, s)[IX − P¯ (s)] ≤ K + eα(t−s) , U (s, t)P¯ (t) ≤ K − eβ(s−t) . Lemma 3.7.5. If Hypothesis 3.7.4 holds, then x = Ak x has a strongly regular exponential 2-splitting on I with growth rates a1 (k) := eαhk , b1 (k) := eβhk , k ∈ I , constants K − , K + and the projector P (k) := P¯ (tk ). Proof. Using Corollary 3.3.8 one sees that P is an invariant projector satisfying the strong regularity condition (3.3j). By Theorem 3.1.9(a) we infer from Hypothesis 1.5.4(i) that (3.1e)
Φ(k, l)[I − P (l)] = Ak−1 · · · Al [I − P (l)] = U (tk , tk−1 ) · · · U (tl+1 , tl )[I − P¯ (tl )] = U (tk , tl )[I − P¯ (tl )] k−1 hn = K + ea1 (k, l) for all l ≤ k, ≤ K + exp(α(tk − tl )) = K + exp α n=l
and the second dichotomy estimate in (3.4g) can be shown analogously.
Corollary 3.7.6. If Hypothesis 1.5.4(i) holds and a(k) := eω(tk+1 −tk ) , k ∈ I , then the evolution operator of x = Ak x satisfies Φ(k, l)L(X) ≤ Kea (k, l) for all l ≤ k. Proof. Apply Lemma 3.7.5 with P¯ (t) ≡ 0 on R.
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3 Linear Difference Equations
3.7.3 Time-Discretized Linear Parabolic Equations Let Ω ⊆ Rd be a bounded domain with Lipschitzian boundary and suppose that aij ∈ C ∞ (cl Ω) are real-valued functions satisfying aij = aji for 1 ≤ i, j ≤ d. Consider an elliptic differential operator in divergence form Bu := −
d
(aij (x)uxi )xj ,
i,j=1
equipped with Dirichlet boundary conditions u(x) = 0 for x ∈ bd Ω, which satisfies a uniform ellipticity condition, i.e., there exists a θ > 0 such that 2
θ ξ ≤
d
aij (x)ξi ξj
for all x ∈ Ω, ξ ∈ Rd .
i,j=1
The symmetric elliptic operator B fulfills (see, e.g., [149, p. 335, Theorem 1]): (i) σ(B) = {λn }n∈N , where every λn is a real eigenvalue of B. If we repeat each λn according to its (finite) multiplicity, then 0 < λ1 ≤ λ2 ≤ λ3 ≤ . . . ,
lim λn = ∞.
n→∞
(ii) There exists an orthonormal basis (φn )n∈N of L2 (Ω), where φn ∈ H01 (Ω) is an eigenfunction corresponding to λn . Furthermore, referring to [432, p. 105ff] we know that the linear operator B is sectorial with domain D(B) = H 2 (Ω) ∩ H01 (Ω), Y = L2 (Ω) and the fractional power space X = Y 1/2 = H01 (Ω). Thus, the assumptions imposed on the linear part of (SE) in Sects. 1.5.2 and 2.6.2 are satisfied. For instance, a compact resolvent of B implies that e−Bt , t > 0, is compact. From standard Sturm–Liouville theory we deduce Example 3.7.7. Let d = 1 and a, b ∈ R satisfying a < b. For Ω = (a, b) and δ > 0 we consider the second order differential operator Bu := −δuxx equipped with Dirichlet boundary conditions u(a) = u(b) = 0. The eigenvalues of B are given !2 nπ by λn = δ b−a for all n ∈ N, and the corresponding orthonormal eigenfunc
! 2 kπ tions read as φn (x) = b−a sin b−a x . Moreover, B has a compact resolvent (cf. [244, p. 187, Example 6.31]). For simplicity let us work with constant stepsizes T = τ . We use the backward Euler method in order to discretize ut + Bu = 0 in time, which yields [IY + τ B]x = x.
(3.7c)
3.7 Applications
175
Note that (3.7c) is a special case of the θ-method known from Sect. 2.6.2 and when B has compact resolvent we deduce Corollary 3.7.8. If (αn−1 )n∈N , (βn )n∈N are positive real sequences with 1 1+τ λn
αn < βn ≤ αn−1 ,
∈ (βn , αn−1 )
for all n ∈ N,
then (3.7c) admits an exponential N -splitting ∞ n=1 (βn+1 , αn ) on Z with constants Kn± = 1, the splitting P consists of the (constant) orthogonal spectral projectors Pn associated to {λn } and dim P1n < ∞, n ∈ N. Proof. The function rτ : (0, ∞) → (0, 1), rτ (t) := (1 + τ t)−1 is strictly decreasing and the sequence (rτ (λn ))n∈N inherits this property. Thus, the assumptions of Theorem 3.4.30 are satisfied with νn = rτ (λn ) and our claim follows.
3.7.4 Fully Discretized Diffusion Equations
×d
On a rectangular domain Ω = i=1 (−ai , ai ) with a1 , . . . , ad > 0 we suppose the elliptic differential operator B from Sect. 3.7.3 is the Laplacian −Δ. Given a continuous function δ : R → F let us consider the diffusion problem ut − δ(t)Δu = 0 in Ω,
u(t, 0) = 0
in bd Ω.
(3.7d)
As carried out in Sect. 2.6.3, a spatial discretization of (3.7d) leads to initial value problems for ordinary differential equations in RN of the form M v˙ = δ(t)Av,
(3.7e)
v(t0 ) = v0 ,
with positive definite matrices A, M ∈ RN ×N .
Finite Differences with Zero Boundary Conditions For the sake of a finite difference spatial discretization, we prescribe integers N1 , . . . , Nd > 1, introduce the rectangular grids G=
×[−N , N ] d
j
j=1
j Z
,
G0 =
×(−N , N ) d
j
j Z
j=1
and the mesh size vector h = (h1 , . . . , hd ) with hi = ai /Ni . For fixed times t, the solution of (3.7e) is represented as vector u = (ui )i∈G0 , where i = (i1 , . . . , id ) is a multi-index to specify the grid point xi = (h1 ii , . . . , hd id ). As discretization of the Laplacian in the diffusion equation (3.7d) we use
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3 Linear Difference Equations
(Δh u)i = −
d 1 2 (ui+ej − 2ui + ui−ej ) for all i ∈ G; h j=1 j
here, e1 , . . . , ed is the canonical basis of Rd . This yields the mass matrix M = IRN and for the stiffness matrix A = Δh we obtain the following properties: Lemma 3.7.9 (eigenvalues and -vectors 8 of Δh ). The matrix Δh ∈ RN ×N is symd metric positive definite and possesses the i=1 (2Ni − 1) eigenvalues λn =
d d 2 4 nj π nj π 2 1 − cos = , sin h2 2Nj h2 4Nj j=1 j j=1 j
where n = (n1 , . . . , nd ) is a multi-index with nj ∈ {1, . . . , 2Nj − 1}. The corresponding eigenvectors are ⎧ ! d ⎨cos i nj π j 2Nj ! φn (i) = ⎩sin ij nj π j=1 2Nj
if nj is odd, if nj is even
for all i ∈ G0 .
Proof. See [324, Lemma 13].
For illustration we focus on the 1d case d = 1. As spatial discretization we subdivide the interval Ω = (−a, a) into 2N1 uniform subintervals of length h = a/N1 . In conclusion, the state space for a finite difference approximation of (3.7d) respecting boundary conditions is FN with N = 2N1 − 1. We introduce the difference operators δh+ , δh− : FN → FN with components (δh+ x)j
1 := h
(δh− x)j :=
1 h
xj+1 − xj
for − N1 < j < N1
−xN
for j = N1 ,
x1
for j = −N1 ,
xj−1 − xj
for − N1 < j ≤ N1 ,
and a finite difference version of the second order spatial derivative uxx is given by the product δh+ δh− : FN → FN with tridiagonal matrix representation ⎞ 2 −1 ⎟ ⎜−1 2 −1 ⎟ 1 ⎜ ⎟ ⎜ . . . .. .. .. Δh := 2 ⎜ ⎟ ∈ RN ×N , ⎟ h ⎜ ⎝ −1 2 −1⎠ −1 2 ⎛
where entries not explicitly stated are assumed to be zero.
(3.7f)
3.7 Applications
177
Lemma 3.7.10 (properties of Δh ). Suppose d = 1 and N = 2N1 − 1. The matrix Δh ∈ RN ×N has eigenvalues λn > 0 with corresponding eigenvectors φn ∈ RN ,
λn =
1 N +1 2 2
a
1 − cos Nnπ +1
!
⎛⎧ ⎞N1 −1 ! ⎨cos j nj π if n is odd, N +1 ! ⎠ , φn = ⎝ ⎩sin j nj π if n is even N +1
j=−N1 +1
for n = 1, . . . , N , where φ1 , . . . , φN are orthonormal in L2N . Moreover, one has: (a) 0 < λn < λn+1 for all n = 1, . . . , N!− 1. √
2 (b) λn+1 − λn ≥ 23 Na+1 sin Nπ+1 , if N ≥ 5 and
N −2 6
≤n≤
2N −1 6 .
Proof. The expression for the eigenvalues λn > 0 follows from Lemma 3.7.9 with d = 1. Assertion (a) is a direct consequence of monotonicity properties for the cosine function. In order to prove (b), elementary trigonometric identities yield λn+1 − λn =
N +1 2 a
sin
π N +1
! sin
π(2n+1) N +1
!
for all 1 ≤ n < N
√ and we have sin x ≥ 23 for all x ∈ π3 , 2π 3 . Provided N ≥ 5 there exist positive 2π integers n such that π3 ≤ (2n+1)π N +1 ≤ 3 and the claim follows.
Finite Differences with Periodic Boundary Conditions Instead (3.7d) we now work with periodic boundary conditions and consider ut − δ(t)uxx = 0 in R,
u(t, 0) = u(t, l)
(3.7g)
with some period l > 0. Concerning the spatial discretization we subdivide the periodicity interval [0, l] into N ≥ 3 uniform subintervals of length h = l/N . Thus, the state space for a finite difference approximation of (3.7g) respecting periodic boundary conditions is the set of N -periodic sequences in F, which will be canonically identified with FN . On this set we introduce the difference operators δh− , δh+ : FN → FN with components δh+ xj
1 := h
xj+1 − xj
x1 − xN 1 x1 − xN δh− xj := h xj−1 − xj
for j = 1, . . . , N − 1, for j = N,
(3.7h)
for j = 1, for j = 2, . . . , N
and a finite difference version of the second order spatial derivative is given by δh+ δh− : FN → FN . The operators δh+ and δh+ δh− allow the respective representation
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3 Linear Difference Equations
⎞
⎞ ⎛ −1 1 2 −1 −1 ⎟ ⎟ ⎜ ⎜−1 2 −1 −1 1 ⎟ ⎟ 1⎜ 1 ⎜ ⎟ ⎟ ⎜ ⎜ + . . . . . .. .. .. .. .. Dh := ⎜ ⎟ , Δh := 2 ⎜ ⎟ ∈ RN ×N , ⎟ ⎟ h⎜ h ⎜ ⎝ ⎝ −1 1 ⎠ −1 2 −1⎠ 1 −1 −1 −1 2 ⎛
where entries not explicitly stated are assumed to be zero. For the stiffness matrix A = Δh we now deduce: Lemma 3.7.11 (properties of Δh ). The matrix Δh ∈ RN ×N has the eigenvalues νn ∈ R with corresponding eigenvectors φn ∈ CN given by νn = 4N 2 sin2
nπ N
N −1 φn = exp 2π ijn N j=0
,
for all n = 1, . . . , N,
respectively, where φ1 , . . . , φN are orthonormal in L2N . Moreover, one has: 3 4 (a) 0 < νn < νn+1 for all n = 1, . . . , N2 . √ π (b) νn+1 − νn ≥ 2 3N 2 sin N , if N ≥ 5 and n ≥ 1 with N 6−3 ≤ n ≤ 2N6−3 . Remark 3.7.12. The function ϕ : [2, ∞) → [4, ∞), ϕ(t) := t2 sin πt is strictly increasing and thanks to limt→∞ ϕ(t) = ∞ the gaps in the spectrum of Δh in both Lemmata 3.7.11 and 3.7.10 become arbitrarily large as N → ∞. Proof. Concerning the eigenvalues νn ∈ R and pair-wise orthogonal eigenvectors φn ∈ CN , we refer to [3, p. 848, Example 12.1.3]. Assertion (a) is a direct consequence of monotonicity properties for the sine function. In order to prove (b), we remark that elementary trigonometric identities yield νn+1 − νn = 4N 2 sin
(2n+1)π N
! sin
π N
for all n = 1, . . . , N − 1
√ and that we have sin x ≥ 23 for all x ∈ π3 , 2π 3 . Under the condition N ≥ 5 there exist positive integers n such that π3 ≤ (2n+1)π ≤ 2π N 3 and the claim follows.
We can also interpret Δh as linear operator on a discrete Lebesgue space. In general, a discrete Lebesgue space LpN is simply FN canonically normed via xLp := N
⎧ ⎨
h
N
⎩sup
p
j=1 |xj |
1≤j≤N
!1/p
|xj |
if p ∈ [1, ∞), if p = ∞
for all x ∈ FN
and in particular, the discrete L2N -product reads as x, yL2 := h N Directly from Jensen’s inequality we deduce xLp ≤ xLq N
N
for all x ∈ FN and 1 ≤ p ≤ q.
N j=1
xj yj .
(3.7i)
3.7 Applications
179
Further, having quantitative information on the operator Δh at hand, one can define 2s discrete Sobolev spaces HN as follows: We equip FN with inner products x, yH 2s := N
N (1 + νj )2s x, φj y, φj for all s > 0, x, y ∈ FN j=1
and define the discrete Dirichlet inner product by x, yΔh := h x, Δh y. If we
introduce a semi-norm |x|Δh := x, xΔh on FN , then xH 1 = N
x2L2 + |x|2Δh
for all x ∈ FN
N
(3.7j)
and using the difference operators δh− , δh+ from (3.7h) one easily deduces x, xΔh = −h
N
xj δh+ δh− xj = h
j=1
N
δh+ xj δh+ xj
for all x ∈ FN .
(3.7k)
j=1
Note that ·L2 is related to the above inner product on FN and ·H 1 via N
xL2 = N
N
h x, x,
xL2 ≤ xH 1 N
for all x ∈ FN ,
N
(3.7l)
respectively. Lemma 3.7.13. If 1 ≤ q ≤ p < ∞, then for all x ∈ FN one has: (a) The interpolation inequality 2
|x|Δh ≤ xL2 Δh xL2 . N
(3.7m)
N
1 → L∞ (b) The continuous embedding HN N with 2
2
xL∞ ≤ 3 xH 1 . N
(3.7n)
N
(c) The discrete Gagliardo–Nirenberg inequality 2
p
xLp ≤ xL2 + 2 xL2 |x|Δh N
N
!(p−q)/2
N
q
xLq
N
(3.7o)
and moreover the following relations hold true: + 2 + 2 D x 2 = |x|2 , D x = Δh x2 2 , Δ LN h h LN Δh h ! + 4 D x 4 ≤ 6 x2 2 + Δh x2 2 |x|2 . h L L Δh L N
N
N
(3.7p) (3.7q)
180
3 Linear Difference Equations
Proof. See [310, Lemmata 3.4–3.7].
Finite Elements with Zero Boundary Conditions We again restrict to the case d = 1 when discretizing the Laplacian in (3.7d) with zero boundary conditions u(−a) = u(a) = 0. Using the terminology from Example 2.6.9, we choose N ∈ N and define h = N2a +1 , xi := −a + ih, i = 0, . . . , N + 1, yielding a triangulation of Ω = (−a, a). Our approximation of L2 (−a, a) is v(−a) = v(a) = 0 and v|[xi ,xi+1 ] V := u ∈ C[−a, a] : is affine-linear for i = 0, . . . , N and we get a basis φ1 , . . . , φN ∈ V by postulating φi (xj ) = δi,j for i, j ∈ [1, N ]Z . The resulting stiffness matrix A is the same as in the above finite difference case and given by (3.7f), whose properties have been summarized in Lemma 3.7.10.
3.8 Remarks Basics: A treatment of explicit finite-dimensional linear difference equations can be found in standard textbooks like [3, p. 49ff, Chap. 2], [133, p. 105ff, Chap. 3] (invertible systems) or [294, p. 63ff, Chap. 3]. Due to the commonly imposed assumption (3.1a) we essentially deal with explicit linear equations as well. Indeed, for an operator Bk+1 with nontrivial kernel, the class (Lg ) or (L0 ) includes difference-algebraic problems. However, in finite-dimensional spaces, a corresponding general theory for such implicit linear difference equations (Lg ) has been developed in [305]. The variation of constants formula in Theorem 3.1.16 is a tool whose importance for the linear and nonlinear theory cannot be overestimated. In fact, it enables us to extend many results on linear equations to semilinear problems by means of a suitable perturbation theory (cf. Chap. 4). A variation of constants formula for nonlinear difference equations with applications to stability problems under nonlinear perturbations can be found in [311]. General results on the stability of linear systems are treated in research monographs like [3, 4, 294], or countless papers from with we only quote [16, 17, 32, 422,452]. Stability and perturbation criteria for evolution operators are addressed in [175, p. 143ff, Chaps. 9–10]. In particular, a converse to the principle of linearized asymptotic stability from Remark 3.5.9(2) is due to [185]. In finite-dimensional state spaces, [298, pp. 24–33] provides various characterizations for autonomous linear difference equations to be asymptotically stable in terms of conditions on their coefficient matrix. Various equivalent characterizations of uniform asymptotic stability for linear infinite-dimensional systems have been given in [287,288]. Keeping
3.8 Remarks
181
in mind that an exponential dichotomy means uniform asymptotic stability in the invariant vector bundles, the conditions of [287] should also yield equivalent characterizations of dichotomies. Another important issue in this context is the problem of reducibility, i.e., finding a uniformly bounded time-varying change of variables of which the inverse is also uniformly bounded (a Lyapunov transformation), which transform an equation into block diagonal form. An exponential dichotomy or trichotomy is sufficient for this (cf. [240]), and N -splittings yield reducibility into N diagonal blocks (cf. [436, Reduction theorem]). A classification of reducible systems via invariant projectors and spectral conditions has been given in [178]. Periodic linear equations: Criteria for the existence of periodic solutions can be found in [294, p. 75ff, Sect. 3.4], a corresponding Fredholm alternative is due to [106, Theorem 2] and a further Fredholm theory (with application to bifurcation problems) has been developed in [202]. In finite dimensions, a Floquet theory for invertible difference equations, can be found in various textbooks (cf. [3, p. 70, Theorem 2.9.2(b)], [294, p. 94, Theorem 4.4.2] or [133, Theorem 3.30]). The paper [340] investigates a Floquet representation of linear inhomogeneous equation. In general, invertibility of the coefficient matrix seems to be a commonly made assumption. In part this can be motivated due to Remark 3.2.7(0). Nonetheless, a corresponding approach for noninvertible equations is due to [203]. Finally, a classification of kinematically similar linear periodic equations (with possibly periodic Lyapunov transformation) has been given by [177] in terms of algebraically verifiable conditions. Invariant splittings and exponential growth: Invariant splittings are a device to extend the autonomous generalized eigenspace decomposition to an infinitedimensional and time-dependent setting. In doing so, finite-dimensional difference systems with a finite invariant splitting can be brought into block-diagonal form (cf. [436, Reduction theorem]) via a Lyapunov transformation. Results on asymptotic diagonalization of finite-dimensional systems are due to [62, 63], and apply to linear equations with full dichotomy spectrum. The use of exponentially weighted sequence spaces in dynamics dates back at least to [230]. In the work of Aulbach and his students, beginning with [19] (see [18] for the case of ODEs) such functions are called quasibounded – a terminology we have adopted in, e.g., [26, 374, 375, 385]. Dichotomies and splittings: The classical paper [304] features a remarkably modern approach to the stability theory of nonautonomous difference equations, which paved the way to the contemporary notion of an exponential dichotomy. Indeed, as shown in Example 3.4.1, Lyapunov exponents do not yield robust stability properties. Indeed, Lyapunov exponents in (0, 1) do not guarantee stability under nonlinear perturbations of order o(x). For this, one needs a so-called regular linear system (cf. [187, pp. 321–322, Theorem 65.3] for the ODE case), where Lyapunov exponents pairwise coincide and one has actual limits instead of lim sup’s in (2.4a) and (2.4b). We, nevertheless, need a spectral notion more appropriate for general
182
3 Linear Difference Equations
nonautonomous equations. For this, suppose the discrete interval I is unbounded above and the linear equation (L0 ) has an invariant projector P . The upper resp. lower forward Bohl exponents of (L0 ) are given by
βu+ (P ) := lim sup sup k Φ(k + κ, κ)P (κ)Xk+κ , k→∞ κ∈I
+ βl (P ) := lim inf sup k Φ(k + κ, κ)P (κ)Xk+κ k→∞ κ∈I
and backward Bohl exponents can be defined analogously for k → −∞. Definition 3.8.1. Suppose I is unbounded above and (3.1a), (3.1j) hold. Given an invariant N -splitting (Pn )N n=1 , the forward Bohl spectrum of (L0 ) is + (A, B; P ) := ΣB
N + βl (Pn ), βu+ (Pn ) . n=1
By relation (3.1r) the forward Bohl spectrum is invariant under linear conjugacy. There are various other spectral notions commonly used and we refer, for instance, in the ODE case to [118] for a discussion of their relationship. In particular, the Lyapunov spectrum is contained in the dichotomy spectrum and the Bohl spectrum is the dichotomy spectrum over a finest splitting of the state space. In various ways, exponential dichotomies (and N -splittings) are the appropriate hyperbolicity notion for nonautonomous equations: Dichotomous equations have robustness properties in a sufficiently large perturbation class (see, e.g., [201, p. 232, Theorem 7.6.7] or [341]), they are exactly the structurally stable linear systems (cf. [32, 291, 292]) and dichotomies allow to establish continuation results for bounded complete solutions (see [379, 380]). Hence, the importance of dichotomies is eminent. Parallel to the corresponding theory for ODEs (see, e.g., [98] or [413, 415–417, 419]) and despite the early contributions in [94], they are often defined for invertible difference equations only (see [240, 351, 353] and most of the other references). Historically there are two definition for an exponential dichotomy of noninvertible linear difference equations (we refer to [420] for the case of evolutionary differential equations). The “discrete dichotomy” popularized by Henry (see [201, p. 229, Definition 7.6.4], see also [94,439]), which seems to be commonly used (cf., e.g., [30, 369]), and the “forward dichotomy” from Kalkbrenner’s dissertation [241] (see also [24]), which even works without a regularity condition a` la (3.3c) and is sufficiently fruitful to infer a dichotomy spectrum (cf. [28]). Already Kalkbrenner recognized that his dichotomy notion lacks ∞ -roughness and therefore introduces a so-called regularity condition (cf. (3.3c) in our notation). With this additional assumption, both dichotomy definitions are equivalent (see [24, 135]). In this context, Examples 3.4.8, 3.5.6 and 3.6.7 are taken from [24, 241]. Our notion of an exponential dichotomy extends the one commonly used in the literature: There is no hyperbolicity condition like ai 1 bi on the growth
3.8 Remarks
183
rates; we are bounding the exponential growth in the invariant vector bundles, rather than dealing with solutions decaying to zero exponentially. In addition, instead of assuming that the growth in the invariant vector bundles is bounded by the usual exponential function eα (n, l) = αn−l , we allow the bound ea (n, l) with a timedependent sequence a(n). This has technical advantages, when it comes to time discretizations with variable stepsizes (see Lemmata 4.9.16 and 4.9.20). Furthermore, it is an approach to describe nonexponential growth occurring naturally in time-variant situations. There exists a close relationship of our approach to the theory of (h, k)-dichotomies (cf. [330,363,364]), where the dichotomy estimates (3.4g) 2 are assumed to hold with a quotient h(n) h(l) instead of ea (n, l). In order to approximate nonhyperbolic heteroclinic orbits, a dichotomy notion allowing polynomial growth was discussed in [217, p. 29ff] and [56]. So-called p -dichotomies and -trichotomies provide a further extension of classical exponential dichotomies and trichotomies (in the sense of [134, 358]). In case the linear part of semilinear equations satisfies such an assumption, results on the asymptotic behavior can be found in [323]. For general nonautonomous difference equations it is difficult to verify an exponential dichotomy. Indeed this is an infinite-dimensional problem, since a dichotomy is effectively related to the spectrum of a weighted shift operator (Lx)k := Bk+1 xk+1 − Ak xk
(3.8a)
on an ambient sequence space (cf. [30, 31, 378]). The situation drastically simplifies for autonomous or p-periodic equations, where knowledge of the spectrum for the evolution operator on a discrete interval of length 1 resp. p yields an exponential dichotomy. Similarly, for almost-periodic equations the knowledge of Φ(k + K, k) for an ample K > 0 yields conditions sufficient for an exponential dichotomy (cf. [354], [7, Theorem 3.1] or also [439, Theorem 2]). In [352] it is shown that if an invertible difference equation on Z has an exponential dichotomy on sufficiently many intervals of sufficiently large fixed length and with uniform constants and exponents, then it has an exponential dichotomy on the whole axis Z. For general time-dependence, [221] develops a numerical approach to compute dichotomy rates and projectors. Also perturbation arguments allow to approximate invariant projectors by solving (large) systems of linear algebraic equations. Precisely, the methods from [384] or Sect. 4.8 also apply to linear homogeneous perturbations of a dichotomous equation. Unfortunately, the notion of an exponential trichotomy is not consistently used in the literature. In our terminology, the center vector bundle contains all bounded complete solutions and is a generalization of the classical autonomous setup with
An equation with an exponential dichotomy and growth rates a, b has a (h, k)-dichotomy with h(n) = ea (n, κ) and k(n) = eb (n, κ). Conversely, a (h, k)-dichotomy implies an exponential 2
dichotomy with a(n) :=
h (n) , h(n)
b(n) :=
k (n) . k(n)
184
3 Linear Difference Equations
semisimple eigenvalues on the unit circle – for instance, [307] or [25] proceed accordingly. This differs from the intrinsically nonautonomous situation from [134, 358], where the center bundle consists of complete solutions which decay exponentially in both directions. The dichotomy spectrum for invertible difference equations has been introduced in [49] (see also [29]), and a generalization to the noninvertible case is due to [28]. The latter approach can be traced back to the work on ODEs of [413, 415– 417, 419]. It has been shown in [436, Corollary 2.2] that the dichotomy spectrum is invariant under linear conjugacy. A different spectral notion (the so-called spectral dichotomy) was developed in [30, 31] using the spectrum σ(L) of the weighted difference operator L from (3.8a). It has been observed in [378] that both concepts are connected by the formula Σ(A, B) = σ(L) ∩ (0, ∞); this relation enables us to use results from [48] in order to compute dichotomy spectra for certain difference equations. Moreover, one can show that Σ(A, B) is invariant under perturbations decaying to 0 (see [378]). Nevertheless, in general the set Σ(A, B) can be approximated only numerically (see [219] and [118] for the corresponding theory of ODEs). We finally mention applications of exponential dichotomies. They allow an elegant proof of the Shadowing Lemma and Smale’s Theorem (see [241, 351]), form the basics for nonautonomous continuation results (see [64, 378]) and a bifurcation theory (see [414], [392, pp. 81–114, Chap. 4], [381]). It turned out that the interplay between forward and backward dichotomy spectra (associated to dichotomies on − Z+ κ and Zκ ) is of crucial importance here. Admittedly, we are not aware of a thorough approach to analyze dichotomy spectra of infinite-dimensional equations. In particular, this refers to the geometric implications concerning spectral manifolds, i.e., invariant vector bundles. In order to circumvent this problem, we are working with exponential splittings. Admissibility: A pair 1 , 2 of sequence spaces is called admissible for a linear homogeneous difference equation (L0 ), if for each linear inhomogeneous perturbation g ∈ 1 there exists a (unique) solution of (Lg ) in the space 2 . Thus, our + Theorems 3.5.3 and 3.5.4 show that the pairs X+ κ,c , Xκ,c and Xc,d , Xc,d are admissible. These results are inspired by [241, 351]. Further admissible sequence spaces have been investigated in [94,332], in [365] for (h, k)-dichotomies and an operatortheoretical approach to such problems can be found in [30]. In particular, almost periodic equations are considered in [215]. We emphasize that admissibility properties also characterize hyperbolic exponential dichotomies. Indeed, a linear equation is hyperbolic on the whole integer axis Z, if and only if for every bounded perturbation there exists a unique bounded solution (cf. [201, p. 230, Theorem 7.6.5], [439, Theorem 1], see also [369] for an approach via discrete skew product flows). A corresponding result for dichotomies on the half-line Z+ κ can be found in [224] – here dichotomies are characterized via the operator L being onto and the space of bounded forward solutions being complemented (which always holds in finite dimensions). Moreover, for finite-dimensional implicit equations (L0 ), the existence of 2 -solutions is characterized in terms of a hyperbolic exponential dichotomy (see [50]).
3.8 Remarks
185
Roughness: The crucial property of exponential dichotomies or more general splittings is their ∞ -robustness. For this reason, corresponding results have an extensive history. Completely ignoring the preceding work on continuous time systems, ∞ perturbation results for dichotomies of invertible difference equations can be found in [356] and [351, Proposition 2.10]; they found generalizations to the noninvertible case in [201, p. 232, Theorem 7.6.7] or [241]. Roughness issues in a discrete skew product setting are addressed in [88, 369]. Corresponding results for exponential trichotomies are due to [134, 358]. For 0 -roughness results we refer to [58, Proposition 2.5] and [217, p. 36, Satz 2.7]. Finally, roughness of (h, k)-dichotomies for invertible difference equations was investigated in [307, Lemma 3.1], [341]. Our fixed point argument for the proof of Theorem 3.6.5 is different from the approaches in [201, 241, 351] and inspired by [217, 271]. As mentioned above, it provides a quite flexible technical framework which is easily adapted to parameter-dependent equations or dichotomies describing nonexponential growth. However, when explicit perturbation bounds are of minor interest, then spectral properties of shift operators yield elegant roughness results using some elementary operator theory (see [30, 31, 378]). Applications: Our discretization scheme for linear FDEs generalizes the Euler method from [155]. In such discretizations one typically meets companion matrices, whose properties have been summarized in [251]; for detecting gaps in the spectrum of companion matrices we refer to [405]. Temporal discretizations of linear FDEs based on a semigroup formulation in the space Cr are investigated in [46]. Dealing solely with discretizations via the evolution family U (tk+1 , tk ), k ∈ I , we neglected approximation schemes of general C 0 -semigroups. For this, we refer to the monograph [144, p. 157ff] or the papers [36, 65]. For the growth of operator powers see [37]. There is an enormous literature on full discretizations of (linear) autonomous parabolic equations, from which we quote [296, p. 129ff] (finite differences), [148, p. 279ff, Chap. 6] or [388, pp. 363, Chap. 11] (finite elements). A theory of discrete Sobolev spaces becomes particularly important when dealing with finite difference discretizations. Corresponding results can be found in [309, 310] or [461].
Chapter 4
Invariant Fiber Bundles
In a broad range of qualitative studies on nonlinear dynamical systems, invariant manifolds are omnipresent and play a crucial role for local as well as global questions: For instance, local stable and unstable manifolds dictate the saddle-point behavior in the vicinity of hyperbolic solutions (or surfaces) of a system. As illustrated by the celebrated reduction principle of Pliss, center manifolds are a paramount tool to simplify given dynamical systems in terms of a reduction of their state space dimension. Concerning a more global perspective, stable manifolds serve as separatrix between different domains of attractions and allow a classification of solutions with a specific asymptotic behavior. Systems with a gradient structure possess global attractors consisting of unstable manifolds (and equilibria). Finally, so-called inertial manifolds are global versions of the classical center-unstable manifolds and yield a global reduction principle for typically infinite-dimensional dissipative equations. The invariant fiber bundles introduced in this chapter generalize invariant manifolds from the well-known autonomous dynamical systems to nonautonomous difference equations. Precisely, we call a nonautonomous set W in the extended state space X a (forward) invariant fiber bundle1 of (D), if it is (forward) invariant and each fiber W(k) is a submanifold of a linear space Xk for k ∈ I. The contents of this chapter can be summarized as follows: •
From a technical perspective it is advantageous to initially work with semilinear implicit difference equations. For this type of systems, we provide an existence and uniqueness criterion for forward and backward solutions, as well as assumptions guaranteeing the existence of a nontrivial global attractor. • In the following section, we present and discuss a fairly general version of an existence theorem for invariant fiber bundles of semilinear equations. It applies to non-invertible implicit nonautonomous difference equations, whose linear part can be pseudo-hyperbolic, i.e., associated to an arbitrary spectral splitting. More detailed, each gap in an exponential splitting (see Fig. 3.4) gives rise to two
1 We refer to [1, p. 184, Definition 3.4.27] for the general notion of a fiber bundle in differential topology. In this sense, our fiber bundles W are trivial with the discrete interval I as base space and submanifolds W(k) as fibers.
C. P¨otzsche, Geometric Theory of Discrete Nonautonomous Dynamical Systems, Lecture Notes in Mathematics 2002, DOI 10.1007/978-3-642-14258-1 4, c Springer-Verlag Berlin Heidelberg 2010
187
188
•
•
• •
•
4 Invariant Fiber Bundles
invariant fiber bundles intersection along a complete (exponentially) bounded solution. These fiber bundles consist of solutions with a particular exponential growth behavior in forward resp. backward time – thus, our approach is based on the Lyapunov–Perron method. Using a less functional analytical argument, we also construct nontrivial intersections of invariant fiber bundles yielding an extended hierarchy. Whereas in Sect. 4.2 we construct invariant fiber bundles, we next investigate the asymptotic behavior of solutions which are not contained in these bundles. Indeed, due to our general pseudo-hyperbolic framework, attractivity properties of invariant fiber bundles need to be generalized to exponential boundedness of solutions approaching the bundle. Here, we work with invariant foliations which are equivalence classes of solutions converging towards a given solution at an exponential rate. In order to obtain an asymptotic phase property, we track a particular solution starting on a fiber bundle. Besides existence we next tackle the smoothness of invariant fiber bundles, which is of fundamental importance in applications. In particular, an elementary yet lengthy proof for the differentiability is presented: Fundamentally based on the classical contraction mapping principle only, none of the classical approaches, i.e., Banach space scale techniques, a Henry-type lemma or the fiber contraction theorem, is involved. As prerequisite for persistence under perturbation or discretization, we also establish the normal hyperbolicity of invariant fiber bundles. The subsequent sections present two applications of this flexible framework. First, we weaken the global assumptions and obtain (pseudo-) stable and unstable fiber bundles, which are related to given (pseudo-) hyperbolic reference solutions and describe the local saddle-point structure around them. W.r.t. the aspects given above, the corresponding Theorem 4.6.4 extends stable manifold theorems commonly found in the literature. Intersections of these pseudo-stable and -unstable bundles yield center-like bundles and in particular the classical hierarchy of the stable, center-stable, center, center-unstable and the unstable bundle. The centerunstable fiber bundle’s asymptotic phase enables us to derive a nonautonomous version of Pliss’ reduction principle. It states that stability properties of nonhyperbolic solutions are determined by their behavior on the center-unstable fiber bundle. In order to apply it, we address local approximation issues of invariant fiber bundles by means of Taylor series. Differing from the autonomous situation, the time-dependent Taylor coefficients are bounded solutions of a linear difference equation rather than solutions of a linear algebraic problem. Furthermore, discrete versions of inertial manifolds are constructed. Despite not having an asymptotic phase, they still possess the beneficial property of being asymptotically complete. Beyond the situation of classical center-unstable manifolds, inertial fiber bundles allow a global reduction principle guaranteeing that the essential dynamics of a possibly infinite-dimensional problem is given by a finite-dimensional difference equation. In particular, inertial bundles contain the global attractor of dissipative equations.
4.1 Semilinear Difference Equations
189
•
Last but not least, we discuss an approximation method for the invariant fiber bundles. It is based on fixed point iteration for the Lyapunov–Perron operator. A corresponding error estimate justifies that one can pass over to the so-called truncated Lyapunov–Perron operator, which involves only finite sums. In a nutshell, this means that invariant fiber bundles can be approximated by solving nonlinear systems of algebraic equations, which can be efficiently and successfully achieved using, e.g., Newton methods from numerical analysis. • Our theoretical results from Sect. 4.1 apply to full discretizations of semilinear FDEs. Using a simple example we show that under corresponding assumptions, attractors of discretized semilinear DDEs can be nontrivial. The following subsection on time-discretized abstract evolutionary equations shows how global integral manifolds can be constructed using an appropriate discretization from Sect. 2.6.2; moreover, we provide criteria for the existence of a global attractor. Using two examples, we illustrate how hierarchies of invariant fiber bundles can be constructed for temporal discretizations of parabolic evolution equation. For full discretizations of scalar RDEs and the complex Ginzburg–Landau equation we prove the existence of inertial fiber bundles – here, our results are quantitative and we obtain explicit dimension estimates. The algorithm from Sect. 4.8 is exemplified in order to approximate the inertial manifold of a scalar RDE of Chafee–Infante type. Throughout the chapter, we suppose that I is an unbounded discrete interval, the extended state space X consists of Banach spaces and Y of linear spaces.
4.1 Semilinear Difference Equations During this opening section and beyond, nonautonomous equations of the form Bk+1 x = Ak x + fk (x, x )
(S)
are in the center of our interest. As opposed to (Sg ) studied in Sect. 3.5, we do not suppose that (S) admits the trivial solution. One denotes (S) as semilinear, when it is studied using perturbation techniques on the basis of an established linear theory applicable to Bk+1 x = Ak x,
(L0 )
where Ak , Bk are as in Definition 3.1.1. For instance, this is possible for globally Lipschitzian or linearly bounded nonlinearities fk . Note that for a linearly implicit equation the functions fk do not depend on their second argument x . Hypothesis 4.1.1. Suppose that the linear homogeneous equation (L0 ) satisfies (3.1a) and fk : Xk × Xk+1 → Yk+1 fulfills fk (Xk , Xk+1 ) ⊆ im Bk+1 , −1 fk < 1 lip2 Bk+1
for all k ∈ I .
(4.1a)
190
4 Invariant Fiber Bundles
Remark 4.1.2. The global Lipschitz condition (4.1a) trivially holds for linearly implicit equation (S). In various discretizations it can be fulfilled for small temporal stepsizes. Especially for the θ-method from Example 2.1.4 or the 2-stage θ-method in Example 2.1.6, small values of θ ∈ [0, 1] ensure (4.1a). Proposition 4.1.3. Let m ∈ N. Under Hypothesis 4.1.1 the following holds: (a) The general forward solution ϕ to (S) exists on X . −1 −1 (b) If Bk+1 Ak ∈ L(Xk , Xk+1 ) and Bk+1 fk (·, x ) : Xk → Xk+1 , x ∈ Xk+1 , is continuous for all k ∈ I , then also ϕ is continuous. −1 −1 (c) If Bk+1 Ak ∈ L(Xk , Xk+1 ) and Bk+1 fk ∈ C m (Xk × Xk+1 , Xk+1 ) for all k ∈ I , then also ϕ(k; κ, ·) ∈ C m (Xκ , Xk ) for all κ ≤ k. Proof. The claims (a) and (b) are an immediate consequence of Theorem 2.3.6, while assertion (c) follows from Theorem 2.3.9. Next we formulate a dual version of Proposition 4.1.3 for backward solutions. Proposition 4.1.4. Let m ∈ N and suppose that (L0 ) satisfies (3.1a) with −1 Bk+1 Ak ∈ GL(Xk , Xk+1 ), k ∈ I . If fk : Xk × Xk+1 → Yk+1 fulfills −1 −1 Ak )−1 L(X ,X ) lip1 Bk+1 fk < 1 fk (Xk , Xk+1 ) ⊆ im Bk+1 , (Bk+1 k+1
k
for all k ∈ I , then the following holds: (a) The general backward solution ϕ to (S) exists on X . −1 (b) In case Bk+1 fk (x, ) : Xk+1 → Xk+1 , x ∈ Xk , is continuous for all k ∈ I , then ϕ is continuous. −1 (c) In case Bk+1 fk ∈ C m (Xk × Xk+1 , Xk+1 ) for all k ∈ I , then one has the inclusion ϕ(k; κ, ·) ∈ C m (Xκ , Xk ) for all k ≤ κ. Proof. In order to construct backward solutions, we proceed as in Proposition 4.1.3 using Theorem B.1.1 for resp. Theorem B.1.5 for (c) applied to the fixed (a), (b), −1 −1 point problem Bk+1 Ak x − Bk+1 fk (x, x ) = x for all k ∈ I , x ∈ Xk+1 . Having the variation of constants formula from Theorem 3.1.16 available, we can prove dissipativity results for semilinear equations (S). While the Lipschitz conditions assumed in Theorem 3.5.8 implied a condition for exponential stability (see Remark 3.5.9(1)), we now are interested in boundedness properties: Proposition 4.1.5. Suppose that beyond Hypothesis 4.1.1 the following holds: −1 (i) Bk+1 Ak ∈ L(Xk , Xk+1 ), k ∈ I , and there exist K ≥ 1 and a : I → (0, ∞) with
Φ(k, l) L(Xl ,Xk ) ≤ Kea (k, l) for all l ≤ k. (ii) There exist sequences βk , γk ≥ 0 and δ ∈ [0, 1/K) such that −1 B f (x, x ) ≤ β + max γ
x
, δ
x
k k k Xk Xk+1 k+1 X k+1
for all k ∈ I , x ∈ Xk and x ∈ Xk+1 .
(4.1b)
(4.1c)
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191
Then the general forward solution of (S) satisfies for all κ ≤ k, ξ ∈ Xκ that k−1 K e a+γK (k, κ) ξ Xκ + e a+γK (k, l + 1)βl .
ϕ(k; κ, ξ) Xk ≤ 1−δK 1−δK 1 − δK l=κ
Proof. Thanks to Corollary 3.1.18(a) the forward evolution operator Φ for (L0 ) exists and is bounded. From Proposition 4.1.3(a) we know that the general forward solution ϕ to (S) exists. For a fixed (κ, ξ) ∈ X we abbreviate ϕ(k) := ϕ(k; κ, ξ). Thus, according to Theorem 3.1.16(a) the sequence ϕ satisfies (3.1h)
ϕ(k) = Φ(k, κ)ξ +
k−1
−1 Φ(k, l + 1)Bl+1 fl (ϕ(l))
l=κ
for all κ ≤ k, where we have abbreviated fl (ϕ(l) := fl (ϕ(l), ϕ (l)) (see p. 54). Passing over to the norms, we obtain (4.1b)
ϕ(k) ≤ Kea (k, κ) ξ + K
k−1
−1 ea (k, l + 1) Bl+1 fl (ϕ(l))
l=κ (4.1c)
≤ Kea (k, κ) ξ + K
k−1
ea (k, l + 1)βl
l=κ
+K
k−1
ea (k, l + 1)γl ϕ(l) + δK
l=κ
k−1
ea (k, l + 1) ϕ (l) ,
l=κ
abbreviating u(k) := ea (κ, k) ϕ(k) yields u(k) ≤ K ξ + K
k−1
ea (κ, l + 1)βl + K
l=κ
k−1 l=κ
γl u(l) + δK u (l) a(l) k−1 l=κ
and from this we finally infer K u(k) ≤ 1 − δK
ξ +
k−1 l=κ
ea (κ, l + 1)βl
k−1
K γl + δ u(l) + 1 − δK a(l) l=κ
for all κ ≤ k. The Gronwall lemma in Proposition A.2.1(a) implies k−1 K K e1+b (k, κ) ξ + e1+b (k, l + 1)ea (κ, l + 1)βl u(k) ≤ 1 − δK 1 − δK (A.2b)
l=κ
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4 Invariant Fiber Bundles
with b(k) :=
K 1−δK
(A.1b)
ϕ(k) ≤
γk a(k)
+ δ and consequently by Proposition A.1.2(b)
k−1 K K ea+ab (k, κ) ξ + ea+ab (k, l + 1)βl 1 − δK 1 − δK l=κ
for all κ ≤ k, which was our claim.
Corollary 4.1.6. Suppose ε > 0 is fixed and I is unbounded below. If the assumptions of Proposition 4.1.5 hold with the summability condition ρk :=
k−1
e a+γK (k, l + 1)βl < ∞
l=−∞
1−δK
for all k ∈ I,
then the nonautonomous set A := (k, x) ∈ X : x Xk ≤ ε + bing for every absorption universe
Kρk 1−δK
ˆ is B-absor-
⎪ ⎪ ⎪ ⎪ Bˆ ⊆ B ⊆ X : lim e a+γK (k, k − n) ⎪ ⎪B(k − n)⎪ ⎪ = 0 for all k ∈ I , n→∞
1−δK
ˆ and B-uniformly absorbing for every absorption universe Bˆ ⊆
⎪ ⎪ ⎪ ⎪ B ⊆ X : lim sup e a+γK (k, k − n) ⎪ ⎪B(k − n)⎪ ⎪=0 . n→∞ k∈I
1−δK
Proof. Using Proposition 4.1.5 this can be shown as Corollary 2.3.19.
Corollary 4.1.7. Let I be unbounded below. If beyond Hypothesis 4.1.1 the map−1 −1 pings Bk+1 Ak , Bk+1 fk satisfy (4.1c) and a Darbo condition with −1 −1 Ak + dar Bk+1 fk ∈ [0, 1] for all k ∈ I , q(k) := dar Bk+1
then the semilinear difference equation (S) is: ˆ (a) B-contracting for every family Bˆ ⊆ B ⊆ S : lim eq (k, k − n)χk−n (B(k − n)) = 0 n→∞
for all k ∈ I .
ˆ (b) B-uniformly contracting, provided limn→∞ supk∈I eq (k, k − n) = 0. Proof. Let k ∈ I . By Proposition 4.1.3 the general forward solution ϕ of (S) exists on X . Referring to the proof of Theorem 2.3.6 one constructs the generator ϕˆk of ϕ using the fixed point equation (2.3e), which in the present situation reads as −1 −1 x = Bk+1 Ak ξ + Bk+1 fk (ξ, x) =: Tk (ξ, x)
for all (k, ξ) ∈ X .
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193
Due to the assumed inequality (4.1c) its right-hand side Tk (·, x) is bounded for every fixed x ∈ Xk+1 and [35, p. 39, Proposition 5.3(c)] guarantees −1 −1 Ak + dar Bk+1 fk = q(k) ≤ 1 for all κ ∈ I . dar Tk ≤ dar Bk+1
Hence, we can apply Corollary 2.3.8 yielding the claim.
This brings us to the main result of this section: Theorem 4.1.8. Let q ∈ [0, 1), suppose that I is unbounded below, the family Bˆ consists of all uniformly bounded nonautonomous sets and that beyond Hypothesis 4.1.1 the following holds for all k ∈ I : −1 Ak ∈ L(Xk , Xk+1 ) satisfies (4.1b). (i) Bk+1 −1 (ii) Bk+1 fk (·, x ) : Xk → Xk+1 , x ∈ Xk+1 , is continuous with (4.1c). k ∈ [0, q]. (iii) supl∈I βl < ∞ and one has the estimate a(k)+Kγ 1−Kδ
ˆ If the semilinear equation (S) is B-contracting, then it possesses a uniformly bounded global attractor A∗ , which additionally satisfies A∗ (k) ⊆ B
K sup β l∈I l (1−q)(1−Kδ)
(0, Xk ) for all k ∈ I.
Proof. First of all, by Proposition 4.1.3 the general forward solution ϕ of (S) exists and is continuous. Choose B ∈ Bˆ and by assumption (iii) with Proposition A.1.2(d) we get ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ e a+γK (k, k − n) ⎪ ⎪B(k − n)⎪ ⎪ −−−−→ 0 uniformly in k ∈ I, ⎪B(k − n)⎪ ⎪ ≤ qn ⎪ 1−δK
n→∞
k−1
sup
β
l∈I l e a+γK (k, l + 1)βl ≤ 1−q for all k ∈ I. So Corollary 4.1.6 1−δK ˆ ˆ i.e., ϕ is uniensures that ϕ has a closed B-uniformly absorbing set A ∈ B, ˆ formly bounded dissipative. We assumed that ϕ is B-contracting. Because A is ˆ B-uniformly absorbing, for each B ∈ Bˆ there exists an N = N (B) ≥ 0 such that (cf. Definition 1.3.6(b))
as well as
l=−∞
γBN (k) =
ϕ(k; k − n, B(k − n)) ⊆ A(k) for all k ∈ I
n≥N
ˆ Thus, Proposition 1.2.30 implies that ϕ is B-asymptotically ˆ and γBN ⊆ A ∈ B. compact. With this we have verified the assumptions of Theorem 1.3.9 and (S) admits a global attractor A∗ , which is uniformly bounded; in particular Theorem 1.3.9(b) implies the claimed bound on the fibers A∗ (k).
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4 Invariant Fiber Bundles
4.2 Existence of Invariant Fiber Bundles Every five years or so, if not more often, someone ‘discovers’ the theorem of Hadamard and Perron, proving it by Hadamard’s method of proof, or by Perron’s. D.V. Anosov (cf. [11]) For linear difference equations admitting an exponential splitting we have a solid understanding of their dynamical behavior. Indeed, by virtue of Remark 3.4.16 it was possible to characterize the set of c± -bounded solutions using kernels and ranges of associated invariant projectors – the invariant vector bundles form the skeleton of the extended state space. Moreover, for autonomous equations these vector bundles become generalized eigenspaces and can be determined using purely algebraic methods. In this section, we aim to extend the above observation to nonlinear equations, yielding invariant fiber bundles. As explicated in the introduction to this chapter, applications for invariant manifolds or fiber bundles cover local questions (behavior near reference solutions in Sect. 4.6, linearization in Chap. 5) as well as global ones (dynamics of dissipative equations in Sect. 4.7). For this reason, we need flexible existence theorems which apply to both cases and carry most of the technical preparations. It is the goal of the present section to tackle this problem for a sufficiently wide class of equations, namely semilinear equations as already considered above. According to this, our intent is centered around semilinear equations Bk+1 x = Ak x + fk (x, x ),
(S)
which are understood as a perturbation of the linear homogeneous system Bk+1 x = Ak x.
(L0 )
Thus, it will be a standing assumption throughout the remaining chapter that (L0 ) has an exponential splitting. The alert reader surely remembers that for autonomous or periodic linear equations (L0 ), one can formulate the following crucial hypothesis in terms of spectral properties for Ak , Bk (cf. Theorems 3.4.28 and 3.4.31). Hypothesis 4.2.1. Suppose that Ak ∈ Hom(Xk , Yk+1 ), Bk ∈ Hom(Xk , Yk ) has −1 an inverse with Bk+1 Ak ∈ L(Xk , Xk+1 ), k ∈ I , and that the linear equation (L0 ) admits a strongly regular exponential N -splitting on I with N > 1, namely S(A, B; P ) =
N −1
(bi+1 , ai ),
i=0
where the sequences bi are bounded above. The associated Green’s functions are abbreviated by Gi := GP1i , 1 ≤ i < N . ˆ Remark 4.2.2. If a linear equation (L0 ) is B-contracting, Bˆ denoting the family of uniformly bounded subsets of X , then the pseudo-unstable vector bundles P1i are finite-dimensional, as soon as bi ≥ 1 (cf. Proposition 3.4.24).
4.2 Existence of Invariant Fiber Bundles
195
In contrast to Sect. 4.1, we additionally impose that the nonlinearity fk satisfies a global Lipschitz condition, as opposed to the linear bound in (4.1c). It will be demonstrated in the upcoming Sects. 4.6–4.7 how such a restrictive assumption can be weakened when it comes to applications. Yet, we try to make our results as explicit as possible, since quantitative results pay off when it comes to domain estimates of locally invariant fiber bundles (see Sect. 4.6) or dimension estimates for inertial fiber bundles (see Sect. 4.7). Conditions for solutions of implicit difference equations to exist have been given throughout Sects. 2.2 and 2.3; moreover, the particular situation of semilinear equations (S) is addressed in Proposition 4.1.3. We, however, explicitly suppose the existence of forward solutions in order to remain flexible: Hypothesis 4.2.3. Let the general forward solution ϕ of equation (S) exist on X . Suppose that fk : Xk × Xk+1 → Yk+1 fulfills fk (Xk , Xk+1 ) ⊆ im Bk+1 for all k ∈ I and that we have the global Lipschitz estimates −1 Lj := sup lipj Bk+1 fk < ∞ k∈I
for j = 1, 2.
(4.2a)
Remark 4.2.4 (spectral gap condition). For an integer 1 ≤ i < N we require the i spectral gap condition that there exists a ςi ∈ 0, bi −a such that 2 max Ki− , Ki+ (L1 + bi L2 ) − + < ςi , 1 + max Ki , Ki L2
(Gi )
choose a fixed real number ς ∈ max Ki− , Ki+ (L1 + bi − ςi L2 ) , ςi and define intervals Γ¯i := [ai + ς, bi − ς]. (1) The gap condition (Gi ) guarantees that neither the real interval for ς nor Γ¯i + bi L2 ) itself is empty. If we introduce the function g : [0, ∞) → R by g(t) := t(L11+tL , 2 − + then (Gi ) can be written as g(max Ki , Ki ) < ςi . For later use we point out that g is strictly increasing on [0, ∞) from 0 to ∞. Thus, if g(t∗ ) < ςi holds for one t∗ > 0, one surely has g(t) ≤ ςi for all t ∈ (0, t∗ ]. (2) For semi-implicit equations (S) the gap condition (Gi ) simplifies to max Ki− , Ki+ L1 < ςi . (3) In order to give an intuition for the crucial condition (Gi ) we observe the following: Assume a more classical situation in which the linear part (L0 ) is autonomous and generates a bounded discrete semigroup ((B −1 A)k )k∈Z+ on a common 0 space X = Xk . Referring to Theorem 3.4.28, an exponential dichotomy holds, ˙ − into provided the spectrum σ(A, B) allows a decomposition σ(A, B) = σ+ ∪σ disjoint spectral sets σ+ , σ− ⊆ C such that maxz∈σ− |z| < ai < bi < inf z∈σ+ |z| with positive reals ai , bi (see Fig. 3.2). Moreover, we suppose (S) is linearly implicit, i.e., one has L2 = 0 and (Gi ) reduces to the above inequality. Hence, we are able
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4 Invariant Fiber Bundles
to fulfill the spectral gap condition (Gi ), if one of the following two conditions is satisfied: i (i) For a given spectral gap bi − ai and ςi ∈ 0, bi −a the nonlinear perturbation 2 fk is so weak that its Lipschitz constant L1 > 0 fulfills (Gi ). (ii) Given a fixed value for L1 > 0, the spectral gap bi −ai > 0 has to be sufficiently i large so that there exists a ςi ∈ 0, bi −a satisfying (Gi ). 2 Which of these perspectives (i) or (ii) is favorable, depends on the application. When dealing with local questions, i.e., in the context of invariant fiber bundles associated to fixed reference solutions (see Sect. 4.6), the first interpretation applies. For inertial fiber bundles (see Sect. 4.7), which are global in nature, the second one is crucial. Admittedly, the situation of (ii) changes for implicit nonlinearities, i.e., L2 > 0. In fact, we still have to require a smallness condition on L2 , no matter how large the gap bi − ai in the spectrum is. Remark 4.2.5 (growth condition). For 1 ≤ i < N we require the growth conditions −1 ∃κ ∈ I : Γκ+ (i) := sup Bk+1 fk (0, 0)X k∈Z+ κ
k+1
k∈Z− κ
k+1
−1 ∃κ ∈ I : Γκ− (i) := sup Bk+1 fk (0, 0)X
eai (κ, k) < ∞,
(Γi+ )
ebi (κ, k) < ∞,
(Γi− )
provided the discrete interval I is unbounded above resp. below: (1) If a constant Γκ± (i) exist for one κ ∈ I, then it exists for all κ ∈ I. (2) Besides in Sect. 4.6, we do not assume that the nonlinearity fk (0, 0) vanishes identically on I . Rather, we weaken this frequently made assumption to an exponential boundedness, i.e., the finite existence of Γκ± (i). More detailed, the condition (Γi+ ) is equivalent to the inclusion f· (0, 0) ∈ X+ κ,ai ,B , i.e., the sequence −1 − B·+1 f· (0, 0) is a+ -bounded and dually (Γ ) means f (0, 0) ∈ X− · i i κ,bi ,B , i.e., the se−1 − quence B·+1 f· (0, 0) is bi -bounded. By Lemma 3.3.26 this yields the implications + (Γi+ ) ⇒ (Γi−1 ),
− (Γi−1 ) ⇒ (Γi− )
for all 2 ≤ i < N.
We construct invariant fiber bundles of (S) using a functional analytical approach. For this, let (κ, ξ) ∈ X , c : I → (0, ∞) and suppose the discrete interval I is unbounded below. We aim to characterize the solutions of (S) which exist in backward time and are c− -bounded. Choose a fixed 1 ≤ i < N . For given φ ∈ X− κ,c , we formally define a sequence-valued mapping – the so-called Lyapunov–Perron operator Tκ− (φ; ξ) := Φ− (·, κ)P1i (κ)ξ + Pi 1
κ−1
−1 Gi (·, n + 1)Bn+1 fn (φ(n))
(4.2b)
n=−∞
resembling the Lyapunov–Perron sums in Theorem 3.5.3(b). The backward evolution operators Φ− of (L0 ) exist on P1i by the strongly regular splitting from P1i Hypothesis 4.2.1.
4.2 Existence of Invariant Fiber Bundles
197
The next lemma establishes a solid part of our notation: Lemma 4.2.6. Assume Hypotheses 4.2.1 and 4.2.3. If (Γi− ) holds and c, d : I → (0, ∞) satisfy c ∈ (ai , bi ),
0d≤c
for one 1 ≤ i < N,
(4.2c)
− then the mapping Tκ− : X− κ,c × Xκ → Xκ,d is well-defined with
− T (φ; ξ)− ≤ K − P i (κ)ξ + Ci (c) Γ − (i) + L(c) φ − , κ 1 κ i κ,c κ,d Xκ + − i Q1 (κ)Tκ− (κ, φ; ξ) ≤ Ki Γκ (i) + + (c) φ − i κ,c Xκ c − ai
(4.2d)
for all (κ, ξ) ∈ X , φ ∈ X− κ,c and we have Lipschitz estimates lip1 Qi1 (κ)Tκ− (κ, ·) ≤ + i (c),
lip1 Tκ− ≤ i (c),
lip2 Tκ− ≤ Ki−
(4.2e)
with the constants Ci (c) from Theorem 3.5.3 and i (c) := Ci (c)(L1 + c L2 ), + i (c) :=
Ki+
c − ai
(L1 + c L2 ),
L(c) := L1 + c L2 , − i (c) :=
Ki− (L1 + c L2 ). bi − c
Proof. Let (κ, ξ) ∈ X be given and choose growth rates c, d : I → (0, ∞) as required in (4.2c). We begin with preparatory estimates. For a sequence φ ∈ X− κ,c , − using the triangle inequality and (Γi ), one has (4.2a)
−1 − Bn+1 fn (φ(n)) ≤ Γκ− (i) + L(c) φ κ,c ec (n, κ) for all n ∈ Z− κ as in the proof of Theorem 3.5.8 (cf. (3.5h)). Using the splitting estimates (3.4g), we obtain almost identically to the proof of Theorem 3.5.3(b) that
Ki− − − Γκ (i) + L(c) φ κ,c bi − c
− Γκ− (i) + L(c) φ κ,c
i P (k)T − (k, φ; ξ) ed (κ, k) ≤ K − P i (κ)ξ + 1 κ 1 i i Q1 (k)Tκ− (k, φ; ξ) ed (κ, k) ≤
Ki+ c − ai
for all k ∈ Z− κ (cf. Proposition A.1.2). Combining these two estimates, one deduces the inclusion Tκ− (φ; ξ) ∈ X− κ,d as well as the first estimate (4.2d). The second relation (4.2d) follows from the latter above estimate by setting k = κ. To prove the ¯ Lipschitz estimates in (4.2e), let φ, φ¯ ∈ X− κ,c and ξ, ξ ∈ Xκ . We get from (3.4g) that i (4.2a) − ¯ ξ) ≤ Ki L(c) φ − φ¯− ec (k, κ) P1 (k) Tκ− (k, φ; ξ) − Tκ− (k, φ; κ,c bi − c
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4 Invariant Fiber Bundles
for all k ∈ Z− κ , and similarly i (4.2a) + ¯ ξ) ≤ Ki L(c) φ − φ¯− ec (k, κ) Q1 (k) Tκ− (k, φ; ξ) − Tκ− (k, φ; κ,c c − ai for all k ∈ Z− κ . Setting k = κ gives us the first relation in (4.2e). Multiplying both − above estimates with ed (κ, k), the definition of the norm · κ,d immediately yields the Lipschitz condition for Tκ− (·; ξ), i.e., the middle relation of (4.2e). Finally, using (4.2b), (3.4g), the remaining Lipschitz estimate in (4.2e) follows from − ¯ ed (κ, k) ≤ K − ξ − ξ¯ Tκ (k, φ; ξ) − Tκ− (k, φ; ξ) i
for all k ∈ Z− κ.
This establishes Lemma 4.2.6.
By virtue of the Lyapunov–Perron operator Tκ− from (4.2b), we will characterize the exponentially bounded solutions of (S) as its fixed points, and solve the corresponding problem using the contraction mapping principle. Lemma 4.2.7. Let (κ, ξ) ∈ X and assume Hypotheses 4.2.1 and 4.2.3 are fulfilled. If (Γi− ) holds, a sequence c : I → (0, ∞) satisfies (4.2c) and φ ∈ X− κ,c , then for the − mapping Tκ− (·; ξ) : X− → X the following statements are equivalent: κ,c κ,c (a) φ solves the difference equation (S) with P1i (κ)φ(κ) = P1i (κ)ξ. (b) φ is a solution of the fixed point equation φ = Tκ− (φ; ξ).
(4.2f)
Proof. Let (κ, ξ) ∈ X and define a sequence gk := fk (φ(k)) in Y for k ∈ I . As in the above proof of Lemma 4.2.6 one has the estimate
−1 − B ≤ Γ (i) + L(c) φ − ec (k, κ) for all k ∈ Z− g k κ κ κ,c k+1 with the constant L(c) from Lemma 4.2.6, and consequently g ∈ X− κ,c,B . − (a) ⇒ (b) Let φ : Z− → X be a c -bounded solution of (S) satisfying κ P1i (κ)φ(κ) = P1i (κ)ξ. Then φ also solves the linear inhomogeneous difference equation Bk+1 x = Ak x + gk and Theorem 3.5.3(b) implies assertion (b). (b) ⇒ (a) Conversely, a fixed point of Tκ− (·, ξ) is a solution of the above linear inhomogeneous equation, and thus of the semilinear equation (S) satisfying the relation P1i (κ)φ(κ) = P1i (κ)ξ. Lemma 4.2.8. Let (κ, ξ) ∈ X and assume Hypotheses 4.2.1 and 4.2.3. If (Gi ), − (Γi− ) hold and c ∈ Γ¯i , then Tκ− (·; ξ) : X− κ,c → Xκ,c has a unique fixed point − φκ (ξ) ∈ Xκ,c . The fixed point mapping φκ : Xκ → X− κ,c satisfies φκ (ξ) = i φκ (P1 (κ)ξ) and:
4.2 Existence of Invariant Fiber Bundles
199
(a) φκ : Xκ → X− κ,c is linearly bounded, i.e., for ξ ∈ Xκ it is −
φκ (ξ) κ,c ≤
− Ki− P1i (κ)ξ + Ci (c)Γκ (i) , X κ 1 − i (c) 1 − i (c)
+ − + i Q1 (κ)φκ (κ, ξ) ≤ Ki Γκ (i) + i (c) · Xκ c − ai 1 − i (c)
· Ki− P1i (κ)ξ Xκ + Ci (c)Γκ− (i) ,
(4.2g)
(b) φκ is globally Lipschitzian with lip φκ ≤
Ki− , 1 − i (c)
lip Qi1 (κ)φκ ≤
Ki− + i (c) , 1 − i (c)
(4.2h)
where the constants Ci (c), i (c) ∈ [0, 1), + i (c) are defined in Lemma 4.2.6. Proof. Let (κ, ξ) ∈ X be given. The spectral gap condition (Gi ) implies i (c) = max
Ki− Ki+ , c − ai bi − c
L(c) ≤
max Ki+ , Ki− L(c) < 1 ς
(4.2i)
for all c ∈ Γ¯i . Thus, from the middle estimate (4.2e) in Lemma 4.2.6 we know that Tκ− (·; ξ) is a contraction on the Banach space X− κ,c (cf. Lemma 3.3.25(a)) and the contraction mapping theorem (see, for example, [295, p. 361, Lemma 1.1]) implies the existence of a unique fixed point φκ (ξ) ∈ X− κ,c . Furthermore, the claimed relation φκ (ξ) = φκ (P1i (κ)ξ) follows from the fact Tκ− (·; ξ) = Tκ− (·; P1i (κ)ξ), and consequently the fixed points of the two contractions coincide. (a) Thanks to i (c) < 1 (see (4.2i)), the first estimate (4.2g) follows from (4.2f) − Tκ (φκ (ξ); ξ)−
φκ (ξ) − κ,c = κ,c
≤ Ki− P1i (κ)ξ + Ci (c) Γκ− (i) + L(c) φκ (ξ) − κ,c
(4.2d)
and the second estimate (4.2g) is a consequence of the above inequality and i Q1 (κ)φκ (κ, ξ) (4.2f) = Qi1 (κ)Tκ− (κ, φκ (ξ); ξ) (4.2d)
≤
Ki+ Γκ− (i) − + + i (c) φκ (ξ) κ,c c − ai
for all ξ ∈ Xκ .
(b) Next we derive the Lipschitz estimates in (4.2h). Let ξ, ξ¯ ∈ Xκ be given and from (4.2f), (4.2e) we obtain using the triangle inequality ¯ − ≤ K − ξ − ξ¯ + i (c) φκ (ξ) − φκ (ξ) ¯ − , φκ (ξ) − φκ (ξ) i κ,c κ,c
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4 Invariant Fiber Bundles
which yields the left relation in (4.2h). From (4.2f), (4.2b), (4.2e) one has i ¯ ≤ + (c) φκ (ξ) − φκ (ξ) ¯ − , Q1 (κ) φκ (κ, ξ) − φκ (κ, ξ) i κ,c and this in connection with the left estimate for φκ in (4.2h) established above, leads to the remaining estimate claimed in (4.2h). After these preparations we formulate and prove the main existence theorem for invariant fiber bundles. It states that the vector bundles Qi1 and P1i guaranteed by Hypothesis 4.2.1 and Remark 3.4.17 resp. Remark 3.4.18 persist globally as invariant fiber bundles Wi+ and Wi− over Qi1 and P1i , resp., under nonlinear perturbations as in Hypothesis 4.2.3. This includes the dynamical characterization from (3.4k) and (3.4l). Theorem 4.2.9 (of Hadamard–Perron). Assume Hypotheses 4.2.1 and 4.2.3 are satisfied and that (Gi ) holds for one 1 ≤ i < N . (a) If I is unbounded above and (Γi+ ) holds, then the nonautonomous set Wi+ := (κ, ξ) ∈ X : ϕ(·; κ, ξ) ∈ X+ κ,c is a forward invariant fiber bundle of (S), which is independent of c ∈ Γ¯i and possesses the representation as graph Wi+ = (κ, η + wi+ (κ, η)) ∈ X : (κ, η) ∈ Qi1 of a uniquely determined mapping wi+ : X → X with wi+ (κ, ξ) = wi+ (κ, Qi1 (κ)ξ) ∈ P1i (κ)
for all (κ, ξ) ∈ X
(4.2j)
and satisfying the invariance equation −1 wi+ (κ + 1, η1 ) = Bκ+1 Aκ wi+ (κ, η) −1 fκ (η + wi+ (κ, η), η1 + wi+ (κ + 1, η1 )), +P1i (κ + 1)Bκ+1 −1 −1 Aκ η + Bκ+1 fκ (η + wi+ (κ, η), η1 + wi+ (κ + 1, η1 )) η1 = Bκ+1
(4.2k) for all (κ, η) ∈ Qi1 , η1 ∈ Qi1 (κ + 1). Furthermore, for all c ∈ Γ¯i it holds: (a1 ) wi+ : X → X is linearly bounded, i.e., for (κ, ξ) ∈ X one has − + −
+ w (κ, ξ) ≤ Ki Γκ (i) + i (c) K + Qi1 (κ)ξ + Ci (c)Γκ+ (i) , i i Xκ Xκ bi − c 1 − i (c) (a2 ) wi+ (κ, ·) is globally Lipschitzian with lip2 wi+ ≤ ˜+ i (c).
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201
(b) If I is unbounded below and (Γi− ) holds, then the nonautonomous set there exists a solution φ : I → X of (S) − Wi := (κ, ξ) ∈ X with φ(κ) = ξ ∈ Xκ and φ|Z− ∈ X− κ,c κ is an invariant fiber bundle of (S), which is independent of c ∈ Γ¯i and possesses the representation as graph Wi− = (κ, η + wi− (κ, η)) ∈ X : (κ, η) ∈ P1i (4.2l) of a uniquely determined mapping wi− : X → X with wi− (κ, ξ) = wi− (κ, P1i (κ)ξ) ∈ Qi1 (κ) for all (κ, ξ) ∈ X and satisfying the invariance equation −1 Aκ wi− (κ, η) wi− (κ + 1, η1 ) = Bκ+1 −1 +Qi1 (κ + 1)Bκ+1 fκ (η + wi− (κ, η), η1 + wi− (κ + 1, η1 )), −1 −1 η1 = Bκ+1 Aκ η + Bκ+1 fκ (η + wi− (κ, η), η1 + wi− (κ + 1, η1 ))
(4.2m) for all (κ, η) ∈ P1i , η1 ∈ P1i (κ + 1). Furthermore, for all c ∈ Γ¯i it holds: (b1 ) wi− : X → X is linearly bounded, i.e., for (κ, ξ) ∈ X one has
+ − + − − i w (κ, ξ) ≤ Ki Γκ (i) + i (c) + Ci (c)Γκ− (i) , K P (κ)ξ 1 i i Xκ Xκ c − ai 1 − i (c) (4.2n) (b2 ) wi− (κ, ·) is globally Lipschitzian with lip2 wi− ≤ ˜− i (c), where the constants Ci (c), ± i (c), i (c) ∈ [0, 1) are defined in Lemma 4.2.6 and Ki± ∓ ± i (c) ˜ . (c) := i
1−i (c)
Remark 4.2.10. (1) The (forward) invariance of Wi+ and Wi− implies for all κ ≤ k the relations P1i (k)ϕ(k; κ, ξ) = wi+ (k, Qi1 (k)ϕ(k; κ, ξ))
for all (κ, ξ) ∈ Wi+ ,
Qi1 (k)ϕ(k; κ, ξ) = wi− (k, P1i (k)ϕ(k; κ, ξ))
for all (κ, ξ) ∈ Wi− .
(4.2o)
(2) Wi+ is an invariant fiber bundle, if the general solution to (S) exists on X . (3) The fiber bundle Wi+ can be considered as set of all c+ -bounded forward solutions, while Wi− consists of c− -bounded backward solutions for (S). In detail, given (κ1 , ξ1 ), (κ2 , ξ2 ) ∈ X we introduce the following equivalence relations on X : •
With κ := max {κ1 , κ2 } define (cf. Remark 3.4.17) there exist solutions φj : Z+ κj → X to (S) with (κ1 , ξ1 ) ∼+ i (κ2 , ξ2 ) :⇔ ¯ φj (κj ) = ξj and φ1 − φ2 ∈ X+ κ,c for all c ∈ Γi .
202 •
4 Invariant Fiber Bundles
With κ := min {κ1 , κ2 } define (cf. Remark 3.4.18) (κ1 , ξ1 ) ∼− i (κ2 , ξ2 ) :⇔
there exist solutions φj : Z− κj → X to (S) with φj (κj ) = ξj and φ1 − φ2 ∈ X− for all c ∈ Γ¯i . κ,c
Provided we have a c± -bounded solution φ∗ : I → X of (S) with c ∈ Γ¯i , the corresponding equivalence classes fulfill +
−
[(κ, φ∗ (κ))]i = Wi+ ,
[(κ, φ∗ (κ))]i = Wi− .
(4) It is possible to apply Theorem 4.2.9 in case of a linear function fk . In this sense, Theorem 4.2.9 resembles our previous roughness result Theorem 3.6.5 for exponential splittings. However, the gap condition (Gi ) appears to be weaker than (3.6j). Proof. Let (κ, ξ) ∈ X , we choose a fixed 1 ≤ i < N , c ∈ Γ¯i and abbreviate P+ (k) := IXk − P1i (k), P+ := Qii , ±
w (κ, x) :=
P− (k) := P1i (k), P− := Pii ,
wi± (κ, x),
W± :=
(4.2p)
Wi±
for all k, κ ∈ I, x ∈ Xκ . This brief and intuitive notation will be useful later as well. (a) Since the present part (a) of Theorem 4.2.9 can be proved along the same lines as part (b) we present only a sketch of the proof. Analogously to Lemma 4.2.7, the c+ -bounded solutions of (S) can be characterized as fixed points of the Lyapunov– + Perron operator Tκ+ : X+ κ,c × Xκ → Xκ,c , Tκ+ (φ; ξ) := Φ(·, κ)P+ (κ)ξ +
∞
−1 Gi (·, n + 1)Bn+1 fn (φ(n))
(4.2q)
n=κ
(cf. Theorem 3.5.3(a)). Here, the forward evolution operator Φ for (L0 ) exists due to Lemma 3.3.6(a). In particular, under (Γi+ ) corresponding counterparts to our preparatory Lemmata 4.2.6, 4.2.7 and 4.2.8 hold true in the Banach space X+ κ,c , where the proof of Lemma 4.2.7 relies on Theorem 3.5.3(a). It follows from the spectral gap condition (Gi ) that Tκ+ (·, ξ), ξ ∈ Xκ , is a contraction on the Banach space X+ κ,c (cf. Lemma 3.3.25(a)) and we denote its unique fixed + + point by φ+ (ξ) ∈ X κ κ,c . Then the function w (κ, ·) : Xκ → Xκ is defined by + + w (κ, ξ) := P− (κ)φκ (κ, ξ). (b) We want to show first that W− is an invariant fiber bundle of (S). By definition, for each pair of initial values (κ, ξ0 ) ∈ W− there exists a solution φ ∈ X− κ,c of (S) with φ(κ) = ξ0 . Due to the uniqueness of forward solutions guaranteed by Hypothesis 4.2.3, we have φ = ϕ(·; l, φ(l)); accordingly ϕ(·; l, φ(l)) is a c− bounded solution and this yields the inclusion ϕ(l; κ, ξ) ∈ W− (l) for all l ∈ Z+ κ. Conversely, for ξ1 ∈ W− (κ) there exists a c− -bounded solution φ : I → X
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203
of (S) with φ (κ) = ξ1 . Obviously, ξ0 := φ(κ) ∈ W− (κ) and since the general forward solutions exists, ξ1 = ϕ(κ + 1; κ, ξ0 ), i.e., we have the inclusion ξ1 ∈ ϕ(κ + 1; κ, W− (κ)). − Our Lemma 4.2.8 implies that the mapping Tκ− (·; ξ) : X− κ,c → Xκ,c has a − unique fixed point φκ (ξ) ∈ Xκ,c . This fixed point is independent of the growth − rate c ∈ Γ¯i because one has the inclusion X− κ,bi −ς ⊆ Xκ,c (cf. Lemma 3.3.26) − − − and every Tκ (·; ξ) : Xκ,c → Xκ,c possesses the same fixed point as the restriction Tκ− (·; ξ)|X− . Furthermore, the fixed point φκ (ξ) is a solution of (S) on Z− κ κ,bi −ς
satisfying P− (κ)φκ (κ, ξ) = P− (κ)ξ (cf. Lemma 4.2.7). Now we define w− (κ, ξ) := P+ (κ)φκ (κ, ξ)
(4.2r)
and immediately conclude w− (κ, ξ) ∈ P+ (κ). In addition, (4.2b) and the relation φκ (ξ) = φκ (P− (κ)ξ) in Lemma 4.2.8 imply (4.2r)
(4.2r)
w− (κ, ξ) = P+ (κ)φκ (κ, ξ) = P+ (κ)φκ (κ, P− (κ)ξ) = w− (κ, P− (κ)ξ). We now verify the representation (4.2l) and the invariance equation (4.2m). (⊆) Let (κ, x0 ) ∈ W− , i.e., there exists a c− -bounded solution φ : I → X of (S) with φ(κ) = x0 . Then φ satisfies P− (κ)φ(κ) = P− (κ)x0 and is consequently the unique fixed point of Tκ− (·; x0 ), i.e., φ = φκ (x0 ) (see Lemma 4.2.7). This, and φκ (ξ) = φκ (P− (κ)ξ) (cf. Lemma 4.2.8), implies x0 = P− (κ)φκ (κ, x0 ) + P+ (κ)φκ (κ, x0 ) = P− (κ)x0 + P+ (κ)φκ (κ, P− (κ)x0 ). So, setting ξ := P− (κ)x0 , we have x0 = ξ + P+ (κ)φκ (κ, ξ) = ξ + w− (κ, ξ) by (4.2r) and finally the first inclusion of (4.2l) is verified. (⊇) On the other hand, let x0 ∈ Xκ be of the form x0 = ξ + w− (κ, ξ) for some ξ ∈ P− (κ). Then (4.2r)
x0 = ξ + P+ (κ)φκ (κ, ξ) = P− (κ)φκ (κ, ξ) + P+ (κ)φκ (κ, ξ) = φκ (κ, ξ) and therefore, φ = φκ (ξ) is a c− -bounded solution of (S) with φ(κ) = x0 . With points (κ, ξ0 ) ∈ W− , κ ∈ I , the invariance of W− (i.e., (4.2o)) implies the relation ϕ(k; κ, ξ0 ) = P− (k)ϕ(k; κ, ξ0 )+w− (k, P− (k)ϕ(k; κ, ξ0 )) and multiplication with P+ (k) yields P+ (k)ϕ(k; κ, ξ0 ) = w− (k, P− (k)ϕ(k; κ, ξ0 )) for k ∈ Z− κ. We set k = κ + 1 and the solution identity for (S) eventually implies the invariance equation (4.2m). (b1 ) We obtain (4.2n) from Lemma 4.2.8(a), since (4.2r) yields + − + − − (4.2g) w (κ, ξ) ≤ ˜− (c) P− (κ)ξ + Ki Γκ (i) + Ci (c)Γκ (i)i (c) . i c − ai 1 − i (c)
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(b2 ) To prove the claimed Lipschitz estimate, consider ξ, ξ¯ ∈ Xκ and corre¯ ∈ X− of T − (·; ξ) and T − (·; ξ), ¯ respectively. sponding fixed points φκ (ξ), φκ (ξ) κ,c κ κ One obtains from Lemma 4.2.8(b) that − (4.2h) ¯ (4.2r) ¯ ≤ ˜− (c) ξ − ξ¯ w (κ, ξ) − w− (κ, ξ) = P+ (κ) φκ (κ, ξ) − φκ (κ, ξ) i
and this finishes the present proof of Theorem 4.2.9. It can be seen that the gap condition (Gi ) is optimal in the following sense:
Example 4.2.11 (optimal spectral gap). Let reals 0 < α < 1 < β and ε be given. For X = Z × R2 we consider an autonomous explicit difference equation (S) with
α0 , Ak :≡ 0β
Bk :≡ IR2 ,
−x2 . fk (x) := ε x1
The linear part (L0 ) is hyperbolic on Z and for the nonlinearity we obtain the Lipschitz constant lip fk = |ε|. The side of x = Ak x + fk (x) can α right-hand ε be written as linear system x = ε β x, whose coefficient matrix has the two 1 eigenvalues 2 α + β ± (β − α)2 − 4ε2 . Thus, the origin is a saddle, precisely as long as |ε| < β−α 2 and this is equivalent to the spectral gap condition (Gi ). For nonlinearities with |ε| ≥ β−α the origin becomes a sink or a source (depending 2 on the value α+β ) and we do not have invariant fiber bundles given by graphs over 2 images of nontrivial projectors as postulated in Theorem 4.2.9. Our next observation is that the invariant fiber bundles Wi± from Theorem 4.2.9 are nested, which means they are ordered w.r.t. the set inclusion. Corollary 4.2.12. Let 1 ≤ i∗ < N and S(A, B; P ) be minimal. (a) In case I is unbounded above, (Γi+∗ ) and (Gi ) for 1 ≤ i ≤ i∗ hold, we obtain the pseudo-stable hierarchy of forward invariant fiber bundles, Wi+∗ ⊂ Wi+∗ −1 ⊂ . . . ⊂ W1+ ⊂ X . (b) In case I is unbounded below, (Γi−∗ ) and (Gi ) for i∗ ≤ i < N hold, we obtain the pseudo-unstable hierarchy of invariant fiber bundles, Wi−∗ ⊂ Wi−∗ +1 ⊂ . . . ⊂ X . Proof. From Remark 4.2.5(2) we see that the growth condition (Γi−∗ ) implies (Γi+ ) for all 1 ≤ i ≤ i∗ . Thus, by Theorem 4.2.9(a) there exist forward invariant fiber bundles Wi+ consisting of c+ i -bounded forward solutions. The growth rates ci ∈ Γ¯i fulfill ci+1 ci and therefore claim (a) follows from the embeddings in Lemma 3.3.26. The pseudo-unstable hierarchy in (b) can be established analogously.
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205
Generally speaking, through a given point (κ, ξ) ∈ X there may exist no or a multiple number of backward solutions of (S). The next result ensures that for (κ, ξ) ∈ Wi− , exactly one of them lies on the fiber bundle Wi− and that ϕ(k; κ, ·) is Lipschitzian between the fibers of Wi− . Moreover, it relates the dynamics of equation (S) to a reduced equation (the so-called Wi− -reduced equation), which is finite-dimensional provided dim P1i < ∞. Corollary 4.2.13 (Wi− -reduced equation). The general solution ϕ of (S) exists on Wi− . Moreover, the so-called Wi− -reduced equation −1 fk (x+wi− (k, x), x +wi− (k+1, x )) (4.2s) Bk+1 x = Ak x+Bk+1 P1i (k+1)Bk+1
in the pseudo-unstable vector bundle P1i has the following properties: (a) Its general solution ϕˆ exists on P1i and ϕ is related to ϕˆ by virtue of ϕ(k; ˆ κ, ξ) = P1i (k)ϕ(k; κ, ξ + wi− (κ, ξ)) for all (k, κ, ξ) ∈ I × P1i .
L1 (b) Under the condition Ki− max 1, ˜− i (c) bi (k) + L2 < 1 for all k ∈ I and a fixed c ∈ Γ¯i , one has the following Lipschitz estimate for all k ∈ Z− , κ
˜− (c) eˆ (k, κ), lip ϕ(k; κ, ·)|W − (κ) ≤ Ki− max 1, ˜− (c) 1 + L max 1, 2 i i bi i (4.2t) where ˆbi (k) := bi (k) − Ki+ max 1, ˜− (c) (L + b (k)L ). 1 i 2 i Proof. Let κ ∈ I be given and choose c ∈ Γ¯i . First of all, we prove that the general solution ϕ of (S) exists on Wi− . Due to the invariance of Wi− we know that the mapping ϕ(κ + 1; κ, ·) : Wi− (κ) → Wi− (κ + 1) is onto. Let us show now that the inverse of this mapping is given by ξ → φκ+1 (κ, ξ), with the φκ+1 (ξ) from the proof of Theorem 4.2.9(b). Indeed, for each ξ ∈ Wi− (κ + 1) there exists a c− bounded solution of (S), given by φκ+1 (ξ) : Z− κ+1 → X, and Lemma 4.2.7 yields ξ = P1i (κ + 1)ξ + wi− (κ + 1, P1i (κ + 1)ξ) (4.2r)
= P1i (κ + 1)ξ + Qi1 (κ + 1)φκ+1 (κ + 1, ξ) = φκ+1 (κ + 1, ξ).
Since the sequence φκ+1 (ξ) is a solution of (S), one has ϕ(κ + 1; κ, φκ+1 (κ, ξ)) = φκ+1 (κ+ 1, ξ) = ξ. It remains to derive the relation φκ+1 (κ, ϕ(κ+ 1, κ, ξ)) = ξ for all ξ ∈ Wi− (κ). For this purpose, we define ψ(κ + 1) := ϕ(κ + 1; κ, ξ) and ψ(k) := − φκ+1 (k + 1, ξ) for k ≤ κ. Then ψ : Z− κ+1 → X is a c -bounded solution of (S) and again Lemma 4.2.7 implies ψ(κ) = φκ+1 (κ, ψ (κ)) = φκ+1 (κ, ϕ(κ; κ + 1, ξ)). Noting that ψ(κ) = φκ+1 (κ + 1, ξ) = ξ holds we are done. Hence, each mapping ϕ(κ + 1; κ, ·) : Wi− (κ) → Wi− (κ + 1) is bijective and therefore ϕ exists on Wi− . (a) By multiplying the solution identity for ϕ with P1i (k + 1), using (3.3h) and the invariance of Wi− given in (4.2o), it is easily seen that the relation between ϕ and ϕˆ holds and that ϕˆ is defined on P1i , yielding the assertion (a).
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4 Invariant Fiber Bundles
(b) Finally, it remains to prove the Lipschitz estimate (4.2t). We pick two points ξ1 , ξ2 ∈ Wi− (κ) and the invariance of Wi− implies (4.2o)
ϕ(k; κ, ξj ) = ϕ(k; ˆ κ, P1i (κ)ξj ) + wi− (k, ϕ(k; ˆ κ, P1i (κ)ξj ))
for all k ∈ I
and j = 1, 2, which, in turn, using Theorem 4.2.9(b2) yields the estimate |ϕ(k; κ, ξ1 ) − ϕ(k; κ, ξ2 )|i ˆ κ, P1i (κ)ξ1 ) − ϕ(k; ≤ max 1, lip2 wi− ϕ(k; ˆ κ, P1i (κ)ξ2 ) ˆ κ, ξ¯2 ) with for all k ∈ I. To obtain an estimate for the difference ϕ(k; ˆ κ, ξ¯1 ) − ϕ(k; i ¯ ¯ ξ1 , ξ2 ∈ P1 (κ), we remark that Φ(k, l) is an isomorphism between the fibers of P1i (cf. Lemma 3.3.6(b)). If we abbreviate ϕˆj (k) := ϕ(k; ˆ κ, ξ¯j ), then the variation of constants formula from Theorem 3.1.16(b) and Remark 3.1.17(1) yields ϕˆj (k) = Φ(k, κ)ξ¯j −
κ−1
−1 Φ(k, n + 1)P1i (n + 1)Bn+1 ·
n=k
· fn ϕˆj (n) + wi− (n, ϕˆj (n)), ϕˆj (n + 1) + wi− (n + 1, ϕˆj (n + 1)) ˆ1 (k) − ϕˆ2 (k) , the relations for all k ∈ Z− κ and j = 1, 2. Thus, setting u(k) := ϕ (3.4g), (4.2a) imply the inequality κ−1 u(k)ebi (κ, k) ≤ Ki− ¯ ξ1 − ξ¯2 + Ki− L1 max 1, lip2 wi− ebi (κ, n + 1)u(n) n=k
κ−1 −
+ Ki− L2 max 1, lip2 wi
ebi (κ, n + 1)u (n)
n=k
1 + L2 max 1, lip2 wi− ¯ ξ1 − ξ¯2 L1 κ−1 + L2 ebi (κ, n)u(n) + Ki− max 1, lip2 wi− bi (n)
≤ Ki−
n=k
for all k ∈ Z− κ , so that our assumption allows us to apply Gronwall’s lemma in backward time (cf. Proposition A.2.1(b)), which leads to u(k) ≤ Ki− 1 + L2 max 1, lip2 wi− eˆbi (k, κ) ¯ ξ1 − ξ¯2 for all k ≤ κ. Thanks to Theorem 4.2.9(b2), this finally implies (4.2t).
Before proceeding, we need a technical result for later purpose in Sect. 4.7. It states that the general forward solution ϕ of (S) satisfies a Lipschitz estimate, if the linear part of the Wi− -reduced equation (4.2s) has bounded forward growth. Note that the required estimate becomes void for explicit equations.
4.2 Existence of Invariant Fiber Bundles
207
Corollary 4.2.14. Let I = Z, Ki ≥ 1 and a ¯i : Z → (0, ∞). If Φ(k, l)P1i (l) i ≤ Ki ea¯i (k, l) for all l ≤ k L(P (l),P i (k)) 1
1
¯ and Ki L2 max 1, ˜− i (c) < 1 for a fixed c ∈ Γi , then the Lipschitz estimate
lip ϕ(k; κ, ·)|W − (κ) i
Ki max 1, ˜− (c) i eaˆi (k, κ) for all k ∈ Z+ ≤ κ − ˜ 1 − Ki L2 max 1, i (c)
holds, where a ˆi (k) := a ¯i (k) + 1 +
Ki max{1,˜− i (c)} 1−Ki L2 max{1,˜− (c)} i
¯i (k)L2 ). (L1 + a
Proof. The argument follows along the lines to infer (4.2t): The difference is to apply results in forward time k ∈ Z+ κ , namely the variation of constants formula for (4.2s) from Theorem 3.1.16(a) and the Gronwall’s lemma in Proposition A.2.1(a). The bundles Wi− and Wi+ intersect along exponentially bounded solutions. Corollary 4.2.15. Let I = Z, 1 ≤ i < N and assume c, d : Z → (0, ∞) satisfy c, d ∈ Γ¯i . If beyond (Γi+ ) and (Γi− ) also the strengthened gap condition 2 max Ki− , Ki+ (L1 + bi L2 ) < ςi 1 + 2 max Ki− , Ki+ L2
(4.2u)
holds, then (S) has a unique c, d-bounded solution φ∗ : Z → X and beyond ± + − [(κ, φ∗ (κ))]± i = Wi , κ ∈ Z, one has Wi ∩ Wi = φ∗ . Remark 4.2.16. (1) From Remark 4.2.4(1) one sees that (4.2u) implies (Gi ). (2) Under the assumption fk (0, 0) ≡ 0 on Z, the fiber bundles Wi+ and Wi− intersect along the trivial solution, i.e., Wi+ ∩ Wi− = Z × {0}. Proof. Given c, d ∈ Γ¯i we proceed in two steps: (I) We first show that Theorem 3.5.10 is applicable to equation (S). For this, we observe that gk := fk (0, 0) satisfies the inclusion g ∈ Xai ,bi ,B ⊆ Xc,d,B (cf. Lemma 3.3.26) due to (Γi± ). Moreover, the constant Di (c, d) defined in Theorem 3.5.4 fulfills the estimate Di (c, d) ≤ 2ς max Ki+ , Ki− and from (L1 + L2 max { c , d })Di (c, d) ≤ (L1 + L2 bi − L2 ςi )Di (c, d)
2 max{Ki− ,Ki+ } ςi
we see that (4.2u) implies the condition (3.5j). Thus, our semilinear equation (S) admits a unique c, d-bounded solution φ∗ : Z → X.
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4 Invariant Fiber Bundles
(II) Let κ ∈ Z be arbitrary. Then the above Theorem 4.2.9 immediately yields that the c, d-bounded complete solution φ∗ must satisfy φ∗ (κ) ∈ Wi− (κ) ∩ Wi+ (κ). Conversely, a point in the intersection ξ ∈ Wi− (κ) ∩ Wi+ (κ) is an initial value for a c, d-bounded solution to (S). Since such solutions are uniquely determined by step (I), we have φ∗ (κ) = ξ and thus the assertion follows. Our next intention is to have a look at nontrivial intersections and to show that also the extended hierarchy from Remark 3.4.19 persists under weak perturbations: Proposition 4.2.17 (intersection of invariant fiber bundles). Let I = Z and assume Hypotheses 4.2.1, 4.2.3 hold. For pairs (i, j) with 1 < i ≤ j < N satisfying + (Γi−1 ), (Γj− ) and the strengthened spectral gap conditions (Kn+ Kn− + max {Kn− , Kn+ }) (L1 + bn L2 ) < ς¯n 1 + (Kn+ Kn− + max Kn− , Kn+ )L2
˜ n) (G
for n ∈ {i − 1, j}, the nonautonomous set Wij :=
(κ, ξ) ∈ X
there exists a solution φ : Z → X of (S) with φ(κ) = ξ ∈ Xκ and φ ∈ Xc,d
is a forward invariant fiber bundle for (S), which is independent of c ∈ Γ¯i−1 , d ∈ Γ¯j and possesses the representation as graph + Wij = Wi−1 ∩ Wj− = (κ, η + wij (κ, η)) ∈ X : (κ, η) ∈ Uij
(4.2v)
of a uniquely determined mapping wij : X → X with wij (κ, ξ) = wij (κ, Pij (κ)ξ) ∈ Qji (κ) for all (κ, ξ) ∈ X . Furthermore, it holds: (a) wij : X → X is linearly bounded, i.e., for all (κ, ξ) ∈ X one has 2ij (c, d) j Pi (κ)ξ 1 − ij (c, d) Xκ Xκ − − Ki−1 2 i−1 (d)Ci−1 (d) max + Γκ+ (i − 1), + 1 − ij (c, d) bi−1 − d 1 − i−1 (d) Kj+ + j (c)Cj (c) − + Γκ (j) . c − aj 1 − j (c)
j wi (κ, ξ)
≤
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209
(b) wij (κ, ·) is globally Lipschitzian with lip2 wij ≤
2ij (c,d) 1−ij (c,d) ,
4.2.6, ˜± where the constants Ci (c), ± i (c), i (c) ∈ [0, 1)are defined in Lemma i (c) + − ˜ ˜ is given in Theorem 4.2.9 and ij (c, d) := max i−1 (c), j (d) ∈ [0, 1). − Remark 4.2.18. (1) Under the conventions W0+ = Q01 = X , WN = P1N = X (provided N < ∞), the intersection (4.2v) extends to the cases i = 1, j = N and + one obtains W1j = Wj− , WiN = Wi−1 (provided N < ∞). ˜ n ) for n ∈ {i − 1, j} implies both the (2) By Remark 4.2.4(1) one sees that (G ˜ i ) is sufficient for (4.2u). spectral gap conditions (Gi−1 ) and (Gj ). Moreover, (G (3) The fiber bundle Wij consists of all c, d-bounded complete solutions to (S). More detailed, given a complete solution φ∗ ∈ Xc,d with c ∈ Γ¯i−1 , d ∈ Γ¯j , as in Remark 3.4.19 we define the equivalence relation
(κ1 , ξ1 ) ∼ji (κ2 , ξ2 )
⎧ ⎪ ⎪ ⎨there exist complete solutions φj : Z → X to
(S) with φj (κj ) = ξj and φ2 − φ1 ∈ Xc,d for ⎪ ⎪ ⎩all a c≤b and a d ≤ b ,
:⇔
i−1
j
i−1
j
j
j
whose equivalence classes [·]i satisfy Wij = [(κ, φ∗ (κ))]i for all κ ∈ Z. (4) The (forward) invariance of Wij guarantees Qji (k)ϕ(k; κ, ξ) = wij (k, Pij (k)ϕ(k; κ, ξ))
for all (κ, ξ) ∈ Wij , κ ≤ k.
(5) Another possibility to construct the desired mapping wij : X → X is using the Lyapunov–Perron operator Tκ± : Xc,d × Xκ → Xc,d, Tκ± (φ, η) := Φ(·, κ)Pij (κ)η +
k−1
−1 Φ(k, n + 1)Qj−1 1 (n + 1)Bn+1 fn (φ(n))
n=−∞
+ −
k−1 n=κ ∞
−1 Φ(k, n + 1)Pij (n + 1)Bn+1 fn (φ(n)) −1 Φ(k, n + 1)P1i−1 (n + 1)Bn+1 fn (φ(n)).
n=κ
Applying methods as previously in this section, Tκ± (·, η) turns out to be a contraction on Xc,d . Having the unique fixed point φκ (η) ∈ Xc,d at hand, Wij is the graph i−1 (κ) φκ (η) over Pij . However, of the unique mapping wij (κ, η) := Qj−1 1 (κ) + P1 our construction of the fiber bundle Wij is more geometrical in nature.
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Proof. We suppose 1 < i ≤ j < N and subdivide the proof into several steps: + (I) Our Theorem 4.2.9 guarantees the existence of two fiber bundles Wi−1 and − ˜ Wj . Here, thanks to our assumption (Gn ), one sees from Theorem 4.2.9(a2 ) and + (b2 ) that the corresponding functions wi−1 and wj− satisfy + lip2 wi−1 ≤ ˜+ i−1 (c) < 1,
lip2 wj− ≤ ˜− j (d) < 1
and consequently ij (c, d) < 1 for all c ∈ Γ¯i−1 , d ∈ Γ¯j . Having this at our disposal, for every κ ∈ Z we define the operator Tκ : Xκ2 × Xκ → Xκ2 by
+ Tκ (x, z; y) := wi−1 (κ, z + y), wj− (κ, x + y) .
(4.2w)
Considering y ∈ Xκ as a fixed parameter, thanks to the estimate + (??) +
Tκ (x, z; y) − Tκ (¯ x, z¯; y) = max wi−1 (κ, z + y) − wi−1 (κ, z¯ + y) , − w (κ, x + y) − w− (κ, x ¯ + y) j j ≤ ij (c, d) x − x ¯, z − z¯ for all x, x¯, z, z¯ ∈ Xκ , the operator Tκ (·, y) : Xκ2 → Xκ2 is a uniform contraction in y ∈ Xκ . Similarly, we deduce from Theorem 4.2.9(a2 ) and (b2 ) that lip3 Tκ ≤ ij (c, d) and the uniform contraction principle in Theorem B.1.1 ensures that there exists a unique fixed point + − Υi,j (κ, y) = (Υi,j , Υi,j )(κ, y) ∈ Xκ2 of Tκ (·, y). (II) Now we infer the representation (4.2v) of Wij as graph of a function wij over Pij . From Theorem 4.2.9(a) we know that a point (κ, x0 ) ∈ X is contained in + + Wi−1 , if and only if there exists a ξ0 ∈ Qi−1 1 (κ) such that x0 = ξ0 + wi−1 (κ, ξ0 ) i−1 i−1 + and accordingly Q1 (κ)x0 = ξ0 + Q1 (κ)wi−1 (κ, x0 ) = ξ0 (cf. (4.2j)). This + + i−1 yields (κ, x0 ) ∈ Wi−1 if and only if x0 = Qi−1 1 (κ)x0 + wi−1 (κ, Q1 (κ)x0 ). Analogously from Theorem 4.2.9(b) we have the inclusion (κ, x0 ) ∈ Wj− if and only if x0 = P1j (κ)x0 +wj− (κ, P1j (κ)x0 ). The unique decomposition x0 = ξ+η+ζ j j into ξ ∈ Qi−1 1 (κ), η ∈ Pi (κ), ζ ∈ P1 (κ) leads to the equivalence (κ, x0 ) ∈ Wij ⇔
+ i−1 x0 = Qi−1 1 (κ)x0 + wi−1 (κ, Q1 (κ)x0 ) and
x0 = P1j (κ)x0 + wj− (κ, P1j (κ)x0 ) + ⇔ ζ = wi−1 (κ, ξ + η) and ξ = wj− (κ, η + ζ) (4.2w)
⇔
(ξ, ζ) = Tκ (ξ, ζ; η),
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211
j i.e., the pair (ξ, ζ) ∈ Qi−1 1 (κ) × P1 (κ) is a fixed point of Tκ (·; η); from the above step (I) it is uniquely determined by Υi,j (κ, η). As a result, if we define wij (κ, x0 ) := + − Υi,j (κ, Pij (κ)x0 )+Υi,j (κ, Pij (κ)x0 ) for (κ, x0 ) ∈ X , then the representation (4.2v) holds. Moreover, by construction one has + wij (κ, Pij (κ)x0 ) = wij (κ, x0 ) = wi−1 (κ, x0 ) + wj− (κ, x0 ) ∈ Qji (κ).
The fiber bundle Wij is forward invariant, because for (κ, x0 ) ∈ Wij we obtain − ϕ(·; κ, x0 ) ∈ X+ κ,c , as well as the existence of a d -bounded solution φ : Z → X of (S) with φ(κ) = x0 for all c ∈ Γ¯i−1 , d ∈ Γ¯j . Then the semigroup property (2.3a) + − implies that ϕ(·; k0 , ϕ(k0 ; κ, x0 )), k0 ∈ Z+ κ , is also c - and d -bounded (in the j sense above), and therefore (k0 , ϕ(k0 ; κ, x0 )) ∈ Wi . (a) Let (κ, x) ∈ X . In order to establish the linear bound for wij (κ, x) we point + (κ, x) ∈ P1i−1 (κ) and wj− (κ, x) ∈ Qj1 (κ) readily imply out that the inclusions wi−1 + − the inclusion Υi,j (κ, x) ∈ P1i−1 (κ) resp. Υi,j (κ, x) ∈ Qj1 (κ). Therefore, the fixed + point relation for Υi,j (κ, x) and Theorem 4.2.9(a1) yields the estimate + + − Υ (κ, y) (4.2w) = wi−1 (κ, Υi,j (κ, y) + y) i,j ≤
≤
− − − Ki−1 Γκ+ (i − 1) i−1 (c) + i−1 + Ki−1 Q1 (κ) Υi,j (κ, y) + y bi−1 − c 1 − i−1 (c) + Ci−1 (c)Γκ+ (i − 1) ,
− − (c) + i−1 Ki−1 Γκ+ (i − 1) Q (κ)y + Υ − (κ, y) + i−1 K 1 i,j bi−1 − c 1 − i−1 (c) i−1 + Ci−1 (c)Γκ+ (i − 1) ,
while correspondingly Theorem 4.2.9(b1) leads to − + Υ (κ, y) (4.2w) = wj− (κ, Υi,j (κ, y) + y) i,j + + Kj+ Γκ− (j) j (d) + Kj− P1j (κ) Υi,j ≤ (κ, y) + y d − aj 1 − j (d) + Cj (d)Γκ− (j) ≤
+ Kj+ Γκ− (j) j (d) + + Kj− P1j (κ)y + Υi,j (κ, y) d − aj 1 − j (d) + Cj (d)Γκ− (j)
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for all y ∈ Xκ . Using (??), this equips us with the relation
+ − (κ, y) Υi,j (κ, y), Υi,j − + Ki−1 Γκ (i−1) ≤ max + bi−1 −c
+ i−1 Ki−1 Q1 (κ)y + Ci−1 (c)Γκ+ (i − 1) , Kj+ Γκ− (j) + − j j (d) − K + (κ)y + C (d)Γ (j) P j κ 1 j d−aj 1−j (d)
+ − +ij (c, d) Υi,j (κ, y), Υi,j (κ, y) for all y ∈ Xκ − i−1 (c) 1−i−1 (c)
and by definition of wij we can use the elementary inequality a + b ≤ 2 max {a, b}
for all a, b ≥ 0,
(4.2x)
in order to deduce the estimate claimed in assertion (a). (b) From Theorem B.1.1(b) we know that Υi,j (κ, ·) : Xκ → Xκ2 fulfills the ij (c,d) Lipschitz estimate lip2 Υi,j ≤ 1− and as a result of (4.2x) the assertion (b) ij (c,d) yields. For the sake of completeness we also state that the extended hierarchy of invariant vector bundles from Corollary 3.3.18 persists under nonlinear perturbations −1) yielding the promised (N +2)(N nontrivial invariant fiber bundles of (S). Using 2 the conventions explained in Remark 4.2.18(1) we consequently have Corollary 4.2.19 (hierarchy of invariant fiber bundles). Let S(A, B; P ) be mini˜ n ) hold for 1 ≤ n < N , mal. If (Γn± ) and the strengthened spectral gap condition (G then: j (a) One has the inclusions Wi−1 ⊃ Wij ⊂ Wij+1 for all 1 < i ≤ j < N . (b) In case N < ∞ one has the extended hierarchy
W11 ⊂ W12 ⊂ . . . ⊂ W1N −1 ⊂ X ∪ ∪ ∪ W22 ⊂ . . . ⊂ W2N −1 ⊂ W2N ∪ ∪ .. .. .. . . . ∪ ∪ N −1 N ⊂ W WN N −1 −1 ∪ N WN Proof. Referring to the notation from Remark 4.2.18(1), the cases i = 1 and j = N have already been shown in Corollary 4.2.12. We thus restrict to indices 1 < i ≤ j < N . Above all, we choose growth rates c ∈ Γ¯i−1 , d ∈ Γ¯j and point out that
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213
the sets Wij are dynamically characterized using c, d-bounded solutions. A growth rate d¯ ∈ Γ¯j+1 satisfies d¯ d and Lemma 3.3.26 yields the inclusion Xc,d ⊆ Xc,d¯ guaranteeing Wij ⊂ Wij+1 . Analogously, for growth rates c¯ ∈ Γ¯i−2 one has c c¯, j . Xc,d ⊂ Xc¯,d by Lemma 3.3.26 and thus Wij ⊂ Wi−1 Corollary 4.2.20. Under the assumption fk (0, 0) ≡ 0 on I one has for κ ∈ I: ∼ + (a) If I is unbounded above, then wi+ (κ, 0) ≡ 0 on I and [(κ, 0)]+ i = Wi . − − ∼ (b) If I is unbounded below, then wi (κ, 0) ≡ 0 on I and [(κ, 0)]i = Wi− . ˜ n ), n ∈ {i − 1, j}, holds, then wj (κ, 0) ≡ 0 on Z and one also (c) If I = Z and (G i j has [(κ, 0)]i = Wij . Proof. The assumption fk (0, 0) ≡ 0 on I yields Γκ± (i) = 0 for every κ ∈ I. Both the claims (a) and (b) follow from the respective assertions (a1 ) and (b1 ) of Theorem 4.2.9, whereas (c) is a consequence of Proposition 4.2.17(a). The following corollary deals with the case where (S) is periodic in time or even autonomous. In this case the fibers of the bundles Wi± and Wij repeat periodically; in the autonomous case they are identical copies of each other. Corollary 4.2.21. Let p ∈ N. (a) If (S) is p-periodic, then one has for all (κ, ξ) ∈ X that wi± (κ + p, ξ) = wi± (κ, ξ),
wij (κ + p, ξ) = wij (κ, ξ),
i.e., the mappings wi± , wij are also p-periodic in their first argument. (b) If (S) is autonomous, then the mappings wi+ , wi− , wij do not depend on their first argument, i.e., the constant fibers Wi± (κ), Wij (κ), κ ∈ Z, are invariant manifolds of (S). Proof. Let κ ∈ Z. The assertions of (b) readily follow from (a) and we restrict to the proof for the mapping wi+ . We choose a growth rate c ∈ Γ¯i and an arbitrary point ξ0 ∈ P1i (κ). By Theorem 4.2.9(a) the solution φ := ϕ(·; κ, ξ0 + wi+ (κ, ξ0 )) of (S) is c+ -bounded and due to the p-periodicity of (S) we know that also ψ : Z+ κ+p → X, ψ(k) := φ(k − p) is a c+ -bounded solution (cf. Proposition 2.5.3). Hence, we have the inclusion (κ + p, ψ(κ)) ∈ Wi+ and consequently, using our convention that periodic equations have periodic invariant projectors (cf. Corollary 3.4.25), (4.2j)
wi+ (κ + p, ξ0 ) = wi+ (κ + p, P1i (κ)φ(κ + p − p)) (4.2j)
= wi+ (κ + p, P1i (κ + p)ψ(κ + p)) = P1i (κ + p)ψ(κ + p) = wi+ (κ, ξ0 ), (4.2o)
i.e., we have established the p-periodicity of wi+ (·, ξ0 ) in case ξ0 ∈ P1i (κ). With this the p-periodicity of wi+ (·, x0 ) for general x0 ∈ X follows from (4.2j).
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4.3 Invariant Foliations and Asymptotic Phase As starting point, consider a linear homogenous equation Bk+1 x = Ak x
(L0 )
for which we have an exponential N -splitting as in Hypothesis 4.2.1. Given a fixed reference solution φ∗ : I → X to (L0 ), we are interested in the nonautonomous set Vφ∗ ⊆ X consisting of all initial pairs (κ, ξ) ∈ X such that the difference Φ(·, κ)ξ − φ∗ is c+ -bounded. This enables us to group points in X according to the above asymptotic behavior as equivalence classes. The latter are easily characterized, since we have the equivalence Φ(·, κ)[ξ − φ∗ (κ)] ∈ X+ κ,ci
(3.4k)
⇔
ξ = φ∗ (κ) + Qi1 (κ)
for all κ ∈ I
and 1 ≤ i < N , yielding Vφ∗ = φ∗ + Qi1 . We expect that this observation persists when passing over from the linear equation (L0 ) to (S). Accordingly, in this section we investigate attraction properties of the invariant fiber bundles Wi± from Theorem 4.2.9 in the generalized framework of c± -boundedness. For this, our main tools will be certain invariant fibers, which serve as leaves for an invariant foliation of the extended state space X . Equivalently, given (κ, ξ) ∈ X we aim to characterize the ± equivalence classes [(κ, ξ)]i defined in Remark 4.2.10(3). This enables us to construct an asymptotic phase property for each Wi± by choosing (κ, ξ) ∈ Wi± . This means that Wi± is not only exponentially attracting in forward resp. backward time, but solutions are also in phase with corresponding solutions on the invariant set Wi± . Our strategy in the first part of this section is parallel to the previous one. Nonetheless, the present assumptions are stronger than in Sect. 4.2, in a sense that continuity of the general forward solution ϕ(k; κ, ·) : Xκ → Xk to Bk+1 x = Ak x + fk (x, x ),
(S)
will play a crucial role. Such issues were addressed in Theorem 2.3.6 for general implicit equations and in Proposition 4.1.3 for semilinear problems (S). Hypothesis 4.3.1. Let the general forward solution ϕ of (S) exist as a continuous mapping. Suppose fk : Xk × Xk+1 → Yk+1 with fk (Xk , Xk+1 ) ⊆ im Bk+1 for all k ∈ I and that we have the global Lipschitz estimates (4.2a). Next, we introduce an appropriate Lyapunov–Perron operator to construct invariant fibers. Choose a fixed 1 ≤ i < N and suppose I is unbounded above. Let (κ, η, ξ) ∈ Qi1 × X and c : I → (0, ∞). Since the general forward solution ϕ of (S) exists, for ψ ∈ X+ κ,c , we can formally define the mapping
4.3 Invariant Foliations and Asymptotic Phase
215
∞ −1 Sκ+ (ψ; η, ξ) := Φ(·, κ) η − Qi1 (κ)ξ + Gi (·, n + 1)Bn+1 · n=κ
· fn (ψ(n) + ϕ(n; κ, ξ)) − fn (ϕ(n; κ, ξ)) . Note that the forward evolution operator Φ and Green’s function Gi for (L0 ) exist by Lemma 3.3.6(a). Due to the particular difference structure of Sκ+ , the growth conditions (Γi± ) will be of minor importance in the following considerations. Lemma 4.3.2. Assume Hypotheses 4.2.1 and 4.3.1. If c, d : I → (0, ∞) satisfy + i (4.2c), then the mapping Sκ+ : X+ κ,c × Q1 (κ) × Xκ → Xκ,d is well-defined with + S (ψ; η, ξ)+ ≤ K + η − Qi (κ)ξ + i (c) ψ + , κ 1 i κ,c κ,d Xκ i + − + P1 (κ)Sκ (κ, ψ; η, ξ) ≤ (c) ψ
i κ,c X
(4.3a)
κ
for all (κ, η, ξ) ∈ Qi1 × X , ψ ∈ X+ κ,d and we have the Lipschitz estimates lip1 P1i (κ)Sκ+ (κ, ·) ≤ − i (c),
lip1 Sκ+ ≤ i (c),
lip2 Sκ+ ≤ Ki+ ,
(4.3b)
where the constants i (c), − i (c) are defined in Lemma 4.2.6. i Proof. Let c ∈ (ai , bi ), ψ ∈ X+ κ,c and (κ, η, ξ) ∈ Q1 × X be given. First, we show + + that the sequence Sκ (ψ; η, ξ) is d -bounded for c ≤ d. For this, from (3.5a), (3.4g), (4.2a) one has using Lemma A.1.5(a) that
i P1 (k)Sκ+ (k, ψ; ξ, η) ed (κ, k) ∞ (A.1d) + + − ≤ Ki L(c) ebi (k, n + 1)ec (n, κ) ψ κ,c ed (κ, k) ≤ − i (c) ψ κ,c n=k
for all k ∈ Z+ κ , where L(c) is defined in Lemma 4.2.6. Accordingly, (3.5a), (3.4g) and (4.2a) also imply with Lemma A.1.5(b) the dual inequality i Q1 (k)Sκ+ (k, ψ; η, ξ) ed (κ, k) ≤ K + e ai (k, κ) η − Qi1 (κ)ξ i d +Ki+ L(c) (A.1e)
≤
Ki+ η
k−1
+
eai (k, n + 1)ec (n, κ) ψ κ,c ed (κ, k)
n=κ
+ − Qi1 (κ)ξ + + i (c) ψ κ,c
for all k ∈ Z+ κ . Combining these estimates, Lemma 3.3.22 leads to + Sκ (k, ψ; η, ξ) ed (κ, k) ≤ K + η − Qi1 (κ)ξ + i (c) ψ + i κ,c i
for all k ∈ Z+ κ.
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4 Invariant Fiber Bundles
This implies Sκ+ (ψ; η, ξ) ∈ X+ κ,d , as well as the first estimate (4.3a). Moreover, if we set k = κ, then the second relation in (4.3a) is a consequence of the above. Next we derive the Lipschitz estimates (4.3b). Let ψ, ψ¯ ∈ X+ ¯ ∈ Qi1 (κ) and fix κ,c , η, η ξ ∈ Xκ . We obtain from (3.5a), (3.4g), (4.2a) and Lemma A.1.5(a), i ¯ η, ξ) ed (κ, k) P1 (k) Sκ+ (k, ψ; η, ξ) − Sκ+ (k, ψ; ∞ (A.1d) + ¯+ ebi (k, n + 1)ec (n, κ)ψ − ψ¯κ,c ed (κ, k) ≤ − ≤ Ki− L(c) i (c) ψ − ψ κ,c n=k
(4.3c) for all k ∈ Z+ κ , and dually (3.5a), (4.2a) have the consequence i ¯ η, ξ) ed (κ, k) Q1 (k) Sκ+ (k, ψ; η, ξ) − Sκ+ (k, ψ; ≤
Ki+ L(c)
k−1
(A.1e) + ¯+ eai (k, n + 1)ec (n, κ) ψ − ψ¯κ,c ed (κ, k) ≤ + i (c) ψ − ψ κ,c
n=κ
for all k ∈ Z+ κ . Again, combining the last two inequalities in order to estimate the ¯ η, ξ), we are using the norm |·| from Lemma 3.3.22. difference Sκ+ (ψ; η, ξ)−Sκ+ (ψ; i This implies the middle relation in (4.3b); moreover, setting here k = κ leads to the first estimate in (4.3b). Finally, the remaining right Lipschitz estimate in (4.3b) follows from Sκ+ (k, ψ; η, ξ) − Sκ+ (k, ψ; η¯, ξ) ed (κ, k) ≤ Ki+ η − η¯ for every k ∈ Z+ κ , which we get from (4.2b), (3.4g). Thus, we are done. The next lemma provides a dynamical interpretation of the operator Sκ+ . Lemma 4.3.3. Let (κ, η, ξ) ∈ Qi1 × X and assume Hypotheses 4.2.1 and 4.3.1. If one chooses c : I → (0, ∞) according to (4.2c) and ψ ∈ X+ κ,c , then the following + statements are equivalent for Sκ+ (·; η, ξ) : X+ → X : κ,c κ,c (a) There exists a ζ ∈ Xκ such that ψ = ϕ(·; κ, ζ) − ϕ(·; κ, ξ) ∈ X+ κ,c and Qi1 (κ)ψ(κ) = η − Qi1 (κ)ξ.
(4.3d)
(b) ψ is a solution of the fixed point equation ψ = Sκ+ (ψ; η, ξ). Proof. Let (κ, ξ) ∈ X and assume ψ ∈ X+ κ,c . We define the inhomogeneity gk := fk (ψ(k) + ϕ(k; κ, ξ)) − fk (ϕ(k; κ, ξ)), which clearly satisfies gk ∈ Yk+1 , k ∈ I. In addition, due to the estimate −1 (4.2a) B gk ≤ (L1 + c L2 ) ψ + ec (k, κ) for all k ∈ Z+ κ κ,c k+1
(4.3e)
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217
one obtains the inclusion g ∈ X+ κ,c,B . From these preparations we get: i (a) ⇒ (b) Let η ∈ Q1 (κ) and assume there exists a ζ ∈ Xκ such that the difference ψ = ϕ(·; κ, ζ) − ϕ(·; κ, ξ) is c+ -bounded and Qi1 (κ)ψ(κ) = η − Qi1 (κ)ξ. Then ψ is a c+ -bounded solution of the inhomogeneous equation Bk+1 x = Ak x + gk and Theorem 3.5.3(a) implies that ψ is a fixed point of Sκ+ (·; ξ, η). i (b) ⇒ (a) Conversely, assume that ψ ∈ X+ κ,c satisfies (4.3e) for some η ∈ Q1 (κ), i ξ ∈ Xκ , define ζ := Q1 (κ) [ξ + ψ(κ)] + η and set φ := ψ + ϕ(·; κ, ξ). Hence, (4.3e)
φ(κ) = ψ(κ) + ξ = P1i (κ)ψ(κ) + Qi1 (κ)Sκ+ (κ, ψ; η, ξ)(κ) + ξ = P1i (κ)ψ(κ) + η − Qi1 (κ)ξ + ξ = P1i (κ) [ψ(κ) + ξ] + η = ζ and φ also solves (S). Due to the uniqueness of forward solutions guaranteed by Hypothesis 4.3.1, this gives us φ = ϕ(·; κ, ζ), i.e., ψ = ϕ(·; κ, ζ) − ϕ(·; κ, ξ). Finally, one has Qi1 (κ)ψ(κ) = Qi1 (κ) [ζ − ξ] = Qi1 (κ) [η − ξ] = η − Qi1 (κ)ξ. Lemma 4.3.4. Let (κ, η, ξ) ∈ Qi1 × X and assume Hypotheses 4.2.1 and 4.3.1. If + (Gi ) holds and c ∈ Γ¯i , then the mapping Sκ+ (·; η, ξ) : X+ κ,c → Xκ,c has a unique i fixed point ψκ (η, ξ) ∈ X+ κ,c . Moreover, the fixed point mapping ψκ : Q1 (κ)× Xκ → + i Xκ,c satisfies ψκ (η, ξ) = ψκ (Q1 (κ)η, ξ) and one has: (a) ψκ : Qi1 (κ) × Xκ → X+ κ,c is linearly bounded, i.e., it is Ki+ + η − Qi1 (κ)ξ ,
ψκ (η, ξ) κ,c ≤ Xκ 1 − i (c) i P (κ)ψκ (κ, η, ξ) ≤ ˜+ (c) η − Qi (κ)ξ . 1 1 i Xκ Xκ
(4.3f)
(b) One has the Lipschitz estimates lip1 ψκ ≤
Ki+ , 1 − i (c)
lip1 P1i (κ)ψκ (κ, ·) ≤ ˜+ i (c)
(4.3g)
and ψκ : Qi1 (κ) × Xκ → X+ κ,c is continuous for c ∈ (ai + ς, bi − ς], where the constants i (c) ∈ [0, 1), ˜+ i (c) are defined in Lemma 4.2.6. Proof. Let (κ, η, ξ) ∈ Qi1 × X . From the middle estimate (4.3b) in Lemma 4.3.2 and (4.2i) we know that Sκ+ (·; η, ξ) is a contraction on X+ κ,c and Banach’s theorem (see, for instance, [295, p. 361, Lemma 1.1]) implies the existence of a unique fixed point denoted by ψκ (η, ξ) ∈ X+ κ,c . (a) Thanks to i (c) < 1 (cf. (4.2i)), the first estimate (4.3f) follows from + + (4.3e)
ψκ (η, ξ) κ,c = Sκ+ (ψκ (η, ξ); η, ξ)κ,c
+ ≤ Ki+ η − Qi1 (κ)ξ + i (c) ψκ (η, ξ) κ,c ,
(4.3a)
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4 Invariant Fiber Bundles
and similarly we get i (4.3a) + P1 (κ)ψκ (κ, η, ξ) (4.3e) = P1i (κ)Sκ+ (κ, ψκ (η, ξ); η, ξ) ≤ − i (c) ψκ (η, ξ) κ,c ; by the inequality shown before, this implies (4.3f). (b) Next we derive the Lipschitz estimates in (4.3g). For this, let η, η¯ ∈ Qi1 (κ), fix ξ ∈ Xκ , and from (4.3e) we obtain +
(4.3b)
+
ψκ (η, ξ) − ψκ (¯ η , ξ) κ,c ≤ Ki+ η − η¯ + i (c) ψκ (η, ξ) − ψκ (¯ η , ξ) κ,c , yielding the left relation in (4.3g). Similarly, using (4.3e), (4.3b) one has i + P (κ) [ψκ (κ, η, ξ) − ψκ (κ, η¯, ξ)] ≤ − (c) ψκ (η, ξ) − ψκ (¯ η , ξ) κ,c 1 i leading to the remaining right relation in (4.3g). To complete the proof of (b), one has to show the continuity of ψκ : Qi1 (κ) × Xκ → X+ κ,c for c ∈ (ai + ς, bi − ς]. Here, our strategy serves as a prototype for various related continuity proofs in the following. Due to (4.3e), in order to prove the continuity of ψκ , it suffices to show for arbitrary fixed (κ, η0 ) ∈ Qi1 the following limit relation: +
lim ψκ (η0 , ξ) − ψκ (η0 , ξ0 ) κ,c = 0
ξ→ξ0
(4.3h)
(cf. Lemma B.1.3). To arrive at a short notation, we suppress the dependence on η0 ∈ Qi1 (κ) from now on and define mappings Hk : Xk × Xk+1 × Xκ → Xk+1 , −1 [fk (x + ϕ(k; κ, ξ), y + ϕ(k + 1; κ, ξ)) − fk (ϕ(k; κ, ξ))] Hk (x, y, ξ) := Bk+1
¯ ¯ k (ζ, ξ) := B −1 Hk (ψκ (k, ζ), ξ) for k ∈ Z+ and H κ . Note that Hk and Hk (ζ, ·) are k+1 continuous due to Hypothesis 4.3.1. For any parameter ξ0 ∈ Xκ we obtain from (4.3e), similarly to (4.3c), the estimate i Q (κ) [ψκ (k; ξ) − ψκ (k; ξ0 )] ≤ Φ(k, κ)Qi (κ) ξ − ξ0
1 1 +Ki+
ψκ (k; ξ) − ψκ (k; ξ0 ) ≤
k−1
¯ n (ξ, ξ) − H ¯ n (ξ0 , ξ0 ) , eai (k, n + 1) H
n=κ ∞ Ki− ebi (k, n n=k
¯ n (ξ, ξ) − H ¯ n (ξ0 , ξ0 ) + 1) H
for all k ∈ Z+ κ , using the dichotomy estimates (3.4g). Subtraction and addition of ¯ n (ξ0 , ξ) in the corresponding norms in connection with Lemma 3.3.22 leads to H |ψκ (k; ξ) − ψκ (k; ξ0 )|i ≤ max {S0 + S2 + S4 , S1 + S3 }
for all k ∈ Z+ κ,
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219
where (cf. (3.4g) and (4.2a)) S0 := Ki+ eai (k, κ) ξ − ξ0 , S1 := Ki− L1
∞
ebi (k, n + 1) ψκ (n, ξ) − ψκ (n, ξ0 )
n=k ∞
+ Ki− L2
ebi (k, n + 1) ψκ (n + 1, ξ) − ψκ (n + 1, ξ0 ) ,
n=k
S2 :=
Ki+ L1
k−1
eai (k, n + 1) ψκ (n, ξ) − ψκ (n, ξ0 )
n=κ
+ Ki+ L2 S3 := Ki−
∞
k−1
eai (k, n + 1) ψκ (n + 1, ξ) − ψκ (n + 1, ξ0 ) ,
n=κ
¯ n (ξ0 , ξ) − H ¯ n (ξ0 , ξ0 ) , ebi (k, n + 1) H
n=k
S4 := Ki+
k−1
¯ n (ξ0 , ξ) − H ¯ n (ξ0 , ξ0 ) . eai (k, n + 1) H
n=κ +
Using well-known arguments we get Sl ec (κ, k) ≤ i (c) ψκ (ξ) − ψκ (ξ0 ) κ,c for every l ∈ {1, 2} and k ∈ Z+ κ . Therefore, we obtain the estimate |ψκ (k; ξ) − ψκ (k; ξ0 )|i ec (κ, k) ≤ max Ki+ ξ − ξ0 + S3 ec (κ, k), S4 ec (κ, k) + i (c) ψκ (ξ) − ψκ (ξ0 ) + κ,c
and by passing over to the least upper bound for k ∈ Z+ κ , we get
ψκ (ξ) −
+ ψκ (ξ0 ) κ,c
max Ki− , Ki+ Ki+
ξ − ξ0 + sup U (k, ξ) ≤ 1 − i (c) 1 − i (c) k∈Z+ κ
with the mapping U (k, ξ) := ec (κ, k)
∞
¯ n (ξ0 , ξ) − H ¯ n (ξ0 , ξ0 ) ebi (k, n + 1) H
n=k
+ec (κ, k)
k−1
¯ n (ξ0 , ξ) − H ¯ n (ξ0 , ξ0 ) . eai (k, n + 1) H
n=κ
Therefore, it suffices to prove the limit relation lim sup U (k, ξ) = 0
ξ→ξ0
k∈Z+ κ
(4.3i)
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4 Invariant Fiber Bundles
in order to establish (4.3h). We proceed indirectly. If (4.3i) does not hold, then there exists an ε > 0 and a sequence (ξj )j∈N in Xκ with limj→∞ ξj = ξ0 and supk∈Z+ U (k, ξj ) > ε for all j ∈ N. This implies the existence of a sequence κ (kj )j∈N in Z+ κ such that U (kj , ξj ) > ε
for all j ∈ N.
(4.3j)
From now on assume ai + ς c, choose a fixed growth rate d ∈ (ai + ς, c) and remark that the inequality d c will play an important role below. Because ¯ of Hypothesis 4.3.1 and the inclusion ψκ (ξ) ∈ X+ κ,d we arrive at Hn (ξ0 , ξ) ≤ + + L(c) ψκ (ξ0 ) κ,d ed (n, κ) for all n ∈ Zκ (cf. (4.2a)) and Lemma A.1.5 leads to +
L(c) L(c) + bi − d d − ai
U (k, ξ) ≤ ψκ (ξ0 ) κ,d
e d (k, κ) for all k ∈ Z+ κ. c
Because of dc 1, passing over to the limit k → ∞ yields limk→∞ U (k, ξ) = 0 uniformly in ξ ∈ Xκ , and taking into account (4.3j) the sequence (kj )j∈N in Z+ κ has to be bounded above, i.e., there exists an integer K > κ with kj ≤ K for all j ∈ N. Hence, we can infer using Proposition A.1.2(d) that U (k, ξj ) ≤ e bi (k, κ) c
∞
¯ n (ξ0 , ξj ) − H ¯ n (ξ0 , ξ0 ) ebi (κ, n + 1) H
n=κ
K ¯ n (ξ0 , ξj ) − H ¯ n (ξ0 , ξ0 ) + e aci (k, κ) eai (κ, n + 1) H !" # n=κ ≤1
≤ e bi (K, κ) c
+
K
∞
¯ n (ξ0 , ξj ) − H ¯ n (ξ0 , ξ0 ) ebi (κ, n + 1) H
n=κ
¯ n (ξ0 , ξj ) − H ¯ n (ξ0 , ξ0 ) eai (κ, n + 1) H
for all j ∈ N,
n=κ
where the finite sum tend to zero for j → ∞ by the continuity of Hn . Continuity ¯ n (ξ0 , ξ0 ) and with the domi¯ n (ξ0 , ξj ) = H properties of Hn also imply limj→∞ H nated convergence theorem of Lebesgue2 the infinite sum tends to zero in the limit j → ∞. Thus, we derived the relation limj→∞ U (kj , ξj ) = 0, which obviously contradicts (4.3j). Consequently, we have shown the continuity of the fixed point mapping ψκ (η0 , ·) : Xκ → X+ κ,c and the proof of (b) is finished. Our preparations yield the first basic result in this section. It guarantees the existence of invariant fiber bundles through given solutions, i.e., invariant fibers: 2
In order to apply this result from integration theory (see, e.g., [295, p. 141, Theorem 5.8]), one has to write the infinite sum as an integral over piecewise-constant functions and use the Lipschitz estimate on Hn , which is implied by (4.2a), to get an integrable majorant.
4.3 Invariant Foliations and Asymptotic Phase
221
Proposition 4.3.5 (existence of invariant fibers). Let (κ, ξ) ∈ X . Assume that Hypothesi 4.2.1, 4.3.1 are satisfied and that (Gi ) holds for one 1 ≤ i < N . (a) If I is unbounded above, then the forward fiber through (κ, ξ), given by Vi+ (κ, ξ) := ζ ∈ Xκ : ϕ(·; κ, ζ) − ϕ(·; κ, ξ) ∈ X+ κ,c is independent of c ∈ Γ¯i , forward invariant w.r.t. (S), i.e. ϕ(k; κ, Vi+ (κ, ξ)) ⊆ Vi+ (k, ϕ(k; κ, ξ))
for all k ∈ Z+ κ
and possesses the representation Vi+ (ξ) = (κ, η + vi+ (κ, η, ξ)) : (κ, η) ∈ Qi1
(4.3k)
(4.3l)
as graph of a unique mapping vi+ : X × X → X satisfying vi+ (κ, η, ξ) ∈ P1i (κ) for all (κ, η, ξ) ∈ Qi1 × X
(4.3m)
and the invariance equation −1 Aκ vi+ (κ, η, ξ) vi+ (κ + 1, η1 , ξ1 ) = Bκ+1
+Qi1 (κ + 1)fκ (η + vi+ (κ, η, ξ), η1 +vi+ (κ + 1, η1 , ξ1 )),
(4.3n)
−1 η1 = Bκ+1 Aκ η + Qi1 (κ + 1)fκ (η + vi+ (κ, η, ξ), η1 + vi+ (κ + 1, η1 , ξ)), −1 −1 ξ1 = Bκ+1 Aκ ξ + Bκ+1 fκ (ξ, ξ1 ) for all (κ, η, ξ) ∈ Qi1 × X .
Furthermore, for all c ∈ Γ¯i it holds: (a1 ) vi+ : Qi1 × X → X is continuous and linearly bounded, i.e., for all triple (κ, η, ξ) ∈ Qi1 × X one has + v (κ, η, ξ) ≤ P i (κ)ξ + ˜+ (c) η − Qi (κ)ξ . 1 1 i i Xκ Xκ Xκ
(4.3o)
(a2 ) vi+ (κ, ·, ξ) is globally Lipschitzian with lip2 vi+ ≤ ˜+ i (c). (b) If I is unbounded below and the general solution ϕ of (S) exists on X as continuous mapping, then the backward fiber through (κ, ξ), given by Vi− (κ, ξ) := ζ ∈ Xκ : ϕ(·; κ, ζ) − ϕ(·; κ, ξ) ∈ X− κ,c is independent of c ∈ Γ¯i , invariant w.r.t. (S), i.e. ϕ(k; κ, Vi− (κ, ξ)) = Vi− (k, ϕ(k; κ, ξ)) for all k ∈ Z+ κ
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4 Invariant Fiber Bundles
and possesses the representation Vi− (ξ) = (κ, η + vi− (κ, η, ξ)) : (κ, η) ∈ P1i as graph of a unique mapping vi− : X × X → X satisfying vi− (κ, η, ξ) ∈ Qi1 (κ)
for all (κ, η, ξ) ∈ P1i × X
and the invariance equation −1 Aκ vi− (κ, η, ξ) vi− (κ + 1, η1 , ξ1 ) = Bκ+1
+ P1i (κ + 1)fκ (η + vi− (κ, η, ξ), η1 + vi− (κ + 1, η1 , ξ1 )), −1 Aκ η + P1i (κ + 1)fκ (η + vi− (κ, η, ξ), η1 + vi− (κ + 1, η1 , ξ)), η1 = Bκ+1 −1 −1 ξ1 = Bκ+1 Aκ ξ + Bκ+1 fκ (ξ, ξ1 )
for all (κ, η, ξ) ∈ P1i × X .
Furthermore, for all c ∈ Γ¯i it holds: (b1 ) vi− : P1i × X → X is continuous and linearly bounded, i.e., for all triple (κ, η, ξ) ∈ P1i × X one has − v (κ, η, ξ) ≤ Qi1 (κ)ξ + ˜− (c) η − P1i (κ)ξ . i i Xκ Xκ Xκ (b2 ) vi− (κ, ·, ξ) is globally Lipschitzian with lip2 vi− ≤ ˜− i (c), where the constants ˜± i (c) are defined in Theorem 4.2.9. Remark 4.3.6. (1) In case fk (0, 0) ≡ 0 on I holds, one has Wi± = Vi± (0) and in this setting, Theorem 4.2.9 can be seen as a special case of Proposition 4.3.5 in the sense that the c, d-bounded solution φ∗ from Corollary 4.2.15 is the trivial one. ± (2) Given a pair (κ, ξ) ∈ X , we have the relation [(κ, ξ)]± i = Vi (ξ) with the equivalence classes from Remark 4.2.10(3). (3) For the existence of a function vi− : X × X → X parametrizing Vi− (ξ) and satisfying both the assertions (b1 ) and (b2 ) it is sufficient to assume that the general backward solution of (S) exists as a continuous mapping. (4) If the general solution to (S) exists on X , then the fibers Vi+ (ξ) are invariant w.r.t. (S), i.e., the inclusion (4.3k) can be strengthened to ϕ(k; κ, Vi± (κ, ξ)) = Vi± (k, ϕ(k; κ, ξ))
for all k ∈ I.
(4.3p)
For instance, a combination of the assumptions for Propositions 4.1.3 and 4.1.4 yields the existence of the general solution ϕ on X .
4.3 Invariant Foliations and Asymptotic Phase
223
Proof. Let c ∈ Γ¯i be given. (a) Let (κ, η, ξ) ∈ Qi1 ×X . First, we show the invariance assertion for the forward fiber Vi+ (ξ). Let x0 ∈ ϕ(k; κ, Vi+ (κ, ξ)) for some k ∈ Z+ κ , and by definition this is equivalent to the existence of a ζ ∈ Xκ with x0 = ϕ(k; κ, ζ) guaranteeing a difference ϕ(·; κ, ζ) − ϕ(·; κ, ξ) ∈ X+ κ,c . Therefore, ϕ(·; k, x0 ) − ϕ(·; k, ϕ(k; κ, ξ)) = ϕ(·; k, ϕ(k; κ, ζ)) − ϕ(·; k, ϕ(k; κ, ξ)) (2.3a)
= ϕ(·; κ, ζ) − ϕ(·; κ, ξ),
i.e., x0 ∈ Vi+ (k, ϕ(k; κ, ξ)) for all k ∈ Z+ κ. Due to the spectral gap condition (Gi ) one has (4.2i) and the middle estimate + (4.3b) in Lemma 4.3.2 shows that Sκ+ (·; η, ξ) : X+ κ,c → Xκ,c is a contraction. Hence, + Lemma 4.3.4 implies that Sκ (·; η, ξ) has a unique fixed point ψκ (η, ξ) ∈ X+ κ,c , which is independent of c ∈ Γ¯i , because due to Lemma 3.3.26 one has the inclusion + + + + X+ κ,ai +ς ⊆ Xκ,c and every mapping Sκ (·; η, ξ) : Xκ,c → Xκ,c possesses the same + fixed point as the restriction Sκ (·; η, ξ)X+ . Furthermore, the fixed point is of κ,ai +ς
the form ψκ (η, ξ) = ϕ(·; κ, ζ) − ϕ(·; κ, ξ) with ζ ∈ Xκ (cf. Lemma 4.3.3). Having this available, we define vi+ (κ, η, ξ) := P1i (κ) [ξ + ψκ (κ, η, ξ)]
(4.3q)
and evidently get vi+ (κ, x0 ) ∈ P1i (κ). Let us verify the representation (4.3l). (⊆) Let ζ ∈ Vi+ (κ, ξ), i.e., ψ = ϕ(·; κ, ζ) − ϕ(·; κ, ξ) ∈ X+ κ,c . By Lemma 4.3.3, ζ = ψ(κ) + ξ = P1i (κ)ψ(κ) + Qi1 (κ)ψ(κ) + ξ (4.3d)
= P1i (κ)ψ(κ) + η − Qi1 (κ)ξ + ξ = P1i (κ)ψ(κ) + η + P1i (κ)ξ,
hence Qi1 (κ)ζ = η, and ζ = Qi1 (κ)ζ +P1i (κ) [ξ + ψκ (κ, η, ξ)]. Thus, ζ is contained in the graph of vi+ (κ, ·, ξ) over Qi1 (κ). (⊇) On the other hand, suppose that ζ ∈ Xκ is of the form ζ = η + vi+ (κ, η, ξ) with some given η ∈ Qi1 (κ). Then (4.3e) implies Qi1 (κ)ψκ (η, ξ) = η − Qi1 (κ)ξ, which yields ζ = η + P1i (κ) [ξ + ψκ (κ, η, ξ)] = ξ + ψκ (κ, η, ξ), and consequently + the inclusion ϕ(·; κ, ζ) − ϕ(·; κ, ξ) ∈ X+ κ,c , i.e., ζ ∈ Vi (κ, ξ). To establish our invariance equation (4.3n), we observe that (4.3l) and the forward invariance of Vi+ (ξ) imply ϕ(k; κ, η + vi+ (κ, η, ξ)) = Qi1 (k)ϕ(k; κ, η + vi+ (κ, η, ξ)) + vi+ (k, Qi1 (k)ϕ(k; κ, η + vi+ (κ, η, ξ)), ϕ(k; κ, ξ)) i i for all k ∈ Z+ κ and η ∈ Q1 (κ). Multiplying this relation with the projection P1 (k), setting k = κ + 1, and keeping the inclusion (4.3m) in mind, this yields (4.3n) using the solution identity for (S).
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4 Invariant Fiber Bundles
(a1 ) We obtain (4.3o) from Lemma 4.3.4, since (4.3q), (4.3f) imply + v (κ, η, ξ) ≤ P1i (κ)ξ + ˜+ (c) η − Qi1 (κ)ξ . i i (a2 ) To prove the claimed Lipschitz estimate, consider η, η¯ ∈ Qi1 (κ), ξ ∈ Xκ + and the corresponding fixed points ψκ (η, ξ), ψκ (¯ η , ξ) ∈ X+ κ,c of Sκ (·; η, ξ) and + Sκ (·; η¯, ξ), respectively. One gets from Lemma 4.3.4(b) that + v (κ, η, ξ) − v + (κ, η¯, ξ) (4.3q) = P1i (κ) [ψκ (κ, η, ξ) − ψκ (κ, η¯, ξ)] i i (4.3g)
≤
Ki− − i (c)
η − η¯ . 1 − i (c)
From Hypothesis 4.3.1 we deduce by Lemma 4.3.4(b) that ψκ : Qi1 (κ)×Xκ → X+ κ,c is continuous, and by definition in (4.3q) we get the continuity of vi+ (κ, ·). (b) The proof of assertion (b) is dual to (a) and we merely present a sketch. Above all, since the general (backward) solution ϕ to (S) exists on X , we can define the − Lyapunov–Perron operator Sκ− (·; η, ξ) : X− κ,c → Xκ,c , κ−1 −1 i (·, κ) η − Q (κ)ξ + Gi (·, n + 1)Bn+1 · Sκ− (ψ; η, ξ) := Φ− 1 Pi 1
n=−∞
· fn (ψ(n) + ϕ(n; κ, ξ)) − fn (ϕ(n; κ, ξ)) for all (κ, η, ξ) ∈ P1i × X . Using dual versions of Lemmata 4.3.2, 4.3.3 and 4.3.4 in − − the linear spaces X− κ,c , one shows that Sκ (·; η, ξ) is a contraction on Xκ,c , uniformly − − in the parameters η, ξ. With its unique fixed point ψκ (η, ξ) ∈ Xκ,c at hand, we define vi− (κ, η, ξ) := Qi1 (κ) [ξ + ψκ− (κ, η, ξ)] and proceed as above. In a descriptive way, the subsequent asymptotic phase property is also referred as “exponential tracking” of the fiber bundle Wi− . It states that convergence to Wi− is actually “in phase” with solutions on the invariant fiber bundle Wi− , and for that reason we speak of an asymptotic phase (see Fig. 4.1). The proof relies on a geometric argument, which demands a stronger spectral gap condition.
Wi− ϕ(k; κ, πκ+(κ, ξ))
ξ
ϕ(k; κ, ξ) I
πi+(κ, ξ)
Xκ
Xκ+1
Xκ+2
Xκ+3
...
Xk
Fig. 4.1 Asymptotic forward phase of Wi− and decaying solutions ϕ(·; κ, ξ) − ϕ(·; κ, πκ+ (κ, ξ))
4.3 Invariant Foliations and Asymptotic Phase
225
Theorem 4.3.7 (existence of an asymptotic phase). Let I = Z and (κ, ξ) ∈ X . Assume Hypotheses 4.2.1 and 4.3.1 are satisfied and define
ˆ i := K
⎧√ + − ⎨ Ki Ki −1 ⎩1
if Ki+ Ki− > 1,
Ki+ Ki− −1
if Ki+ Ki− = 1.
2
Moreover, suppose the strengthened spectral gap condition max Ki+ , Ki− (L1 + bi L2 ) < ςi ˆ i + max K + , K − L2 K i
ˆi) (G
i
holds for some 1 ≤ i < N and choose c ∈ Γ¯i . (a) If the growth condition (Γi− ) is satisfied, then the invariant fiber bundle Wi− from Theorem 4.2.9(b) possesses an asymptotic forward phase, i.e., there exists a mapping πi+ : X → X with the property that for all k ∈ Z+ κ, ϕ(k; κ, ξ) − ϕ(k; κ, π + (κ, ξ)) ≤ i X k
Ki+ ec (k,κ) 1−i (c)
Qi1 (κ)ξ + X κ
˜ + (ξ,c) C κ 1−˜i (c)
.
(4.3r) Geometrically, πi+ (κ, ξ) is given as unique intersection Wi− (κ) ∩ Vi+ (κ, ξ) = πi+ (κ, ξ)
for all (κ, ξ) ∈ X
(4.3s)
and one has: (a1 ) πi+ (κ, ·) : Xκ → Wi− (κ) is a continuous retraction onto the κ-fiber Wi− (κ), linearly bounded, i.e.
˜ + (ξ, c) + C + κ i π (κ, ξ) ≤ 1 + ˜ (c) Q1 (κ)ξ + i i Xκ Xκ 1 − ˜i (c) and, thus, maps bounded subsets of Xκ on bounded subsets of Wi− (κ), (a2 ) ϕ(k; κ, ·) ◦ πi+ (κ, ·) = πi+ (k, ·) ◦ ϕ(k; κ, ·) for all k ∈ Z+ κ. (b) If the growth condition (Γi+ ) is satisfied and the general solution ϕ of (S) exists on X as a continuous mapping, then the invariant fiber bundle Wi+ from Theorem 4.2.9(a) possesses an asymptotic backward phase, i.e., there exists a mapping πi− : X → X with the property that for all k ∈ Z− κ, ϕ(k; κ, ξ) − ϕ(k; κ, π − (κ, ξ)) ≤ i X k
Ki− ec (k,κ) 1−i (c)
P1i (κ)ξ
Xκ
+
˜ − (ξ,c) C κ 1−˜i (c)
.
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4 Invariant Fiber Bundles
Geometrically, πi− (κ, ξ) is given as unique intersection for all (κ, ξ) ∈ X Wi+ (κ) ∩ Vi− (κ, ξ) = πi− (κ, ξ) and one has: (b1 ) πi− (κ, ·) : Xκ → Wi+ (κ) is a continuous retraction onto the κ-fiber Wi+ (κ), linearly bounded, i.e.
− ˜ − (ξ, c) C π (κ, ξ) ≤ 1 + ˜− (c) P1i (κ)ξ + κ i i Xκ Xκ 1 − ˜i (c) and, thus, maps bounded subsets of Xκ on bounded subsets of Wi+ (κ), (b2 ) ϕ(k; κ, ·) ◦ πi− (κ, ·) = πi− (k, ·) ◦ ϕ(k; κ, ·) for all k ∈ Z+ κ, where the constants i (c) ∈ [0, 1) are defined in Lemma 4.2.6, ˜± i (c) are given in ˜+ (c) ∈ [0, 1), (c) Theorem 4.2.9 and ˜i (c) := ˜− i i
+ −
+ Ki Γκ (i) i (c)Ci (c)Γκ− (i) ˜− (c) 1 + ˜+ (c) Qi (κ)ξ , C˜κ+ (ξ, c) := c−a + + 1 i i 1− (c) Xκ i i
− +
+ Ki Γκ (i) − + − − i i (c)Ci (c)Γκ (i) ˜ ˜ ˜ Cκ (ξ, c) := bi −c + + i (c) 1 + i (c) P1 (κ)ξ Xκ . 1−i (c) ˆ i ≤ 1 one sees that (G ˆ i ) implies Remark 4.3.8 (spectral gap condition). Thanks to K (Gi ). Indeed, our different gap conditions are related as follows: ˜i) (G
⇒
(4.2u)
⇒
(Gi )
⇐
ˆ i ). (G
˜ i ) in Theorem 4.3.7. Actually, the conseˆ i ) can be replaced by (G Moreover, (G quences of our three different spectral gap conditions are as follows: •
Condition (Gi ) guarantees i (c) < 1 (cf. (4.2i)) and therefore the operators Tκ± from the proof of Theorem 4.2.9 and Sκ± needed to prove Proposition 4.3.5 satisfy lip1 Tκ± < 1,
•
lip1 Sκ± < 1.
˜ i ) yields ˜± (c) < 1 and so the mappings w± from Theorem 4.2.9 Condition (G i i ∓ and vi in Proposition 4.3.5 satisfy lip2 wi± < 1,
•
lip2 vi∓ < 1.
ˆ i ) finally ensures lip2 w± · lip2 v ∓ < 1 (see (4.3t) below). Condition (G i i
Proof. Let c ∈ Γ¯i and fix a pair (κ, ξ) ∈ X . We begin with a preliminary remark ˆ i ). From (G ˆ i ) one easily deduces the inequalities illustrating the consequences of (G + − K K ˆ i and − (c) ≤ i L(bi − ς) < K ˆ i , which guarantee + (c) ≤ i L(bi − ς) < K i
ς
i
+ 2 2 Ki+ Ki− + i (c) < (1 − i (c)) ,
ς
− 2 2 Ki+ Ki− − i (c) < (1 − i (c))
4.3 Invariant Foliations and Asymptotic Phase
227
and these two relations imply: •
+ If − i (c) ≤ i (c), then − + − + + 2 2 2 Ki+ Ki− + i (c)i (c) ≤ Ki Ki i (c) < (1 − i (c)) = (1 − i (c)) ,
•
− if + i (c) ≤ i (c), then − + − − − 2 2 2 Ki+ Ki− + i (c)i (c) ≤ Ki Ki i (c) < (1 − i (c)) = (1 − i (c)) .
We can therefore conclude that the functions wi± : X → X from Theorem 4.2.9, as well as vi∓ : X × X → X from Proposition 4.3.5 satisfy lip2 wi± · lip2 vi∓ ≤
+ Ki+ Ki− + i (c)i (c)
(1 − i (c))2
= ˜i (c) < 1.
(4.3t)
(a) We show that there exists one and only one ζ ∈ Wi− (κ) ∩ Vi+ (κ, ξ). For this, note that ζ ∈ Wi− (κ) ∩ Vi+ (κ, ξ) holds if and only if ζ = P1i (κ)ζ + wi− (κ, P1i (κ)ζ) and ζ = Qi1 (κ)ζ + vi+ (κ, Qi1 (κ)ζ, ξ), which is equivalent to Qi1 (κ)ζ = wi− (κ, P1i (κ)ζ)
and P1i (κ)ζ = vi+ (κ, Qi1 (κ)ζ, ξ).
Due to the contraction condition (4.3t) we can apply Corollary B.1.4(a) to the above equations. Thus, there exist two uniquely determined functions qκ : Xκ → Qi1 (κ), pκ : Xκ → P1i (κ) satisfying the identities qκ (ξ) ≡ wi− (κ, pκ (ξ))
and pκ (ξ) ≡ vi+ (κ, qκ (ξ), ξ)
on Xκ .
(4.3u)
Therefore, πi+ (κ, ξ) := pκ (ξ) + qκ (ξ) is the unique element in the intersection Wi− (κ) ∩ Vi+ (κ, ξ). We now derive the estimate (4.3r). From (4.3u) we get (4.2n)
qκ (ξ) ≤
(4.3o)
≤
+ (c) − Ki+ Γκ− (i) + i Ki vi+ (κ, qκ (ξ), ξ) + Ci (c)Γκ− (i) c − ai 1 − i (c) (c)Ci (c)Γκ (i) Ki+ Γκ− (i) + + i c − ai 1 − i (c)
− − + i (c) i i ˜ qκ (ξ) − Q1 (κ)ξ +i (c) Q1 (κ)ξ + Ki 1 − i (c)
and the triangle inequality together with (4.3t) yields
qκ (ξ) ≤
C˜κ+ (ξ, c) . 1 − ˜i (c)
(4.3v)
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4 Invariant Fiber Bundles
Since by construction one has the inclusion πi+ (κ, ξ) ∈ Vi+ (κ, ξ) for all ξ ∈ Xκ , it follows from Lemma 4.3.3 that ϕ(·; κ, ξ) − ϕ(·; κ, πi+ (κ, ξ)) = ψκ (Qi1 (κ)πi+ (κ, ξ), ξ) and Lemma 4.3.4 together with (4.3f) implies ϕ(k; κ, ξ) − ϕ(k; κ, π + (κ, ξ))+ ≤ i κ,c
Ki+
qκ (ξ) + Qi1 (κ)ξ . 1 − i (c)
Thanks to (4.3v) this gives us (4.3r). (a1 ) Proposition 4.3.5(a1 ) yields the continuity of vi+ (κ, ·) : Qi1 (κ) × Xκ → i P1 (κ) and accordingly Corollary B.1.4(b) implies that also πi+ (κ, ·) : Xκ → Xκ is continuous. It remains to estimate the norm of πi+ (κ, ξ). From (4.3u) we get (4.3o) i
pκ (ξ) ≤ Qi1 (κ)ξ + ˜+ i (c) qκ (ξ) − Q1 (κ)ξ
˜+ (4.3v) Qi (κ)ξ + ˜+ (c) Cκ (ξ, c) ≤ 1 + ˜+ 1 i (c) i 1 − ˜i (c)
and together with (4.3v) this yields the claimed inequality. (a2 ) The (forward) invariance of Wi− and Vi+ (κ, ξ) implies for all k ∈ Z+ κ, (4.3s)
ϕ(k; κ, πi+ (κ, ξ)) ∈ ϕ(k; κ, Wi− (κ) ∩ Vi+ (κ, ξ)) ⊆ ϕ(k; κ, Wi− (κ)) ∩ ϕ(k; κ, Vi+ (κ, ξ)) (4.3s) ⊆ Wi− (k) ∩ Vi+ (k, ϕ(k; κ, ξ)) = πi+ (k, ϕ(k; κ, ξ)) . (b) In order to construct the asymptotic backward phase πi− one proceeds analogously as in (a). For this, one again uses Corollary B.1.4 in order to obtain unique functions pκ : Xκ → P1i (κ), qκ : Xκ → Qi1 (κ) such that pκ (ξ) ≡ wi+ (κ, qκ (ξ))
and qκ (ξ) ≡ vi− (κ, pκ (ξ), ξ)
on Xκ ,
where the mappings wi+ and vi− had been constructed in Theorem 4.2.9(a) resp. in Proposition 4.3.5(b). Then πi− (κ, ξ) := pκ (ξ) + qκ (ξ) fulfills the above assertions. As consequence of Proposition 4.3.5 and Theorem 4.3.7 we obtain that for arbitrary pairs (κ, ξ) ∈ Wi∓ the fibers Vi± (κ, ξ) are mutually disjoint. In conclusion, the nonautonomous sets Vi± (ξ) form a foliation of the extended state space X . Corollary 4.3.9 (invariant foliation over Wi± ). The nonautonomous sets Vi± (ξ) from Proposition 4.3.5 are leaves of a forward invariant foliation over each fiber of the bundle Wi∓ from Theorem 4.2.9, i.e., for κ ∈ Z and ξ1 , ξ2 ∈ Wi∓ (κ), ξ1 = ξ2 we have Vi± (κ, ξ), Vi± (κ, ξ1 ) ∩ Vi± (κ, ξ2 ) = ∅. (4.3w) Xκ = ξ∈Wi∓ (κ)
4.3 Invariant Foliations and Asymptotic Phase
229
Remark 4.3.10. The fibers Vi+ (κ, ξ), ξ ∈ Wi− (κ), constitute the pseudo-stable foliation over the invariant fiber bundle Wi− of (S). A dual result holds in the sense that Vi− (κ, ξ), ξ ∈ Wi+ (κ), is the pseudo-unstable foliation over the fiber bundle Wi+ . Proof. Let (κ, ξ) ∈ X be given. The forward invariance of Vi± (κ, ξ) has been established in Proposition 4.3.5. By relation (4.3r) we get ϕ(·; κ, ξ) − ϕ(·; κ, πi± (κ, ξ)) ∈ ± ± X± κ,c and thus Proposition 4.3.5 implies ξ ∈ Vi (κ, πi (κ, ξ)). Since ξ ∈ Xκ was arbitrary, we established the left relation in (4.3w). The pair-wise disjointness in (4.3w) follows from ∅ = {ξ1 } ∩ {ξ2 } = Vi± (κ, ξ1 ) ∩ Vi± (κ, ξ2 ) for all ξ1 , ξ2 ∈ Wi∓ (κ) with ξ1 = ξ2 . In case ai + ς 1, the asymptotic forward phase πκ+ from Theorem 4.3.7(a) implies forward convergence of every solution to the nonautonomous set Wi− , but it does not instantly imply the convergence to a specific fiber Wi− (k). In order to achieve this, one needs to start “progressively earlier” leading to the concept of attraction discussed in Chap. 1. Under an additional assumption we can prove such an attraction property of the invariant fiber bundle Wi− . Corollary 4.3.11 (attraction). Assume ai + ς 1. If the sequence (Γκ− (i))κ∈Z from (Γi− ) is backward tempered, then the invariant fiber bundle Wi− from ˆ Theorem 4.2.9(b) is exponentially B-attracting, i.e., for all B ⊆ X one has exponential convergence lim hXk (ϕ(k; k − n, B(k − n)), Wi− (k)) = 0 for all k ∈ Z,
n→∞
with an attraction universe Bˆ consisting of uniformly bounded subsets of X . Proof. By assumption there exists a real γ ∈ (0, 1) with γ ∈ Γ¯i . Let B ⊆ X be uniformly bounded and w.l.o.g. we can assume B ⊆ BR for some R > 0. For an arbitrary k ∈ Z we choose a sequence ξn ∈ B(k − n), n ∈ N. The dichotomy estimates (3.4g) imply that the sequences ( P1i (k − n)ξn )n∈N , ( Qi1 (k − n)ξn )n∈N are bounded by Ki+ R resp. Ki− R. Hence, they are backward tempered. More+ over, the assumption on (Γκ− (i))κ∈Z ensures that the sequence (C˜k−n (ξn , γ))n≥0 , + ˜ where Cκ (ξ, c) is given in Theorem is backward tempered uniformly in 4.3.7, ξn ∈ BR (0). Thus, if we choose ∈ 1, γ1 there exists an integer K = K(γ, , R) such that C˜κ+ (ξ, γ) (4.3x) ≤ −κ for all κ ≤ K, ξ ∈ BR (0). 1 − ˜i (γ) For each ξn ∈ B(k − n) the invariance of Wi− and (4.3r) imply dist(ϕ(k; k − n, ξn ), Wi− (k)) = dist ϕ(k; k−n, ξn ), ϕ(k; k − n, Wi− (k − n)) ≤ ϕ(k; k − n, ξn ) − ϕ(k; k − n, πi+ (k − n, ξ)) + (4.3r) C˜k−n (ξn , c) Ki+ + γn Ki R + ≤ 1 − i (γ) 1 − ˜i (γ)
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4 Invariant Fiber Bundles
for all n ∈ Z+ 0 , and together with (4.3x) this guarantees dist(ϕ(k; k − n, ξn ), Wi− (k)) ≤
Ki+ + n Ki Rγ + ε−k (γ)n 1 − i (γ)
for all ξn ∈ B(k − n) and n ≥ k − K. Since the right-hand side of this estimate does not depend on ξn we get dist(ϕ(k; k − n, ξn ), Wi− (k)) −−−−→ 0 n→∞
for all k ∈ Z,
where the choice of γ implies convergence at an exponential rate.
Corollary 4.3.12. Let p ∈ N. (a) If (S) is p-periodic, then one has πi± (κ + p, ξ) = πi± (κ, ξ) for all (κ, ξ) ∈ X , i.e., the mappings πi+ , πi− are also p-periodic in their first argument. (b) If (S) is autonomous, then the mappings πi+ , πi− do not depend on their first argument. Proof. Let (κ, ξ) ∈ X and choose a growth rate c ∈ Γ¯i . By construction, the solu+ + tion φ : Z+ κ → X, φ(k) := ϕ(k; κ, πi (κ, ξ)) fulfills φ − ϕ(·; κ, ξ) ∈ Xκ,c . Since our equation (S) is p-periodic, also the shifted sequence ψ := φ(· − p) : Z+ κ+p → X solves (S) and the difference ψ − ϕ(· − p; κ, ξ) is c+ -bounded. The p-periodicity of (S) implies ϕ(k − p; κ, ξ) = ϕ(k; κ + p, ξ) for all k ∈ Z+ κ+p (cf. Proposition 2.5.3) and Wi− (κ + p) ∩ Vi+ (κ + p, ξ) = {ψ(κ + p)} . This yields πi+ (κ + p, ξ) = ψ(κ + p) = φ(κ) = πi+ (κ, ξ) and thus the sequence πi+ (·, ξ) is p-periodic. The claim for the asymptotic backward phase πi− can be shown similarly. Finally, assertion (b) is an immediate consequence of (a).
4.4 Smoothness of Fiber Bundles and Foliations At first glance it seems to be a straight forward task to derive continuous differentiability of the invariant fiber bundles Wi± constructed in the Hadamard–Perron Theorem 4.2.9, provided the nonlinearities are sufficiently smooth. One is tempted to apply the uniform C m -contraction principle from Theorem B.1.5 to the fixed point equation (4.2f) with ξ ∈ Xκ as parameter. In fact, this approach is successful for the fiber bundles associated to a hyperbolic splitting of the linear part (L0 ) – and also for weakly nonhyperbolic situations (cf. [26]). Yet, for arbitrary splittings the situation is different, since exponentially bounded sequences need not to be bounded in the classical way. As a result, substitution operators on spaces of such sequences need not to be differentiable
4.4 Smoothness of Fiber Bundles and Foliations
231
(see [26, Examples 4.7 and 4.9]) and thus Theorem B.1.5 is unfortunately not applicable. This necessitates another more flexible proof strategy and various other techniques have been developed, where we refer to Sect. 4.10 for a survey. We carefully try to give a clear and accessible “ad hoc” proof for the maximal smoothness class of invariant fiber bundles. Moreover, we give an example which shows that our necessary gap conditions are sharp. Instead of applying Theorem B.1.5 directly, we formally differentiate the fixed point equation (4.2f), obtain a new fixed point relation and show that its unique solutions are the desired derivatives of wi± . This approach needs no technical tools beyond the contraction mapping principle and Lebesgue’s theorem. The C m -smoothness of invariant fiber bundles is proved by induction over m. The induction over the smoothness class m is the key for understanding the structure of the problem. Our focus is not to hide the core of the proof by omitting the technical induction argument as it is frequently done in the literature. To our understanding this is one of the reasons why the “Hadamard– Perron-Theorem” has been reproven by so many authors for similar situations over the years. The induction argument of the proof is crucial because it is needed to rigorously compute the higher order derivatives of compositions of maps, the so-called “derivative tree”. It turned out to be advantageous to use two different representations of the derivative tree: First, a “totally unfolded derivative tree” to show that a fixed point operator is well-defined and to compute explicit global bounds for the higher order derivatives of the fiber bundles. Second, a “partially unfolded derivative tree” to elaborate the induction argument in a recursive way. After this foretaste for the things to come, we again deal with semilinear equations Bk+1 x = Ak x + fk (x, x )
(S)
and begin our analysis with a technical lemma valid in the setting from Hypothesis 4.2.3 of only Lipschitzian nonlinearities. Lemma 4.4.1. Let κ ∈ I, φ, φ¯ : Z± κ → X be solutions to (S) and suppose Hypothesis 4.2.1 and 4.2.3 hold true. If c ∈ (ai , bi ) and i (c) < 1 for one 1 ≤ i < N , then: (a) For I unbounded above and φ − φ¯ ∈ X+ κ,c one has ¯ ≤ φ(k) − φ(k) i
Ki+ ¯ ec (k, κ) for all k ∈ Z+ . φ(κ) − φ(κ) κ i 1 − i (c)
(b) For I unbounded below and φ − φ¯ ∈ X− κ,c one has ¯ ≤ φ(k) − φ(k) i
Ki− ¯ ec (k, κ) for all k ∈ Z− , φ(κ) − φ(κ) κ i 1 − i (c)
where the constant i (c) is given in Lemma 4.2.6.
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4 Invariant Fiber Bundles
Proof. Using Theorem 3.5.3(a) the present assertion (a) can be shown similarly to the following proof of (b). Above all, we remark that φ − φ¯ ∈ X− κ,c solves the linear ¯ inhomogeneous equation Bk+1 x = Ak x + gk with gk := fk (φ(k)) − fk (φ(k)). − −1 gk ≤ L(c) φ − φ¯ ec (k, κ) By the Lipschitz conditions (4.2a) one gets B k+1
κ,c −
and therefore the inclusion g ∈ X− for all k ∈ Z− κ,c,B with g κ,c,B ≤ − κ L(c) φ − φ¯κ,c . Consequently, we can apply Theorem 3.5.3(b) in order to infer the estimate ¯ + Ci (c)L(c) φ − φ¯− , φ − φ¯− ≤ K − φ(κ) − φ(κ) i κ,c κ,c i where we made use of Remark 3.5.9(3). Thus, our assumption Ci (c)L(c) = i (c) < 1 (cf. (4.2i)) finally implies the claimed inequality. From now on we strengthen Hypothesis 4.2.3 by imposing globally bounded −1 Fr´echet-differentiable nonlinearities Bk+1 fk . −1 Hypothesis 4.4.2. Let m ∈ N. Suppose that Bk+1 fk : Xk ×Xk+1 → Xk+1 , k ∈ I , m are of class C and for all 1 ≤ n ≤ m one has
δi+ (n) :=
sup
δi− (n) :=
sup
(k,x,x )∈X ×X (k,x,x )∈X ×X
i Q1 (k)Dn B −1 fk (x, x ) k+1 L i P1 (k)Dn B −1 fk (x, x ) k+1 L
n (Xk ×Xk+1 ;Xk+1 )
n (Xk ×Xk+1 ;Xk+1 )
< ∞, < ∞.
(4.4a)
Remark 4.4.3. Under Hypothesis 4.2.3 it follows using Proposition C.1.1 that sup
(k,x,x )∈X ×X
DB −1 fk (x, x ) k+1 L(X
k ×Xk+1 ;Xk+1 )
≤ L1 + L2
(4.4b)
and the constants δ1± (i) exist per se. Furthermore, if we suppose Hypothesis 4.1.1 −1 and Bk+1 Ak ∈ L(Xk , Xk+1 ), k ∈ I , then Proposition 4.1.3(c) guarantees that the general forward solution of (S) exists with ϕ(k; κ, ·) ∈ C m (Xκ , Xk ), κ ≤ k. A similar statement holds for the general backward solution under the assumptions of Proposition 4.1.4. Remark 4.4.4 (spectral gap condition). For an integer 1 ≤ i < N we define $ % & a + b bi − ai i i , ai m , := min − ai 2 a i + am i % $ & ai + b i bi − ai − m , bi − bi ςi (m) := min 2 bi + bm i
ςi+ (m)
4.4 Smoothness of Fiber Bundles and Foliations
233
and strengthen (Gi ) to the spectral gap condition am i bi , ai b m i ,
∃ςi ∈ 0, ςi+ (m) : (Gi ) holds, ∃ςi ∈ 0, ςi− (m) : (Gi ) holds,
(G+ i,m ) (G− i,m )
choose a fixed real number ς ∈ max Ki− , Ki+ (L1 + bi − ςi L2 ) , ςi and de+ fine intervals Γ¯i := [ai + ς, bi − ς]. The condition am i bi guarantees ςi (m) > 0, b − ± i −ai while ai bm and i ensures that ςi (m) > 0. In addition, one has ςi (1) = 2 − ) and (G ) coincide for m = 1. the conditions (G+ i,m i,m In order not to interrupt our later argument, we insert the elementary Lemma 4.4.5. Let α, β, ς > 0 and m ∈ N: ' α+β m < β − ς. (a) If αm < β and ς < α m α+α m − α, then (α + ς) ' α+β m (b) If α < β m and ς < β − m β+β . m , then α + ς < (β − ς) ' α+β Proof. The assumption αm < β implies 0 < α m α+α m − α, which in turn is m ς m and consequently equivalent to α + ς < (β − β) 1 − β m m
ς ς ς α + β < (β m + β) 1 − ≤ βm 1 − +β 1− β β β = (β − ς)m + β − ς, i.e., α + ς < (β − ς)m . The assertion (b) can be shown along the same lines.
Theorem 4.4.6 (smoothness of invariant fiber bundles). Assume Hypotheses 4.2.1, 4.2.3 and 4.4.2 are satisfied and choose 1 ≤ i < N , c ∈ Γ¯i . + (a) If I is unbounded above and (Γi+ ), (G+ i,m ) hold, then the map wi (κ, ·) : Xκ → P1i (κ) from Theorem 4.2.9(a) is of class C m with globally bounded derivatives sup D2n wi+ (κ, ξ)Ln (Xκ ) ≤ Cn for all 1 ≤ n ≤ m, (κ,ξ)∈X
where in particular C1 := ˜+ i (c). − (b) If I is unbounded below and (Γi− ), (G− i,m ) hold, then the map wi (κ, ·) : Xκ → i m Q1 (κ) from Theorem 4.2.9(b) is of class C with globally bounded derivatives sup D2n wi− (κ, ξ)Ln (Xκ ) ≤ Cn
(κ,ξ)∈X
where in particular C1 := ˜− i (c).
for all 1 ≤ n ≤ m,
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4 Invariant Fiber Bundles
(c) The global bounds C2 , . . . , Cm ≥ 0 can be determined recursively using Cn := max
j n ( Ki− j − δ (j) max {1, bi − ς } C#Nν , ς(1 − i (c)) j=2 i < ν=1 (N1 ,...,Nj )∈Pj (n)
j n ( Ki+ j + δ (j) max {1, bi − ς } C#Nν ς(1 − i (c)) j=2 i < ν=1
(N1 ,...,Nj )∈Pj (n)
(4.4c) for all 2 ≤ n ≤ m, where the constants i (c) ∈ [0, 1) are defined in Lemma 4.2.6 and ˜± i (c) is given in Theorem 4.2.9. Remark 4.4.7. In case ai ≤ 1 or 1 ≤ bi it makes sense to investigate the behavior of − the conditions (G+ i,m ) resp. (Gi,m ) for arbitrarily large values of m. Having constant rates ai (k) ≡ αi , bi (k) ≡ βi on I, the asymptotic behavior of the sequences ςi± (m) is as follows: lim ς + (m) m→∞ i
lim ς − (m) m→∞ i
= 0, if αi ≤ 1,
= βi − 1, if βi ≥ 1.
+ Hence, the spectral gap condition (G+ i,m ) for Wi becomes increasingly restrictive for growing m ∈ N.
Proof. Let (κ, ξ) ∈ X and c ∈ Γ¯i for a fixed 1 ≤ i < N . Above all, we remark that the assumptions of Theorem 4.2.9 are fulfilled and we use the brief notation introduced in (4.2p). So there exist invariant fiber bundles W± which are graphs of globally Lipschitzian mappings w± : X → X over the vector bundles P± . These mappings are given by the relation w± (κ, ξ) = P∓ (κ)φ± κ (κ, ξ),
(4.4d)
± + where φ± κ (ξ) ∈ Xκ,c is the unique fixed point of the Lyapunov–Perron operators Tκ − and Tκ defined in (4.2q) resp. (4.2b). Here, thanks to Lemma 4.2.6 the operators ± Tκ± : X± κ,c × Xκ → Xκ,c are uniform contractions in their first argument with (4.2e)
(4.2i)
lip1 Tκ± ≤ i (c) < 1.
(4.4e)
(a) Since the arguments for the operator Tκ+ are analogous, we only sketch the higher order smoothness case. We formally differentiate the fixed point identity for the operator Tκ+ defined in (4.2q) w.r.t. ξ ∈ Xκ and obtain another fixed point equation φlκ (ξ) = Tκl,+ (φlκ (ξ); ξ) with the right-hand side Tκl,+ (φl ; ξ) :=
∞ n=κ
Gi (·, n + 1) Dfˆn (φκ (n, ξ))φl (n, ξ) + Rnl (ξ)
(4.4f)
4.4 Smoothness of Fiber Bundles and Foliations
235
for all l ∈ {2, . . . , m}. The remainder Rnl allows representations analogous to (4.4p) and (4.4q) below. We refer to the following for further details. (b) Our induction argument is involved and subdivided into two main steps. First, we address the case of continuous differentiability in step (I) and show the general C m -situation on the foundation of induction over l ∈ {1, . . . , m} in step (II). −1 fk , k ∈ I . By formal differentiation (I) We conveniently abbreviate fˆk := Bk+1 of the fixed point equation (cf. (4.2f) and (4.2b)) φκ (k, ξ) = Φ− P− (k, κ)P− (κ)ξ +
κ−1
Gi (k, n + 1)fˆn (φκ (n, ξ))
n=−∞
for all k ∈ Z− κ , w.r.t. the parameter ξ ∈ Xκ we obtain another fixed point equation φ1κ (ξ) = Tκ1,− (φ1κ (ξ); ξ)
for all ξ ∈ Xκ
(4.4g)
for the formal derivative φ1κ of the fixed point mapping φκ : Xκ → X− κ,c from Lemma 4.2.8, where the right-hand side of (4.4g) is given by Tκ1,− (φ1 ; ξ)
:=
Φ− P− (·, κ)P− (κ)
+
κ−1
Gi (·, n + 1)Dfˆn (φκ (n, ξ))φ1 (n).
n=−∞
Here, the sequence φ1 (k), k ∈ Z− κ , has values in L(Xκ , Xk ) and in the following we investigate this operator Tκ1,− . 1,− (I1 ) Claim: For every c ∈ Γ¯i the operator Tκ1,− : X1,− κ,c × Xκ → Xκ,c is welldefined and satisfies the estimate 1,− 1 − Tκ (φ ; ξ) ≤ K − + i (c) φ1 − i κ,c κ,c
for all φ1 ∈ X1,− κ,c , ξ ∈ Xκ .
(4.4h)
Choose φ1 ∈ X1,− κ,c . Using Lemma A.1.5 we argue as in the proof of Theorem 3.5.3 in order to show the two inequalities − P− (k)Tκ1,− (k, φ1 ; ξ) ≤ K − ebi (k, κ) + Ki L(c) φ1 − , i κ,c bi − c + P+ (k)T 1,− (k, φ1 ; ξ) ≤ Ki L(c) φ1 − for all k ∈ Z− κ κ κ,c c − ai
and with the norm from Lemma 3.3.22 this yields 1,− Tκ (k, φ1 ; ξ) ec (κ, k) ≤ K − + i (c) φ1 − i i κ,c
for all k ∈ Z− κ.
Consequently, we have the inclusion Tκ1,− (φ1 ; ξ) ∈ X1,− κ,c and passing over to the least upper bound over k ∈ Z− implies the linear bound (4.4h). κ
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4 Invariant Fiber Bundles
1,− (I2 ) Claim: For every c ∈ Γ¯i the operator Tκ1,− (·; ξ) : X1,− κ,c → Xκ,c is a uniform 1 contraction in ξ ∈ Xκ ; moreover, the unique fixed point φκ (ξ) ∈ X1,− κ,c does not ¯ depend on c ∈ Γi and satisfies
1 − φ (ξ) ≤ κ,c
Ki− 1 − i (c)
for all ξ ∈ Xκ .
(4.4i)
Analogous to the estimate deduced in step (I1 ) we obtain using Lemma A.1.5, 1,− 1 Tκ (φ ; ξ) − Tκ1,− (φ¯1 ; ξ)− ≤ i (c) φ1 − φ¯1 − κ,c κ,c
for all φ1 , φ¯1 ∈ X1,− κ,c .
Taking the estimate (4.4e) into account, Banach’s fixed point theorem (cf. [295, p. 361, Lemma 1.1]) guarantees the unique existence of a fixed point φ1κ (ξ) ∈ X1,− κ,c 1,− 1 for Tκ1,− (·; ξ) : X1,− κ,c → Xκ,c . This fixed point φκ (ξ) is independent of the growth 1,− rate c ∈ Γ¯i since with Lemmata 3.3.26 and 3.3.27 we have X1,− κ,bi −ς ⊆ Xκ,c and thus 1,− every mapping Tκ1,− (·; ξ) : X1,− κ,c → Xκ,c has the same fixed point as the restriction 1,− Tκ (·; ξ)|X1,− . Finally, the fixed point property (4.4g) together with (4.4h) imply κ,bi −ς
the global bound for φ1κ (ξ). (I3 ) Claim: For every c ∈ [ai + ς, bi − ς) the mapping φκ : Xκ → X+ κ,c is differentiable with derivative Dφκ = φ1κ : Xκ → X1,− κ,c .
(4.4j)
In relation (4.4j), as well as in all subsequent considerations, we use the isomor− phism between the Banach spaces X1,− κ,c and L(Xκ , Xκ,c ) from Lemma 3.3.27 and identify them. To show the differentiability assertion, we derive the quotients 1 φκ (k, ξ + h) − φκ (k, ξ) − φ1κ (k, ξ)h for all ξ ∈ Xκ , |h|i
h ¯ ˆ ˆ ˆ fk (x + h, y + h) − fk (x, y) − Dfk (x, y) ¯ h ¯ := Δfˆk (x, y; h, h)
(h1 , h2 )
Δφ(k, h) :=
¯ y ∈ Xκ+1 , where h, h ¯ = 0. Thereby, the inclusion for all k ∈ I, h, x ∈ Xκ and h, − Δφ(·, h) ∈ Xκ,c holds due to (I2 ) and Lemma 4.2.8. To prove differentiability, we have to show the limit relation limh→0 Δφ(·, h) = 0 in X− κ,c . For this, consider growth rates c bi − ς, d ∈ (c, bi − ς) and from Lemma 4.4.1(b) we obtain 1 Ki− ed (n, κ) for all n ∈ Z− |φκ (n, ξ + h) − φκ (n, ξ)|i ≤ κ. |h|i 1 − i (c)
(4.4k)
4.4 Smoothness of Fiber Bundles and Foliations
237
Using the fixed point relation (4.2f) for φκ (ξ) and (4.4g) for φ1κ (ξ) it results Δφκ (k, h) =
κ−1 1 Gi (k, n + 1) |h|i n=−∞ · fˆn (φκ (n, ξ + h)) − fˆn (φκ (n, ξ)) − Dfˆn (φκ (n, ξ))φ1κ (n, ξ)h
for all k ∈ Z− κ , where subtraction and addition of the expression
Dfˆn (φκ (n, ξ)) φκ (n, ξ + h) − φκ (n, ξ) in the above parenthesis implies the estimate
P− (k)Δφ(k, h)
(3.4g)
≤ Ki−
κ−1
ebi (k, n + 1) Δfˆn (φκ (n, ξ), φκ (n, ξ + k) − φκ (n, ξ))
n=k
1 · φκ (n, ξ + h) − φκ (n, ξ) |h|i +Ki− L(d)
κ−1
ebi (k, n + 1) |Δφ(n, h)|i
for all k ∈ Z− κ
n=k
and together with (4.4k) we infer
P− (k)Δφκ (k, h) ≤ Ki−
κ−1 Ki− max {1, d } ebi (k, n + 1) 1 − i (d) n=k
· Δfˆn (φκ (n, ξ), φκ (n, ξ + k) − φκ (n, ξ)) +Ki− L(d)
κ−1
ebi (k, n + 1) |Δφ(n, h)|i
for all k ∈ Z− κ.
n=k
Analogously, also using (4.4k) we can derive a similar estimate
P+ (k)Δφκ (k, h) ≤ Ki+
k−1 Ki− max {1, d } eai (k, n + 1) 1 − i (d) n=−∞
· Δfˆn (φκ (n, ξ), φκ (n, ξ + k) − φκ (n, ξ)) +Ki+ L(d)
k−1
eai (k, n + 1) |Δφ(n, h)|i
n=−∞
and thanks to the norm from Lemma 3.3.22 we obtain
for all k ∈ Z− κ
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4 Invariant Fiber Bundles
|Δφκ (k, h)|i ≤ max {S1 + S2 , S3 + S4 }
for all k ∈ Z− κ
with the abbreviations S1 :=
κ−1 (Ki− )2 max {1, d } ebi (k, n + 1)ed (n, κ) 1 − i (d) n=k · Δfˆn (φκ (n, ξ), φκ (n, ξ + h) − φk (n, ξ)) ,
S2 := Ki− L(d)
κ−1
ebi (k, n + 1) |Δφκ (n, k)|i ,
n=k
S3 :=
k−1 Ki− Ki+ max {1, d } eai (k, n + 1)ed (n, κ) 1 − i (d) n=−∞ · Δfˆn (φκ (n, ξ), φκ (n, ξ + h) − φk (n, ξ)) ,
S4 := Ki+ L(d)
k−1
eai (k, n + 1) |Δφκ (n, k)|i .
n=−∞
The elementary estimate max {S1 + S2 , S3 + S4 } ≤ S1 + S3 + max {S2 , S4 } together with Lemma A.1.5 yields |Δφκ (k, h)|i ec (κ, k) ≤ (S1 + S3 )ec (κ, k) Ki+ Ki− − ,
Δφκ (h) κ,c + L(d) max bi − d d − ai
for all k ∈ Z− κ
and passing over to the supremum over k ∈ Z− κ ensures (cf. (4.4e)) K − max Ki− , Ki+ max {1, d } − sup V (k, h)
Δφκ (h) κ,c ≤ i (1 − i (d))2 k∈Z− κ with V (k, h) := ec (κ, k)
κ−1
ebi (k, n + 1)ed (n, κ)
n=k
· Δfˆn (φκ (n, ξ), φκ (n, ξ + h) − φk (n, ξ)) , +ec (κ, k)
k−1
ebi (k, n + 1)ed (n, κ)
n=−∞
· Δfˆn (φκ (n, ξ), φκ (n, ξ + h) − φk (n, ξ))
for all k ∈ Z− κ.
4.4 Smoothness of Fiber Bundles and Foliations
239
Thus, in order to prove claim (I3 ), we only have to show the limit relation (4.4l)
lim sup V (k, h) = 0,
h→0
k∈Z− κ
which will be done indirectly. Supposing (4.4l) is not true, there exists an ε > 0 and a sequence (hj )j∈N in Xκ with limit 0 such that supk∈Z− V (k, hj ) > ε for all κ j ∈ N. This, in turn, implies the existence of a further sequence (kj )j∈N in Z− κ with V (kj , hj ) > ε for all j ∈ N.
(4.4m)
Using the crude estimate Δfˆn (x, y, h1 , h2 ) ≤ 2(L1 + L2 ), which results from (4.2a) and (4.4b), it follows using Lemma A.1.5 that
V (k, h) ≤
L 1 + L2 L1 + L2 + d − ai bi − d
e d (k, κ) for all k ∈ Z− κ c
and due to Lemma A.1.3(b) the right-hand side of this estimate converges to 0 for k → ∞, i.e., we have limk→∞ V (k, h) = 0 uniformly in h ∈ Xκ . Because of (4.4m) the sequence (kj )j∈N has to be bounded in Z− κ , i.e., there exists an integer K ≤ κ with kj ∈ [K, κ]Z for all j ∈ N. We consequently obtain from Proposition A.1.2(a) that κ−1 V (kj , hj ) ≤ e bc (κ, k) ebi (κ, n + 1)ed (n, κ) i !" # n=K
(4.4n)
≤1
· Δfˆn (φκ (n, ξ), φκ (n, ξ + hj ) − φk (n, ξ)) , κ−1
+ ec (κ, K)
eai (K, n + 1)ed (n, κ)
n=−∞
· Δfˆn (φκ (n, ξ), φκ (n, ξ + hj ) − φk (n, ξ))
for all j ∈ N
and due to the continuity of the fixed point mapping φκ (n, ·) : Xκ → Xn guaranteed by both Lemmata 4.3.4(b) and 3.3.28, lim φκ (n, ξ + hj ) = φκ (n, ξ)
j→∞
for all n ∈ Z− κ,
as well as using the differentiability of fˆn , required in Hypothesis 4.4.2, lim
¯ (h,h)→(0,0)
ˆ Δfn (x1 , x2 , h, ¯h) = 0,
240
4 Invariant Fiber Bundles
which leads to the limit relation lim Δfˆn (φκ (n, ξ), φκ (n, ξ + hj ) − φκ (n, ξ) = 0 for all n ∈ Z− κ. j→∞
We can conclude that the finite sum in (4.4n) tends to 0 in the limit j → ∞. As in the proof of Lemma 4.3.4(b), Lebesgue’s theorem ensures that also the infinite sum in (4.4n) converge to 0 for j → ∞. In conclusion, limj→∞ V (kj , hj ) = 0, which contradicts (4.4m). Hence, the claim (I3 ) is true, where (4.4j) follows by the uniqueness of Fr´echet derivatives. (I4 ) Claim: For every c ∈ [ai + ς, bi − ς) the mappings Dφκ : Xκ → X1,− κ,c and D2 w− (κ, ·) : Xκ → L(Xκ ) are continuous. With a view to (4.4j) it is sufficient to show the continuity of the fixed point mapping φ1κ : Xκ → X1,− κ,c . In order to do this, fix ξ0 ∈ Xκ and choose ξ ∈ Xκ . Using the − fixed point equation (4.4g) for φ1 , we can estimate the norm φ1 (ξ) − φ1 (ξ0 ) κ
κ
κ
κ,c
and as in the continuity proof of Lemma 4.3.4(b) it follows limξ→ξ0 φ1κ (ξ) = φ1κ (ξ0 ) − in X1,− κ,c . By the identity (4.4d) and Lemma 3.3.28 also D2 w (κ, ·) is continuous. Hence, we have shown the assertion (b) for m = 1, where the given bound C1 is a consequence of Theorem 4.2.9(b2) interplaying with Proposition C.1.1. (II) Now let m ≥ 2. By formal differentiation of the fixed point equation (4.2f) w.r.t. ξ ∈ Xκ , using the higher order chain rule from Theorem C.1.3, we obtain another fixed point equation φlκ (ξ) = Tκl,− (φlκ (ξ); ξ)
(4.4o)
for the formal derivative φlκ of φκ : Xκ → X+ κ,c of order l ∈ {2, . . . , m}, where the right-hand side of (4.4o) is given by Tκl,− (φl ; ξ) :=
Gi (·, n + 1) Dfˆn (φκ (n, ξ))φl (n, ξ) + Rnl (ξ) .
κ−1 n=−∞
l Here, φl (k) ∈ Ll (Xκ , Xk ), k ∈ Z− κ , and the remainder Rn has the representations:
•
As partially unfolded derivative tree (C.1a) Rnl (ξ) =
•
l−1
l − 1 dj Dfˆn (φκ (n, ξ))φl (n, ξ), j dξ j j=1
(4.4p)
which is appropriate for the induction in the subsequent step, and as totally unfolded derivative tree (C.1b)
Rnl (ξ) =
l
j=2
(N1 ,...,Nj )∈Pj< (l)
Dj fˆn (φκ (n, ξ))φ#N1 (n, ξ) · · · φ#Nj (n, ξ),
(4.4q) which enables us to get explicit global bounds for higher order derivatives.
4.4 Smoothness of Fiber Bundles and Foliations
241
For our forthcoming considerations it is crucial that Rnl does not depend on φlκ . In the next steps, we will solve the fixed point equation (4.4o) for the operator Tκl,− . As preparation, for every l ∈ {1, . . . , m} we introduce the growth rates cl (k) :=
bi (k) − ς
if bi (k) − ς ≥ 1,
(bi (k) − ς)
l
if bi (k) − ς < 1
for all k ∈ Z− κ
and c1 , . . . , cl ∈ (ai + ς, bi − ς] holds, which in case bi (k) − ς ≥ 1 follows i from the relation ς < bi −a and otherwise results from ai + ς (bi − ς)m 2 (cf. Lemma 4.4.5(b)). We formulate for m ¯ ∈ {1, . . . , m} the induction hypothesis: ⎧ For every l ∈ {1, . . . , m} ¯ and growth rates c ∈ (ai + ς, cl ] the operator ⎪ ⎪ ⎪ l l,− l,− ⎪ T : X × X → X κ ⎪ κ κ,c κ,c satisfies: ⎪ ⎪ ⎪ ⎪ ⎪ (a) It is well-defined. ⎪ ⎪ ⎪ ⎨(b) Tκl,− (·; ξ) is a uniform contraction in ξ ∈ Xκ . A(m) ¯ : (c) The unique fixed point φlκ (ξ) of Tκl,− (·; ξ) is globally bounded in the l ⎪ + ⎪ φκ (n, ξ) ≤ Cl ec (n, κ) for all n ∈ Z− ⎪ c -norm ⎪ κ , ξ ∈ Xκ with the l l ⎪ ⎪ ⎪ constants C ≥ 0 given in (4.4c). l ⎪ ⎪ ⎪ ⎪ (d) If c cl then φl−1 : Xκ → Xl,− ⎪ κ κ,c is continuously differentiable ⎪ ⎩ = φlκ : Xκ → Xl,− w.r.t. ξ ∈ Xκ and derivative Dφl−1 κ κ,c . Ki− 1−i (c) even choose C1 = ˜− i (c).
For m ¯ = 1 our step (I) establishes the induction hypothesis A(1) with C1 =
(cf. (4.4i)). Actually, thanks to Theorem 4.2.9(b2) one can Now we assume A(m ¯ − 1) holds true for some m ¯ ∈ {2, . . . , m} and we are going to prove A(m) ¯ in the following steps: m,− ¯ ¯ m,− ¯ (II1 ) Claim: For every c ∈ (ai +ς, cm : Xm,− ¯ ] the operator Tκ κ,c ×Xκ → Xκ,c is well-defined and satisfies the estimate m,− − ¯ − ¯ T ¯ (φm + C¯m ; ξ)κ,c ≤ i (c) φm ¯, κ κ,c
(4.4r)
with the constant
j m ¯ ( Ki− − j ¯ Cm δ (j) max {1, bi − ς } C#Nν , ¯ := max ς j=2 i < ν=1 (N1 ,...,Nj )∈Pj (m) ¯
m ¯ Ki+ + δi (j) max {1, bi ς j=2 (N1 ,...,Nj )∈Pj< (m) ¯
i.e., the assertion A(m)(a) ¯ holds true.
j
− ς }
j ( ν=1
C#Nν ,
242
4 Invariant Fiber Bundles
Let l ∈ {2, . . . , m} ¯ and choose c ∈ (ai + ς, cl ]. Using the estimate c#N1 · . . . · c#Nj ≥ cl for any ordered partition (N1 , . . . , Nl ) ∈ Pj< (l) of length j ∈ {2, . . . , l}, from (3.4g), (4.4a), (4.4q) and A(m ¯ − 1)(c), we obtain the inequalities κ−1 − l ΦP− (k, n + 1)Rn (ξ) P− (k) n=k
(A.1d)
≤
Ki− ec (k, κ) − δ (j) bi − cl j=2 i l
j (
C#Nν max {1, c#Nν }
(N1 ,...,Nl )∈Pj< (l) ν=1
using Lemma A.1.5(b) and analogously using Lemma A.1.5(a) one has k−1 Φ(k, n + 1)Rnl (ξ) P+ (k) n=−∞ Ki+ ec (k, κ) + ≤ δ (j) cl − ai j=2 i l
(A.1e)
j (
C#Nν max {1, c#Nν }
(N1 ,...,Nl )∈Pj< (l) ν=1
m ¯ m,− ¯ for all k ∈ Z− κ . Given φ ∈ Xκ,c the above estimates yield
m,− ¯ − ¯ Tκ¯ (k, φm + C¯m ; ξ)i ec (κ, k) ≤ i (c) φm ¯ κ,c
for all k ∈ Z− κ
with the constants C¯m ¯ ≥ 0 defined above. As usual, passing over to the supremum m,− ¯ ¯ ¯ for k ∈ Z− implies T (φm ; ξ) ∈ Xm,− κ κ κ,c . In particular, the estimate (4.4r) follows due to the inclusion c ∈ (ai + ς, bi − ς]. m,− ¯ ¯ m,− ¯ (II2 ) Claim: For every c ∈ (ai + ς, cm (·; ξ) : Xm,− ¯ ] the operator Tκ κ,c → Xκ,c m ¯ m,− ¯ is a uniform contraction in ξ ∈ Xκ ; moreover, the unique fixed point φκ (ξ) ∈ Xκ,c does not depend on c ∈ (ai + ς, cm ¯ ] and satisfies m φ ¯ (ξ)− ≤ Cm ¯ κ κ,c
for all ξ ∈ Xκ ,
(4.4s)
i.e., the assertions A(m)(b) ¯ and A(m)(c) ¯ hold true. m ¯ ¯m ¯ m,− ¯ Choose c ∈ (ai + ς, cm ¯ ] and let φ , φ ∈ Xκ,c . Keeping in mind that the remain¯ ¯m , φ ¯ , resp., from (3.4g) and (4.2a) der in (4.4p) and (4.4q) does not depend on φm we obtain the Lipschitz estimates m,− ¯ ¯ ¯ Tκ¯ (k, φm ; ξ) − Tκm,− (k, φ¯m ; ξ)i ec (κ, k) κ−1 ebi (k, n + 1)ec (n, κ), ≤ L(c) max Ki− n=k
Ki+
k−1 n=−∞
¯ ¯ ¯ − ¯ − eai (k, n + 1)ec (n, κ) φm − φ¯m ≤ i (c) φm − φ¯m κ,c κ,c
4.4 Smoothness of Fiber Bundles and Foliations
243
− for all k ∈ Z− κ using Lemma A.1.5. Passing over to the supremum over k ∈ Zκ m,− ¯ with (4.4e) implies the contraction property for Tκ (·; ξ) and, e.g., [295, p. 361, ¯ m,− ¯ Lemma 1.1]) implies the existence of a unique fixed point φm κ (ξ) ∈ Xκ,c . It can be m ¯ seen along the same lines as in (I2 ) that φκ (ξ) does not depend on c ∈ (ai + ς, cm ¯ ]. The fixed point property (4.4o) with (4.4r) implies the bound (4.4s). m−1 ¯ ¯ (II3 ) Claim: For every c ∈ (ai + ς, cm : Xκ → Xm,− ¯ ) the mapping φκ κ,c is differentiable with derivative ¯ ¯ m,− ¯ Dφm−1 = φm κ κ : Xκ → Xκ,c .
(4.4t)
m−1 ¯ is differentiable and then Let c ∈ (ai + ς, cm ¯ ) be fixed. First we show that φκ m ¯ ¯ ¯ we prove that the derivative is given by φκ : Xκ → L(Xκ , Xm−1,− ) ∼ = Xm,− κ,c κ,c (cf. Lemma 3.3.27). Thereto choose ξ ∈ Xκ arbitrarily, but fixed. Using the fixed ¯ point equation (4.4o) for φm−1 we get for h ∈ Xκ the identity κ ¯ ¯ (k; ξ + h) − φm−1 (k; ξ) φm−1 κ κ κ−1 ¯ m−1 ¯ = Gi (k, n + 1) Dfˆn (φκ (n, ξ + h))φm−1 (n, ξ + h) + R (ξ + h) κ n
n=−∞ κ−1
−
¯ ¯ Gi (k, n + 1) Dfˆn (φκ (n, ξ))φm−1 (n, ξ) + Rnm−1 (ξ) κ
n=−∞
for all k ∈ Z− κ . This leads to ¯ ¯ (k; ξ + h) − φm−1 (k; ξ) φm−1 κ κ κ−1 ¯ ¯ − Gi (k, n + 1)Dfˆn (φκ (n, ξ + h))φm−1 (n, ξ + h) − φm−1 (n, ξ) κ κ n=−∞
=
κ−1
¯ Gi (k, n + 1) Dfˆn (φκ (n, ξ + h)) − Dfˆn (φκ (n, ξ)) φm−1 (n, ξ + h) κ
n=−∞
+
κ−1
¯ ¯ Gi (k, n + 1) Rnm−1 (ξ + h) − Rnm−1 (ξ)
(4.4u)
n=−∞ ¯ ¯ φm−1 ∈ Xm−1,− and h ∈ Xκ we define the for all k ∈ Z− κ . With sequences κ,c m−1,− ¯ m−1,− ¯ ¯ , E ∈ L Xκ , Xκ,c , J : Xκ → Xm−1,− operators H ∈ L Xκ,c as follows κ,c
¯ Hφm−1 :=
κ−1
¯ Gi (·, n + 1)Dfˆn (φκ (n, ξ))φm−1 (n),
n=−∞
Eh :=
κ−1 n=−∞
¯ Gi (·, n + 1)R1m (n, ξ)h
244
4 Invariant Fiber Bundles
and J(h) :=
Gi (·, n + 1) Dfˆn (φκ (n, ξ + h)) − Dfˆn (φκ (n, ξ))
κ−1 n=−∞
¯ (n, ξ ·φm−1 κ
+ h) +
¯ Rnm−1 (ξ
+ h) −
¯ Rnm−1 (ξ)
−
¯ Rnm (ξ)h
(4.4v)
for all k ∈ Z− κ . In the subsequent lines we will show that H, E and J are well¯ ¯ defined. Using (3.4g) and (4.4a) it is easy to see that H : Xm−1,− → Xm−1,− is κ,c κ,c − − m−1 ¯ m−1 ¯ ≤ i (c) φ , which in turn gives us linear and satisfies Hφ κ,c
κ,c
(4.4e)
m−1,− ¯
H L(Xκ,c ) < 1.
(4.4w)
¯ ¯ (0; ξ)h, our Step (II1 ) yields Eh ∈ Xm−1,− , Keeping in mind that Eh = Tκm−1,− κ,c m−1,− ¯ while E is obviously linear and continuous, hence E ∈ L(Xκ , Xκ,c ). Argu¯ ments similar to those in Step (II1 ) lead to the inclusion J(h) ∈ Xm−1,− for any κ,c h ∈ Xκ . Because of (4.4u) we obtain
¯ m−1 ¯ ¯ (ξ) − H φm−1 (ξ + h) − φm−1 (ξ) = Eh + J(h) φκ¯ (ξ + h) − φm−1 κ κ κ for all h ∈ Xκ . Using the Neumann series (cf., e.g., [295, p. 74, Theorem 2.1] or ¯ ¯ Theorem B.3.1) and (4.4w), the linear mapping IXm−1,− − H ∈ L(Xm−1,− ) is κ,c κ,c invertible and this implies −1 ¯ m−1 ¯ (ξ + h) − φ (ξ) = I m−1,− ¯ − H [Eh + J(h)] φm−1 κ κ Xκ,c Thus, it remains to show limh→0
J(h) h
for all h ∈ Xκ .
¯ = 0 in Xm−1,− , because then one gets κ,c
− −1 1 ¯ m−1 ¯ φm−1 = 0, m−1,− ¯ (ξ + h) − φ (ξ) − I − H Eh κ κ X κ,c h→0 h
κ,c lim
i.e., the claim of the present Step (II3 ) follows. Nevertheless the proof of the limit J(h)− relation limh→0 h κ,c = 0 needs a certain technical effort. Thereto we use the fact that due to the induction hypothesis A(m ¯ − 1)(d) the remainder (4.4p)
¯ (ξ) = Rnm−1
m−2 ¯ j=1
m ¯ −2 j
∂j ˆ ¯ D f (φ (n, ξ)) φm−1−j (n, ξ) κ n κ ∂ξ j
is differentiable w.r.t. ξ ∈ Xκ , where the derivative is given by the product rule (cf. [295, p. 336]) as (4.4p)
¯ ¯ ¯ DRnm−1 (ξ) = Rnm (ξ) − D2 fˆn (φκ (n, ξ))φ1κ (n, ξ)φm−1 (n, ξ). κ
4.4 Smoothness of Fiber Bundles and Foliations
245
Using the abbreviation ¯ ΔRnm−1 (ξ, h) :=
1 ¯ ¯ Rnm−1 (ξ + h) − Rnm−1 (ξ)
h
¯ ¯ − Rnm (ξ) − D2 fˆn (φκ (n, ξ))φ1κ (n, ξ)φm−1 (n, ξ) h κ
¯ (ξ, h) = 0 for n ∈ Z− we obtain limh→0 ΔRnm−1 κ . Now we prove estimates for the components J− and J+ of J in P− resp. P+ , separately. For k ∈ Z− κ we get
J− (k, h) (4.4v)
=
κ−1
Φ− P− (k, n
¯ + 1) Dfˆn (φκ (n, ξ + h))−Dfˆn (φκ (n, ξ)) φm−1 (n, ξ + h) κ
n=k
¯ m−1 ¯ (n, ξ)h + ΔR (ξ, h)
h
, −D2 fˆn (φκ (n, ξ))φ1κ (n, ξ)φm−1 κ n where subtraction and addition of the expression ¯ (n, ξ + h) D2 fˆn (φκ (n, ξ))φκ (n, ξ + h) − φκ (n, ξ) − φ1κ (n, ξ)hφm−1 κ
leads to
J− (k, h) =
κ−1
Φ− P− (k, n
) + 1) Dfˆn (φκ (n, ξ + h)) − Dfˆn (φκ (n, ξ))
n=k
* ¯ ˆ (n, ξ + h) −D fn (φκ (n, ξ))φκ (n, ξ + h) − φκ (n, ξ) φm−1 κ 2
¯ +D2 fˆn (φκ (n, ξ))φκ (n, ξ + h)−φκ (n, ξ)−φ1κ (n, ξ)hφm−1 (n, ξ + h) κ ¯ ¯ +D2 fˆn (φκ (n, ξ))φ1κ (n, ξ)φm−1 (n, ξ + h) − φm−1 (n, ξ)h κ κ ¯ (ξ, h) h
for all k ∈ Z− +ΔRnm−1 κ.
Using the quotient
h 2ˆ ¯ ˆ ˆ Dfn (x + h, y + h) − Dfn (x, y) − D fn (x, y) ¯ h ¯ := ΔDfˆn (x, y, h, h) ¯ (h, h)
246
4 Invariant Fiber Bundles
¯ for all n ∈ Z− κ , x ∈ Xκ , y ∈ Y, h ∈ Xκ \ {0} and h ∈ Xκ+1 \ {0}, we obtain
J− (k, h) ≤
) − ΦP− (k, n+1) ΔDfˆn (φκ (n, ξ), φκ (n, ξ + h)−φκ (n, ξ))
κ−1 n=k
¯ (n, ξ + h) + D2 fˆn (φκ (n, ξ)) · φκ (n, ξ + h)−φκ (n, ξ)φm−1 κ ¯ · φκ (n, ξ + h) − φκ (n, ξ) − φ1κ (n, ξ)h φm−1 (n, ξ + h) κ ¯ ¯ + D2 fˆn (φκ (n, ξ)) φ1κ (n, ξ)φm−1 (n, ξ + h) − φm−1 (n, ξ)h κ κ * m−1 ¯ + ΔR1 (n, ξ, h) h
for all k ∈ Z− κ. With Hypotheses 4.2.1 and 4.4.2 (cf. (3.4g), (4.2a)) and A(m ¯ − 1)(c), we therefore get
J− (k, h) ≤
Ki−
κ−1
) ebi (k, n + 1) ΔDfˆn (φκ (n, ξ), φκ (n, ξ + h)−φκ (n, ξ))
n=k
Cm−1 ¯ · (n, κ) φκ (n, ξ + h) − φκ (n, ξ) ecm−1 ¯
h
+Cm−1 δi− (2) Δφκ (n, h) ecm−1 (n, κ) ¯ ¯ ¯ m−1 ¯ +C1 δi− (2) φm−1 (n, ξ + h) − φ (n, ξ) ec1 (n, κ) κ κ * ¯ + ΔRnm−1 (ξ, h) h
for all k ∈ Z− κ . Rewriting this estimate and using Lemma 4.4.1(b) we obtain −
J− (h) κ,c
h
≤
(Ki− )2 max {1, c } Cm−1 ¯ sup V1 (k, h) 1 − i (c) k∈Z− κ +Cm−1 Ki− δi− (2) sup V2 (k, h) ¯ k∈Z− κ
+Ki− C1 δi− (2) sup V3 (k, h) + Ki− sup V4 (k, h) k∈Z− κ
k∈Z− κ
with V1 (k, h) := ec (κ, k)
κ−1
ebi (k, n + 1)ecm−1 (n, κ) ¯
n=k
· ΔDfˆn (φκ (n, ξ), φκ (n, ξ + h) − φκ (n, ξ)) , V2 (k, h) := ec (κ, k)
κ−1 n=k
ebi (k, n + 1)ecm−1 (n, κ) h) Δφ(n, , ¯
4.4 Smoothness of Fiber Bundles and Foliations
V3 (k, h) := ec (κ, k) V4 (k, h) := ec (κ, k)
κ−1 n=k κ−1
247
¯ m−1 ¯ ebi (k, n + 1)ec1 (n, κ) φm−1 (n, ξ + h) − φ (n, ξ) , κ κ ¯ ebi (k, n + 1) ΔRnm−1 (ξ, h) .
n=k
Vl (k, h) = 0 for l ∈ {1, . . . , 4}, Similarly to Step (I4 ) we deduce limh→0 supk∈Z− κ proving the limit relation limh→0
J− (h)− κ,c h
= 0. Completely analogous one shows
J+ (h)− κ,c the relation limh→0 = 0 and therefore we h m−1 ¯ ¯ bility of the mapping φκ : Xκ → Xm−1,− . Finally κ,c
have verified the differentiawe derive that the derivative
¯ ¯ ¯ Dφm−1 : Xκ → L(Xκ , Xm−1,− )∼ = Xm,− κ κ,c κ,c ¯ m,− ¯ m−1,− ¯ is the fixed point mapping φm . From the fixed point κ : Xκ → Xκ,c of Tκ m−1 ¯ equation (4.4o) for φκ we obtain by differentiation w.r.t. ξ ∈ Xκ the identity ¯ D2 φm−1 (k; ξ) κ
=
κ−1
¯ Gi (k, n + 1)Dfˆn (φκ (n, ξ))D2 φm−1 (n, ξ) κ
n=−∞
+
κ−1
¯ Gi (k, n + 1)Rnm (ξ)
for all k ∈ Z− κ.
n=−∞ ¯ ¯ ¯ ∼ Xm,− (ξ) ∈ L(Xκ , Xm−1,− )= Hence, the derivative Dφm−1 κ κ,c κ,c (cf. Lemma 3.3.27) m ¯ is a fixed point of Tκ (·; ξ), which in turn is unique by Step (II2 ), and so (4.4t) holds. m ¯ m,− ¯ (II4 ) Claim: For every c ∈ (ai + ς, cm ¯ ) the mappings D φκ : Xκ → Xκ,c and m ¯ − D2 w (κ, ·) : Xκ → Lm ¯ holds. ¯ (Xκ ) are continuous, i.e., also A(m)(d) ¯ m,− ¯ Due to the relation (4.4t) it suffices to prove the continuity of φm κ : Xκ → Xκ,c and this is analogous to Step (I4 ) by adding and subtracting the expressions ¯ Dfˆn (φκ (n, ξ))φm ¯ κ (n, ξ0 ) in the corresponding estimates. We established A(m). (II5 ) In the preceding four steps we saw that φκ : Xκ → X− κ,c is ms -times continuously differentiable. With the identity wi− (κ, ξ) = P1i (κ)φκ (κ, ξ) (see (4.4d)) the claim follows from properties of the evaluation map (see Lemma 3.3.28) and the global bound for the derivatives can be obtained using the fact
n − (4.4s) D w (κ, ξ) = Dn P i (κ)φκ (κ, ξ) ≤ P i (κ)φn (ξ)− ≤ Cn 2 i 1 1 κ κ,c for all 1 ≤ n ≤ ms . The expression for C1 is a consequence of Theorem 4.2.9(b2 ). The given (c) recursion for the global bounds Cn ≥ 0 of the partial derivatives Dn w− (κ, ξ) for n ∈ {2, . . . , m} in (4.4c) is a consequence of the estimate (4.4i) 2 i from step (II2 ) in the present proof of (b). A dual argument shows that the solution of the fixed point equation for (4.4f) is globally bounded by Cn as well, and an estimate analogous to (4.4r) gives us the global bounds for the derivatives of wi+ . Hence, we have shown assertion (c) and the proof of Theorem 4.4.6 is finished.
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4 Invariant Fiber Bundles
For our nest result we impose strengthened spectral gap conditions ˜ i ) holds, ∃ςi ∈ 0, ςi+ (m) : (G − ˜ i ) holds. ∃ςi ∈ 0, ςi (m) : (G
am i bi , ai b m i ,
˜+ ) (G i,m ˜ (G− ) i,m
Proposition 4.4.8 (smooth intersection of invariant fiber bundles). Assume I = Z and that Hypotheses 4.2.1, 4.2.3 and 4.4.2 holds If pairs (i, j) with 1 < i ≤ + j < N satisfy (Γi−1 ) and (Γj− ), as well as the strengthened spectral gap conditions j j + − ˜ ˜ (G i−1,m ) and (Gj,m ), then the function wi : Pi → X from Proposition 4.2.17 is of class C m with globally bounded partial derivatives D2n wij (κ, ·) : Xκ → Ln (Xκ ) for 1 ≤ n ≤ m. Proof. Let κ ∈ Z be fixed. Referring Theorem 4.4.6, the map Tκ : Xκ2 × Xκ → Xκ2 introduced in (4.2w) is m-times continuously differentiable and fulfills the contraction condition lip1 Tκ ≤ ij (c, d) < 1 for all c ∈ Γ¯i−1 , d ∈ Γ¯j . Thus, one can show as in Proposition 4.2.17 that Tκ satisfies the assumptions of the uniform C m contraction principle in Theorem B.1.5. We conclude that Tκ (·, y), y ∈ Xκ , has a unique fixed point Υij (y) ∈ Xκ2 and the fixed point mapping Υij : Xκ → Xκ2 is of class C m – for the sake of a convenient notation we suppressed the dependence of Υij on κ ∈ Z. By construction, the smoothness of Υij carries over to wij (κ, ·) : Xκ → Xκ . It remains to show that wij has globally bounded partial derivatives. From Theorem 4.4.6 we see that Tκ : Xκ2 × Xκ → Xκ2 has globally bounded derivatives up to order m, where (4.2w)
Dn Tκ (x1 , x2 ; y) ≤ Cn
for all n ∈ [1, m]Z , x1 , x2 , y ∈ Xκ .
In order to show that this property carries over to Υij , we proceed by induction. For m = 1 we differentiate the fixed point identity Tκ (Υij (y), y) ≡ Υij (y) on Xκ and obtain using the chain rule (cf. Theorem C.1.3) DΥij (y) ≡ D1 Tκ (Υij (y), y)DΥij (y) + D2 Tκ (Υij (y), y) on Xκ . Since Proposition C.1.1 guarantees D1 Tκ (Υij (y), y) ≤ ij (c, d) for y ∈ Xκ , ij (c,d) we get the estimate DΥij (y) ≤ 1− from (4.2w). Now suppose that ij (c,d) m > 1 and choose n ∈ [2, m]Z . Our induction hypothesis is that there exist reals K1 , . . . , Kn−1 ≥ 0 such that Dl Υij (y) ≤ Kl for all y ∈ Xκ , l ∈ [1, n)Z . With this we can define C m -mappings Υ¯ij : Xκ → Xκ , Υ¯ij (y) := (y, Υij (y)), whose derivatives for l = 1, (y1 , DΥij (y)y1 ) l¯ D Υij (y)y1 · · · yl = l for l > 1 (0, D Υij (y)y1 · · · yl )
4.4 Smoothness of Fiber Bundles and Foliations
249
for all y1 , . . . , yl ∈ Xκ satisfy (cf. (??) and our induction hypothesis) DΥ¯ij (y) ≤ max 1, ij (c, d) , Dl Υ¯ij (y) ≤ Kl for all l ∈ [2, n)Z . 1 − ij (c, d) Thus, relation (C.1b) in the higher order chain rule from Theorem C.1.3 implies n Cl
Dn Υij (y) ≤ ij (c, d) Υijn (y) + 2 l=2
l ( #N D ν Υ¯ij (y)
(N1 ,...,Nl )∈Pl< (n) ν=1
and since components Nν ⊆ {1, . . . , n} of a partition (N1 , . . . , Nl ) ∈ Pl< (n), l ∈ [2, n]Z , have a smaller cardinality than n, we conclude from our hypothesis
Dn Υij (y) ≤ ·
n l=2
2 1 − ij (c, d)
Cl
⎧ l ⎨ ( max 1,
(N1 ,...,Nl )∈Pl< (n) ν=1
⎩K ν
ij (c,d) 1−ij (c,d)
for #Nν = 1, for #Nν > 1.
Therefore, the mapping Dn Υij : Xκ → Ln (Xκ ; Xκ2 ) is globally bounded and by definition of wij , this implies our assertion. An even more delicate question is the smooth dependence of the invariant fibers Vi± (ξ) ⊆ X on the initial point ξ ∈ Xκ . Here, the C m -smoothness of the 2-parameter semigroup generated by (S) carries over to the mappings vi± from Proposition 4.3.5 only for m = 0. For higher order differentiability also the growth behavior of D3n ϕ(·; κ, ξ), (κ, ξ) ∈ X and 1 ≤ n ≤ m plays an important rule and due to the resulting technical complexity we waive a corresponding statement and proof. Yet, corresponding references can be found in Sect. 4.10: Proposition 4.4.9 (smoothness of invariant fibers). Let (κ, ξ) ∈ X and for given 1 ≤ i < N choose c ∈ Γ¯i . Assume that Hypotheses 4.2.1, 4.3.1 and 4.4.2 hold. + (a) If I is unbounded above and (G+ i,m ) holds, then the mapping vi (κ, ·, ξ) : Xκ → P1i (κ) from Proposition 4.3.5(a) is m-times differentiable with continuous partial derivatives D2n vi+ (κ, ·) : Xκ2 → Ln (Xκ ) and the global bounds n + D2 v (κ, η, ξ) ≤ Cn for all 1 ≤ n ≤ m. sup i Ln (Xκ ) (κ,η,ξ)∈X ×X
(b) If I is unbounded below, the general solution ϕ of (S) exists on X as a contin− i uous mapping and (G− i,m ) holds, then the mapping vi (κ, ·, ξ) : Xκ → Q1 (κ) from Proposition 4.3.5(b) is m-times differentiable with continuous partial derivatives D2n vi− (κ, ·) : Xκ2 → Ln (Xκ ) and the global bounds n − D2 v (κ, η, ξ) ≤ Cn for all 1 ≤ n ≤ m. sup i Ln (Xκ ) (κ,η,ξ)∈X ×X
The constants Cn ≥ 0 are recursively given by (4.4c) in Theorem 4.4.6(c).
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4 Invariant Fiber Bundles
Proof. Let the pair (κ, ξ) ∈ X be fixed. (a) In order to show the continuous differentiability of vi+ (κ, ·, ξ) : Xκ → Xκ , −1 we proceed as follows: Again abbreviate fˆk := Bk+1 fk , k ∈ I , and formally differentiate the fixed point equation (cf. (4.3e)) ψκ (k, η, ξ) = Φ(k, κ)[η − Qi1 (κ)ξ] ∞ Gi (k, n + 1) fˆn (ψκ (n, η, ξ) + ϕ(n; κ, ξ)) − fˆn (ϕ(n; κ, ξ)) + n=κ
for all k ∈ Z+ κ w.r.t. the variable η ∈ Xκ . Suppressing the dependence on ξ ∈ Xκ from now on, this leads to a further fixed point equation ψκ1 (η) = Sκ1,+ (ψκ1 (η); η)
for all η ∈ Qi1 (κ),
(4.4x)
yielding the formal derivative ψκ1 w.r.t. η ∈ Xκ for the mapping ψκ : Qi1 (κ) → X+ κ,c from Lemma 4.3.4. Precisely, the operator Sκ1,+ is given by Sκ1,+ (ψ 1 ; η, ξ) := Φ(·, κ) +
∞
+ Gi (k, n + 1) Dfˆn (ψκ (n, η) + ϕ(n; κ, ξ))
n=κ
, ·
ψ 1 (n, η)
+ D3 ϕ(n; κ, ξ) ,
where the variable ψ 1 (k), k ∈ I, is a sequence with values in L(Xκ , Xk ). The analysis of this operator Sκ1,+ strongly resembles the one for Tκ1,− given in the above proof of Theorem 4.4.6. We consequently omit the corresponding further details. (b) Since the argument is analogous to (a), we skip the details.
4.5 Normal Hyperbolicity To motivate our further considerations, we return to the linear equation (L0 ). As we have seen, the invariant fiber bundle Wi− for (S), as formulated in Theorem 4.2.9(b), is a perturbation of the pseudo-unstable bundle P1i , and the linear spectral gap condition ai bi implies that P1i is normally hyperbolic in the sense that (L0 ) possesses an exponential dichotomy. Now we tackle the problem whether this normal hyperbolicity persists under nonlinear perturbations. As important result from the previous section, we know that the invariant fiber bundle Wi− and its invariant foliation Vi+ (ξ) are of class C 1 , if (S) has this property. Hence, for each (κ, x, y) ∈ Wi− × X we can define the tangent bundles
4.5 Normal Hyperbolicity
251
Tx Wi− := (κ, ξ + D2 wi− (κ, x)ξ) ∈ X : ξ ∈ P1i (κ) , Ty Vi+ (x) := (κ, η + D2 vi+ (κ, Qi1 (κ)y, x)η) ∈ X : η ∈ Qi1 (κ) to Wi− resp. Vi+ . For simplicity reasons, we assume for the remaining section that (S) is semi-implicit, i.e., instead of (S) we consider Bk+1 x = Ak x + fk (x).
(S )
As a consequence, under (3.1a) the general forward solution to (S ) exists, for in˜ i ) simplifies to stance the strengthened gap condition (G
i : Ki+ Ki− + max Ki+ , Ki− L1 < ςi ∃ςi ∈ 0, bi −a 2
(4.5a)
and, in particular, both the invariance equations (4.2m), (4.3n) become easier to handle. The sets Γ¯i are defined as in the previous sections. The subsequent lemma roughly states that the two tangential bundles defined above provide a splitting of each fiber Xκ of the extended state space X . Before delving into preparations, let us point out that we focus on the invariant fiber bundles Wi− in this section and that we suppose I = Z from now on. Lemma 4.5.1. Assume Hypotheses 4.2.1, 4.3.1, 4.4.2 with m = 1 and (Γi− ) and (4.5a) hold for one 1 ≤ i < N . Then for each κ ∈ Z we have the decomposition Xκ = Tx Wi− (κ) ⊕ Ty Vi+ (κ, x)
for all x ∈ Wi− (κ), y ∈ Xκ
(4.5b)
and the splitting is continuous in (x, y) ∈ Wi− (κ) × Xκ . Proof. Fix a triple (κ, x, y) ∈ Wi− × X . In order to prove that the tangent spaces Tx Wi− (κ) and Ty Vi+ (κ, x) satisfy (4.5b) we show that each ζ ∈ Xκ possesses the representation ζ = ξ¯ + η¯ with unique ξ¯ ∈ Tx Wi− (κ) and η¯ ∈ Ty Vi+ (κ, x). This is equivalent to the unique existence of points ξ ∈ P1i (κ), η ∈ Qi1 (κ) such that ζ = ξ + D2 wi− (κ, x)ξ + η + D2 vi+ (κ, Qi1 (κ)y, x)η, which holds if and only if P1i (κ)ζ = ξ + D2 vi+ (κ, Qi1 (κ)y, x)η,
Qi1 (κ)ζ = η + D2 wi− (κ, x)ξ
and this, in turn, is equivalent to ξ = P1i (κ)ζ − D2 vi+ (κ, Qi1 (κ)y, x)Qi1 (κ)ζ + D2 vi+ (κ, Qi1 (κ)y, x)D2 wi− (κ, x)ξ, η = Qi1 (κ)ζ − D2 wi− (κ, x)P1i (κ)ζ + D2 wi− (κ, x)D2 vi+ (κ, Qi1 (κ)y, x)η. By Theorem 4.2.9(b2) and Proposition 4.3.5(a2) the Lipschitz constants lip2 wi− , lip2 vi+ , respectively, exist and their product is less than 1 (cf. (4.3t)), so that the operators I −D2 vi+ (κ, Qi1 (κ)y, x)D2 wi− (κ, x) and I −D2 wi− (κ, x)vi+ (κ, Qi1 (κ)y, x) are invertible in the Banach algebra L(Xκ ) (cf. [295, p. 74, Theorem 2.1] or
252
4 Invariant Fiber Bundles
Theorem B.3.1). Therefore, one can represent ζ ∈ Xκ uniquely as ζ = ˜ i (κ, x, y)ζ, where P˜ i (κ, x, y) ∈ L(Xκ ) is the projection of Xκ P˜1i (κ, x, y)ζ + Q 1 1 − onto Tx Wi (κ) along Ty Vi+ (κ, x), ˜ i (κ, x, y) := I − D2 v + (κ, Qi (κ)y, x)D2 w− (κ, x) −1 · Q 1 1 i i · P1i (κ) − D2 vi+ (κ, Qi1 (κ)y, x) , ˜ i1 (κ, x, y) ∈ L(Xκ ) is the projection of Xκ onto Ty V + (κ, x) and accordingly Q i − along Tx Wi (κ) given by −1 i Q1 (κ) − D2 wi− (κ, x) . P˜1i (κ, x, y) := I − D2 wi− (κ, x)D2 vi+ (κ, Qi1 (κ)y, x) Due to both our Theorem 4.4.6(b), Proposition 4.4.9(a) and the fact that the inversion operator ·−1 : L(Xκ ) → L(Xκ ) is of class C ∞ (cf. [1, p. 117, Lemma 2.5.5]), ˜ i (κ, x, y) depend continuously on (x, y) ∈ W − (κ) × Xκ . we see that P˜1i (κ, x, y), Q 1 i Thus, the splitting (4.5b) is continuous. Consider the difference equation in X × X given by (S) and the corresponding variational equation Bk+1 x = Ak x + fk (x) ; (4.5c) Bk+1 z = [Ak + Dfk (x)] z its general solution will be denoted by (ϕ, φ) and exists due to (3.1a). In the following it is our aim to show that the invariant fiber bundle Wi− is normally hyperbolic; that is to say that the tangential and normal bundle for Wi− are invariant under (4.5c), and that we have an exponential dichotomy w.r.t. these bundles. To be more precise, we have Lemma 4.5.2 (tangent bundle). Assume Hypotheses 4.2.1, 4.3.1, 4.4.2 with m = 1 and (Γi− ) and (4.5a) hold for one 1 ≤ i < N . If
Ki− L1 1 + ˜− (c) < bi (k) i
for all k ∈ Z
(4.5d)
is satisfied for one c ∈ Γ¯i , then the tangent bundle T Wi− := (κ, ξ, ζ) ∈ X × X : (κ, ξ) ∈ Wi− , ζ ∈ Tξ Wi− (κ) is invariant w.r.t. (4.5c), the general solution (ϕ, φ) of (4.5c) exists on T Wi− and one has the backward estimate
φ(k; κ, ξ, ζ) Xk ≤ Ki− eˆbi (k, κ) P1i (κ)ζ Xκ
for all k ∈ Z− κ
and (κ, ξ, ζ) ∈ T Wi− , where ˆbi (k) := bi (k) − Ki L1 1 + ˜− i (c) .
(4.5e)
4.5 Normal Hyperbolicity
253
−1 Proof. First, we abbreviate fˆk := Bk+1 fk . Choose any triple (κ, ξ, ζ) ∈ T Wi− and consequently we have representations ξ = ξ0 +wi− (κ, ξ0 ), ζ = ζ0 +D2 wi− (κ, ξ0 )ζ0 for some points ξ0 , ζ0 ∈ P1i (κ). Then Corollary 4.2.13(a) implies that the general solution ϕˆ of the Wi− -reduced equation (4.2s) is defined on Pi− . The further proof is subdivided into four steps: (I) Claim: The general solution ϕ˜ of the variational equation for (4.2s),
Bk+1 x = Ak x + P1i (k + 1)Dfˆk ϕ(k; ˜ κ, ξ0 ) + wi− (k, ϕ(k; ˜ κ, ξ0 )) · x + D2 wi− (k, ϕ(k; ˜ κ, ξ0 ))x
(4.5f)
is defined on P1i . We differentiate the solution identity for ϕˆ w.r.t. ξ0 . Then D3 ϕ(·; ˆ κ, ξ0 ) is an operator solution of (4.5f) satisfying the initial condition x(κ) = I, and ϕ(k; ˜ κ, ξ0 ) := D3 ϕ(k; ˆ κ, ξ0 )ξ0 defines the general solution of (4.5f) for k ∈ Z, (κ, ξ0 ) ∈ P1i . (II) Claim: The tangent bundle T Wi− is forward invariant w.r.t. (4.5c). Define the sequence ψ1 : Z+ ˆ κ, ξ0 ) + wi− (k, ϕ(k; ˆ κ, ξ0 )) and κ → X, ψ1 (k) := ϕ(k; i due to the inclusion ϕ(k; ˆ κ, ξ0 ) ∈ P1 (k) one obviously has ψ1 (k) ∈ Wi− (k) for all k ∈ Z+ κ . In addition, from the invariance equation (4.2m) we see that ψ1 is a solution of the first equation in (4.5c) with ψ1 (κ) = ξ and this yields ϕ(k; κ, ξ) = + ψ1 (k) ∈ Wi− (k) for all k ∈ Z+ κ . Next we define the sequence ψ2 : Zκ → X, − ψ2 (k) = ϕ(k; ˜ κ, ζ0 ) + D2 wi (k, ϕ(k; κ, ξ))ϕ(k; ˜ κ, ζ0 ). Observing the inclusion ϕ(k; ˜ κ, ζ0 ) ∈ P1i (k) one has ψ2 (k) ∈ Tϕ(k;κ,ξ) Wi− (k) for all k ∈ Z+ κ . Using an identity obtained by differentiating the invariance equation (4.2m) w.r.t. the variable in P1i (k), one verifies that ψ2 solves the second equation in (4.5c) and satisfies ψ2 (κ) = ζ0 + D2 wi− (k, ξ)ζ0 . Hence, φ(k; κ, ξ, ζ) = ψ2 (k) ∈ Tϕ(k;κ,ξ) Wi− (k) and the tangent bundle T Wi− is forward invariant. (III) The fact that ϕ is defined on Wi− is given in Corollary 4.2.13(a) and we will show that the second component φ is defined on T Wi− . For this, let k ∈ Z. From Step (II) we have φ(k + 1; k, ·) : Tϕ(k;κ,ξ) Wi− (k) → Tϕ(k+1;κ,ξ) Wi− (k + 1) is well-defined and it suffices to show that this mapping is bijective. Let η ∈ Tϕ(k+1;κ,ξ) Wi− (k + 1), i.e., η = η1 + D2 wi− (k + 1, ϕ(k + 1; k, ξ))η1 for some η1 ∈ P1i (k + 1); note that lip2 wi− < 1 (cf. Remark 4.3.8) and consequently η1 uniquely determines the point η. We show that the endomorphism −1 Ak + P1i (k + 1)Dfˆk (ϕ(k; κ, ξ)) I + D2 wi− (k, ϕ(k; κ, ξ)) Bk+1 is actually an isomorphism between the linear spaces P1i (k) and P1i (k + 1). i We abbreviate Φk := P1 (k + 1)Dfˆk (ϕ(k; κ, ξ)) I + D2 wi− (k, ϕ(k; κ, ξ)) and from Theorem 4.2.9(b2) with (4.2a) one derives Φk ≤ L1 1 + ˜− (c) for i
all k ∈ Z. On the other hand, from Hypothesis 4.2.1 we know that the inverse −1 −1 −1 B Bk+1 Ak |−1 exists (see Lemma 3.3.6(b)) and (3.4g) implies i k+1 Ak |P i (k) ≤ P (k) Ki− bi (k−1)
1
for all k
1
∈
Z. Then our assumption (4.5d) and Theorem B.3.1
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4 Invariant Fiber Bundles
−1 shows the invertibility of Bk+1 Ak + Φk ∈ L(P1i (k), P1i (k + 1)), and η0 := −1 −1 η1 is the unique point in P1i (k) satisfying the relation φ(k + 1; Bk+1 Ak + Φk k, ξ, η0 + D2 wi− (k, ϕ(k; κ, ξ))η0 ) = η. In particular, (4.5d) ensures that the Gronwall estimate Proposition A.2.1(b) can be applied. (IV) Referring to Step (I) we know that the general solution ϕ(k; ˜ κ, ·) of the variational equation (4.5f) exists for k ∈ Z− κ . So, the variation of constants formula in backward time from Theorem 3.1.16(b) (see also Remark 3.1.17(1)) implies the relation
ϕ(k; ˜ κ, ζ0 ) = Φ− (k, κ)ζ0 − Pi 1
κ−1 n=k
Φ− (k, n + 1)P1i (n + 1) Pi 1
˜ κ, ξ0 ) + wi− (n, ϕ(n; ˜ κ, ξ0 )) ·Dfˆn ϕ(n; · I + D2 wi− (n, ϕ(n; ˜ κ, ξ0 )) ϕ(n; ˜ κ, ζ0 ) for all k ∈ Z− κ and analogous to the proof of (4.2t) in Corollary 4.2.13 one gets the estimate (4.5e). Lemma 4.5.3 (normal bundle). Assume Hypotheses 4.2.1, 4.3.1, 4.4.2 with m = 1 and (Γi− ) and (4.5a) hold for one 1 ≤ i < N . Then the normal bundle N Wi− := (κ, ξ, ζ) ∈ X × X : (κ, ξ) ∈ Wi− , ζ ∈ Tξ Vi+ (κ, ξ) is forward invariant w.r.t. (4.5c), and one has the forward estimate
i
φ(k; κ, ξ, ζ) Xk ≤ Ki+ 1 + ˜+ ˆ i (k, κ) Q1 (κ)ζ X i (c) ea
κ
for all k ∈ Z+ κ
and (κ, ξ, ζ) ∈ N Wi− , where a ˆi (k) := ai (k) + Ki+ L1 1 + ˜+ (c) . i −1 Proof. Let (κ, ξ, ζ) ∈ N Wi− and fˆk := Bk+1 fk . We proceed in two steps: − (I) To show the forward invariance of N Wi we choose an arbitrary η ∈ Wi− (κ) and let η0 ∈ Qi1 (κ) be such that η = η0 + vi+ (κ, η0 , ξ). From the forward invariance of Vi+ (ξ) guaranteed by Proposition 4.3.5(a) we know that there exists a sequence of points ψ1 (k) ∈ Qi1 (k) satisfying
ϕ(k; κ, η) = ψ1 (k) + vi+ (k, ψ1 (k), ϕ(k))
for all k ∈ Z+ κ,
(4.5g)
where we abbreviate ϕ(k) = ϕ(k; κ, ξ) from now on, since ξ ∈ Wi− (κ) remains fixed. If we multiply the solution identity for ϕ with Qi1 (k + 1), we see that the sequence ψ1 = Qi1 (·)ϕ(·; κ, η) : Z+ κ → X solves the equation −1 x = Bk+1 Ak x + Qi1 (k + 1)fˆk x + vi+ (k, x, ϕ(k)) .
(4.5h)
4.5 Normal Hyperbolicity
255
Let ψ denote the general solution of (4.5h). Then the partial derivative D3 ψ exists and D3 ψ(·; κ, η0 )Qi1 (κ)ζ is a solution of the variational equation (cf. (4.5g)) −1 x = Bk+1 Ak x + Qi1 (k + 1)Dfˆk ψ(k; κ, η0 ) + vi+ (k, ψ(k; κ, η0 ), ϕ(k; κ, η)) · x + D2 vi+ (k, Qi1 (k)ϕ(k; κ, η), ϕ(k))x (4.5i) satisfying the initial condition x(κ) = Qi1 (κ)ζ. On the other hand, the invariance equation (4.3n) yields the identity −1 vi+ (k + 1, ψ(k + 1; κ, η0 ), ϕ (k)) ≡ Bk+1 Ak vi+ (k; ψ(k; κ, η0 ), ϕ(k)) + Qi1 (k + 1)fˆk ψ(k; κ, η0 ) + vi+ (k, ψ(k; κ, η0 ), ϕ(k)) on Z+ κ
and if we differentiate this identity w.r.t. η0 and apply Qi1 (κ)ζ one gets D2 vi+ (k + 1, ψ(k + 1; κ, η0 ), ϕ (k))D3 ψ(k + 1; κ, η0 )Qi1 (κ)ζ −1 Ak D2 vi+ (k; ψ(k; κ, η0 ), ϕ(k))D3 ψ(k; κ, η0 )Qi1 (κ)ζ ≡ Bk+1 +Qi1 (k + 1)Dfˆk ψ(k; κ, η0 ) + vi+ (k, ψ(k; κ, η0 ), ϕ(k)) · D3 ψ(k; κ, η0 ) + D2 vi+ (k, ψ(k; κ, η0 ), ϕ(k))D3 ψ(k; κ, η0 ) Qi1 (κ)ζ i on Z+ κ . From this, and the solution identity for D3 ψ(·; κ, η0 )Q1 (κ)ζ (cf. (4.5i)) we see that the sum
σ(k) : = D3 ψ(k; κ, η0 )Qi1 (κ)ζ + D2 vi+ (k; ψ(k; κ, η0 ), ϕ(k))D3 ψ(k; κ, η0 )Qi1 (κ)ζ = D3 ψ(k; κ, η0 )Qi1 (κ)ζ +D2 vi+ (k; Qi1 (k)ϕ(k; κ, η0 ), ϕ(k))D3 ψ(k; κ, η0 )Qi1 (κ)ζ
∈ Tϕ(k;κ,η) Vi− (k, ϕ(k))
for all k ∈ Z+ κ
is a solution of the linear difference equation Bk+1 x = Ak x + Dfk (ϕ(k; κ, η))x satisfying σ(κ) = Qi1 (κ)ζ + D2 vi+ (κ, Qi1 (κ)η, ζ)Qi1 (κ)ζ. Since η ∈ Vi+ (κ, ξ) was arbitrary, we can choose η = πi+ (κ, ξ) now, and ξ ∈ Wi− (κ) yields η = πi+ (κ, ξ) = ξ (cf. Theorem 4.3.7(a)). Hence, σ(k) ∈ Tϕ(k) Vi+ (k, ϕ(k)) for all k ∈ Z+ κ and σ(κ) = ζ. The uniqueness of forward solutions implies φ(k; κ, ξ, ζ) = σ(k), i.e. φ(k; κ, ξ, ζ) = D3 ψ(k; κ, η0 )Qi1 (κ)ζ (4.5j) + i i +D2 vi (k; Q1 (k)ϕ(k; κ, ξ), ϕ(k))D3 ψ(k; κ, η0 )Q1 (κ)ζ and due to the invariance of Wi− we have (ϕ, φ)(k; κ, ξ, ζ) ∈ N Wi− (k), k ∈ Z+ κ. (II) It remains to deduce the claimed forward estimate for φ. The variation of constants formula from Theorem 3.1.16(a), applied to (4.5i), gives us
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4 Invariant Fiber Bundles
D3 ψ(k; κ, η0 )Qi1 (κ)ζ = Φ(k, κ)Qi1 (κ)ζ +
k−1
Φ(k, n + 1)Qi1 (n + 1)Dfˆn ψ(n; κ, η0 ) + vi+ (n, ψ(k; κ, η0 ), ϕ(n; κ, η))
n=κ
· D3 ψ(k; κ, η0 ) + D2 vi+ (k, Qi1 (k)ϕ(k; κ, η), ϕ(k))D3 ψ(k; κ, η0 ) Qi1 (κ)ζ
for all k ∈ Z+ κ , and from (3.4g), (3.5a) and Proposition 4.3.5(a2 ) we get D3 ψ(k; κ, η0 )Qi1 (κ)ζ eai (κ, k) k−1
ea (κ, n) i D3 ψ(n; κ, η0 )Qi1 (κ)ζ (c) L ≤ Ki+ Qi1 (κ)ζ +Ki+ 1 + ˜+ 1 i ai (n) n=κ for all k ∈ Z+ κ . The Gronwall lemma from Theorem A.2.1(a) implies D3 ψ(k; κ, η0 )Qi (κ)ζ ≤ K + eaˆi (k, κ) Qi (κ)ζ 1 1 i
for all k ∈ Z+ κ
and (4.5j) together with Proposition 4.3.5(a2 ) leads to our assertion.
Theorem 4.5.4 (normal hyperbolicity). Assume Hypotheses 4.2.1, 4.3.1, 4.4.2 with m = 1 and (Γi− ) and (4.5a) hold for one 1 ≤ i < N . If (4.5d) is satisfied for one c ∈ Γ¯i , then the invariant fiber bundle Wi− is normally hyperbolic: (a) One has the Whitney sum X × X = T Wi− ⊕ N Wi− , where the splitting is continuous in each fiber. (b) The nonautonomous sets T Wi− and N Wi− possess the properties stated in Lemma 4.5.2 and Lemma 4.5.3, resp. (c) In particular, the pseudo-contraction in the normal direction of Wi− is stronger than in the tangential direction. Proof. The claim follows readily from the above Lemmata 4.5.1, 4.5.2 and 4.5.3. ˜ i ) we have Here, thanks to the strengthened spectral gap condition (G (4.5a) bi − ai (Ki+ + Ki− )L1 ≤ Ki+ Ki− + max Ki+ , Ki− L1 < ςi < 2 and from this one easily derives a ˆi ˆbi , i.e., the normal pseudo-contraction rate a ˆi is stronger than the corresponding tangential rate ˆbi .
4.6 Pseudo-stable and Pseudo-unstable Fiber Bundles In this section, we make the first attempt to weaken the global assumptions in form of Hypotheses 4.2.3, 4.3.1 or 4.4.2. Indeed, we return to general equations Hk+1 (x ) = Fk (x, x )
(D)
4.6 Pseudo-stable and Pseudo-unstable Fiber Bundles
257
as in Definition 2.1.1, where X consists of Banach spaces. We are interested in the local behavior of (D) near a fixed reference solution φ∗ : I → X, which, for instance, might be a constant, a periodic or a general bounded solution. In particular, we want to provide a local description of the stable set corresponding to φ∗ , Wφ+∗
:=
there exists a solution φ : Z+ κ → X of (D) with , (κ, ξ) ∈ X φ(κ) = ξ ∈ Xκ and limk→∞ φ(k) − φ∗ (k) Xk = 0
when I is unbounded above, as well as of the unstable set corresponding to φ∗ , Wφ−∗ :=
(κ, ξ) ∈ X
there exists a solution φ : Z− κ → X of (D) with φ(κ) = ξ ∈ Xκ and limk→−∞ φ(k) − φ∗ (k) = 0 , Xk
provided I is unbounded below. Let us suppose the difference equation (D) is defined on a nonautonomous set S containing a convex neighborhood of the reference solution φ∗ , i.e., there exists a ρ0 > 0 such that Bρ0 (φ∗ ) ⊆ S. It is advantageous to subtract the solution identity Hk+1 (φ∗ (k)) ≡ Fk (φ∗ (k), φ∗ (k)) on I for φ∗ from the equation of φ∗ -perturbed motion (D)φ∗ yielding the equation Hk+1 (x +φ∗ (k))−Hk+1 (φ∗ (k)) = Fk (x+φ∗ (k), x +φ∗ (k))−Fk (φ∗ (k), φ∗ (k)) with the nonautonomous set S − φ∗ as state space. This equation has the trivial solution and under appropriate assumptions on the mappings Hk+1 , Fk , k ∈ I , we can write it in the form (S). Indeed, this is possible in each of the settings: •
Provided Hk+1 , Fk are continuously differentiable, one introduces Bk+1 := DHk+1 (φ∗ (k)) − D2 Fk (φ∗ (k), φ∗ (k)),
Ak := D1 Fk (φ∗ (k), φ∗ (k)), fk (x, x ) := −Hk+1 (x + φ∗ (k)) + Hk+1 (φ∗ (k)) + DHk+1 (φ∗ (k))x + Fk (x + φ∗ (k), x + φ∗ (k)) − Fk (φ∗ (k), φ∗ (k)) − D1 Fk (φ∗ (k), φ∗ (k))x − D2 Fk (φ∗ (k), φ∗ (k))x
•
and the linear part (L0 ) is the linearization of (D) along φ∗ . Provided Hk+1 satisfies the invertibility condition (2.2c) and the composition −1 gk := Hk+1 ◦ Fk is of class C 1 , one defines Bk+1 := IXk+1 − D2 gk (φ∗ (k), φ∗ (k)), Ak := D1 gk (φ∗ (k), φ∗ (k)),
fk (x, x ) := gk (x + φ∗ (k), x + φ∗ (k)) − D1 gk (φ∗ (k), φ∗ (k))x − gk (φ∗ (k), φ∗ (k)) and the linear part (L0 ) is the variational equation as in Corollary 2.3.11.
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4 Invariant Fiber Bundles
In conclusion, in order to describe the above sets Wφ+∗ and Wφ−∗ locally, it is possible to formulate (D) in the familiar form Bk+1 x = Ak x + fk (x, x ).
(S)
Yet, differing from our previous analysis, the nonlinearities fk are not assumed to be globally Lipschitzian or to possess globally bounded derivatives as required for instance in Theorem 4.2.9 resp. Theorem 4.4.6. Hypothesis 4.6.1. Let ρ0 > 0 and let the general forward solution ϕ of (S) exist on Bρ0 . Suppose that fk : Xk × Xk+1 → Yk+1 with fk (Xk , Xk+1 ) ⊆ im Bk+1 , k ∈ I , and: (i) fk (0, 0) ≡ 0 on I. (ii) The following limit relations hold −1 lim sup lipj Bk+1 fk |Br (0,Xk )×Br (0,Xk+1 ) = 0 for j = 1, 2.
r0 k∈I
(4.6a)
When interested in differentiability results, it is reasonable to demand a smooth right-hand side of (S). However, the use of cut-off functions in order to derive local results from global ones additionally requires smooth norms and the concept of a C m -Banach space. For a survey on such results we refer to Sect. C.2. Hypothesis 4.6.2. Let m ∈ N. Suppose that X consists of C m -Banach spaces, the −1 mappings Bk+1 fk : Xk × Xk+1 → Xk+1 are of class C m for all k ∈ I and that −1 the derivatives Dn Bk+1 fk : Xk × Xk+1 → Ln (Xk × Xk+1 ; Xk+1 ) are uniformly bounded, i.e., for each uniformly bounded B ⊆ X one has −1 sup sup Dn Bk+1 fk (x, y)L
k∈I x∈B(k) y∈B (k)
n (Xk ×Xk+1 ;Xk+1 )
< ∞.
Remark 4.6.3. From Proposition C.1.1 and (4.6a) we obtain the limit relation lim
(x,y)→(0,0)
−1 DBk+1 fk (x, y) = 0
uniformly in k ∈ I.
(4.6b)
Before we formulate our first result, a weaker version of the invariance notion established Definition 1.2.1 is due, which is tailor-made for fiber bundles. Given a vector bundle X0 ⊆ X and an open neighborhood U ⊆ X of 0, we say the graph W := {(κ, ξ + w(κ, ξ)) ∈ X : ξ ∈ X0 (κ) ∩ U(κ)} of a given mapping w : X0 ∩ U → X is a locally forward invariant fiber bundle of (S), if the implication (k0 , x0 ) ∈ W
⇒
(k, ϕ(k; k0 , x0 )) ∈ W
4.6 Pseudo-stable and Pseudo-unstable Fiber Bundles
259
holds for all k ≥ k0 as long as ϕ(k; k0 , x0 ) ∈ U(k). Accordingly, one speaks of a locally invariant fiber bundle W, if it is locally forward invariant and for each initial pair (k0 , x0 ) ∈ W the solution ϕ(·; k0 , x0 ) has a backward continuation in W as long as (k, ϕ(k; k0 , x0 )) ∈ U. In this context, in case U = X we say W is a global (forward) invariant fiber bundle of (S), if the above conditions holds for all k ≥ k0 resp. all k ∈ I. One speaks of a C m -fiber bundle of (S), provided the partial derivatives D2n w exist and are continuous for n ∈ {1, . . . , m}. Theorem 4.6.4 (pseudo-stable and -unstable fiber bundles). Let m ∈ N. If both Hypotheses 4.2.1 and 4.6.1 are satisfied for some 1 ≤ i < N , then there exist reals ρ ∈ (0, ρ0 ), γ0 , . . . , γm ≥ 0 such that the following holds: (a) For I unbounded above there exists a locally forward invariant bundle Wi+ := (κ, η + wi+ (κ, η)) ∈ X : (κ, η) ∈ Bρ of (S), where wi+ : Bρ → X is a Lipschitzian mapping with wi+ (κ, ξ) = wi+ (κ, Qi1 (κ)ξ) ∈ P1i (κ) for all (κ, ξ) ∈ Bρ which satisfies the invariance equation (4.2k) for all (κ, η) ∈ Bρ ⊆ Qi1 and also η1 ∈ Bρ (0, Xκ+1 ) ⊆ Qi1 (κ + 1). Moreover, one has (a1 ) wi+ (κ, 0) ≡ 0 on I and wi+ (κ, ξ)Xκ ≤ ρ for all (κ, ξ) ∈ Bρ , (a2 ) lip2 wi+ < 1 and limr0 lip2 wi+ |Br = 0, (a3 ) if additionally Hypothesis 4.6.2 holds and am i bi ,
(4.6c)
then the nonautonomous set Wi+ is a C m -fiber bundle, i.e., wi+ : Bρ → X is of class C m in the second variable, D2 wi+ (κ, 0) ≡ 0 on I and n + D2 w (k, x) ≤ γn i
for all (k, x) ∈ Bρ , 0 ≤ n ≤ m.
(4.6d)
One denotes Wi+ as pseudo-stable fiber bundle of (S). (b) For I unbounded below there exists a locally invariant bundle Wi− := (κ, η + wi− (κ, η)) ∈ X : (κ, η) ∈ Bρ of (S), where wi− : Bρ → X is a Lipschitzian mapping with wi− (κ, ξ) = wi− (κ, P1i (κ)ξ) ∈ Qi1 (κ)
for all (κ, ξ) ∈ Bρ
(4.6e)
which satisfies the invariance equation (4.2m) for all (κ, η) ∈ Bρ ⊆ P1i and also η1 ∈ Bρ (0, Xκ+1 ) ⊆ P1i (κ + 1). Moreover, one has: (b1 ) wi− (κ, 0) ≡ 0 on I and wi− (κ, ξ)Xκ ≤ ρ for all (κ, ξ) ∈ Bρ .
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4 Invariant Fiber Bundles Wi+(κ)
ϕ(l; κ,·)
Wi+(l)
Wi+(k)
ϕ(k; l,·) I Wi−(k) Xκ
Wi−(κ)
Wi−(l) Xl
Xk
Fig. 4.2 Pseudo-stable and -unstable fiber bundles Wi+ and Wi−
(b2 ) lip2 wi− < 1 and limr0 lip2 wi− |Br = 0. (b3 ) If additionally Hypothesis 4.6.2 holds and ai b m i ,
(4.6f)
then the nonautonomous set Wi− is a C m -fiber bundle, i.e., wi− : Bρ → X is of class C m in the second variable, D2 wi− (κ, 0) ≡ 0 on I and n − D w (k, x) ≤ γn 2 i
for all (k, x) ∈ Bρ , 0 ≤ n ≤ m.
One denotes Wi− as pseudo-unstable fiber bundle of (S). (c) For I = Z one has Wi+ ∩ Wi− = Z × {0}. The pseudo-stable and -unstable fiber bundles Wi+ and Wi− intersecting along the trivial solution are illustrated in Fig. 4.2. In the C 1 -case, by D2 wi± (κ, 0) ≡ 0 on I, they are tangential to the invariant vector bundles Qi1 resp. P1i . Remark 4.6.5. (1) If the condition ai∗ 1 holds for a minimal 1 ≤ i∗ < N , then Ws := Wi+∗ is called stable fiber bundle of (S) and every fiber bundle Wi− , i∗ < i, in the stable hierarchy I × {0} ⊂ . . . ⊂ Wi+∗ +1 ⊂ Wi+∗ = Ws is denoted as a strongly stable fiber bundle. One speaks of a center-stable fiber bundle Wcs := Wi+∗ , provided 1 ≤ ai∗ and thus 1 bi∗ . Under our Hypothesis 4.6.2 + m with am i∗ bi∗ , all members Wi , i∗ ≤ i, fulfill (4.6c) and are C -fiber bundles. (2) The dual situation occurs, if there exists a maximal index j∗ with 1 bj∗ . Then Wu := Wj−∗ is the unstable fiber bundle of (S) and every fiber bundle of the unstable hierarchy I × {0} ⊂ . . . ⊂ Wj−∗ −1 ⊂ Wj−∗ = Wu is a strongly unstable fiber bundle. A center-unstable fiber bundle Wcu := Wj−∗ occurs for a spectral gap with bj∗ ≤ 1 and hence aj∗ 1. Under Hypothesis 4.6.2 − m with aj∗ bm j∗ , all members Wj , j ≤ j∗ , fulfill (4.6f) and are C -fiber bundles.
4.6 Pseudo-stable and Pseudo-unstable Fiber Bundles
261
(3) The above fiber bundles Wi± are associated to the trivial solution of (S). Consequently, the nonautonomous sets φ∗ + Wi+ , φ∗ + Wi− are denoted as pseudostable resp. -unstable fiber bundle of the solution φ∗ . (4) For a p-periodic equation (S) the fiber bundles Wi± are also p-periodic. In particular, for autonomous equations (S) the fibers are constant and one calls Wi+ (κ) a pseudo-stable manifold and every Wi− (κ) a pseudo-unstable manifold. (5) In a differentiable setting of Hypothesis 4.6.2 one has explicit constants γ0 = ρ and γ1 = 1. In general, the radius ρ > 0 depends on m ∈ N as well. This is due to the fact that ςi+ (m) might decay to 0 as m increases (cf. Remark 4.4.7), which makes the spectral gap condition (G+ i,m ) increasingly restrictive. ¯ 1 (0) denote the radial retraction on Xk , Proof. (I) Above all, let rXk : Xk → B k ∈ I (cf. Lemma C.2.1). In order to obtain Lipschitzian extensions we define the ∗ constant rX := supk∈I lip rXk and remark that Lemma C.2.1 guarantees the relation ∗ rX ∈ [1, 2]. For ρ ∈ (0, ρ0 ) we define the Lipschitz constants (4.6a)
−1 fk |Bρ (0,Xk )×Bρ (0,Xk+1 ) < ∞ for all i = 1, 2 Li (ρ) := sup lipi Bk+1 k∈I
and observe from Hypothesis 4.6.1(ii) that the limit relations limr0 Li (r) = 0 hold. We consider the modified difference equation Bk+1 x = Ak x + fkρ (x, x ),
˜ (S)
−1 ρ −1 where the nonlinearities Bk+1 fk are globally Lipschitzian extensions of Bk+1 fk as ρ provided in Proposition C.2.5. Due to fk (0, 0) ≡ fk (0, 0) ≡ 0 on I the growth conditions (Γi± ) are trivially fulfilled for all 1 ≤ i < N . Furthermore, it is possible to choose ρ > 0 so small that L2 (ρ) < 1 holds and Proposition 4.1.3 guarantees ˜ exists on X . Since (S) and (S) ˜ coincide that the general forward solution ϕ˜ to (S) on the set Bρ , one has ϕ(k; ˜ κ, ξ) = ϕ(k; κ, ξ) as long as ϕ(k; ˜ κ, ξ) ∈ Bρ (0), where (κ, ξ) ∈ Bρ .
(II) Let 1 ≤ i < N and choose ςi ∈ 0, bi −ai , c ∈ Γ¯i as defined in 2
Hypothesis 4.2.3. Via a further downsizing of ρ > 0 we can enforce ∗ rX
2 max Ki− , Ki+ (L1 (ρ) + bi L2 (ρ)) < ςi , ∗ max K − , K + L (ρ) 1 + 2rX 2 i i
˜± i (c) < 1;
note here that in the definition of ∓ i (c), i (c) in Lemma 4.2.6 one has to replace the ˜ satisfies constants L1 , L2 by L1 (ρ), L2 (ρ), respectively. Firstly, this ensures that (S) the spectral gap condition (Gi ); actually it fulfills even the strengthened spectral gap condition (4.2u). In conclusion, all the assumptions of Theorem 4.2.9 are satisfied ˜ and there exist (forward) invariant fiber bundles W ˜ ± given as graph of a for (S) i ± mapping w ˜i over the vector bundles P1i resp. Qi1 . We now show that the mapping wi± := w ˜i± |Bρ fulfills the assertions claimed in Theorem 4.6.4:
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4 Invariant Fiber Bundles
˜ coincide on Bρ , the nonautonomous sets Indeed, since the two equations (S) and (S) Wi± are locally (forward) invariant and the invariance equations (4.2k) resp. (4.2m) hold near the trivial solution. From Corollary 4.2.20 one deduces wi± (κ, 0) ≡ 0 on I. In case I = Z, the assertion (c) follows directly using Corollary 4.2.15. Since both the mappings wi± and w ˜i± share the same Lipschitz constant ˜i (c) < 1 (cf. Theorem 4.2.9 (a2 ) and (b2 )), we infer from limr0 Li (r) = 0 that the limit relations in assertion (a2 ) and (b2 ) hold true. In addition, the estimate ± w (κ, ξ) ≤ ˜± (c) ξ ≤ ρ for all (κ, ξ) ∈ Bρ i i implies that wi± (κ, 0) has values in Bρ (0). (III) Instead of using Proposition C.2.5 in order to modify the nonlinear−1 ity Bk+1 fk , under Hypothesis 4.6.1 we obtain a C m -smooth extension via ˜ satisfies the assumptions of Theorem 4.4.6, provided Proposition C.2.17. Then (S) ρ > 0 is sufficiently small. More precisely, we have to choose ςi ∈ 0, ςi± (m) , c ∈ Γ¯i , and ρ > 0 so small that max Ki− , Ki+ (L1 (ρ) + bi L2 (ρ)) < ςi 3 1 + 3 max Ki− , Ki+ L2 (ρ) and this relation also holds with ρ replaced by sρ with s > 1 close to 1.
The following example shows that the gap condition (4.6c) is sharp, i.e., the invariant fiber bundle Wi+ from Theorem 4.6.4(a) is not of class C m in general, even if the nonlinearity fk is a C ∞ -function. Example 4.6.6. Let X = Z × R2 and e = exp(1). Given an integer m ≥ 2, let us consider the planar autonomous difference equation x = ex , (4.6g) y = em y + em xm satisfying the assumptions of Theorem 4.6.4(a) in form of an exponential 2-splitting with a1 = e, b1 = em and K1± = 1. Thus, there exists a pseudo-stable fiber bundle W1+ ⊆ Z × R2 given as graph of a function w1+ : Z × Bρ (0) → R2 for some ρ > 0. On the other hand, for every γ ∈ R the sets m Wγ := (ξ, η) ∈ Bρ (0) \ {0} : η = ξ2 ln ξ 2 + γξ m ∪ {0} contain the origin and are (locally) forward invariant w.r.t. (4.6g), i.e., Z × Wγ is a forward invariant fiber bundle. Additionally, each point (ξ, η) ∈ Bρ (0), ξ = 0, 2 is contained in exactly one of the sets Wγ , namely for γ = ξηm − ln2ξ . Thus, the pseudo-stable fiber bundle W1+ from Theorem 4.6.4(a) has the form Z × Wγ ∗ for some γ ∗ ∈ R (see Fig. 4.3). Every fiber Wγ is graph of a C m−1 -function wγ (ξ) = η, but wγ fails to be m-times continuously differentiable. Note that in the present exs ample the gap condition a1 < bm 1 is only fulfilled for 1 ≤ ms < m.
4.6 Pseudo-stable and Pseudo-unstable Fiber Bundles
263
wγ(ξ)
wγ∗(ξ)
0
0
ξ
Fig. 4.3 Graphs of the functions wγ from Example 4.6.7
Next we illustrate that center-unstable fiber bundles as postulated above in Theorem 4.6.4 need not to be uniquely determined. Example 4.6.7. For X = Z × R2 consider the two-dimensional autonomous equation x = e−1 x, (4.6h) y2 , y = y + 1−y satisfying the assumptions of Theorem 4.6.4(b) with an exponential 2-splitting, where K1+ = K1− = 1, a1 = e−1 and b1 = 1. It is easy to verify that Wγ := (κ, ξ, η) ∈ Z × R × (−∞, 1) : ξ = γe1/η for η < 0 and ξ = 0 for η ≥ 0 is a center-unstable fiber bundle of (S) for any parameter γ ∈ R in the sense that Wγ is a locally invariant graph containing the zero solution and being tangential to the center-unstable vector bundle. Our following result shows that compact 2-parameter semigroups induce unstable fiber bundles of finite-dimension. More precisely, one has Corollary 4.6.8. Suppose that Hypothesis 4.6.2 holds and let Bˆ be the family of all ˆ uniformly bounded subsets of X . Provided ϕ is B-contracting with q(k) := dar ϕˆk ,
lim eq (k, k − n) = 0
n→∞
for all k ∈ I ,
then the pseudo-unstable fiber bundles Wi− are finite-dimensional, if bi ≥ 1.
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4 Invariant Fiber Bundles
Proof. Thanks to Hypothesis 4.6.1(i) and (4.6b) the general forward solution ϕ to (S) and the evolution operator of (L0 ) are related by D3 ϕ(k; κ, 0) = Φ(k, κ), κ ≤ k; for the corresponding generators this means Dϕˆk = Φˆk , k ∈ I . Hence, our assumptions and Proposition C.1.2 imply dar Φˆk ≤ q(k), k ∈ I ˆ and consequently Corollary 1.2.29(a) guarantees that (L0 ) is B-contracting. Thus, Proposition 3.4.24(b) implies dim P1i < ∞. We continue with an asymptotic description of the stable and center-stable fiber bundles, as well as of their unstable counterparts. This requires Hypothesis 4.6.9. Let I be a discrete interval, ρ > 0 as in Theorem 4.6.4 and suppose that φ∗ : I → X is a reference solution of (D) such that the corresponding equation (Dφ∗ ) can be brought into the form (S) satisfying Hypotheses 4.2.1 and 4.6.1. Corollary 4.6.10. If Hypothesis 4.6.9 holds and φ : I → X is a solution of (D), then: (a) For I unbounded above: (a1 ) If φ − φ∗ decays exponentially in forward time, then there exists a k0 ∈ I such that (k, φ(k)) ∈ φ∗ + Ws for all k ≥ k0 . (a2 ) There exists a ρ1 ∈ (0, ρ) such that every forward solution of (D) starting in φ∗ + (Ws ∩ Bρ1 ) decays exponentially in forward time. (b) For I unbounded below: (b1 ) If φ − φ∗ decays exponentially in backward time, then there exists a k0 ∈ I so that (k, φ(k)) ∈ φ∗ + Wu for all k ≤ k0 . (b2 ) There exists a ρ1 ∈ (0, ρ) such that every backward solution of (D) starting in φ∗ + (Wu ∩ Bρ1 ) decays exponentially in backward time. Proof. W.l.o.g. we can assume that φ∗ is the trivial solution of (S). (a) We choose 1 ≤ i < N minimal with ai 1 ≤ bi and growth rates a, b with ai a b b i ,
a+b 1. 2
Thus, (L0 ) admits an exponential 2-splitting Σ(A, B; P ) = (0, a) ∪ (b, ∞) and as ˜+ in the proof of Theorem 4.6.4(a) there exists a forward invariant fiber bundle W ˜ consisting of forward solutions to (S) ˜ in X+ with c a+b . of (S), κ,c 2 (a1 ) Since the solution φ is exponentially decaying, there exists a positive sequence d 1 such that φ ∈ X+ κ,d , κ ∈ I; by an appropriate choice of a, b one has d ≤ c. Consequently, there exists an entry time k0 ∈ I such that φ(k) ∈ Bρ (0) for ˜ + coincide on Bρ , one has k ≥ k0 . Because the stable fiber bundle Ws of (S) and W (k, φ(k)) ∈ Ws for all k ≥ k0 . ˜ + of the (a2 ) Every initial pair (κ, ξ) ∈ Ws ∩ Bρ is contained in a fiber bundle W + ˜ ˜ ˜ κ, ξ) of (S). modified equation (S) and moreover yields a c -bounded solution ϕ(·; Due to c 1 this solution decays exponentially in forward time. Accordingly, for
4.6 Pseudo-stable and Pseudo-unstable Fiber Bundles
265
sufficiently small ρ1 ∈ (0, ρ) one has (k, ϕ(k; ˜ κ, ξ)) ∈ Ws ∩ Bρ and ϕ(·; ˜ κ, ξ) coincides with a solution of (S) starting in (κ, ξ). (b) For 1 ≤ i < N minimal with ai ≤ 1 bi this can be shown analogously. Corollary 4.6.11. If Hypothesis 4.6.9 holds and φ : I → X is a solution of (D), then: (a) If I is unbounded above and there exists a k0 ∈ I with (k, φ(k)) ∈ Bρ (φ∗ ) for all k0 ≤ k, then (k, φ(k)) ∈ φ∗ + Wcs for all k0 ≤ k. (b) If I is unbounded below and there exists a k0 ∈ I with (k, φ(k)) ∈ Bρ (φ∗ ) for all k ≤ k0 , then (k, φ(k)) ∈ φ∗ + Wcu for all k ≤ k0 . Proof. W.l.o.g. we again suppose that φ∗ is the trivial solution of (S). (a) First, choose 1 ≤ i < N minimal with 1 ≤ ai and growth rates a, b such that ai a b bi . Then the exponential 2-splitting Σ(A, B; P ) := (0, a) ∪ (b, ∞) ˜ + of the and the proof of Theorem 4.6.4(a) guarantees an invariant fiber bundle W + + ˜ ˜ modified system (S). We know that W consists of c -bounded solutions for some 1 c. If a solution φ : Z+ κ → X of (S) stays in Bρ for all k ≥ k0 , then it also ˜ and c+ -bounded (cf. Lemma 3.3.26). Hence, the solution is contained in solves (S) ˜ + ∩ Bρ . ˜ + for k ≥ k0 and therefore on Wcs = W W (b) One proceeds analogously with a minimal 1 ≤ i < N with bi ≤ 1. Proposition 4.6.12 (pseudo-center fiber bundles). Let m ∈ N and I = Z. Assume that Hypotheses 4.2.1 and 4.6.1 are satisfied. If (i, j) is a pair satisfying 1 < i ≤ j < N , then there exists a ρ ∈ (0, ρ0 ) such that the intersection + ∩ Wj− Wij := Wi−1
is a locally forward invariant fiber bundle of (S), representable as graph Wij = (κ, η + wij (κ, η)) ∈ X : (κ, η) ∈ Bρ of a Lipschitzian mapping wij : Bρ → X with wij (κ, ξ) = wij (κ, Pij (κ)ξ) ∈ Qji (κ)
for all (κ, ξ) ∈ Bρ .
Furthermore, it holds:
(a) wij (κ, 0) = 0 on I and wij (κ, ξ) lip2 wij
Xκ limr0 lip2 wij |Br =
≤ ρ for all (κ, ξ) ∈ Bρ .
< 1 and 0. (b) m (c) If additionally Hypothesis 4.6.2 and the conditions am i−1 bi−1 , aj bj hold, then wij : Bρ → X is a C m -mapping in the second argument, D2 wij (κ, 0) ≡ 0 on I and the derivatives D2n wij (κ, ·) : Bρ (0, Xκ ) → Ln (Xκ ) are globally bounded for n ∈ {2, . . . , m} (uniformly in κ ∈ Z). One denotes Wij as pseudo-center fiber bundle of (S).
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4 Invariant Fiber Bundles
Remark 4.6.13 (classical hierarchy). For (S) possessing a linear part admitting an exponential splitting with bi∗ ≤ 1 ≤ ai∗ −1 and growth rates an cn bn , we get the following classical invariant fiber bundles. As long as solutions stay in Bρ one can describe them asymptotically as follows: •
•
•
•
•
+1 Stable fiber bundle Ws = Wii∗∗+1 : Because of ci∗ bi∗ and the dynamical characterization all solutions on Ws converge to 0 exponentially for k → ∞. m Due to am i∗ bi∗ it is of class C . Center-stable fiber bundle Wcs = Wii∗∗ +1 : All solutions which are not growing too fast as k → ∞ (in the sense that they are c+ i∗ −1 -bounded with ci∗ −1 ≤ bi∗ −1 ) are contained in Wcs , like e.g., solutions bounded in forward time. Center-unstable fiber bundle Wcu = W1i∗ : All solutions which exist and are not growing too strong as k → −∞ (in the sense of c− i∗ -boundedness with ai∗ ≤ ci∗ ) lie on Wcu , like e.g., solutions bounded in backward time. Unstable fiber bundle Wu = W1i∗ −1 : All solutions on the unstable fiber bundle exist in backward time and converge exponentially to 0 as k → −∞. It is of class C m , since ai∗ −1 bm i∗ −1 holds. Center fiber bundle Wc := Wii∗∗ : The center fiber bundle consists of those solutions which are contained both in the center-stable and the center-unstable fiber bundle. Particularly, bounded complete solutions in Bρ lie on Wc .
These invariant fiber bundles form the classical hierarchy depicted in Fig. 4.4. If ai∗ −1 can be chosen close to 1, then the center-stable bundle Wcs is of class C m . The same holds for the center-unstable bundle Wcu if bi∗ is near 1. In this sense, the classical hierarchy inherits its smoothness from (S). Remark 4.6.14. For a p-periodic equation (S) the fiber bundles Wij are also p-periodic. In particular, for autonomous equations (S) one speaks of invariant manifolds and the above classical special cases are denoted as stable, center-stable, center-unstable, unstable resp. center manifold.
Wu ⊂ Wcu ⊂ X ∪ ∪ Wc ⊂ Wcs ∪ Ws
Ws Wu Wcu
Wc
Wcs
Fig. 4.4 Classical hierarchy of invariant fiber bundles (left) and classical invariant manifolds Wcs (dotted), Wcu (dashed) and Wu , Ws , Wc (right)
4.6 Pseudo-stable and Pseudo-unstable Fiber Bundles
267
Proof. The argument is parallel to the proof of Theorem 4.6.4. The conditions − (Γi−1 ) and (Γj+ ) are trivially fulfilled and instead of Theorem 4.2.9 one applies ˜ Here, ρ ∈ (0, ρ0 ) the global result Theorem 4.2.17 to the modified equation (S). ˜ has to be chosen so small that in particular (Gn ) holds for n ∈ {i − 1, j}. We omit further details. In the remaining section, we discuss an application of locally invariant fiber bundles to stability theory. The simplest situation is given for solutions φ∗ admitting a hyperbolic variational equation. In absence of an unstable vector bundle, the principle of linearized stability indicated in Remark 3.5.9(2) yields (exponential) stability of φ∗ . Conversely, if there is an unstable vector bundle, Theorem 4.6.4(b) guarantees an unstable fiber bundle and consequently the instability of φ∗ . Between these two cases is the situation of a nonhyperbolic variational equation, where a center-unstable vector bundle exists. Then stability properties are determined by the behavior on the center-unstable fiber bundle Wi−∗ and therefore by the lowerdimensional Wi−∗ -reduced equation (4.2s). Theorem 4.6.15 (reduction principle). Let I = Z. Suppose that Hypotheses 4.2.1 and 4.6.1 are satisfied for some 1 ≤ i∗ < N with ai∗ 1. The zero solution of (S) is stable (uniformly stable, asymptotically stable, uniformly asymptotically stable, exponentially stable, pullback stable, uniformly pullback stable, asymptotically pullback stable, uniformly asymptotically pullback stable, or unstable), if and only if the zero solution of the Wi−∗ -reduced equation (4.2s) has the respective stability property. Remark 4.6.16. For difference equations (D) generating a compact general forward solution and 1 ≤ bi∗ , we know from Corollary 4.6.8 that the Wi−∗ -reduced equation (4.2s) is finite-dimensional. Proof. Our assumptions guarantee that one can choose a growth rate c ∈ Γ¯i−∗ with c 1. In addition, there exists a pseudo-unstable fiber bundle Wi−∗ associated to the trivial solution of (S); it is graph of a function wi−∗ defined on a neighborhood Bρ in P1i∗ . By construction (cf. the proof of Theorem 4.6.4(b)), Wi−∗ is the restric˜ − for the modified equation (S) ˜ as in the proof of tion of global fiber bundle W i∗ − − Theorem 4.6.4, which is graph of a mapping w ˜i∗ and wi∗ = w ˜i−∗ |Bρ . Thanks to − Theorem 4.3.7(a) the invariant fiber bundle Wi∗ has an asymptotic forward phase satisfying (4.3r), where our assumptions yield that C˜κ+ (ξ, c) simplifies to
˜+ Ki∗ (c) := ˜− C˜κ+ (ξ, c) = Ki∗ (c) Qi1∗ (κ)ξ , i∗ (c) 1 + i∗ (c) which, due to Qi1∗ (κ) ≤ Ki+∗ , does not depend on κ ∈ Z. (⇒) By virtue of Corollary 4.2.13, the reduced equation (4.2s) describes the dynamics of (S) on the locally invariant pseudo-unstable fiber bundle Wi−∗ . This local invariance yields that stability properties of the zero solution for (S) carry over to (4.2s).
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4 Invariant Fiber Bundles
(⇐) Conversely, if the zero solution of the reduced equation (4.2s) is unstable, then by invariance of Wi−∗ , also the zero solution of (S) is unstable (cf. Corollary 4.2.13). Now, let ε > 0, κ ∈ Z be given. We suppose the zero solution of (4.2s) is stable, i.e., by Definition 2.4.11 there exists a δ ∈ (0, ρ) so that
φ0 (k) < ε
for all k ∈ Z+ κ
(4.6i)
i∗ and any solution φ0 : Z+ κ → X of (4.2s) with φ0 (κ) ∈ Bδ (0, P1 (κ)). In the + following, let φ : Zκ → X be an arbitrary solution of (S) with
φ(κ) < min
ε δ , 3C1 2C2
,
(K + )2 Ki+ Ki (c) Ki (c) where C1 := 1−ii (c) 1 + 1− and C2 := 1− . Due to the asymptotic ˜ i (c) ˜i (c) forward phase from Theorem 4.3.7(a), we establish that there exists a corresponding solution φ˜0 : Z+ κ → X of the global equation −1 ρ fk (x + w ˜i−∗ (k, x), x + w ˜i−∗ (k + 1, x )) Bk+1 x = Ak x + Bk+1 P1i∗ (k + 1)Bk+1
(cf. (4.2s)) in the pseudo-unstable vector bundle P1i∗ with (4.3r) ˜ κ, φ(κ)) − ϕ˜ k; κ, φ˜0 (κ) + w ˜i−∗ (κ, φ˜0 (κ)) ≤ C1 φ(κ) ec (k, κ) ϕ(k; ˜ We have from ˜ is the general solution of (S). for all k ∈ Z+ κ , where ϕ Theorem 4.3.7(a), (4.3v) ˜ φ0 (κ) = Qi1∗ (κ)πi+∗ (κ, φ(κ)) ≤ C2 φ(κ) < δ and thus (4.6i) gives us φ˜0 (k) <
ε 2(1+˜− i∗ (c))
for all k ∈ Z+ κ . But this yields (note
ec (k, κ) ≤ 1 for k ∈ Z+ κ ) with the triangle inequality and Theorem 4.2.9(b2 ),
ϕ(k; ˜ κ, φ(κ))
≤ ϕ(k; ˜ κ, φ(κ)) − ϕ˜ k; κ, πi+∗ (κ, φ(κ)) + ϕ˜ k; κ, πi+∗ (κ, φ(κ)) ≤ C1 φ(κ) ec (k, κ) + φ˜0 (k) + w ˜i−∗ (k, φ˜0 (k))
˜ + ≤ C1 φ(κ) + 1 + ˜− i∗ (c) φ0 (k) < ε for all k ∈ Zκ ˜ However, since the systems (S) and (S) ˜ coincide on and 0 is a stable solution of (S). + the ball Bρ , and due to ϕ(k; ˜ κ, φ(κ)) ∈ Bρ (0) for all k ∈ Zκ , it is φ = ϕ(·; ˜ κ, φ(κ)).
4.6 Pseudo-stable and Pseudo-unstable Fiber Bundles
269
Thus, the zero solution is also stable w.r.t. (S). Keeping in mind that Wi−∗ is uniformly exponentially attracting (cf. (4.3r)) with constants independent of κ ∈ Z, a similar reasoning gives us the assertion on the remaining stability properties. Taylor Approximation of Invariant Fiber Bundles The striking advantage of Theorem 4.6.15 is that stability investigations can be performed using the lower-dimensional Wi−∗ -reduced equation (4.2s), which for compact semigroups is even finite-dimensional (cf. Corollary 4.6.8). Yet, since its linear part is critical, stability depends on the nonlinearity, which in turn involves the center-unstable fiber bundle Wi−∗ . In fact, it suffices to know the Taylor coefficients of the corresponding mapping wi−∗ up to a certain order. For the remaining, we tackle this problem and describe a procedure to compute Taylor approximations of locally invariant C m -fiber bundles and in particular of center-unstable bundles. Here, a convenient and compact notation is advisable and as in Sect. 4.6 we restrict to the case where (S) is semi-implicit Bk+1 x = Ak x + fk (x).
(S )
Since Taylor approximations only make sense for smooth functions, we suppose that beyond Hypotheses 4.2.1, 4.6.1 also Hypothesis 4.6.2 is satisfied for (S ). In particular, its linear part admits an exponential splitting and thus the strong regularity condition (3.3j) holds. Hence, (S ) is equivalent to the explicit problem x = Ck x + fˆk (x)
(Sfˆ )
−1 −1 Ak ∈ L(Xk , Xk+1 ) and fˆk := Bk+1 fk : Xk → Xk+1 , k ∈ I . with Ck := Bk+1 In the subsequent considerations we choose a fixed 1 ≤ i∗ < N and use the brief notation introduced in (4.2p). The existence of the locally forward invariant fiber bundles W± for (S ) or (Sfˆ ) is guaranteed by Theorem 4.6.4. It furthermore yields that the mappings w± : U → X are defined on an open convex neighborhood U of 0, are continuously differentiable in the second argument and satisfy
w± (k, 0) ≡ 0 on I, lim D2 w± (k, x) = 0 uniformly in k ∈ I,
x→0 ±
(4.6j)
w± (k, x) = w (k, P± (k)x) ∈ P∓ (k) for all (k, x) ∈ U. In our present semi-implicit setting the invariance equations (4.2k) and (4.2m) for w± postulated in Theorem 4.6.4 simplify to Ck w± (k, ξ) + P∓ (k)fˆk (ξ + w± (ξ, k)) = w± k + 1, Ck ξ + P± (k)fˆk (ξ + w± (ξ, k)) for all (k, ξ) ∈ P± ∩ U so that Ck ξ + P± (k)fˆk (ξ + w± (k, ξ)) ∈ U(k).
(4.6k)
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4 Invariant Fiber Bundles
As demonstrated in Example 4.6.7, locally invariant fiber bundles are not unique in general. However, they can be obtained as restrictions of uniquely determined global fiber bundles of appropriately modified difference equations, and calculated using Taylor expansions. We will show this under the mild assumption −1 sup Bk+1 Ak L(X k∈I
k ,Xk+1 )
< ∞.
(4.6l)
Proposition 4.6.17. Suppose that Hypotheses 4.2.1, 4.6.1 and 4.6.2 with (4.6l) hold. If W± denotes a locally forward invariant C m -fiber bundle of (S ), where the corresponding mapping w± : U → X possesses uniformly bounded derivatives D2n w± and (4.6c) (when W+ is considered) resp. (4.6f) (when W− is considered) holds, then there exists a ρ > 0 and mappings g¯ρ : X → X, wρ± : X → X such that: (a) A global forward invariant C m -fiber bundle is given by the graph ρ := (κ, ξ + wρ± (κ, ξ)) ∈ X : ξ ∈ im P± (κ) . W± (b) g¯ρ (k, x) = fˆk (x) for all (k, x) ∈ Bρ . ρ (c) wρ± (k, x) = w± (k, x) for (k, x) ∈ Bρ and W± ∩ Bρ = W± ∩ Bρ .
Proof. See [383, Proposition 3.3].
Our next theorem states that all locally forward invariant C m -fiber bundles W± of (S ) have the same Taylor series up to order m. Moreover, it enables us to calculate them using solutions of the invariance equation (4.6k). Theorem 4.6.18 (Taylor expansion). Suppose that Hypotheses 4.2.1, 4.6.1, 4.6.2 and (4.6l) hold. Assume that W± denotes a locally forward invariant C m -fiber bundle of (S ), where the corresponding mapping w± : U → X possesses uniformly bounded derivatives D2n w± and one has (4.6c) (when W+ is considered) resp. (4.6f) (when W− is considered). If a mapping ω : X → X is m-times continuously differentiable in the second variable and satisfies: (i) ω(k, 0) ≡ 0 on I, limx→0 D2 ω(k, x) = 0 uniformly in k ∈ I, D2n ω are uniformly bounded and ω(k, x) = ω(k, P± (k)x) ∈ P∓ (k) for (k, x) ∈ X . (ii) With r > 0 so small that x + ω(k, x) ∈ Bρ0 (k) holds for all (k, x) ∈ Br , the mapping υk ω : Br (0) → Xk+1 , (υk ω)(x) := Ck ω(k, x) + P∓ (k)fˆk (x + ω(k, x)) − ω k + 1, Ck P± (k)x + P (k)fˆk (x + ω(k, x)) ±
satisfies Dn (υk ω)(0) = 0
for all n ∈ {1, . . . , m} , k ∈ I,
then we have D2n ω(k, 0) = D2n w± (k, 0) for all k ∈ I, n ∈ {1, . . . , m}.
(4.6m)
4.6 Pseudo-stable and Pseudo-unstable Fiber Bundles
271
Remark 4.6.19. The assumption (i) of Theorem 4.6.18 holds for polynomials ω(k, x) =
m
ωn (k)P± (k) xn
n=2
with bounded ωn : I → Ln (Xκ ) satisfying the inclusion coefficient sequences ωn (k) ∈ Ln Xk ; im P∓ (k) for all n ∈ {2, . . . , m}, k ∈ I. Proof. Define a C m -diffeomorphism Ψk : Xk → Xk , k ∈ I, by Ψk (x) := x − ω(k, x) and the change of variables x → Ψk (x) transforms (S ) into (SF ) with Fk (x) := Ck ω(k, x) + fˆk (x + ω(k, x)) − ω k + 1, Ck P± (k)x + P± (k + 1)fˆk (x + ω(k, x)) . From our assumption (i) we have Fk (0) ≡ 0 on I , and a consequence of (4.6b) with (4.6l) is limx→0 DFk (x) = 0 uniformly in k ∈ I. Moreover, it follows from (3.5a) that P∓ (k)Fk (x)P± (k) = (υk ω)(x) and (4.6m) yields P∓ (k)Dn Fk (0) ≡ 0 on I and for all n ∈ {1, . . . , m} . Also the graph {(κ, ξ + (w± − ω)(κ, ξ)) ∈ X : ξ ∈ im P± (κ)} is a locally invariant fiber bundle for (SF ). An application of Proposition 4.6.17 to (SF ) then guarantees the existence of a ρ > 0 and a mapping wρ± : X → X with wρ± (k, x) ≡ (w± − σ)(k, x) on the ball Bρ . The construction of the mapping wρ± in Theorems 4.2.9 and 4.4.6 in connection with the above identity implies D2n (w± − σ)(k, 0) ≡ D2n wρ± (k, 0) ≡ 0 on I for n ∈ {2, . . . , m}. This proves the assertion. We are interested in local approximations of a mapping w± : U → X defining a forward invariant C m -fiber bundle for (S ). For this purpose, Taylor’s Theorem (cf. [295, p. 350]) together with (4.6j) implies the representation w± (k, x) =
m 1 ± ± wn (k)xn + Rm (k, x) n! n=2
(4.6n)
with coefficient functions wn± (k) ∈ Ln (Xk ) given by wn± (k) := D2n w± (k, 0) and R± ± m (k,x) satisfying limx→0 x = 0. Theorem 4.6.18 guarantees that a remainder Rm m ± wn (k) is uniquely determined by the mapping from Theorem 4.6.4. In addition, the latter result yields that the sequences wn± are bounded, i.e., one has wn± (k) ≤ γn for all k ∈ I, n ∈ {2, . . . , m} with reals γ2 , . . . , γm ≥ 0. The following notation is helpful: •
We introduce W ± : U → X, W ± (k, x) := P± (k)x + w± (k, x), satisfying (4.6j)
D2 W ± (k, 0) = P± (k),
D2n W ± (k, 0) = D2n w± (k, 0)
(4.6o)
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4 Invariant Fiber Bundles
for all k ∈ I and n ∈ {2, . . . , m}. Hence, for the corresponding derivatives Wn± (k) := D2n W ± (k, 0) we have the estimates ± (3.4g) W (k) ≤ K± , 1
± (4.6d) Wn (k) ≤ γn for all n ∈ {2, . . . , m} . (4.6p) • We abbreviate Γ ± (k, x) := P± (k) Ck x + fˆk (P± (k)x + w± (k, x)) and the chain rule from Theorem C.1.3 yields that the corresponding partial derivatives Γn± (k) := D2n Γ ± (k, 0) are given by (cf. (4.6b) and (4.6j)) (3.5a)
Γ1± (k)x1 = Ck P± (k)x1 , Γn± (k)x1 · · · xn n = l=2
± ± P± (k)Dl fˆk (0)W#N (k)P± (k) xN1 · · · W#N (k)P± (k) xNl 1 l
(N1 ,...,Nl )∈Pl< (n)
for all x1 , . . . , xn ∈ Xk and n ∈ {2, . . . , m}. Moreover, the uniform boundedness assumption for Dl fˆk (cf. Hypothesis 4.6.2) and the estimates (??), (3.4g), (4.6p) imply that Γn± (k) ∈ Ln (Xk ; Xk+1 ) are bounded sequences for n ∈ {2, . . . , m}. Note that both the mappings W ± and Γ ± satisfy (cf. (4.6j)) W ± (k, x) = W ± (k, P± (k)x), Γ ± (k, x) = Γ ± (k, P± (k)x) for all (k, x) ∈ Br , where r > 0 is chosen so small that W ± (k, x) ∈ Bρ (0), Γ ± (k, x) ∈ U(k) for every (k, x) ∈ Br . From the invariance equation (4.6k) and (4.6j), Ck w± (k, x) + P∓ (k)fˆk (P± (k)x + w± (k, x)) = w± (k + 1, Ck P± (k)x + P± (k)fˆk (P± (k)x + w± (k, x))) and using the notation introduced above, this reads as Ck w± (k, x) + P∓ (k)fˆk (W ± (k, x)) ≡ w± (k + 1, Γ ± (k, x))
on Br .
If we differentiate this identity using Theorem C.1.3 and set x = 0, one gets wn± (k + 1)Ck P± (k) x1 · · · xn +
n−1
± ± wl± (k + 1)Γ#N (k)P± (k) xN1 · · · Γ#N (k)P± (k) xNl 1 l
l=2 (N1 ,...,Nl )∈Pl< (n)
= Ck wn± (k)P± (k) x1 · · · xn + P∓ (k + 1) Dn fˆk (0)P± (k) x1 · · · xn
(C.1b)
+
n−1
± ± Dl fˆk (0)W#N (k)P± (k) xN1 · · · W#N (k)P± (k) xNl 1 l
l=2 (N1 ,...,Nl )∈Pl< (n)
4.6 Pseudo-stable and Pseudo-unstable Fiber Bundles
273
for every n ∈ {2, . . . , m} and x1 , . . . , xn ∈ Xk . Therefore, we see that the Taylor coefficient wn± : I → Ln (Xκ ) is a solution of the linear difference equation XC k P± (k) = Ck XP± (k) + Hn± (k)P± (k) ,
(4.6q)
denoted as homological equation for W± with Hn± : I → Ln (Xκ ) defined by Hn± (k)x1 · · · xn :=P∓ (k + 1) Dn fˆk (0)P ± (k) x1 · · · xn +
n−1
± ± Dl fˆk (0)W#N (k)P± (k) xN1 · · · W#N (k)P± (k) xNl 1 l
l=2 (N1 ,...,Nl )∈Pl< (n)
± ± − wl± (k + 1)Γ#N (k)P± (k) xN1 · · · Γ#N (k)P± (k) xNl 1 l
.
(4.6r)
Obviously, one has H2± (k) = P∓ (k)D2 fˆk (0)P± (k) and for n ∈ {3, . . . , m} the ± . This leads to the following values Hn± (k) only depend on w2± , . . . , wn−1 Theorem 4.6.20. Suppose that Hypotheses 4.2.1, 4.6.1 and 4.6.2 with (4.6l) are satisfied. If w± : U → X is a mapping as in Theorem 4.6.4, then one has: (a) For I unbounded above, the coefficients wn+ : I → Ln (Xκ ) in the Taylor expansion (4.6n) of the mapping w+ : U → X can be determined recursively from the Lyapunov–Perron sums wn+ (k) = −
∞
−1 + Φ− P− (k, j + 1)Bj+1 Hn (j)Φ(j,k)P+ (k)
for all 2 ≤ n ≤ m.
j=k
(b) For I unbounded below, the coefficients wn− : I → Ln (Xκ ) in the Taylor expansion (4.6n) of the mapping w− : U → X can be determined recursively from the Lyapunov–Perron sums wn− (k) =
k−1 j=−∞
−1 Φ(k, j + 1)Bj+1 Hn− (j)Φ−
P− (j,k)P− (k)
for all 2 ≤ n ≤ m.
Remark 4.6.21. If (S ) is autonomous, the above Lyapunov–Perron sums are constant and the stationary solutions of the homological equation (4.6q). Proof. In the explanations preceding Theorem 4.6.20 we have seen that the sequence wn± : I → Ln (Xκ ) is a bounded solution of the homological equation (4.6q). It follows recursively from Hypothesis 4.6.2, (4.6p), (3.4g) and (4.6r) that each inhomogeneity Hn± is bounded, i.e., 1-bounded. Consequently, due to the gap conditions (4.6c) and (4.6f), it yields from Lemma 3.5.12 that wn± has the claimed appearance.
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4 Invariant Fiber Bundles
4.7 Inertial Fiber Bundles In the first instance, the goal of this section is to provide a discrete counterpart for the concept of an inertial manifold, which is applicable to equations Bk+1 x = Ak x + fk (x, x )
(S)
as studied above. Regarding this, our approach so far lacks certain features. Thus, let us reconsider the theory developed in this chapter from an applied point of view. In this context, two aspects need to be addressed: •
At least in the autonomous or periodic situation, classical spectral or Floquet theory provides sufficient criteria that the linear part of (S) meets the exponential splitting assumption Hypothesis 4.2.1. The global Lipschitz condition Hypothesis 4.2.3 on the nonlinearity fk , however, will hardly be satisfied in relevant applications. More often the nonlinear term fk is only Lipschitzian on bounded sets. • The existence of inertial manifolds relies on a certain kind of dissipativity. Hence, we need appropriate counterparts of notions like absorbing sets or attractors in our nonautonomous framework. Here, the concept of pullback convergence as discussed throughout Chap. 1, will serve as the right tool. To incorporate these two points into our theory, we weaken Hypothesis 4.2.3 or 4.3.1 by imposing the following Hypothesis 4.7.1. Let L1 , L2 : [0, ∞) → [0, ∞) denote nondecreasing uppersemicontinuous functions, suppose the general forward solution ϕ of (S) exists as a continuous mapping, fk : Xk × Xk+1 → Yk+1 fulfills fk (Xk , Xk+1 ) ⊆ im Bk+1 for all k ∈ Z and that the following local Lipschitz estimates are satisfied: For each r > 0 one has −1 Lj (r) := sup lipj Bk+1 fk |Br (0,Xk )×Br (0,Xk+1 ) < ∞
for j = 1, 2.
(4.7a)
k∈Z
Next we suppose (S) is uniformly bounded dissipative as follows: Hypothesis 4.7.2. Let ρ > 0 and Bˆ be the family of uniformly bounded subsets ˆ of X . Suppose that (S) has a B-uniformly absorbing set A ⊆ X , i.e., for every B ∈ Bˆ there exists an M = M (B) ∈ Z+ with 0 ϕ(k; k − n, B(k − n)) ⊆ A(k)
for all k ∈ Z, n ≥ M.
In order to obtain Lipschitzian extensions we define the constants ∗ := sup lip rXk rX k∈Z
∗ and remark that Lemma C.2.1 guarantees rX ∈ [1, 2].
4.7 Inertial Fiber Bundles
275
Theorem 4.7.3 (inertial fiber bundles). Let I = Z. Assume that Hypotheses 4.2.1, 4.7.1 and are satisfied 4.7.2 for some 1 ≤ i < N and that the boundedness condition −1 supk∈Z Bk+1 Ak P1i (k)L(X ,X ) < ∞ holds. Beyond that suppose k
k+1
∗ rX L2 (ρ) < 1, ∗ rX Ki− (L1 (ρ)
(4.7b) for all k ∈ Z,
(4.7c)
Ki+ Ki− + max Ki+ , Ki− (L1 (ρ) + bi L2 (ρ)) < ςi . ∗ K + K − + max K + , K − L2 (ρ) 1 + rX i i i i
(4.7d)
+ bi (k)L2 (ρ)) < bi (k)
and the following strengthened spectral gap condition ∗ rX
If (Γi− ) holds, then there exists a nonautonomous set Wi ⊆ X , which is forward invariant w.r.t. (S), and possesses the following properties: (a) Wi is graph of a mapping wi : O → X over a nonempty open set O ⊆ P1i , i.e., Wi = {(κ, η + wi (κ, η)) : (κ, η) ∈ O}, wi (κ, ·) : O(κ) → Qi1 (κ) is welldefined, globally Lipschitzian with lip2 wi < 1 and the invariance equation (4.2m) holds for all (κ, η) ∈ O such that η1 ∈ O (κ). (b) The nonautonomous set Wi is asymptotically complete, i.e., for every initial value pair (κ, ξ) ∈ X there exists a point (κ0 , η) ∈ Wi with κ ≤ κ0 such that
ϕ(k; κ, ξ) − ϕ(k; κ0 , η) Xk ≤ Ci (κ, ξ)ec (k, κ) for all k ∈ Z+ κ0 , where the constant Ci (κ, ξ) ≥ 0 depends boundedly on κ, ξ and c ∈ Γ¯i . Under the assumption dim P1i < ∞ we call Wi inertial fiber bundle of (S). Remark 4.7.4. (1) Note that the condition (4.7b) becomes void for semi-implicit equations (S ) and (4.7c), (4.7d) degenerate in this case to ∗ Ki− L1 (ρ) < bi (k), rX
+ − ∗ Ki Ki + max Ki− , Ki+ L1 (ρ) < ςi , rX
∗ = 1 (cf. Lemma C.2.1). respectively. If X consists of Hilbert spaces, one has rX (2) A typical situation, where the pseudo-unstable vector bundle P1i is finiteˆ dimensional, arises when the linear part (L0 ) is B-contracting, bi ≥ 1 and the family Bˆ is as in Hypothesis 4.7.2. This is a consequence of Proposition 3.4.24(b). (3) The inertial fiber bundle Wi can be seen as a nonautonomous discrete counterpart of an inertial manifold (cf. [432, p. 569ff]). If dim P1i (κ) < ∞ for one κ ∈ Z, then Lemma 3.3.6(b) guarantees that all fibers Wi (k), k ∈ Z, possess the same finite dimension, they are Lipschitzian, Wi is forward invariant w.r.t. (S) and in case ai + ς 1 also exponentially attractive. In conclusion, the forward dynamics of (S) is equivalent to the finite-dimensional Wi -reduced equation (4.2s), which is called inertial form in this context.
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4 Invariant Fiber Bundles
The usual procedure to prove Theorem 4.7.3 is to replace (S) by an appropriately modified difference equation and to apply our previous global results from both Sects. 4.2 and 4.3 to the modified equation. It then remains to show that this modification does not affect the long term dynamics. Proof. Let Bˆ be the family of uniformly bounded subsets of X and ρ > 0 be the radius of the ball Bρ containing the absorbing set required in Hypothesis 4.7.2. Above all, we use Proposition C.2.5 in order to obtain a globally Lipschitzian extension −1 ρ −1 Bk+1 fk of Bk+1 fk , where both functions coincide on Bρ ×Bρ . By Hypothesis 4.7.1 this yields −1 ρ ∗ sup lip1 Bk+1 fk ≤ rX L1 (ρ), k∈Z
−1 ρ ∗ sup lip2 Bk+1 fk ≤ rX L2 (ρ). k∈Z
Having this at hand, we can focus on the modified equation Bk+1 x = Ak x + fkρ (x, x ),
˜ (S)
which fulfills Hypothesis 4.2.1 and the Lipschitz conditions required in Hypothe˜ sis 4.2.3. In addition, by Proposition 4.1.3 the general forward solution ϕ˜ to (S) exists as a continuous mapping, since (4.7b) holds. This ensures Hypothesis 4.3.1 as well. ˜ Indeed, by the gap condition Our next goal is to apply Theorem 4.2.9(b) to (S). (4.7d), Theorem 4.2.9(b) and Theorem 4.3.7(a) apply and there exists an invariant ˜ − of the modified equation (S), ˜ which is graph of a mapping w fiber bundle W ˜i− over i + − i P1 with an asymptotic forward phase πi . In particular, (4.7d) yields lip2 w ˜i < 1. ˜ − a forward invariant nonautonomous We now demonstrate how to derive from W i ˆ absorbing, there exists set Wi for the initial equation (S). Since A is B-uniformly an M = M (Bρ ) ∈ N such that ϕ(k; k − n, Bρ (k − n)) ⊆ A(k)
for all k ∈ Z, n ≥ M
(4.7e)
and we define the nonautonomous set B∗ ⊆ X by its fibers B∗ (k) :=
ϕ(k; k − n, Bρ (k − n))
for all k ∈ Z.
n≥M
Then (4.7e) implies B∗ ⊆ A, cl B∗ ⊆ Bρ and the inclusion
ϕ(k; l, B∗ (l)) = ϕ k; l, ϕ(l; l − n, Bρ (l − n)) (2.3a)
⊆
n≥M
ϕ(k; l − n, Bρ (l − n))
n≥M
=
ϕ(k; k − n, Bρ (k − n)) ⊆ B∗ (k) for all l ≤ k,
n≥M+k−l
(4.7f)
4.7 Inertial Fiber Bundles
277
which yields ϕ(k; l, ·)|B∗ (l) = ϕ(k; ˜ l, ·)|B∗ (l) for all l ≤ k, and B∗ is also attracting ˜ − ∩ B∗ and we obtain for the initial equation (S). Now define Wi∗ := W i ˜ − (κ)) ∩ ϕ(k; ˜ κ, Wi∗ (κ)) ⊆ ϕ(k; ˜ κ, W ˜ κ, B∗ (κ)) ϕ(k; κ, Wi∗ (κ)) = ϕ(k; i − ∗ ˜ (k) ∩ B∗ (k) = Wi (k) for all k ∈ Z+ ⊆W κ, i
˜ so that Wi∗ is forward invariant w.r.t. the initial equation (S), as well as (S). Choose ε > 0 so small that the open ε-neighborhood Bε (B∗ ) of B∗ is contained ˜ − ∩ Bε (B∗ ). Then W ε is an open neighborhood of W ∗ in in Bρ and set Wiε := W i i i ˜ − and due to the uniform continuity of ϕ(k; ˜ κ, ·) in k − κ ≤ M (see the Lipschitz W i estimate (4.2t) in Corollary 4.2.13, which can be applied due to (4.7c)), we obtain a ˜ − satisfies δ > 0 such that the open δ-neighborhood Wiδ of Wi∗ in W i ϕ(k; ˜ κ, Wiδ (κ)) ⊆ Wiε (k) for all k − κ ≤ M. Thus, using the above inclusion (4.7e) we obtain ϕ(k; κ, Wiδ (κ)) ⊆ Wiε (k) and ϕ(k; κ, Wiδ (κ)) = ϕ(k; ˜ κ, Wiδ (κ)) for all k ∈ Z+ κ . Let us show that Wi , defined by Wi (k) :=
ϕ(k; k − n, Wiδ (k − n)) for all k ∈ Z
n≥0
is the desired forward invariant nonautonomous set for (S). By definition, we readily see the inclusion ϕ(k; κ, Wi (κ)) ⊆ Wi (k) for all k ∈ Z+ κ , i.e., Wi is forward invariant w.r.t. (S). (a) Thanks to Corollary 4.2.13 and Corollary 4.2.14, which apply due to (4.7c), ˜ − (κ) → W ˜ − (k) we are able to deduce that the restriction ϕ(k; ˜ κ, ·)|W˜ − (κ) : W i i i is a homeomorphism (indeed a Lipeomorphism), so that it sends open subsets of ˜ − (k). Thus, ϕ(k; κ, W δ (κ)) = ϕ(k; ˜ − (κ) into open sets of W ˜ κ, Wiδ (κ)) is open W i i i δ + ˜ − (k) and W ˜ −, in Wi (k) for k ∈ Zκ , and therefore Wi (k) and Wi are open in W i i − − i ˜ respectively. Due to the fact that I + w ˜i (k, ·) : P1 (k) → Wi (k) is a homeomorphism (see Theorem B.3.1 and note lip2 w ˜i− < 1), also the set O ⊆ X , fiber-wise given by −1 O(k) := I + w ˜i− (k, ·) (Wi (k)) for all k ∈ Z is open in P1i . Now, if we define wi := w ˜i− |O , then wi (κ, ·) : O(κ) → Qi1 (κ) satisfies lip2 wi < 1 and also the further claims instantly follows from the corresponding properties for w ˜i− guaranteed by Theorem 4.2.9(b). (b) Let (κ, ξ) ∈ X and c ∈ Γ¯i . Choose B ∈ Bˆ such that (κ, ξ) ∈ B and we see that there exists a N1 = N1 (B) ∈ Z+ 0 with ϕ(k; k − n, B(k − n)) ⊆ B∗ (k) for all k ∈ Z, n ≥ N1 . In particular, this yields ξ0 := ϕ(κ + N1 ; κ, ξ) ∈ B∗ (κ + N1 ), thanks to (4.7f) one has ϕ(k; κ + N1 , ξ0 ) = ϕ(k; ˜ κ + N1 , ξ0 ), (2.3a)
ϕ(k; κ, ξ) = ϕ(k; κ + N1 , ξ0 ) = ϕ(k; ˜ κ + N1 , ξ0 ) for all k ≥ κ + N1 . (4.7g)
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4 Invariant Fiber Bundles
˜ − (cf. Theorem 4.3.7(a)) there exists a Due to the asymptotic forward phase of W i − ˜ (κ + N1 ) such that point η0 ∈ W i ˜ κ + N1 , η0 ) Xk ≤ Cec (k, κ)
ϕ(k; ˜ κ + N1 , ξ0 ) − ϕ(k;
(4.7h)
for all k ≥ κ + N1 , where the real constant C ≥ 0 depends boundedly on κ, ξ, as well as c. Now we choose another set C ∈ Bˆ such that (κ + N1 , η0 ) ∈ C. Again, there exists N2 = N2 (B) ∈ Z+ 0 with ϕ(k; k − n, C(k − n)) ⊆ B∗ (k) for all k ∈ Z, n ≥ N2 , and in particular η := ϕ(κ + N1 + N2 , η0 ) ∈ B∗ (κ + N1 + N2 ). Then, the forward invariance of B∗ from (4.7f) implies ϕ(k; κ + N1 , η0 ) = ϕ(k; ˜ κ + N1 , η0 ) and therefore (2.3a)
ϕ(k; κ + N1 + N2 , η0 ) = ϕ(k; κ + N1 , η0 ) = ϕ(k; ˜ κ + N1 , η0 )
(4.7i)
for all k ≥ κ + N1 + N2 . Setting κ0 := κ + N1 + N2 , inserting (4.7g) and (4.7i) into the estimate (4.7h) gives us the claim (b). This finishes the proof. Another important feature of inertial fiber bundles is that they contain the attractor of a dissipative equation. As we know from Chap. 1, under compactness assumptions on ϕ, the existence of an attractor is implied by a more easily determinable absorbing set. As we will see next, this feature fits well into our theory. Corollary 4.7.5 (attractors). Assume ai + ς 1. If the sequence (Γκ− (i))κ∈Z from (Γi− ) is backward tempered, then every w.r.t. (S) invariant nonautonomous set B ∈ Bˆ with B(k) ⊆ n≥M ϕ(k; k − n, Bρ (k − n)) for all k ∈ Z satisfies B ⊆ Wi . In particular, the fiber bundle Wi contains the global attractor A∗ of (S), i.e., A∗ ⊆ Wi . Proof. Let k ∈ Z be arbitrary. Then B ⊆ B∗ and the invariance of B leads to ˜ − (k)) ˜ − (k)) = h(ϕ(k; k − n, B(k − n)), W h(B(k), W i i ˜ − (k)) −−−−→ 0 = h(ϕ(k; ˜ k − n, B(k − n)), W i n→∞
˜ − (k), but B ⊆ B∗ and the indue to Corollary 4.3.11. Consequently, B(k) ⊆ cl W i − ˜ ∩Bδ (B∗ ) implies the desired relation B(k) ⊆ Wi (k). Obviously, clusion Wi ⊇ W i this holds for the special case B = A∗ . Corollary 4.7.6 (smoothness of inertial fiber bundles). Suppose that Hypothesis 4.6.2 holds. Under the strengthened spectral gap condition (4.7d) replaced by ai b m i ,
Ki+ Ki− + max Ki+ , Ki− (L1 (ρ) + bi L2 (ρ)) < ςi− (m) 3 1 + 3 Ki+ Ki− + max Ki+ , Ki− L2 (ρ)
(4.7j)
the mapping wi (κ, ·) : O(κ) → Qi1 (κ) is of class C m with globally bounded derivatives (uniformly in κ ∈ Z).
4.8 Approximation of Invariant Fiber Bundles
279
Proof. Let s > 1 and ρ > 0 as in the proof of Theorem 4.7.3. Instead of Proposition C.2.5 we use it differentiable version Proposition C.2.17 in order to −1 ρ −1 obtain a C m -smooth modification Bk+1 fk of Bk+1 fk . It satisfies the global Lipschitz conditions −1 ρ sup lip1 Bk+1 fk ≤ (1 + 2s)L1 (sρ), k∈Z
−1 ρ sup lip2 Bk+1 fk ≤ (1 + 2s)L2 (sρ) k∈Z
˜ fulfills Hypotheses 4.2.1, 4.2.3, and the global and the modified equation (S) smoothness assumption Hypothesis 4.4.2. Thanks to (4.7j) there exists a s > 1 close to 1 such that + − Ki Ki + max Ki+ , Ki− (L1 (sρ) + bi L2 (sρ)) (1 + 2s) < ςi− (m) 1 + (1 + 2s) Ki+ Ki− + max Ki+ , Ki− L2 (sρ) holds and we can apply Theorem 4.4.6(b). It yields a mapping w ˜i− : O → X as in − the proof of Theorem 4.7.3, but now w ˜i (κ, ·) is m-times continuously differentiable with globally bounded derivatives (uniformly in κ ∈ Z). As above, we see that the restriction wi := w ˜i− |O is the desired C m -mapping.
4.8 Approximation of Invariant Fiber Bundles The reduction principle from Theorem 4.6.15 is local in nature and a Taylor approximation of the center-unstable bundle is sufficient in order to apply it. The inertial fiber bundles constructed in the previous Sect. 4.7, however, are global objects having a dynamical meaning on the whole absorbing set of a semilinear equation Bk+1 x = Ak x + fk (x, x ).
(S)
Therefore, and for various other reasons mentioned in the introduction to this chapter, it is important to provide more global approximation techniques for invariant fiber bundles, which also work for merely Lipschitzian nonlinearities fk . In order to meet the requirements of our nonautonomous framework, we provide an approach based on the Lyapunov–Perron method. For given κ ∈ I and K ∈ Z+ 0 ∪ {∞} define discrete intervals I+ κ (K)
:=
[κ, κ + K]Z
for K < ∞,
[κ, ∞)Z
for K = ∞,
I− κ (K)
:=
[κ − K, κ]Z
for K < ∞,
(−∞, κ]Z
for K = ∞
and impose our standing assumptions for this section: Hypothesis 4.8.1. Suppose that the linear part (L0 ) satisfies Hypothesis 4.2.1, for the nonlinearity fk : Xk × Xk+1 → Yk+1 we require Hypothesis 4.7.1 and we assume the growth condition (Γi± ) holds for one 1 ≤ i < N .
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4 Invariant Fiber Bundles
For a convenient notation, we keep 1 ≤ i < N fixed as required in Hypothesis 4.8.1 and use the abbreviations established in (4.2p). Having this at hand, we can establish our necessary functional analytical framework. Given a sequence c : I → (0, ∞), κ ∈ I and K ∈ (0, ∞]Z such that κ − K ∈ I or κ + K ∈ I, respectively, it is not difficult to see that the following spaces of exponentially bounded sequences X± κ,c (K)
:= φ : I± κ (K) → X
sup
k∈I± κ (K)
ec (κ, k) φ(k) Xk < ∞
(4.8a)
in X become Banach spaces w.r.t. the respective norms
φ ± κ,c :=
sup
k∈I± κ (K)
ec (κ, k) max P− (k)φ(k) Xk , P+ (k)φ(k) Xk .
± Indeed, for K = ∞ we can briefly write X± κ,c := Xκ,c (∞) in correspondence with Definition 3.3.19. Clearly, the condition supk∈I± ec (κ, k) φ(k) Xk < ∞ κ (K) is always fulfilled for finite K < ∞. Our idea is to find fixed points of the Lyapunov–Perron operators (4.2q) and (4.2b) in the product space X± Xk , after we passed over to finite κ,c (K) = k∈I± κ (K) Xk instead of in the sequence space X± sums. This yields a problem in k∈I± κ,c . κ (K) As we will see in Proposition 4.8.3, a spectral gap condition still guarantees that the Lyapunov–Perron operators are contractions when passing over to finite sums. Of central importance for our approximation purposes are the following trun± ± cated Lyapunov–Perron operators Tκ,K : X± κ,c (K) × Xκ → Xκ,c (K), which, for a given pair (κ, ξ) ∈ X , read as
×
×
+ Tκ,K (φ, ξ)
= Φ(·, κ)P+ (κ)ξ +
κ+K−1
−1 Gi (·, l + 1)Bl+1 fl (φ(l)),
l=κ − Tκ,K (φ, ξ) = Φ− P− (·, κ)P− (κ)ξ −
κ−1
−1 Gi (·, l + 1)Bl+1 fl (φ(l)),
l=κ−K + respectively. Note that the case K = ∞ is explicitly allowed here and Tκ,∞ is the + − − operator Tκ defined in (4.2q), whereas Tκ,∞ has been denoted as Tκ in (4.2b). The corresponding respective fixed point problems
φ = Φ(·, κ)P+ (κ)ξ +
κ+K−1
−1 Gi (·, l + 1)Bl+1 fl (φ(l)),
+ (LPK )
l=κ
φ = Φ− P− (·, κ)P− (κ)ξ −
κ−1 l=κ−K
−1 Gi (·, l + 1)Bl+1 fl (φ(l))
− (LPK )
4.8 Approximation of Invariant Fiber Bundles
281
− in X+ κ,c (K) resp. Xκ,c (K), are denoted as truncated Lyapunov–Perron equations. Their relation to the dynamical behavior of (S) is described in the following counterpart to Lemma 4.2.7:
Lemma 4.8.2. Let (κ, ξ) ∈ X and suppose Hypothesis 4.8.1. If φ ∈ X± κ,c , ai c bi , is a sequence satisfying
± < ∞, L sup
φ(k)
φ(k)
L1 supk∈I± 2 Xk Xk < ∞, k∈Iκ κ then the following assertions are equivalent: (a) φ solves the difference equation (S) with P± (κ)φ(κ) = ξ. ± (b) φ is a fixed point of the Lyapunov–Perron equation (LP∞ ). −1 Proof. Referring to (4.7a), we assumed that Bk+1 fk fulfills a Lipschitz condition in a ball containing φ. Hence, the proof is essentially identical to the one of Lemma 4.2.7, since the sequence g defined there satisfies g ∈ X± κ,c,B .
Under stronger global conditions, we can establish the existence of unique solutions for the Lyapunov–Perron equations: Proposition 4.8.3. Let (κ, ξ) ∈ X , K ∈ N and assume Hypothesis 4.8.1 holds with lj := sup Lj (r) < ∞ for j = 1, 2.
(4.8b)
r≥0
If the spectral gap condition
max {K− , K+ } (l1 + bi l2 ) bi − ai : < ς¯, ∃¯ ς ∈ 0, 2 1 + max {K− , K+ } l2
(4.8c)
is fulfilled and we have chosen a real ς ∈ (max {K− , K+ } (l1 + bi − ς¯ l2 ) , ς¯), then for all c ∈ Γ¯i one has: ± (a) The truncated Lyapunov–Perron equations (LPK ) have a uniquely determined ± solution φκ,K (ξ) ∈ X± κ,c (K), which moreover satisfy ± ± φκ,K (ξ) ≤ κ,c
ςK±
P± (κ)ξ Xκ ς − max {K− , K+ } L(bi − ς) max {K− , K+ } Γκ± (i) . + ς − max {K− , K+ } L(bi − ς)
(b) For every initial sequence ψ0± ∈ X± κ,c (K) and n ≥ 1 the recursively defined ± ± sequence ψn± := Tκ,K (ψn−1 , ξ) ∈ X± κ,K (K) satisfies ± ± qn ± ± ± ± (ξ) ≤ − T (ψ , ξ) ψ ψn − φ± , 0 0 κ,K κ,K 1−q κ,c κ,c where q := max{Kς− ,K+ } L(bi − ς) ∈ (0, 1), and the sequences φ± κ,K (ξ) do not depend on c, where L(c) := l1 + c l2 .
(4.8d)
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4 Invariant Fiber Bundles
Proof. Let (κ, ξ) ∈ X and K ∈ N. (a) We only sketch the proof, since it resembles the one of Lemmata 4.2.6 and 4.2.8. For this, the spectral gap condition (Gi ) holds due to (4.8c). Consider ± ± the truncated Lyapunov–Perron operator Tκ,K : X± κ,c (K) × Xκ → Xκ,c (K). It ± can be verified as (4.2h) that Tκ,K is well-defined and satisfies the two Lipschitz estimates (4.2i)
± lip1 Tκ,K ≤ i (c) ≤ q < 1,
± lip2 Tκ,K ≤ K± .
(4.8e)
± (·, ξ) is a contraction on X± By the first inequality in (4.8e) we get that Tκ,K κ,c (K), uniformly in ξ ∈ Xκ , and Banach’s theorem (see, e.g., [295, p. 361, Lemma 1.1]) ± implies that there exists a unique fixed point φ± κ,K (ξ) ∈ Xκ,c (K). Moreover, the ± second inequality in (4.8e) yields the bound on φκ,K (ξ). (b) The stated inequality (4.8d) is just the standard a priori error estimate (cf., e.g., [465, p. 17, Theorem 1.A]) for the successive iterates in Banach’s fixed ± ± point theorem applied to the contraction Tκ,K (·, ξ) : X± κ,c (K) → Xκ,c (K).
The relations (cf. (4.2r)) − − w+ (κ, ξ) = P− (κ)φ+ κ (κ, ξ), w (κ, ξ) = P+ (κ)φκ (κ, ξ)
for all (κ, ξ) ∈ X (4.8f)
are central in our approach to compute the invariant fiber bundles W± ⊆ X . Actually, in order to compute the functions w± defining W± , we solve the ± Lyapunov–Perron equations (LPK ) for K < ∞. The corresponding error estimate ± for the distance between the fixed points φ± κ,K (ξ) and φκ (ξ) is deduced in Proposition 4.8.4. Let (κ, ξ) ∈ X , K ∈ N, suppose Hypothesis 4.8.1 holds with (4.8c) and choose ς as in Proposition 4.8.3. Then the mapping w± : X → X defining the fiber bundle W± satisfies ± w (κ, ξ) − P∓ (κ)φ± κ,K (κ, ξ)
Xκ
≤ C(κ, ξ)e ai +ς (κ, κ − K), bi −ς
(4.8g)
where the constant C(κ, ξ) is linearly bounded in P± (κ)ξ and Γi± (κ). Remark 4.8.5 (spectral ratio condition). Since the Lipschitz constant i (c) from +ς (4.8e) is supposed to be small, one can choose ς close to 0 and the decay rate abii−ς ai in (4.8g) basically depends on the ratio bi . Thus, we get a good approximation for small values of K > 0 in (4.8g), if abii is near 0. In the autonomous situation, this means that consecutive spectral points have moduli with small quotients. Proof. To avoid redundancy, we only prove the assertion in the pseudo-unstable situation of w− and φ− κ,K . We choose (κ, ξ) ∈ X fixed, a finite integer K > 0 i , we can select a d ∈ [ai + ς, c). and c ∈ (ai + ς, bi − ς]. Thanks to ς < bi −a 2 − − − Suppressing the dependence on ξ, let φκ,K ∈ X− κ,c (K), φκ ∈ Xκ,c be the unique − − solutions of the respective Lyapunov–Perron equations (LPK ) and (LP∞ ). Then,
4.8 Approximation of Invariant Fiber Bundles
283
− on the finite interval [κ − K, κ]Z , one evidently has φ− ∈ X− κ,K , φκ |I− κ,d (K), κ (K) and we derive two preparatory estimates. First, using the triangle inequality it is
κ−1−K −1 Φ(k, n + 1)P+ (n)Bn+1 fn (φ− κ (n)) ed (κ, k) n=−∞ κ−1−K −1 − ≤ Φ(k, n + 1)P+ (n)Bn+1 fn (φκ (n)) − fn (0, 0) ed (κ, k) n=−∞ κ−1−K −1 + Φ(k, n + 1)P+ (n)Bn+1 fn (0, 0) ed (κ, k) n=−∞
(3.4g)
≤ K+
κ−1−K
−1 fn (φ− eai (k, n + 1) Bn+1 (n)) − f (0, 0) ed (κ, k) n κ
n=−∞
+K+
κ−1−K
−1 eai (k, n + 1) Bn+1 fn (0, 0) ed (κ, k)
n=−∞ (4.8b)
≤ K+ L(c)
κ−1−K
eai (k, n + 1) φ− κ (n) ed (κ, k)
n=−∞
+K+ Γκ− (i)
κ−1−K
eai (k, n + 1)ebi (n, κ)ed (κ, k)
n=−∞
− κ−1−K ≤ K+ L(c) φ− eai (k, n + 1)ec (n, κ)ed (κ, k) κ κ,c n=−∞
+K+ Γκ− (i)
κ−1−K
eai (k, n + 1)ec (n, κ)ed (κ, k)
n=−∞
(A.1d)
≤
− K+ − L(c) φ− (κ, κ − K) for all k ∈ [κ − K, κ]Z κ κ,c + Γκ (i) e d c c − ai
and second, also using (4.8b) one has k−1 −1 − − Φ(k, n + 1)P+ (n)Bn+1 fn (φκ (n)) − fn (φκ,K (n)) ed (κ, k) n=κ−K (3.4g)
≤ K+ L(d)
k−1
− eai (k, n + 1) φ− (n) − φ (n) ed (κ, k) κ κ,K
n=κ−K
(A.1d)
≤
− K+ L(d) − φκ − φ− κ,K d − ai κ,d
for all k ∈ [κ − K, κ]Z ,
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4 Invariant Fiber Bundles
where the sums have been evaluated using Lemma A.1.5(a). Having these two estimates at hand, we can conclude − P+ (k) φ− κ (k) − φκ,K (k) ed (κ, k) ≤ κ−1−K −1 − Φ(k, n + 1)P+ (n)Bn+1 fn (φκ (n)) ed (κ, k) ≤ n=−∞ k−1 −1 − + Φ(k, n + 1)P+ (n)Bn+1 · fn (φ− κ (n)) − fn (φκ,K (n)) ed (κ, k) n=κ−K −
− K+ L(d) K+ − − L(c) φ− (κ, κ − K) + − φ− ≤ φ κ κ,c + Γκ (i) e d κ κ,K c c − ai d − ai κ,d for all k ∈ [κ − K, κ]Z , and similarly by (3.4g) and (4.8b) we get − K− L(d) − − φκ − φ− P− (k) φ− κ (k) − φκ,K (k) ed (κ, k) ≤ κ,K bi − d κ,d −
for all k ∈ [κ − K, κ]Z . By definition of the · κ,d -norm and due to the inclusion c, d ∈ Γ¯i , we arrive at −
− − K+ − − L(c) φ ≤ + Γ (i) e d (κ, κ − K) φκ,K − φ− κ κ κ,c κ c ς κ,d − L(d) − max {K− , K+ } φ− + κ − φκ,K ς κ,d and consequently (note the inequality L(d) max {K− , K+ } < ς), − P+ (k) φ− κ (k) − φκ,K (k) ed (κ, k)
− K+ − L(c) φ− ≤ κ κ,c + Γκ (i) e dc (κ, κ − K) ς − L(d) max {K− , K+ } for all k ∈ [κ − K, κ]Z . Therefore, the claim follows from Lemma 4.2.8, if we use (4.2g), (4.8f) and set k = κ, d = ai + ς, c = bi − ς in the above estimate. Having these error estimates at hand, we are in a position to solve the truncated ± ± ) instead of (LP∞ ) for some fixed length K > 0 and fixed point equations (LPK ± an initial pair (κ, ξ) ∈ X . So we reduce the infinite-dimensional problem (LP∞ ) to ± a nonlinear algebraic equation Tκ,K (ψ, ξ) = ψ in k∈I± X . For Lipschitzian k κ (K)
×
± (ψn−1 , ξ) for nonlinearities this can be done using successive iterations ψn = Tκ,K ± n ∈ N and an arbitrary starting sequence ψ0 ∈ Xκ,c (K). Hence, a combination of the error estimates in Propositions 4.8.3(b) and 4.8.4 yields the following
4.8 Approximation of Invariant Fiber Bundles
285
Algorithm 4.8.6 (approximation of w± (κ, ξ)). Choose a desired accuracy ε > 0, (κ, ξ) ∈ P± , a value ς as in Proposition 4.8.3 and ϑ ∈ (0, 1). (1) Set n := 0, ψ0 := 0 ∈ X± κ,c (K) with an integer K > 0 so large that C(κ, ξ)e ai +ς (κ, κ − K) < ϑε. bi −ς
(2) Set q :=
max{K− ,K+ } L(bi ς
− ς) ∈ (0, 1) and choose n∗ ∈ N so large that
± K± q n∗ ± Tκ,K (ψ0 , ξ) < (1 − ϑ)ε. 1−q κ,c ± (3) For n = 1, . . . , n∗ compute ψn := Tκ,K (ψn−1 , ξ).
Remark 4.8.7. (1) By construction of this algorithm, the distance between the approximate invariant fiber bundle P∓ (κ)ψn∗ (κ) and w± (κ, ξ) satisfies ± w (κ, ξ) − P∓ (κ)ψn∗ (κ) < ε.
(4.8h)
For small values of the accuracy ε > 0, the constant K becomes large and therefore one has to iterate the truncated Lyapunov–Perron operator in a high-dimensional product space k∈I± Xk . This might make our approach numerically difficult. κ (K) Hence, we have introduced the parameter ϑ ∈ (0, 1) in order to balance between the iteration depth n∗ and the dimension of the problem. (2) A further possible strategy to pick the initial sequence ψ0 ∈ X± κ,c (K) is as follows: One considers the nonlinear problem (S) as perturbation of the linear equation (L0 ) and starts the iteration with the exact solution of the unperturbed equation. This offers one of the respective choices:
×
ψ0 (k) = Φ(k, κ)ξ
− − for all k ∈ I+ κ (K), ψ0 (k) = ΦP− (k, κ)ξ for all k ∈ Iκ (K).
When it comes to concrete implementations on a computer, further simplifications are advisable. First, in order to tackle (S) numerically, X has to consist of finite-dimensional spaces and an initial spatial discretization is indispensable. Sec− ond, multiplying the Lyapunov–Perron equation (LPK ) with projections P+ (k) and P− (k) implies ψ+ (k) =
k−1
−1 Φ(k, n + 1)P+ (n)Bn+1 fn (ψ+ (n) + ψ− (n)),
n=κ−K
ψ− (k) = Φ(k, κ)P− (κ)ξ −
κ−1 n=k
−1 Φ(k, n + 1)P− (n)Bn+1 fn (ψ+ (n) + ψ− (n)),
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4 Invariant Fiber Bundles
resp., where we abbreviated ψ± (k)=P± (k)φ− κ,K (k, ξ). In particular, we have the relation ψ− (κ) = P− (κ)ξ. The variation of constants formula from Theorem 3.1.16(b) guarantees that ψ− is a backward solution of the equation (k)), Bk+1 x = Ak P− (k)x + P− (k)fk (x + ψ+ (k), x + ψ+ − ) to the following system of nonlinear equations and we simplified (LPK
ψ+ (k) =
k−1
−1 Φ(k, n + 1)P+ (n)Bn+1 fn (ψ+ (n) + ψ− (n)) = 0
n=κ−K (k) Bk+1 ψ−
for all k ∈ [κ − K, κ]Z , = Ak ψ− (k) +
P− (k)fk (ψ+ (k)
(4.8i) + ψ− (k)) = 0
for all k ∈ [κ − K, κ − 1]Z , ψ− (κ) = P− (κ)ξ. The first equation in (4.8i) degenerates into ψ+ (κ − K) = 0 for k = κ − K, which causes no confusion, since (4.8i) is used to obtain P± (k)φ− κ (k, ξ) only for k = κ. For the corresponding dual approximation method of the fiber bundle W+ , we set ψ± (k) = P± (k)φ+ κ,K (k, ξ), and using Theorem 3.1.16(a) the Lyapunov–Perron + equation (LPK ) reduces to ψ+ (κ) = P+ (κ)ξ, (k) = Ak ψ+ (k) + P+ (k)fk (ψ+ (k) + ψ− (k)) Bk+1 ψ+ for all k ∈ [κ, κ + K − 1]Z , ψ− (k) = −
κ+K−1
(4.8j)
−1 Φ(k, n + 1)P− (n)Bn+1 fn (ψ+ (n) + ψ− (n))
n=k
for all k ∈ [κ, κ + K]Z . Both the (4.8i) resp. (4.8j) are nonlinear systems of algebraic equations depending on the parameter (κ, ξ) ∈ P± . Hence, they can be solved using various methods from numerical analysis: −1 For merely Lipschitzian nonlinearities Bk+1 fk , successive iteration is practicable and yields linear convergence (cf., e.g., [465, p. 17, Theorem 1.A]). −1 • Newton methods lead to quadratic convergence, provided the mappings Bk+1 fk are of class C 2 . However, since the algebraic equations (4.8i) and (4.8j) are typically high-dimensional, we made better experiences using Quasi-Newton methods (cf. [384]) without an update of the Jacobian in every step.
•
4.9 Applications
287
4.9 Applications As before, our starting point is a discretization mesh (tk )k∈I satisfying hk := tk+1 − tk ∈ [ T, T ]
for all k ∈ I
and a given stepsize bound T > 0 and the balancing factor
∈ (0, 1].
4.9.1 Discretized Functional Differential Equations Let r > 0 and d ∈ N. This subsection deals with semilinear FDEs u(t) ˙ = L(t)ut + f (t, ut ),
(4.9a)
whose linear part L(t) : Cr → Rd , t ∈ R, fulfills the assumptions made in Sect. 3.7.1. For the nonlinearity we suppose Hypothesis 4.9.1. Let b, c ≥ 0 be reals and suppose that f : R × Cr → Rd is continuous and linearly bounded f (t, ψ) ≤ b + c |ψ|r for all t ∈ R, ψ ∈ Cr . Next we apply the full discretization scheme established in both Sects. 2.6.1 and 3.7.1 to (4.9a). Given θ ∈ [0, 1] and h = Nr , this immediately yields a semilinear difference equation x = Ak (θ, h)x + fk (x, x ) (4.9b) with the operators Ak (θ, h) ∈ L(Cr,N ) given in Sect. 3.7.1 and inducing an evo2 lution operator Φ(k, l) on Cr,N . Moreover, the nonlinearity fk : Cr,N → Cr,N is defined as in (2.6d). Since (4.9b) is an equation in S = I × Cr,N this leads us to Proposition 4.9.2. Let q ∈ [0, 1) and suppose that I is unbounded below. If beyond Hypothesis 4.9.1 the following holds for all k ∈ I : (i) The general forward solution of (4.9b) exists as a continuous mapping. (ii) There exist α ∈ (0, 1), K ≥ 1 such that |Φ(k, l)| ≤ Kαk−l for all l ≤ k. (iii) Khθc < 12 and one has the estimate α+2h(1−θ)Kc ∈ [0, q], 1−2hθKc then the semilinear equation (4.9b) possesses a uniformly bounded global attractor A∗ ⊆ I × Cr,N , which also satisfies A∗ (k) ⊆ B hKb (0, Cr,N ), k ∈ I. (1−q)(1−2hθKc)
Remark 4.9.3. (1) Proposition 2.6.1 yields conditions for assumption (i) to hold. (2) Since (4.9b) is uniformly bounded dissipative (see the proof below), we can prove the existence of an inertial fiber bundle along the lines of Theorem 4.7.3, provided there exists a gap in the dichotomy spectrum of (LΔE) and the function f : R × Cr → Rd is Lipschitzian in the second argument with sufficiently small constant. Caused by similar investigations in the following Sects. 4.9.4 and 4.9.5, we neglect the technical details.
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4 Invariant Fiber Bundles
Proof. Let the family Bˆ consists of all uniformly bounded nonautonomous sets ˆ in S. Since S consists of finite-dimensional spaces, (4.9b) is B-contracting. From Hypothesis 4.9.1 we see for all x, x ∈ Cr,N that
fk (x, x ) ≤ h(1 − θ)b + hθb + hc(1 − θ) x + hcθ x
≤ hb + 2h max {(1 − θ)c x , cθ x }
for all k ∈ I
and thus the condition (4.1c) holds. We conclude our assertion from Theorem 4.1.8, since the estimate (4.1a) in Hypothesis 4.1.1 was only required to obtain a continuous general forward solution. We conclude the subsection with an example illustrating that the assumptions of Theorem 4.1.8 do not enforce a trivial attractor for (S). Example 4.9.4 (discrete Krisztin–Walther equation). Let a ∈ (0, 1), h > 0, N ∈ Z+ 0 and suppose g : R → R is a strictly increasing odd C 1 -function fulfilling the limit relation limx→±∞ g (x) = 0. We consider a scalar delay difference equation (4.9c)
xk+1 = axk + hg(xk−N ),
which can be interpreted as an explicit Euler discretization of the Krisztin–Walther equation x(t) ˙ = −αx(t) + g(x(t − r)) (choose a = 1 − hα and h, N according to hN = r for positive delays r > 0 and hα ∈ (0, 1)). Above all, we suppose ∗ ∗ g (0) > 1−a h , which ensures that (4.9c) admits three equilibria −x < 0 < x . N 1−a Moreover, there exists a ξ > 0 such that hg (ξ) = a 2 and we infer N 1−a N 1−a h |g(x)| ≤ hg(ξ) − aN 1−a 2 ξ+a 2 |x| ≤ hg(ξ) + a 2 |x|
for all x ∈ R
(cf. Fig. 4.5). Following the procedure from Example 2.1.10, we write (4.9c) as an explicit autonomous difference equation x = Ax + f (x)
(4.9d) (1 − a)x
hg(x)
x∗
Fig. 4.5 Graph of the function g in Example 4.9.4
ξ
x
4.9 Applications
289
in the space RN +1 with ⎞
⎛
a0 ⎜1 0 ⎜ ⎜ A := ⎜ 1 0 ⎜ .. .. ⎝ . . 1
⎟ ⎟ ⎟ ⎟, ⎟ ⎠ 0
⎛ ⎞ g(xN +1 ) ⎜ ⎟ 0 ⎜ ⎟ f (x) := h ⎜ ⎟ .. ⎝ ⎠ . 0
and verify the assumptions of Theorem 4.1.8, where RN +1 is equipped with the max-norm. It is not difficult to show that An ≤ a−N an for all n ∈ Z+ 0 and consequently the evolution operator for x = Ax satisfies Φ(k, l) ≤ a−N ak−l for all l ≤ k. For the nonlinearity one deduces
f (x) ≤ h |g(x)| ≤ hg(ξ) + aN 1−a 2 |xN +1 |
for all x ∈ RN +1
and we apply Theorem 4.1.8 with K = a−N , β = hg(ξ) and γ = aN 1−a 2 . Due to a + Kγ = 1+a < 1 we infer that (4.9d) is uniformly bounded dissipative with 2 2h g(ξ). Hence, there exists a global attractor for an absorbing set of radius a−N 1−a (4.9d) and (4.9c), which is nontrivial, since it contains the three equilibria of (4.9c).
4.9.2 Time-Discretized Abstract Evolution Equations Assume here that X, Y are Banach spaces with X ⊆ Y . We consider a temporal discretization for abstract nonautonomous evolutionary equations (AE)
ut + B(t)u = f (t, u)
in the space Y as discussed at the end of Sect. 2.6.2. Being based on mild solutions, we suppose that Hypothesis 1.5.4 is satisfied throughout. As explained in Sect. 1.5.2, for each pair (t0 , u0 ) ∈ R × X there exists a unique mild solution u(·; t0 , u0 ) : [t0 , ∞) → X of (AE) and u : {(t, s, x) ∈ R2 × X : s ≤ t} → X is continuous. The linear part of (AE) yields an evolution family (U (t, s))s≤t on X. Motivated by Sect. 2.6.2 let us investigate the explicit equation x = Ak x + fk (x)
(AΔE)
in X = I × X with mappings Ak := U (tk+1 , tk ) ∈ L(X) and a nonlinearity 4 fk : X → X, fk (x) :=
tk+1
U (tk+1 , s)f (s, u(s; tk , x)) ds
for all k ∈ I .
tk
Working with semilinear equations, we first apply results from Sect. 4.1 to (AΔE).
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4 Invariant Fiber Bundles
Lemma 4.9.5. If Hypothesis 1.5.4 holds, then the function fk : X → X is continuous and fulfills the linear growth bound (4.1c) with δ = 0, Kb βk := 1−r
KcT 1−r K 2c 1+ E1−r (μT ) h1−r E1−r (μT )h1−r k , γk := k 1−r 1−r
for all k ∈ I , where μ > 0 is the constant from Lemma 1.5.5(a). Proof. The proof is based on Lemma 1.5.5(a) and a direct estimate for fk : X → X using the linear growth of f : R × X → Y . Details are left to the reader. Proposition 4.9.6. Let q ∈ [0, 1) and suppose that I is unbounded below. If beyond Hypothesis 1.5.4 with ω < 0 also (i) for every bounded S ⊆ X one has lims→∞ χ(u(t; t − s, S)) = 0, t ∈ R, (ii) the real c ≥ 0 is so small that eωhk + Kγk ∈ (0, q] for all k ∈ I hold, then the semilinear difference equation (AΔE) has a uniformly bounded global attractor A∗ ⊆ I × X, which additionally satisfies A∗ (k) ⊆ B K supk∈I βk (0, X) for all k ∈ I, (1−q)
where the sequences βk , γk ≥ 0 are defined in Lemma 4.9.5. Proof. We will successively verify the assumptions of Theorem 4.1.8. Above all, our Corollary 3.7.6 implies that (4.1b) holds with a(k) = eωhk and Lemma 4.9.5 guarantees the linear growth bound (4.1c), where the sequence (βk )k∈I is clearly ˆ bounded. Thanks to (1.5e) and Lemma 1.5.5(a) we also see that ϕ is B-contracting, where Bˆ is the family of all uniformly bounded nonautonomous sets in X . Corollary 4.9.7 (parabolic case). The assumption (i) in Proposition 4.9.6 can be replaced by U (t, s) ∈ L(X) is compact for all s < t. Proof. We show that the assumptions of Corollary 4.1.7 are fulfilled and Proposition 4.9.6 can be applied. First, Ak = U (tk+1 , tk ) ∈ L(X), k ∈ I , is compact and thus dar Ak = 0. Next we show that also fk : X → X is compact. For this purpose, if B ⊆ X is bounded, then Lemma 1.5.5(a) yields that u(t; s, ·) is a bounded mapping; more precisely, there exists a C(B) ≥ 0 (we suppress the dependence on T ) with u(t; s, x) X ≤ C(T ) for all x ∈ B and reals s ≤ t with t − s ≤ T . From the decomposition 4 fk (x) = U (tk+1 , tk − ε) 4 +
tk+1 −ε
U (tk+1 − ε, s)f (s, u(s; tk , x)) ds
tk tk+1
tk+1 −ε
U (tk+1 , s)f (s, u(s; tk , x)) ds
4.9 Applications
291
one has fk (B) ⊆ U (tk+1 , tk − ε)B1 + B2 , where B1 , B2 ⊆ X are defined by 4
tk+1 −ε
B1 := t
4 k
tk+1
B2 :=
tk+1 −ε
U (tk+1 − ε, s)f (s, u(s; tk , x)) ds ∈ X : x ∈ B ,
U (tk+1 , s)f (s, u(s; tk , x)) ds ∈ X : x ∈ B
and ε > 0 is arbitrary satisfying ε < hk . Similarly to the argument below, one can show that B1 ⊆ X is bounded. For the set B2 we deduce from 4 tk+1 U (tk+1 , s)f (s, u(s; tk , x)) ds tk+1 −ε X 4 tk+1 K(b + cC(B)) 1−r ε ≤ K(b+cC(B)) eω(tk+1 −s) (tk+1 −s)−r ds ≤ 1−r tk+1 −ε for all x ∈ B that diam B2 ≤ 2 K(b+cC(B)) ε1−r . We equip the Banach space X 1−r with a measure of noncompactness χ satisfying χ(B2 ) ≤ diam B2 (cf. (B.0b)) and χ(fk (B)) ≤ χ(U (tk+1 , tk − ε)B1 ) + χ(B2 ) = χ(B2 ) ≤ 2
K(b + cC(B)) 1−r ε . 1−r
In the one-sided limit ε ! 0 this implies χ(fk (B)) = 0 and thus dar fk = 0. ˆ Consequently, Corollary 4.1.7 yields that ϕ is B-contracting. Example 4.9.8 (sectorial evolutionary equation). A typical example where the above Corollary 4.9.7 can be applied, are sectorial evolutionary equation (SE), when B has a compact resolvent. Here, X is a fractional power space X = Y r . In the remaining subsection, we illustrate how a discrete equation (AΔE) can be used to construct integral manifolds of the nonautonomous differential equation (AE). To shorten our explanations, we restrict to the pseudo-stable situation. Theorem 4.9.9. Let I be unbounded above, suppose that Hypotheses 1.5.4 and 3.7.4 are satisfied with 4 L := sup (ρ), sup ρ>0 k∈I
tk+1
tk
αh k U (tk+1 , s)f (s, u(s; tk , 0)) ds < ∞ (4.9e) e X
√ 1−r and that max eωT , E1−r (μT ), bT1−r ≤ 2. If the spectral gap condition μT ) max eωT , eωT T 1−r β−α max {K + , K − } K 2 E1−r (¯ L< 1−r min { T eαT , T eαT } 2
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4 Invariant Fiber Bundles
holds and we choose a fixed
1−r ς ∈ max K + , K − K 2 LE1−r (¯ μT ) max eωT , eωT T1−r , inf
eβhk −eαhk 2 k∈I
,
then the nonautonomous set W + := (κ, ξ) ∈ X : ϕ(·; κ, ξ) ∈ X+ κ,c is a forward invariant fiber bundle of (AΔE), which is independent of c ∈ [a1 + ς, b1 − ς] and has the representation W + = (κ, η + w+ (κ, η)) ∈ X : η ∈ ker P¯ (tκ ) as graph of a unique mapping w+ : X → X, globally Lipschitzian in the second argument with w+ (κ, ξ) = w+ (κ, [I − P¯ (tκ )]ξ) ∈ im P¯ (tκ ) for all (κ, ξ) ∈ X . Here, one has μ ¯ := (KLΓ (1 − r))1/(1−r) , μ is from Lemma 1.5.5 and the growth rates a1 , b1 are defined in Lemma 3.7.5. Proof. We aim to verify the assumptions of the Hadamard–Perron Theorem 4.2.9(a). Above all, due to Lemma 3.7.5 the linear part of (AΔE) admits a strongly regular 2-splitting with a1 (k) = eαhk , b1 (k) := eβhk and constants K + , K − . Therefore, Hypothesis 4.2.1 is satisfied. In order to show that also Hypothesis 4.2.3 holds, we choose k ∈ I and x, x ¯ ∈ X. Then Lemma 1.5.5(b) guarantees
fk (x) − fk (¯ x)
4 tk+1 ≤ KL (tk+1 − s)−r eω(tk+1 −s) u(s; tk , x) − u(s; tk , x ¯) ds tk
≤ K 2 LE1−r (¯ μT )eωhk
4
tk+1
(tk+1 − s)−r ds x − x ¯
tk
for all k ∈ I and consequently fk : X → X satisfies a global Lipschitz condition 1−r (4.2a) with constant L1 = K 2 LE1−r (¯ μT ) max eωT , eωT T1−r . ad (Γ1+ ): Due to (4.9e) we have the growth condition. ad (G1 ): Since (AΔE) is an explicit equation, the gap condition simplifies to
1 : max K + , K − L1 < ς, ∃ς ∈ 0, b1 −a 2 which we establish as follows: Using the elementary mean value theorem for scalar functions one has b1 (k) − a1 (k) = eβhk − eαhk ≥ eαhk hk (β − α) ≥ min
T eαT , T eαT (β − α)
for all k ∈ I and consequently b1 − a1 ≥ min T eαT , T eαT (β − α). Thus, our assumptions imply that (G1 ) holds and Theorem 4.2.9(a) yields the assertion. For simplicity we suppose next that (AE) has the trivial solution: Hypothesis 4.9.10. Suppose that f (t, 0) ≡ 0 holds on R.
4.9 Applications
293
Corollary 4.9.11. Let γ := W+ =
α+β 2 .
If Hypothesis 4.9.10 holds, then
(τ, ξ) ∈ X : sup u(t; τ, ξ) X eγ(τ −t) < ∞ . τ ≤t
Proof. We define c(k) := eγhk and obtain c ∈ Γ¯1 . Abbreviating the right-hand side of the claimed set equality by WR+ , we need to show two inclusions: (⊆) Let (τ, x) ∈ WR+ with τ = tκ for some κ ∈ I. Then the relation (1.5e)
ϕ(k; κ, x) ec (κ, k) ≤ u(tk ; τ, x) eγ(tκ −tk ) ≤ sup u(t; τ, x) eγ(τ −t) < ∞ τ ≤t
+ for all k ∈ Z+ κ implies that ϕ(·; κ, x) is c -bounded and the dynamical characterization of W + yields the inclusion (τ, x) ∈ W + . (⊇) Conversely, let (κ, x) ∈ W + and so ϕ(·; κ, x) must be c+ -bounded. Thanks to Lemma 1.5.5(a) and the identity f (t, 0) ≡ 0 there exists a constant C(T ) ≥ 0 with u(t; s, u0 ) ≤ C(T ) u0 for all u0 ∈ X and s ≤ t with t − s ≤ T . We define τ = tκ and thus, given t ≥ τ for k ∈ I maximal with tk ≤ t, we obtain
u(t; τ, x) eγ(τ −t) = u(t; tk , u(tk , τ, x)) eγ(τ −t) ≤ C(T )eγ(tk −t) u(tk , τ, x) eγ(τ −tk ) ≤ C(T ) max eγT , eγT ϕ(k; κ, x) ec (κ, k) ≤ C(T ) max eγT , eγT ϕ(·; κ, x) + for all τ ≤ t. κ,c Therefore, we arrive at the inclusion (τ, x) ∈ WR+ .
4.9.3 Time-Discretized Parabolic Evolution Equations Our set-up is as follows: Let Ω ⊆ Rd be a bounded domain with Lipschitzian boundary. We consider a sectorial evolution equation ut + Bu = f (t, u),
(SE)
where the sectorial operator B on the Banach space Y = L2 (Ω) is a symmetric uniformly elliptic differential operator B as in Sect. 3.7.3; under Dirichlet boundary conditions we obtain the domain D(B) = H 2 (Ω)∩H01 (Ω) and the fractional power space X = Y 1/2 = H01 (Ω). Concerning the nonlinearity, we suppose Hypothesis 4.9.10 and that the mapping f : R × X → Y is m-times, m ∈ N, continuously Frech´et differentiable in the second argument such that the derivatives map bounded subsets of X into bounded sets – uniformly in t ∈ R.
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4 Invariant Fiber Bundles
If B has compact resolvent, then Example 4.9.8 and Corollary 4.9.7 apply. However, our focus is different and using two examples, we briefly illustrate how Theorem 4.6.4 can be employed in order to construct a pseudo-stable and -unstable hierarchy of invariant fiber bundles for temporal discretizations of (SE). Our first simplifying assumption is that A :≡ D2 f (t, 0) ∈ L(X, Y ) is independent of t ∈ R. We need this in order to obtain information on an exponential splitting for ut + (B − A)u = 0 on the basis of the spectrum σ(B − A) alone, instead of using roughness arguments (cf. [201, pp. 237–238, Theorem 7.6.10] or [432, p. 216ff]). As a second simplification we restrict to a constant stepsize T , linearly implicit Euler method applied to (SE), which yields a semi-implicit difference equation Bk x = Ak x + fk (x)
(S )
with Ak x := x, Bk := IY + T (B − A) and fk (x) := T [f (tk , x) − D2 f (tk , 0)x]. Instead of deriving a general corollary from Theorem 4.6.4, we rather focus on two examples where the above setting is applicable: Example 4.9.12 (Chafee–Infante equation). Given Ω = (−a, a), a, α1 > 0 and δ > 0, as before in Example 1.5.8 we consider a nonautonomous Chafee–Infante equation ut − δuxx = u(α1 − α2 (t)u2 ) subject to the boundary conditions u(−a) = u(a) = 0, with a continuous bounded reaction function α2 : R → (0, ∞) which is uniformly bounded away from 0. Using the above notation we obtain (B − A)u := −α1 u − δuxx and f (t, u) := −α2 (t)u3 . On the one hand, Example 3.7.7 implies the discrete spectrum −1 σ(Bk+1 Ak )
= {0} ∪
4a2 4a2 − T (4a2 α1 + δπ 2 n2 )
⊆R
for all k ∈ I ,
n∈N
whose only accumulation point is 0 and outside every neighborhood of 0 are only finitely many eigenvalues. Every interval (¯ αi , β¯i ) with reals 0 < α ¯ 1 < β¯i and −1 disjoint from σ(Bk+1 Ak ) yields an exponential dichotomy; thus, one employs Theorem 3.4.30 in order to deduce an exponential N -splitting for the linear part Bk+1 x = Ak x, whose pseudo-unstable vector bundles are finite-dimensional and independent of k; hence, Hypothesis 4.2.1 holds. On the other hand, H01 (−a, a) is a Hilbert space and thus a C ∞ -Banach space (cf. Proposition C.2.10). Following the reasoning of [432, p. 270ff] one shows that −1 −1 −1 Bk+1 fk : H01 (−a, a) → H01 (−a, a), Bk+1 fk (u) = T [IY + T (B − A)] u3 does 2 define a C -mapping satisfying Hypotheses 4.6.1 and 4.6.2 with m = 2. Thus, the linearly implicit Euler discretization (S ) of the above Chafee–Infante equation has C 1 -smooth pseudo-stable and unstable-hierarchies of invariant fiber bundles associated to the trivial solution. Provided the respective conditions (4.6c) or (4.6f) hold for m = 2 also, one obtains fiber bundles of class C 2 .
4.9 Applications
295
Example 4.9.13 (scalar Ginzburg–Landau equation). Let Ω ⊆ Rd , d ∈ {1, 2}, be a bounded domain as above. We consider a nonautonomous complex Ginzburg– Landau equation with cubic nonlinearity 2
ut − μ1 u − (1 + iν)Δu + (1 + iμ2 (t)) |u| u = 0
in (t0 , ∞) × Ω
(GL)
under Dirichlet or periodic boundary conditions (cf. [86, p. 118]), and we assume ν, μ1 ∈ R and that μ2 : R → R is bounded and continuous. In the present setting it is (B − A)u = −μ1 u − (1 + iν)Δu and f (t, u) = −(1 + iμ2 (t))|u|2 u. If λn ≥ 0 denote the eigenvalues of the Laplacian Δ arranged in decreasing order of magnitude, we obtain the spectrum −1 σ(Bk+1 Ak )
= {0} ∪
1 1 − T [(1 + iν)λn + μ1 ]
for all k ∈ I , n∈N
which induces an exponential splitting for Bk+1 x = Ak x by virtue of Theo−1 rem 3.4.30, since every annulus in C centered in 0 and disjoint from σ(Bk+1 Ak ) yields an exponential dichotomy. As in Example 4.9.12 one deduces both pseudostable and pseudo-unstable hierarchies of invariant C 1 -fiber bundles for the linearly implicit Euler discretization of (GL).
4.9.4 Fully Discretized Reaction-Diffusion Equations Let I be a discrete interval unbounded below. In this subsection, we continue our investigations begun in Sect. 2.6.3 on scalar nonautonomous parabolic initialboundary value problems ut − δ(t)Δu = f (t, x, u) u(t, x) = 0 u(t0 , x) = u0 (x)
for t > t0 , x ∈ Ω, for t ≥ t0 , x ∈ bd Ω, for x ∈ Ω
(RDE)
equipped with homogeneous Dirichlet boundary conditions. The corresponding spatial discretization (2.6g) is supposed to fulfill Hypothesis 2.6.6 and for the sake of a full discretization we use the θ-method M
v − v + δ(tθk )A [(1 − θ)v + θv ] = M F (tθk , (1 − θ)v + θv ) hk
(4.9f)
with tθk := (1 − θ)tk + θtk+1 , k ∈ I , as in (2.6j). Under appropriate stepsize restrictions, we know from Proposition 2.6.11(b) that the general forward solution ϕ to (4.9f) exists as a C m -function. Moreover, by Proposition 2.6.12 it is uniformly
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4 Invariant Fiber Bundles
bounded dissipative (if θ ∈ ( 12 , 1]) and Proposition 2.6.14 yields a global attractor, which is uniformly bounded in the discrete Lebesgue space L2N . Additionally to Hypothesis 2.6.6 we impose Hypothesis 4.9.14. Suppose that we have: (i) σ(A, M ) = {λ1 , . . . , λN } with 0 ≤ λn < λn+1 for all 1 ≤ n < N , where φn ∈ RN are the eigenvectors corresponding to λn forming an orthonormal basis of L2N . (ii) supt∈R F (t, 0) < ∞ and for every r ≥ 0 there exists a real L(r) ≥ 0 with sup
¯ r (0,L2 ) (t,u)∈R×B N
D2 F (t, u) L(L2 ) ≤ L(r). N
Remark 4.9.15. Explicit eigenvalues λn , 1 ≤ n ≤ N , and eigenvectors φn for various spatial discretizations schemes have been given in Sect. 3.7.4. For the purpose of this section it is offered to write (4.9f) as semilinear equation Bk+1 x = Ak x + fk (x, x )
(S)
in L2N , with the mappings Ak := IL2N − (1 − θ)hk δ(tθk )M −1 A,
Bk+1 := IL2N + θhk δ(tθk )M −1 A
and the nonlinearity fk (x, x ) := hk F (tθk , (1 − θ)x + θx ) for all k ∈ I . −1 Ak ∈ RN ×N , k ∈ I , Lemma 4.9.16. If Hypothesis 4.9.14 holds, then Bk+1 , Bk+1 are invertible and with complementary orthogonal projections P1n , Qn1 ∈ RN ×N ,
Qn1 x :=
N
"x, φj # φj ,
P1n x := x − Pn x,
j=n+1
one has the following properties for all integers k ∈ I , n = 1, . . . , N − 1: −1 −1 Ak P1n = P1n Bk+1 Ak , (a) Bk+1 −1 1−(1−θ)hk δ(tθk )λn+1 (b) Bk+1 Ak Qn1 L(L2 ) ≤ 1+θh δ(t , θ )λ n+1 k k N −1 θ 1−(1−θ)hk δ(tk )λn (c) B Ak P1n 2 ≥ , θ k+1
L(LN )
1+θhk δ(tk )λn
where φ1 , . . . , φN ∈ L2N are the orthonormal eigenvectors from Hypothesis 4.9.14. Proof. Let k ∈ I be fixed, choose n ∈ [1, N )Z and define the strictly decreasing function ψ : [0, ∞) → (0, 1], ψ(x) := 1−(1−θ)x 1+θx . Using the spectral mapping theorem (cf., e.g., [96, p. 204]) one derives the explicit relations σ(Bk+1 ) = 1 + θhk δ(tθk )λj > 0 : j = 1, . . . , N , −1 σ(Bk+1 Ak ) = {υj (k) ∈ R : j = 1, . . . , N }
4.9 Applications
297
with eigenvalues υj (k) := ψ(hk δ(tθk )λj ), which are strictly decreasing in j. In −1 particular, both Bk+1 , Bk+1 Ak are invertible. For an arbitrary x ∈ RN with repreN sentation x = j=1 xj φj and xj = "x, φj # we get claim (a) from −1 Bk+1 Ak P1n x =
n
−1 xj Bk+1 Ak φj =
j=1 −1 = P1n Bk+1 Ak
n
υj (k)xj φj = P1n
j=1 N
N
xj υj (k)φj
j=1
−1 xj φj = P1n Bk+1 Ak x
for all k ∈ I .
j=1
As in the proof of Theorem 3.4.30 one shows the estimates in (b) and (c).
¯r (0) Lemma 4.9.17. If Hypothesis 4.9.14 holds, then for every r > 0 and u, v ∈ B N N N the nonlinearity fk : R × R → R satisfies for all k ∈ I and θ ∈ [0, 1], lip1 fk |B¯r (0,L2N )×B¯r (0,L2N ) ≤ (1 − θ)hk L(r), lip2 fk |B¯r (0,L2N )×B¯r (0,L2N ) ≤ θhk L(r). ¯r (0). Referring to the convexity of the ¯1 , x ¯2 ∈ B Proof. Let r > 0 and x1 , x2 , x 2 ¯ LN -ball Br (0) also the convex combinations xθi := (1 − θ)xi + θxi are contained ¯r (0), as well as xθ + h(xθ − xθ ) ∈ B ¯r (0) for θ, h ∈ [0, 1]. With this, the mean in B 1 2 1 value theorem (see [295, p. 341, Theorem 4.2]) implies
fk (x1 , x1 ) − fk (x2 , x1 ) = hk F (tθk , xθ1 ) − F (tθk , xθ1 ) 4 1 θ θ θ θ D2 F (tk , x1 + h(x2 − x1 )) dh ≤ hk (1 − θ) x1 − x2
0
for all k ∈ I and thus the first Lipschitz estimate follows. Analogously we show the second claimed inequality. Theorem 4.9.18 (fully discretized RDEs). Let I = Z. Suppose that Hypothe ses 2.6.6 and 4.9.14 hold true, choose ω ∈ (0, 1), θ ∈ 12 , 1 and ρ > ρ0 , where ρ0 is the radius of the absorbing ball from Remark 2.6.13. If there exists an n ∈ [1, N )Z with 2L(ρ) < λn+1 − λn (4.9g) ω inf t∈R δ(t) and if the stepsizes hk satisfy the conditions (2.6k), θL(ρ)T < 1, [(1 − θ) + θbn (k)] L(ρ) < bn (k),
0 < inf hk (b + δ(tθk )λ1 ), k∈Z
θhk δ(tθk )λn+1 <
1−ω ω
for all k ∈ Z,
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4 Invariant Fiber Bundles
then the following holds: (a) The full discretization (4.9f) of (RDE) has an n-dimensional inertial fiber bun¯ρ (0, L2 ) as in Theorem 4.7.3. dle W ⊆ I × B N (b) There exists a unique global attractor A∗ for (4.9f) with A∗ ⊆ W. Proof. Let θ ∈ 12 , 1 , choose ρ > ρ0 as required above and let Bˆ be the family of all uniformly bounded subsets of Z × L2N . We have to verify successively the assumptions of Theorem 4.7.3. ad Hypothesis 4.2.1: We deduce from Lemma 4.9.16 that the linear part of (4.9f) resp. (S) admits a strongly regular N -splitting. More precisely, for each n ∈ [1, N )Z we deduce an exponential dichotomy on Z with constant projectors P1n , constants Kn± = 1 and growth rates an (k) := ψ(hk , δ(tθk )λn+1 ),
bn (k) := ψ(hk , δ(tθk )λn )
with the strictly decreasing function ψ : [0, ∞) → (0, 1] already defined in the proof of Lemma 4.9.16. We get an bn and Lemma 4.9.16 yields the estimates
Φ(k, l)Qn1 L(L2 ) ≤
k−1 (
N
−1 B Aj Qn1 2 ≤ ean (k, l) for all l ≤ k j+1 L(L ) N
j=l
and also Φ(k, l)P1n L(L2 ) ≤ ebn (k, l) for all k ≤ l. N ad Hypothesis 4.7.1: We have established in Proposition 2.6.11(b) that the general forward solution of (4.9f) exists as a C m -function. Moreover, from Lemma 4.9.17 we deduce the local Lipschitz constants L1 (r) := (1 − θ)T L(r),
L2 (r) := θT L(r).
ad Hypothesis 4.7.2: From Proposition 2.6.12 and Remark 2.6.13 we know ¯ ρ (0, L2 ) is a B-uniformly ˆ that Z × B absorbing set. Since L2N is a Hilbert N ∗ space, the Lipschitz constant of the associated radial retraction is lim rL = 1 2 N (cf. Lemma C.2.1). Thus, our stepsize assumptions guarantee that (4.7b) and (4.7c) are fulfilled. It remains to verify both the growth and the spectral gap condition. ad (Γκ− (n)): Thanks to Hypothesis 4.9.14(ii) and bn (k) ≤ 1 we obtain that the growth condition holds. ˆ n ): In the present setting the spectral gap condition simplifies to the relation ad (G (cf. Lemma 4.9.17)
2[(1 − θ) + b θ]T L(ρ) n n < ς. (4.9h) ∃ς ∈ 0, bn −a : 2 1 + 2T θL(ρ) In order to verify (4.9h), we abbreviate α := hk δ(tθk λn+1 ), β := hk δ(tθk λn ) and observe that our stepsize restrictions imply (1 + θα)−1 , (1 + θβ)−1 ≥ ω. Hence, bn (k) − an (k) = ψ(β) − ψ(α) =
α−β ≥ ω 2 hk δ(tθk )(λn+1 − λn ) (1 + θβ)(1 + θα)
4.9 Applications
299
for all k ∈ Z, and on the other hand, due to bn ≤ 1 we have 2[(1 − θ) + bn θ]T L(ρ) ≤ 2[(1 − θ) + bn θ]T L(ρ) ≤ 2L(ρ)T. 1 + 2T θL(ρ) Consequently, our assumption (4.9g) ensures that (4.9h) holds true. (a) Since we have verified all assumptions of Theorem 4.7.3 we know that the implicit difference equation (S) in Z × L2N admits an inertial fiber bundle W. Every fiber W(k), k ∈ Z, is a graph over P1n L2N and by definition of the projector P1n in Lemma 4.9.16 can conclude dim W = dim P1n L2N = n. (b) By Proposition 2.6.14 there exists a of unique global attractor A∗ ⊆ Z × Bρ (0, L2N ) and due to bn 1 we can apply Corollary 4.7.5 yielding A∗ ⊆ W. Our final comprehensive example in this subsection illustrates how to approximate an inertial manifold of a scalar nonautonomous RDE. For this, we apply the algorithm from Sect. 4.8 to a finite-dimensional difference equation, which has been obtained from the original evolutionary PDE by a spectral Galerkin method for spatial, and a linearly implicit Euler scheme for temporal discretization. Error estimates for such full discretizations have been obtained in [121, 236]. As special case of (RDE) we now study the following nonautonomous problem ut − uxx = f (t, u),
(4.9i)
subject to homogeneous Dirichlet boundary conditions u(t, 0) = u(t, π) = 0 and an initial condition u(τ, x) = u0 (x) for given data τ ∈ R, u0 ∈ L2 (0, π). This problem fits into the framework of Sect. 1.5.3 with N = d = 1, Ω = (0, π), if f : R × R → R is continuous, the partial derivative D22 f : R × R → R exists as a continuous mapping and that there exist reals C1 , C2 , C3 , γ > 0, p ≥ 2 such that p
p
p
f (t, v)v ≤ C1 − γ |v| , |f (t, v)| p−1 ≤ C2 (1 + |v| ), D2 f (t, v) ≤ C3 (4.9j) for all t, v ∈ R (cf. Hypothesis 1.5.6). Moreover, choose K1 , K2 : [0, ∞) → R such that i D2 f (t, v) Ki (r) ≥ sup sup √ t∈R |v|≤ πr
for all i = 1, 2, r ≥ 0.
In Sect. 1.5.3 we have seen that (4.9i) generates a dissipative 2-parameter semiflow on the space L2 (0, π). On the other hand, following [432, Sect. 5.1], we can formulate (4.9i) as abstract nonautonomous evolutionary equation u˙ + Bu = g(t, u)
(4.9k)
with linear part B := −Dxx and substitution operator g(t, u)(x) := f (t, u(x)). Referring to [432, p. 272, Theorem 51.1], the mild solutions of (4.9k) generate a
300
4 Invariant Fiber Bundles
dissipative 2-parameter semiflow on H01 (0, π), and in [86, p. 290, Proposition 3.5] it is shown that the radius of the associate absorbing set in H01 (0, π) is bounded by r0 := 2 2C1 C3 . Thanks to Example 3.7.7 the eigenvalues of B equipped with zero boundary conditions u(0) = u(π) = 0'are λn = n2 , n ∈ N, with pair-wise L2 -orthonormal
2 2 2 eigenfunctions φn (x) = π sin(nx) for n ∈ N. Let Pi : L (0, π) → L (0, π) be the orthogonal projection onto the i-dimensional space span {φ1 , . . . , φi } and Qi := I − Pi be the complementary projector. Under our above assumptions and provided i ∈ N satisfies
i>
√ 2L(r0 ) − 1/2,
L(r) :=
2K1 (r)2 + r2 K2 (r)2 ,
(4.9l)
the RDE (4.9i) has an i-dimensional inertial manifold WR− = (τ, ξ + wR− (τ, ξ)) ∈ R × H01 (0, π) : ξ ∈ im Pi with a smooth function wR− : R × im Pi → im Qi (cf. [384, Proposition 4]). Now we describe our discretization strategy for (4.9i). First, the spatial approximation with N Fourier modes, N ≥ 1, is obtained by inserting the ansatz u(t, x) =
N
vi (t)φi (x)
i=1
into (4.9i) and taking the L2 -inner product with φj , j ∈ [1, N ]Z , leads to an initial value problem in im PN . We canonically identify this linear space with RN and arrive at the N -dimensional ODE v˙ j = −j 2 vj + fj (t, v)
for all j ∈ [1, N ]Z
(4.9m)
with the nonlinearities fj : R × RN → R, 4 fj (t, v) =
0
π
N f t, vi (t)φi (x) φj (x) dx
(4.9n)
i=1
5π and initial condition v(τ ) = η, ηj = 0 u0 (x)φj (x)dx. Respecting the stiffness of the matrix − diag(j 2 )N j=1 , we apply a linearly implicit Euler discretization (with stepsize T > 0) to (4.9m) and arrive at the nonautonomous difference equation v = AT v + FT (k, v)
(4.9o)
4.9 Applications
301
N with linear part AT := diag 1+T1 j 2 j=1 and a nonlinearity FT : Z × RN → RN , whose components are given by FT (k, v)j :=
T fj (τ + T k, v) for all j ∈ [1, N ]Z . 1 + T j2
Henceforth, we deduce the existence of an attractive invariant fiber bundle for the difference equation (4.9o). Choosing an integer i according to (4.9l), the linear part of (4.9o) satisfies Hypothesis 4.2.1 with constants Ki± = 1, growth rates ai (k) :≡
1 , 1 + T (i + 1)2
bi (k) :≡
1 1 + T i2
on Z
and projectors P1i = diag(1, . . . , 1, 0, . . . , 0). Moreover, one can verify Hypothesis 4.7.1 and we employ the methods from Sect. 4.8 to approximate the invariant fiber bundle − − WT,N = (k, ξ + wT,N (tk , ξ)) ∈ Z × RN : k ∈ Z, ξ ∈ im P1i of the discretization (4.9o). An error estimate relating the inertial manifold WR− of the full reaction-diffusion equation (4.9i) to the finite-dimensional invariant fiber − bundles WT,N , can be found in [236, Theorem 5.3] and is of the form − ¯ 1 λN T + K ¯2 wT,N (k, ξ) − wR− (tk , ξ) ≤ K
% λi+1 λN +1
(4.9p)
¯ 1, K ¯ 2 > 0, sufficiently large N and small T (cf. also [121]). with constants K Example 4.9.19 (Chafee–Infante equation). We intent to compute fibers of the non− autonomous set WT,N . As in Example 1.5.8, we retreat to a Chafee–Infante equation with time-dependent coefficients ut − uxx = u(α1 (t) − α2 (t)u2 ),
(4.9q)
under the above initial-boundary conditions. For continuous bounded functions α1 , α2 : R → (0, ∞) we know from Example 1.5.8 that (4.9q) fulfills Hypothesis 1.5.6 with C1 :=
α1 (t)2 1 sup , 2 t∈R α2 (t)
C3 := sup α1 (t) < ∞. t∈R
Furthermore, we can choose the functions K1 , K2 : [0, ∞) → R as √ K1 (r) := max C3 , 3πr2 sup α2 (t) − inf α1 (t) , K2 (r) := 6 πr sup α2 (t). t∈R
t∈R
t∈R
302
4 Invariant Fiber Bundles
In the next step we compute a Galerkin approximation for (4.9q). Unfortu¯ 1, K ¯ 2 > 0 in the mentioned error estimate (4.9p) nately, the constants K from [236, Theorem 5.3] are not immediately accessible. For this reason, we heuristically choose a spatial approximation of order N = 6. With help of some computer algebra to evaluate the integrals (4.9n), the resulting nonlinearities f1 , . . . , f6 read as follows: f1 (t, v) =
f2 (t, v) =
α2 (t) − 6v2 v3 v4 + 3v22 v5 − 6v1 v52 − 6v1 v62 − 6v2 v4 v5 − 6v1 v32 2π − 3v32 v5 + 6v1 v3 v5 − 3v22 v3 + 6v2 v3 v6 + 6v1 v4 v6 − 6v2 v5 v6 − 6v1 v42 − 6v1 v22 + 6v1 v2 v4 − 3v13 − 6v3 v4 v6 + 3v12 v3 +α1 (t)v1 , α2 (t) − 6v3 v4 v5 − 6v1 v2 v3 + 3v12 v4 + 6v1 v3 v6 − 6v1 v4 v5 − 3v23 2π − 6v12 v2 − 6v1 v5 v6 − 3v42 v6 − 6v1 v3 v4 − 3v32 v4 − 6v2 v42 + 3v22 v6 − 6v2 v62 − 6v3 v5 v6 − 6v2 v52 + 6v1 v2 v5 − 6v2 v32 +α1 (t)v2 ,
f3 (t, v) =
α2 (t) 3 v1 − 3v33 − 6v2 v5 v6 − 3v1 v22 − 6v1 v3 v5 − 6v1 v2 v4 2π + 6v1 v2 v6 − 6v1 v4 v6 − 6v2 v4 v5 − 6v4 v5 v6 − 6v2 v3 v4 − 6v3 v42 + 3v12 v5 − 6v3 v52 − 6v3 v62 − 6v12 v3 − 3v42 v5 − 6v22 v3 +α1 (t)v3 ,
f4 (t, v) =
f5 (t, v) =
f6 (t, v) =
α2 (t) − 6v3 v4 v5 − 3v2 v32 − 6v4 v52 − 3v43 − 6v22 v4 − 6v2 v4 v6 2π + 3v12 v6 − 6v32 v4 − 6v1 v3 v6 + 3v12 v2 − 6v1 v2 v3 − 6v3 v5 v6 − 6v1 v2 v5 − 6v2 v3 v5 − 6v12 v4 − 3v52 v6 − 6v4 v62 + α1 (t)v4 , α2 (t) − 6v1 v2 v6 − 6v4 v5 v6 − 3v3 v42 − 6v42 v5 − 3v1 v32 + 3v12 v3 2π − 6v1 v2 v4 − 6v3 v4 v6 + 3v1 v22 − 6v22 v5 − 6v2 v3 v6 − 6v5 v62 − 6v2 v3 v4 − 6v12 v5 − 3v53 − 6v32 v5 + α1 (t)v5 , α2 (t) − 6v1 v3 v4 − 3v63 − 3v4 v52 + 6v1 v2 v3 − 3v2 v42 − 6v22 v6 + v23 2π + 3v12 v4 − 6v52 v6 − 6v1 v2 v5 − 6v12 v6 − 6v32 v6 − 6v42 v6 − 6v3 v4 v5 − 6v2 v3 v5 + α1 (t)v6 .
To perform actual computations, we choose α2 constant and define α1 : R → R by α1 (t) := α2
π 2
+ sin t .
4.9 Applications
303
Hence, the radius of the absorbing set for the RDE (4.9q) is bounded above by r0 = 2π 5/2 α2 . Consequently, (4.9q) admits a nonautonomous inertial manifold WR− , whose dimension is the minimal integer i ≥ 0 satisfying ' 2 i > 2πα2 max {12π 3 α22 , 1} + 288π 5 α42 − 1/2.
0.1
0.1
0.05
0.05
wT−,6 (−10,ξ)
wT−,6 (−31,ξ)
471 and the evolutionary equation (4.9i) We are fixing the parameter value α2 = 5000 admits a two-dimensional inertial manifold, i.e., we can choose d = 2 and also − obtain a two-dimensional invariant fiber bundle WT,6 for the spectral Galerkin Euler − discretization (4.9o). In particular, this nonautonomous set WT,6 is given as graph − 2 4 of a function wT,6 : Z × R → R . − We used Algorithm 4.8.6 to approximate wT,6 over the square [−1, 1] × [−1, 1] with a uniform grid of 21 × 21 points, for an Euler stepsize T = 0.1, a truncation length K = 15 and accuracy ε = 10−5 . Yet, the nonlinear equations (4.8i) have been solved numerically by the inexact Newton method K NSoli1 (see [384] for details). The results of this computation are visualized: Fig. 4.6 depicts how the − second component wT,6 (k, ξ)2 changes under varying fibers, while Fig. 4.7 shows all four components of wT,6 (k, ξ) at the fixed instant k = 20.
0
−0.05 −0.1 1
0.5
0
−1 −1
−0.5
0.5
−0.1 1
1
0
−0.5 ξ2
0.1
0.1
0.05
0.05
0
−0.05 −0.1 1
0.5
ξ1
wT−,6 (7,ξ)
wT−,6 (0,ξ)
−0.5 ξ2
0
0 −0.05
−1 −1
−0.5
0
0.5
1
ξ1
0
−0.05
0.5
0
−0.5 ξ2
−1 −1
−0.5
0
0.5
1
−0.1 1
ξ1
− Fig. 4.6 Graphs of wT,6 (k, ξ)2 for k ∈ {−31, −10, 0, 7}
0.5
0 ξ2
−0.5
−1 −1
−0.5
0 ξ1
0.5
1
4 Invariant Fiber Bundles
0.1
0.1
0.05
0.05
wT−,6 (20,ξ)2
w− (20,ξ)1 T ,6
304
0
−0.05
−0.05
−0.1 1
0.5
0
−0.5
ξ2
−1 −1
−0.5
0
0.5
− 0.1 1
1
0
−0.5 ξ2 −1 −1
0.2
0.4
0.1
0.2
0
−0.5
0
0.5
1
ξ1
0
− 0.2
−0.1
−0.2 1
0.5
ξ1
wT−,6 (20,ξ)4
wT−, 6 (20,ξ)3
0
0.5
0 ξ2
− 0.5
−1 −1
− 0.5
0
0.5
1
− 0.4 1
0.5
0 ξ2
ξ1
−0.5
−1 −1
−0.5
0
0.5
1
ξ1
− Fig. 4.7 Graphs of wT,6 (20, ξ)i for i ∈ {1, 2, 3, 4}
4.9.5 Fully Discretized Finite Difference Ginzburg–Landau Equation Let I be a discrete interval unbounded below. We now return to the Ginzburg– Landau equation (GL) previously considered in Example 1.5.9 and Sect. 2.6.4, and rely on the notation introduced there. As full discretization of (GL) we investigated the implicit difference equation x − x ˜h x = F (tk+1 , x ) +Δ hk in I × CN , but differing from Sect. 2.6.4 where it was advantageous to consider it as one-step method, we now write it in the form x = Ak x + fk (x )
(ΔGL)
known from (S) with abbreviations −1 −1 ˜h Ak := ICN + hk Δ˜h , fk (x ) := hk ICN + hk Δ F (tk+1 , x ) for all k ∈ I . On basis and terminology of Lemma 3.7.11 we obtain
4.9 Applications
305
Lemma 4.9.20. The matrices Ak ∈ CN ×N , k ∈ I , are invertible and with complementary orthogonal projections P1n , Qn1 ∈ CN ×N , N −(n+1)
Qn1 x :=
"x, φj # φj ,
P1n x := x − Pn x,
j=n+1
one has the following properties for all integers k ∈ I , n = 1, . . . ,
6 N −2 7 : 2
(a) Ak P1n = P1n Ak , −1 (b) Ak Qn1 L(H 1 ) ≤ |1 + (1 + iν)hk (1 + νn+1 )| , N
−1
(c) Ak P1n L(H 1 ) ≥ |1 + (1 + iν)hk (1 + νn )| N
,
where φ1 , . . . , φN ∈ CN are the orthonormal eigenvectors from Lemma 3.7.11. 7 6 Proof. Let k ∈ I and n be an integer with 1 ≤ n ≤ N2−2 . Referring to the spectral mapping theorem (cf., e.g., [96, p. 204]) one derives the explicit relation σ(Ak ) = {υj (k) ∈ C : j = 1, . . . , N } with eigenvalues υj (k) := [1 + hk (1 + iν)(1 + νj )]−1 and consequently 0 ∈ σ(Ak ). In conclusion, Ak ∈ CN ×N is an invertible matrix. For later use we introduce the discrete intervals I+ {n + 1, . . . , N − (n + 1)}, n := N + N := {1, . . . , N } \ I and choose x ∈ C with x = I− n n j=1 xj φj and xj = "x, φj #. We get claim (a) from Ak P1n x =
xj Ak φj =
j∈I− n
υj (k)xj φj = P1n
j∈I− n
= P1n Ak
N
N
xj υj (k)φj
j=1
xj φj = P1n Ak x.
j=1
In addition, due to |υj (k)| ≤ |υn+1 (k)| for all j ∈ I+ n one has the forward estimate 2
Ak Qn1 x H 1 = N
2
2
2
2
(1 + νj ) |υj (k)| |xj | ≤ |υn+1 (k)| x H 1 , N
j∈I+ n
which implies (b), and claim (c) follows by the corresponding backward estimate
Ak P1n x 2H 1 = N
(1 + νj ) |υj (k)|2 |xj |2 ≥ |υn (k)|2 x 2H 1 ,
j∈I− n
since we have |υj (k)| ≥ |υn (k)| for all j ∈ I− n.
N
306
4 Invariant Fiber Bundles
¯r (0, H 1 ) the nonlinearity fk : CN → CN Lemma 4.9.21. For r > 0 and u, v ∈ B N satisfies fk (u) − fk (v) H 1 ≤ L(r)T u − v H 1 for all k ∈ I with N
N
' L(r) := 2(1 + R12 + ν 2 ) + 360(1 + R22 )r4 . 7 6 ¯r (0, H 1 ) and n ∈ 1, . . . , N −2 . Proof. Let r > 0, u, v ∈ B N 2 (I) We derive a Lipschitz condition for the function FN : CN → CN . Here, our approach is based on relation (3.7j). The mean value inequality (see [295, p. 342, Corollary 4.3]) leads to 2 2 2 |uj | uj − |vj | vj ≤ 2 sup |uj + t(vj − uj )| |uj − vj |
for all j = 1, . . . , N.
t∈[0,1]
√ From Lemma 3.7.13(b) we borrow the relation |uj | ≤ 3 u H 1 , j = 1, . . . , N , N (cf. (3.7n)) and arrive at 2 2 (4.9r) |uj | uj − |vj | vj ≤ 6r2 |uj − vj | for all j = 1, . . . , N ¯ r (0, H 1 ), which, in turn, equips us with the first L2 -estimate and u, v ∈ B N N 2
FN (u) − FN (v) L2 = N
N 2 1 2 2 2 |uj | uj − |vj | vj ≤ 36r4 u − v L2N (4.9s) N j=1
¯r (0, H 1 ). Moreover, for notational convenience we identify uN +1 for all u, v ∈ B N with u1 (and vN +1 with v1 ) to obtain 2 + 2 2 δh (|uj | uj ) − δh+ (|vj | vj ) 2 ≤ |uj+1 |2 uj+1 − |vj+1 |2 vj+1 + |uj |2 uj − |vj |2 vj
(4.9r) 2 2 2 ≤ 6r2 |uj+1 − vj+1 | + 6r2 |uj − vj | ≤ 72r4 |uj+1 − vj+1 | + |uj − vj | for all j = 1, . . . , N from the elementary inequality (x + y)2 ≤ 2x2 + 2y 2
for all x, y ∈ R.
(4.9t)
Therefore, with relation (3.7k) for the seminorm |·|Δh we get |FN (u) − FN (v)|Δh =
N 2 1 + δh (FN (u) − FN (v))j δh+ (FN (u) − FN (v))j N j=1
=
N 2 1 + 2 δh (FN (u) − FN (v))j ≤ 144r4 u − v L2 N N j=1
2
4.9 Applications
307
¯r (0, H 1 ) that and combining this with (4.9s) we obtain from (3.7j) for all u, v ∈ B N 2
2
FN (u) − FN (v) H 1 ≤ 180r4 u − v L2 N
N
(3.7l)
2
≤ 180r4 u − v H 1
N
(4.9u)
(II) Now we aim at a Lipschitz estimate for the full nonlinearity F resp. fk . By definition, adopting the notation from Lemma 4.9.20 and its proof one has
fk (u) − fk (v) 2H 1 ≤ T 2 N
N
(1 + νj ) |υj (k)|2 |"F (tk+1 , u) − F (tk+1 , v), φj #|2
j=1
and referring again to the basic inequality (4.9t) we proceed to
fk (u) −
fk (v) 2H 1 N
≤ 2T
2
1+
R12
+ν
2
N
(1 + νj ) |"u − v, φj #|2
j=1
+2T 2 1 + R22
N
2
(1 + νj ) |"FN (u) − FN (v), φj #|
j=1 (4.9u)
≤ 2T
2
2 (1 + R12 + ν 2 ) + 180(1 + R22 )r4 u − v H 1 . N
Taking the square root of this estimate yields the assertion.
After all these preparations we eventually arrive at Theorem 4.9.22 (fully discretized Ginzburg–Landau equation). Let I = Z. Choose reals ω ∈ (0, 1) and ρ > ρ1 , where ρ1 is the radius of the absorbing ball from (2.6p). If N ≥ 5 fulfills N 2 sin
π N
L(ρ) > √ 3 ω 3/2
and if the stepsize bound T ∈ (0, 1] is so small that L(ρ)T < 1, (3 + ν 2 )(1 + νn+1 )2 T < ω −1 − 1 8 9 with n := N 6−3 , then the following holds:
(4.9v)
(4.9w)
(a) The full finite difference discretization (ΔGL) of (GL) has a (2n + 1)-dimen¯ρ (0, H 1 ) as in Theorem 4.7.3. sional inertial fiber bundle W ⊆ I × B N (b) Under the stepsize restrictions (2.6m), (2.6q) and (2.6s) there exists a unique global attractor A∗ for (ΔGL) with A∗ ⊆ W, where the constant L(ρ) > 0 is defined in Lemma 4.9.21. Remark 4.9.23. The formulation of Theorem 4.9.22 (as well as of Theorem 4.9.18) is quantitative in the sense that the dimension of the inertial fiber bundle W can actually be computed for given values of ν and bounds R1 , R2 > 0 (Fig. 4.8). Related estimates for the continuous problem (GL) (and constant μ1 , μ2 ) are given in [122].
308
4 Invariant Fiber Bundles
60 50
60
40
50
30 20 10 7.0 6.5 6.0 5.5 5.0 4.5 4.0 3.5 10 −2
0.5 40 0.4 0.3 30 0.2 50 0.1 0.0
40
30
20
10
0
0.5 0.4 0.3 0.2 0.1 0.0
Fig. 4.8 Dimension of the inertial fiber bundle W from Theorem 4.9.22. Left: dim W over (R1 , R2 )-plane for ν = 1, R1 ∈ [0.035, 0.07], R2 ∈ [0, 0.5]. Right: dim W over (R1 , ν)-plane for parameters ν ∈ [0, 0.5], R1 = 0.07 and R2 = [0, 50]
Proof. Unfortunately, we cannot apply Theorem 4.7.3 directly, since the forward solutions of (ΔGL) need not to be unique (cf. Lemma 2.6.18). However, this problem can be circumvented as follows: Choose ρ > ρ1 as required above. We modify the nonlinearity of (ΔGL) as in the proof of Theorem 4.7.3 and directly employ Theorems 4.2.9 and 4.3.7 to equation x = Ak x + fkρ (x ).
(4.9x)
1 and the extended For this, we verify the corresponding assumptions with Yk = HN 1 state space X = Z× HN for an appropriate spatial discretization with N ≥ 5. Since 1 HN is a Hilbert space, we are in the scope of Remark 4.7.4(1) and radial retractions 1 on HN have Lipschitz constant 1. ad Hypothesis 4.2.1: Thanks6to Lemma 4.9.20 the linear part of (4.9x) possesses 7 a strongly regular exponential N2+1 -splitting. More precisely, for each positive integer n ≤ N2+1 we obtain an exponential dichotomy with constant projectors P1n , −1 constants Kn± = 1 and the growth rates an (k) := |1 + (1 + iν)hk (1 + λn+1 )| , bn (k) := |1 + (1 + iν)hk (1 + λn )|−1 ; from an elementary calculation we obtain an bn . Indeed, Lemma 4.9.20 implies the desired dichotomy estimates
Φ(k, l)Qn1 L(H 1 ) ≤
k−1 (
N
j=l
Aj Qn1 L(H 1 ) ≤ ean (k, l) for all l ≤ k N
and analogously Φ(k, l)P1n L(H 1 ) ≤ ebn (k, l) for all k ≤ l. N ad Hypothesis 4.3.1: Referring to Lemma 4.9.21 and Proposition C.2.5 one has the Lipschitz condition lip fkρ ≤ L(ρ)T . Hence, Proposition 4.1.3 ensures that the
4.9 Applications
309
general forward solution of (4.9x) exists as a continuous mapping. Moreover, also the global Lipschitz conditions (4.2a) hold with L1 = 0 and L2 = L(ρ)T . ad (Γκ− (n)): This growth condition trivially holds due to fk (0) ≡ 0 on Z. ˆ n ): In the present setting the spectral gap condition reads as ad (G
2 b T L(ρ) n n : ∃ς ∈ 0, bn −a <ς 2 1 + 2T L(ρ) and it is slightly more involved to verify it. For this, we abbreviate α := 2hk (1 + νn+1 ) + h2k (1 + νn+1 )2 + h2k ν 2 (1 + νn+1 )2 , β := 2hk (1 + νn ) + h2k (1 + νn )2 + h2k ν 2 (1 + νn )2 and observe an (k) = (1 + α)−1/2 , bn (k) = (1 + β)−1/2 . Our conditions (4.9w) ensure β < α ≤ 2(1 + νn+1 ) + (1 + νn+1 )2 + ν 2 (1 + νn+1 )2 T ≤ (3 + ν 2 )(1 + νn+1 )2 T < ω −1 − 1, one has
√ √ √ √ √ 1 + β ≤ 1 + α ≤ ω −1/2 , thus 1 + α( 1 + α + 1 + β) ≤ √ω (α 2 1+β
− β) ≤
√
α−β √ √ √ 1+α 1+β( 1+α+ 1+β)
Since we also have β < ω −1 − 1, one deduces
√1 1+β
= ≥
√1 1+β
−√
2 ω
and
1 . 1+α
√ ω and we assemble
1 ω 3/2 ω 1 (α − β). −√ (α − β) ≥ ≥ √ bn (k) − an (k) = √ 2 1+β 2 1+β 1+α 9 8 Directly from the definition one has α − β ≥ 2 T (λn+1 − λn ) and for n = N6−2 √ π using Lemma 3.7.11(b) it is bn (k)−an (k) ≥ 2 3 ω 3/2 N 2 sin N . Thanks to the √ bn −an π 3/2 2 estimate bn (k) ≤ 1 this ensures bn ≥ 2 3 ω N sin N , which implies π (4.9v) √ 2T L(ρ) bn − an ≤ 2T L(ρ) < 2 3 ω 3/2 N 2 sin N T ≤ 1 + 2T L(ρ) bn 8 9 for n = N 6−2 . Note that by Remark 3.7.12 we can satisfy (4.9v). (a) Having verified all assumptions of Theorem 4.3.7 we know that the implicit ˜ if N ≥ 5 difference equation (4.9x) in CN has a global inertial fiber bundle W, ˜ satisfies (4.9v). Each fiber W(k), k ∈ Z, is a graph over P1n CN and by definition ˜ of the projector P1n in Lemma 4.9.20 we have dim W(k) = dim P1n CN = 2n + 1, 1 ¯ρ (0, H ) is uniformly absorbing for k ∈ Z. Since the nonautonomous set Z × B N
310
4 Invariant Fiber Bundles
the original equation (ΔGL) (see Lemma 2.6.20), all solutions of (ΔGL) eventually ¯ρ (0, H 1 ). We define W := W ˜ ∩ Bρ ⊆ Z × H 1 and the positive invariance enter B N N 1 ¯ ρ (0, H ) implies that W is an inertial fiber bundle for (ΔGL). of Z × B N (b) By Proposition 2.6.21 there exists a of unique global attractor A∗ ⊆ Z × 1 Bρ (0, HN ). Clearly, bn 1 and fk (0) ≡ 0 on Z implies that Corollary 4.7.5 can be applied, which yields A∗ ⊆ W. Therewith, Theorem 4.9.22 is established.
4.10 Remarks Semilinear difference equations: Semilinear (also called quasilinear) difference equations inherit their dynamical properties from the linear part and consequently, features like stability might be global in nature. In fact, the dominant linear part invites to apply various perturbation techniques, and we strongly benefit from this observation when it comes to the construction of invariant fiber bundles and foliations in Sects. 4.2 resp. 4.3. Nevertheless, as illustrated by Theorem 4.1.8, semilinear equations can also possess properties typical for nonlinear problems, like nontrivial attractors. Boundedness criteria for quasilinear problems are given in [175, p. 174ff]. Concerning stability questions, there is an enormous literature and for example, issues like asymptotic equivalence of solutions have been studied in [330,331,363,364]. In discretization theory, approximations to dissipative linear parts like elliptic differential operators, yield uniformly stable linear difference equations (see Sects. 3.7.3– 3.7.4). Existence of invariant fiber bundles: As indicated by the quotation in the beginning of Sect. 4.2, the literature on the existence of invariant manifolds for various kinds of evolutionary equations is vast. Hence, in the following (understandably incomplete) survey on methods for the construction of invariant manifolds, we skip important contributions like [91, 247] and essentially restrict to discrete systems: •
Graph transform: This historically first method due to [186] is of geometric nature. It characterizes the graph of an invariant manifold using a functional equation in an appropriate space of bounded Lipschitz functions, whose graph is the desired manifold – a contraction mapping argument enables a constructive proof for the existence of a solution. The references [138,141,210,211,227,230, 236, 254, 319, 343, 434, 447, 462] use this approach. A graph transform technique for nonautonomous problems can be found in [245, 277]. • In this book, we excessively use of the Lyapunov–Perron method (see [316, 361, 362]) with a rather functional analytical flavor. One relies on a dynamical characterization of stable manifolds as set of initial points for solutions decaying exponentially to zero. Such solutions can be characterized as fixed points of so-called Lyapunov–Perron operators in sequence or function spaces (cf. [77, 83, 90, 110, 114, 152, 232, 233, 456]). This abstract method lifts to nonautonomous problems (see [20, 26, 82, 205, 355, 374, 385, 458]). In fact, the occurring Lyapunov–Perron operators are omnipresent throughout the nonautonomous theory.
4.10 Remarks
311
•
In the deformation method from [320] one formulates the problem of finding an invariant manifold as solution of a family of vector fields. These differential equations can be integrated and their solutions define the invariant manifold. • The parametrization method (cf. [71–73]) allows to establish the existence of smooth invariant manifolds associated to linear subspaces, invariant by linearization, which satisfy non-resonance conditions; the basic idea is to construct a conjugacy between a map and its linearization. • The scheme of [214] is much more geometrical than the approaches relying on abstract fixed point theorems. It is based on the convergence of a canonical sequence of “finite time local stable manifolds”, which are related to the dynamics of a finite number of iterations. For differential equations there are further methods to construct invariant manifolds. They can be traced back to the work of [411, 412] (based on PDE techniques like elliptic regularization in order to solve the invariance equation), or [270] (see also [306], where the manifolds are constructed via appropriate boundary value problems). However, to the authors knowledge, these approaches have not been applied to difference equations yet. A fairly flexible approach to construct attractive invariant manifolds of autonomous difference equations based on the graph transform method is due to [254, p. 207ff] and [343], which also has a variety of applications. It is remarkable that already the early contribution [191] constructed invariant manifolds for nonautonomous equations. More contemporary results of this kind have been successively developed in [110, 355], [458, p. 86, Satz 2.3.1], [20, Theorem 4.1], [23, 233] and finally also in the monograph [245, pp. 242–243, Theorem 6.2.8]. Invariant fiber bundles for difference equations with an almost periodic or recurrent timedependence are investigated in [82, Theorems 6.10–6.11]. Invariant manifolds under the assumption of a nonuniformly exponentially dichotomic linear part have been investigated in [41] (dealing with ODEs). For the origins of the pseudo-stable and -unstable hierarchies of invariant fiber bundles we refer to [19, 430]; in [12, p. 339, Corollary 7.3.12] these hierarchies are denoted as flag of (un)stable manifolds. Further references will be given below. Finally, conditions for smooth finite-dimensional mappings precluding the existence of invariant Lipschitz compact submanifolds, are given in [158]. Our goal in Sect. 4.2 was to provide a flexible existence result for invariant fiber bundles, which carries the main technical load in global (inertial manifolds) as well as local situations (pseudo-stable and -unstable manifolds). Due to our nonautonomous and implicit setting we followed the Lyapunov–Perron method of [374]. Nonetheless, our Lyapunov–Perron construction differs from the one in [20, 114] or the continuous counterpart in [432, p. 569ff, Chap. 8]. As a benefit, we require a weaker spectral gap condition and our smoothness proofs (see Sect. 4.4) are less involved. Moreover, differing from the previous approaches [20, 458], our invariant fiber bundles are not associated to the trivial solution (the growth condition (Γi± ) circumvents this) and our method works for equations without a decoupled linear part.
312
4 Invariant Fiber Bundles
An example that even in the classical situation of an exponentially trichotomic autonomous equations, not all invariant subspaces persist under nonlinear perturbations as Lipschitzian manifolds, is given in [18, Sect. 8] in the ODE case. Invariant foliations and asymptotic phase: In the autonomous situation, a construction of invariant foliations has been given in [83, 157, 253] and smoothness issues have been addressed in [153]. We remark that the construction of invariant fiber bundles, as well as of invariant foliations, can be put in a common framework of general “Lyapunov–Perron equations” (cf. [83]). The nonautonomous case is investigated in [458, p. 99ff, Abschnitt 2.5], as well as in [33]. Invariant foliations near normally hyperbolic invariant manifolds of diffeomorphisms have been constructed in [211] and generalizations to maps on Banach spaces are due to [44]. Finally, a basic application of invariant foliations is the decoupling and linearization theory studied in Chap. 5. Using geometrical arguments, invariant foliations yield a crucial asymptotic phase property of an invariant manifold resp. fiber bundle. Instead of an asymptotic phase one also speaks of an exponential tracking. An invariant fiber bundle W is called hyperbolic, if for all pairs (κ, ξ) ∈ W, the variational equation along the solution ϕ(·; κ, ξ) admits an appropriate exponential trichotomy. For discrete dynamical systems, the asymptotic phase property of such invariant manifolds consisting of equilibria has been investigated in [16, p. 59ff, Chap. 2] and was extended to infinite-dimensional problems in [197, Theorem 2.8], [69]. The related nonautonomous situation is addressed in [25, 307, 308]. Smoothness of fiber bundles and foliations: At least the graph transform and the Lyapunov–Perron method work for equations, whose right-hand side is merely Lipschitz in the state space variable. The corresponding proofs are based on the contraction mapping or the Lipschitz inverse function theorem (cf. Theorem B.3.1). For smooth mappings with hyperbolic linear part, the classical stable and unstable manifolds inherit their differentiability properties from the equation – including being C ∞ or analytic. Here, the implicit function theorem (see [228]) or the uniform contraction principle is used (see [227, p. 132ff]). For more general splittings of the linear part, i.e., particularly for pseudo-stable and -unstable manifolds, C 1 -smooth right-hand sides yield continuously differentiable invariant manifolds. The question for a higher-order smoothness is more subtle, since substitution operators on spaces of exponentially bounded sequences need not to be as smooth as the mappings inducing them (see Example 4.6.6). Whence, more delicate techniques and tools come into play, which fall into the following categories (see [21, Remark 8] for another survey): •
Uniform contraction principles on scales of continuously embedded Banach spaces (see [205, 406, 409, 457]). • The fiber contraction principle is due to [209] and applications to derive smoothness assertions of invariant manifolds of difference equations can be found in [138, 141, 456]. • A lemma of Henry (cf. [87, p. 324, Lemma 2.1] or [21, Lemma 1]) yielding a condition when locally Lipschitzian mappings are of class C 1 .
4.10 Remarks
313
Our argument to prove smoothness assertions for invariant fiber bundles does not rely on the sophisticated tools mentioned above. Rather, it is based on a formal differentiation of the Lyapunov–Perron equation and has its origins in [385]. As advantage we point out that the only nonelementary tools involved in the proof are the Neumann series and Lebesgue’s convergence theorem. Smoothness results for invariant foliations can are due to [83, 153], and the nonautonomous differential equations case is treated in [89, 435]. Normal hyperbolicity: The property of normal hyperbolicity is a key issue in the general theory of invariant and inertial manifolds, since it guarantees their robustness, which in turn, is essential for discretizations matters. Indeed, a normally hyperbolic invariant manifold is stable under small perturbations of the right-hand side in the problem (see [160, 370] or [432, p. 494ff] for the differential equations situation); in case of inertial manifolds this is guaranteed by the spectral gap condition (see, e.g., [404]). Pseudo-stable and pseudo-unstable fiber bundles: Stable and unstable invariant manifolds for maps with hyperbolic linear part are considered in [228], or in the infinite-dimensional case in [227, p. 132ff] and [462]. Stable and unstable manifolds for time discretizations of PDEs were constructed in [8]. A generalization to arbitrary spectral splittings of the linear part, i.e., to pseudo-stable and -unstable manifolds, can be traced back to monographs like, e.g., [200, pp. 234–236, Lemma 5.1 and Example 5.2] for C 1 -mappings on Rn , [211, pp. 53–54, Theorem 5.1], [434] for C m -diffeomorphisms, m ≥ 1, and to [430] for the almost periodic ODE case. In a pseudo-hyperbolic context, we also refer to the research papers [229, 293, 460]. Finally, in a globally Lipschitzian setting, the nonautonomous pseudo-hyperbolic case has been addressed in [20]. On the basis of these results, [392, p. 116ff, Sect. 5.1] constructed forward resp. backward invariant fiber bundles meeting the requirements of pullback attraction. Applications of invariant fiber bundles have been given in [218, 220]. Rigorous smoothness proofs for invariant manifolds are due to the contributions [90,138,139,205]. In the nonautonomous situation, we refer to [26,385] for (sharp) differentiability assertions. Our Example 4.6.6 illustrating sharpness of the gap condition (4.6c) implying smoothness of the pseudo-stable fiber bundle is taken from [385, Example 4.1]; a similar example for the pseudo-unstable fiber bundle may be found in [26, Example 4.13]. Pointing at their relevance in stability and bifurcation theory, dynamical properties of center manifolds for maps are discussed at various sources like [319, p. 28, Theorem (2.1)], [227, p. 145ff], [77, p. 33ff] or [275]. In particular, see [232] for center manifolds of mappings with an unbounded linear part. The celebrated reduction principle of Pliss (see [366]) found its discrete counterpart in [77, p. 35, Theorem 8]. A reduction principle for nonautonomous difference equations is due to [458, p. 104, Satz 2.6.1], [34, Theorem 2.1] and [234,399] under global assumptions on the nonlinearities. Using an ad hoc approach, these restrictive conditions have been removed in [373] yielding an applicable form. Note that our approach to deduce Theorem 4.6.15 differs from [373] and does not require a finite-dimensional center-unstable bundle.
314
4 Invariant Fiber Bundles
Taylor approximations of invariant manifolds are discussed in [59] with a focus on the homological equation as an algebraic problem in spaces of multilinear mappings. A nonautonomous generalization has been given in [382, 383]. Here, the homological equation (4.6q) is a linear difference equation and finding Taylor coefficients becomes a dynamical, rather than an algebraic problem. Such nonautonomous Taylor approximations make the reduction principle from Theorem 4.6.15 applicable. We illustrated this via critical nonautonomous stability problems in [383]. We have tackled the lacking uniqueness of locally invariant manifolds in form of Proposition 4.6.17. A different approach has been given in [68, Theorem 4.1], where a single manifold is picked by requiring it to contain the graph of a particular function defined on a sphere in the range of the pseudo-stable spectral projector. An interesting topic is the connection between center manifolds of (semi-) flows and their time-h-map. In general, a center manifold of a time-h-map is not flowinvariant (see [285, Sect. 3] for an explicit example) and it is demonstrated in [285] that invariance depends of the correct choice of an appropriate cut-off function. On the other hand, once a center manifold of a time-h-map is unique, then it is also flow-invariant (cf. [275]). The lack of differentiable cut-off functions on C[−r, 0] (see Example C.2.8(4)) makes it difficult to apply our results on locally invariant fiber bundles, e.g., to temporal discretizations of DDEs or FDEs. Here, one has to employ more subtle techniques elaborated in [286, p. 173ff] and [152, Theorem 5.1]. Inertial fiber bundles: The theory of inertial manifolds for evolutionary differential equations is wide and we refer to [453, p. 498ff, Chap. 8], [432, pp. 569, Chap. 8] for a survey and to [164] for an early contribution. In general, attractive invariant manifolds are a theoretically useful tool in numerical analysis. For instance, [252, 444] show that multistep methods (or general linear methods) to solve ODE initial value problems are asymptotically equivalent to one-step methods. An extension to methods involving varying stepsizes, as well as applications to delay difference equations, is due to [376]. Discrete inertial manifolds in discretization theory have been investigated in [114, 216, 236, 237, 272–274, 309, 433] for autonomous equations. In addition, a linearly implicit temporal Euler discretization of an autonomous RDE was considered in [464]. The nonautonomous case as well as issues like normal hyperbolicity is due to [375]. Inertial manifolds and also fiber bundles from Theorem 4.7.3 satisfy a weaker form of an asymptotic forward phase. In fact, they are asymptotically complete, which means that forward solutions have to enter the absorbing set first, before they are grasped by a solution on the inertial bundle via its asymptotic phase. The main obstacle for the existence of inertial fiber bundles is the spectral gap condition, which is hardly satisfied in spatial discretizations of partial differential equations in more than one dimension. To avoid this problem, the concept of an exponential attractor (cf. Definition 1.6.3) has been introduced in [124, pp. 9–24, Chap. 2]. In a way, exponential attractors are “realistic” objects intermediate between the two “ideal” ones which are global attractors and inertial manifolds.
4.10 Remarks
315
Approximation of invariant fiber bundles: As summarized above, most methods for the construction of invariant manifolds are iterative and consequently in some sense algorithmic. Hence, there is a tremendous literature on the approximation of invariant manifolds for autonomous dynamical systems – for a survey we refer to [283]. As usual, the situation is different in the nonautonomous case. Unstable fiber bundles have been characterized as pullback attractors and approximated using setvalued numerics in [27]. The method presented here, generalizes our approach from [384] to implicit equations; this paper illustrates that both (4.8i) and (4.8j) can be solved efficiently using Newton-methods. Numerical tests showed that the best performance is obtained for inexact Quasi-Newton techniques. Moreover, for 1-dimensional fibers, pseudo-arc length continuation methods have been applied in order to compute relatively long arcs of global fiber bundles for nonlinear problems. Applications: First of all, the construction of invariant manifolds for autonomous evolutionary differential equations by applying discrete results to their time-h-map has been exemplified in [83, Sects. 5–6]. Invariant foliations over inertial manifolds have been investigated in [157]. Concerning flexible C 1 -perturbation results, which apply to general normally hyperbolic invariant manifolds, we quote [368] – this forms the basics for a corresponding discretization theory. For numerical methods to solve DDEs we refer to [47, 467] and [172] gives a survey on the corresponding numerical dynamics. Unstable manifolds of discretized retarded FDEs are the topic of [155] and see [156] for small delay inertial manifolds and numerical structural stability. The behavior of invariant manifolds for ODEs under numerical discretization has a long history. Pioneering papers are [53] (stable and unstable manifolds), [60] (center manifolds), [158], and [171] (pseudo-stable and -unstable case). The results of [53] were generalized to parabolic PDEs in [8]. For periodic ODEs we refer to [468] and the general nonautonomous situation has been investigated in [246]. When discretizing PDEs one is confronted with the problem of nonsmooth initial data, which have an effect on the convergence rates under approximation. The corresponding error estimates to show convergence results can be found, for instance, in [55, 314, 315]. Evolutionary equations having an inertial manifold and corresponding discretizations are the topic of various papers. Early contributions on temporal discretizations are [114, 272], general perturbation results applicable to a variety of (full) discretizations have been considered in [236], and we furthermore refer to [216], [121] (temporal and full Runge–Kutta schemes), [273], [237,238,274] (C 1 -convergence). Abstract convergence results of inertial manifolds are due to [400]. Moreover, [464] analyzes the inertial manifold of a linearly-implicit Euler discretization of a sectorial evolution equation. While many papers restrict to one-step methods, see [433] for generalizations to multistep schemes. Finally, time-discrete inertial manifolds for nonautonomous equations have been discussed in [374, 375]. Inertial manifold for the fully discretized Ginzburg–Landau equations are constructed in [309], which we generalized to the time-variant case. A similar analysis to
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4 Invariant Fiber Bundles
ours from Sect. 4.9.5, but for the Kuramoto–Sivashinsky equation, can be found in [162, 163, 165, 273, 274]. A numerical computation scheme for inertial manifolds based on approximations of the continuous Lyapunov–Perron method has been suggested in [235], while we have preferred an initial time discretization in Sect. 4.9.4. Two further approximation schemes are due to [401].
Chapter 5
Linearization
The advantage of this method is (besides yielding global invariant manifolds and their regularity) that it also gives the Hartman–Grobman theorem. . . . L. Arnold (cf. [12, p. 305])
Decoupling and linearization of dynamical models given by differential or difference equations are very successful and frequently used simplification concepts in applied sciences. On the one hand, provided a system can be decoupled, due to possibly different growth rates, each part might be approached using methods more suitable for a subsystem than for the whole problem. On the other hand, linearization includes the advantage that linear equations are mathematically wellunderstood and problems can be approached on an analytical level. Indeed, the basic reason why such linear models oftentimes yield a realistic and successful description of real nonlinear problems, is the Hartman–Grobman theorem. For that reason, the theorem of Hartman–Grobman, also known as linearization theorem, is one of the central results in the local theory of dynamical systems. At its heart this result states that the behavior of a given dynamical system near a hyperbolic fixed point is qualitatively the same as the behavior of its linearization. Using a more technical terminology, this means that the nonlinear flow is topologically conjugated to the corresponding linear flow, i.e., both flows can be transformed into each other using a homeomorphism and the corresponding phase portraits are homeomorphic images of each other. Thus, when dealing with such fixed points the linearization of the system is sufficient to analyze its behavior. The central and generic assumption in the standard Hartman–Grobman setting is hyperbolicity of the linearization. If we have eigenvalues on the unit circle, though, then the situation changes drastically due to the existence of a center manifold. As discussed in Remark 4.6.13 this invariant manifold contains more complex dynamical objects, like e.g., periodic motions or homoclinic orbits. Such an extended set-up ˇ sitaˇiˇsvili theorem, which states that the dynamiis the playground for the Palmer–Soˇ cal behavior near a nonhyperbolic equilibrium is topologically conjugated to a linear saddle times the nonlinear flow on the center manifold. The above linearization problem remains relevant and of interest in order to understand increasingly general classes of equations for which decoupling or linearization problems can be solved. In particular, studies on the behavior near
C. P¨otzsche, Geometric Theory of Discrete Nonautonomous Dynamical Systems, Lecture Notes in Mathematics 2002, DOI 10.1007/978-3-642-14258-1 5, c Springer-Verlag Berlin Heidelberg 2010
317
318
5 Linearization
nonconstant solutions require corresponding nonautonomous results. To achieve this goal, we proceed as follows: •
In the first section, we briefly discuss problems occurring with the classical notion of a topological conjugation in a nonautonomous setting, and introduce a meaningful substitute. Having this available, we strongly benefit from our preparations on invariant foliations and asymptotic phases in Sect. 4.3. Using these foliations as coordinates we obtain a complete decoupling for semilinear equations, whose linear part admits an exponential splitting. • Having completely decoupled a difference equation, we will linearize the resulting subsystems successively. This is possible for hyperbolic systems, while otherwise the equation reduced to the center fiber bundle remains. In doing so, we obtain a generalized Hartman–Grobman result in Theorem 5.2.5. • It can be illustrated using examples that noninvertible difference equations cannot be linearized in general via topological conjugations. We compensate this problem using the concept of a solution conjugation. It establishes a continuous correspondence between solution spaces instead of individual fibers of the extended state space. • Two brief applications illustrate the main result (Theorem 5.3.4) of Sect. 5.3. Both for the solutions of abstract evolution equations, and for temporal discretizations of parabolic equations, we establish a solution conjugation to their corresponding linear parts. The whole chapter is concerned with semilinear equations, where uniform Lipschitz conditions on the nonlinearities yield quantitative global results. Without presenting details, we remark that local results under less restrictive conditions on the nonlinearities can be deduced as in Sects. 4.6–4.7 using cut-off techniques. Rather than having a global picture, in doing so one still obtains relevant dynamical information in a uniform neighborhood of a reference solution. As usual, let I be a discrete interval. Unless otherwise noted, suppose that X consists of Banach spaces and (Yk )k∈I is a sequence of metric linear spaces.
5.1 Topological Conjugation and Decoupling For the sake of generality, we suppose in the beginning of this section that (Xk )k∈I ˜ k )k∈I , as well as (Yk )k∈I , (Y˜k )k∈I are sequences of metric spaces. Extendand (X ing the concept of a Lyapunov transformation from Definition 3.1.30, we establish a transformation suitable for nonautonomous nonlinear equations Hk+1 (x ) = Fk (x, x )
(D)
as in Definition 2.1.1. Alongside (D) we consider the equation ˜ k+1 (x ) = F˜k (x, x ) H
˜ (D)
5.1 Topological Conjugation and Decoupling
319
˜k : X ˜ k → Y˜k , k ∈ I, and F˜k : X ˜k × X ˜ k+1 → Y˜k+1 , k ∈ I . Here, with mappings H ˜ lives in the extended state space X˜ . (D) is an equation on X , whereas (D) In an autonomous framework, the notion of topological conjugacy states that two ˜ → X ˜ on metric spaces X and X ˜ are semigroups ϕ(k) : X → X and ψ(k) : X ˜ with topologically conjugated, if there exists a homeomorphism T : X → X T ◦ ϕ(k) = ψ(k) ◦ T
for all k ≥ 0
(cf., e.g., [230, p. 28ff]). This notion of conjugacy provides a bijection between periodic motions or more general invariant sets of the two semigroups ϕ(k) and ψ(k). Furthermore, it preserves stability properties of constant and periodic motions. Yet, topological conjugacy as above might not preserve stability of unbounded solutions as illustrated by an example in [437, Sect. 2]. When extending the above concept to nonautonomous problems, it is natural to work with time-depending homeomorphisms Tk acting between some respective k-fibers X (k) and X˜ (k), k ∈ I. Without additional assumptions this causes some problems: Example 5.1.1. (1) Let ϕ and ψ be 2-parameter groups on a common extended state space X , with generators (ϕˆk )k∈I resp. (ψˆk )k∈I . For κ ∈ I fixed, both groups are conjugated via the homeomorphism Tk := ψ(k, κ)ϕ(κ, k) : Xk → Xk , k ∈ I, in the sense that Tk+1 ◦ ϕˆk = ψˆk ◦ Tk for every k ∈ I holds. As a consequence, every difference equation generating a 2-parameter group is conjugated to the trivial identical equation (see below). (2) Suppose that all the fibers Xk ≡ X on I are identical and the general solution ϕ to (D) exists as a continuous mapping. As observed in Remark 2.3.3(2) the general solutions of (D) and of the explicit equation x = ϕ(k + 1; k, x)
(5.1a)
coincide. If we keep a κ ∈ I fixed and define Tk := ϕ(κ; k, ·) : X → X, k ∈ I, then Tk is a homeomorphism (cf. Remark 2.3.3(3)). Due to (2.3b) Tk+1 ϕ(k + 1; k, Tk−1 (x)) = x
for all (k, x) ∈ X
the homeomorphism Tk transforms (5.1a) into the trivial equation x = x. Example 5.1.2. Let X = N × R and α, β ∈ R. Then the scalar linear equation x =
k k+1
α
x
(5.1b)
α for all k, l ∈ N. Referring to has the evolution operator Φ(k, l) = kl Proposition 3.1.23 it is asymptotically stable for α > 0, uniformly stable for α = 0 and unstable for parameters α < 0. The conjugacy Tk (x) := k β x transforms (5.1b) α−β α−β k into the equation x = k+1 x with evolution operator Ψ (k, l) = kl .
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5 Linearization
Reasoning as above, the transformed equation is asymptotically stable for α > β, uniformly stable for α = β and unstable for α < β. Depending on the choice of β, we can invert stability properties. We consequently need additional assumptions beyond Tk being a homeomorphism in order to obtain a meaningful notion of topological conjugacy of nonautonomous equations. This includes the postulation that solution properties of at least a fixed reference solution φ∗ to (D) are preserved. Definition 5.1.3. Let φ∗ : I → X be a solution of (D). A sequence of homeomor˜ k , k ∈ I, is said to be a topological conjugation between (D) phisms Tk : Xk → X ˜ along φ∗ , if following properties hold: and (D) ˜ ˜ (i) For every solution φ of (D) the sequence φ(k) := Tk (φ(k)) solves (D). −1 ˜ ˜ ˜ (ii) For every solution φ of (D) the sequence φ(k) := Tk (φ(k)) solves (D). (iii) The following limit relations hold uniformly in k ∈ I, lim
x→φ∗ (k)
Tk (x) = φ˜∗ (k),
lim
˜∗ (k) x→φ
Tk−1 (x) = φ∗ (k).
(5.1c)
If such a sequence of homeomorphisms Tk , k ∈ I, exists, then the difference ˜ are called topologically conjugated. equations (S) and (S) Remark 5.1.4. (1) If the respective general forward solutions ϕ and ψ of (D) and ˜ exist, then their generators (ϕˆk )k∈I resp. (ψˆk )k∈I fulfill the identities (D) ˜ k+1 Tk+1 (ϕˆk (ξ)) ≡ F˜k Tk (ξ), Tk+1 (ϕˆk (ξ)) on Xk , H −1 −1 ˆ ˜k Hk+1 Tk+1 (ψˆk (ξ)) ≡ Fk Tk−1 (ξ), Tk+1 (ψk (ξ)) on X and also the conjugation relation Tk+1 ◦ ϕˆk = ψˆk ◦ Tk for all k ∈ I (see Fig. 5.1). From this we immediately deduce ϕ(k; κ, ·) = Tk−1 ◦ ψ(k; κ, ·) ◦ Tκ for all κ ≤ k. (2) Topological conjugation is an equivalence relation on the set of equations (D). (3) Given a solution φ∗ : I → X, the various notions of attraction from Definition 2.4.4 and of stability for φ∗ introduced in Definition 2.4.11 are preserved under topological conjugation – thanks to the uniform limit relations (5.1c). (4) Our notion of a topological conjugation is global in nature. Naturally, the concept of a local topological conjugation can be defined via homeomorphisms between neighborhoods Bρ (φ∗ ) and Bρ (φ˜∗ ). ϕ ˆk−2
ϕ ˆk−1
ˆ ψ
ˆ ψ
. . . −−−−−→ Xk−1 −−−−−→ ⏐ ⏐T k−1
ϕ ˆ
ϕ ˆk+1
ˆ
ˆ ψ
k Xk −−−− −→ Xk+1 −−−−−→ . . . ⏐ ⏐ ⏐T ⏐T k k+1
ψk k−1 k−1 ˜ k−1 −−− ˜ k −−− ˜ k+1 −−−k+1 . . . −−−−−→ X −−→ X −−→ X −−→ . . .
˜ k , k ∈ I, and generators ϕ Fig. 5.1 The topological conjugation Tk : Xk → X ˆk , ψˆk , k ∈ I
5.1 Topological Conjugation and Decoupling
321
Example 5.1.5. (1) A difference equation (D) along with the associated equation of φ∗ -perturbed motion (Dφ∗ ) from Definition 2.4.14 are topologically conjugated via the homeomorphism Tk (x) := x + φ∗ (k). (2) Linearly conjugated difference equations (L0 ) and (3.1q) as in Definition 3.1.30 are topologically conjugated via the homeomorphism Tk (x) := Λk x. In order to employ the geometrical tools developed in the previous Chap. 4, we retreat to difference equations, where the extended state space X consists of Banach spaces. We are interested in general equations (D) with a given fixed reference solution φ∗ : I → X. Since the transformation to the equation of perturbed motion (Dφ∗ ) is a topological conjugation, we can restrict to equations possessing the trivial solution. In doing so, we consider two equations Bk+1 x = Ak x + fk (x, x )
(S)
Bk+1 x = Ak x + f˜k (x, x ),
˜ (S)
and ˜ are assumed to have the same extended state space as in Sect. 4.2. Both (S) and (S) X and for their common linear part (L0 ) we assume (cf. Hypothesis 4.2.1): Hypothesis 5.1.6. Let I = Z. Suppose Ak ∈ Hom(Xk , Yk+1 ), Bk ∈ Hom(Xk , Yk ) −1 has an inverse with Bk+1 Ak ∈ L(Xk , Xk+1 ), k ∈ Z, and that the linear equation (L0 ) admits a strongly regular exponential N -splitting on Z with N > 1, namely N −1 S(A, B; P ) = (bi+1 , ai ), (5.1d) i=0
where the sequences bi are bounded above. Moreover, we use the abbreviations −1 , P i1 (k) := Bk+1 P1i (k)Bk+1
−1 Qi1 (k) := Bk+1 Qi1 (k)Bk+1
for all k ∈ Z.
˜ and Beyond that we are concerned with semilinear difference equations (S), (S) strengthen Hypothesis 4.3.1 as follows: Hypothesis 5.1.7. Let the general solution ϕ of (S) exist on X as a continuous mapping. Suppose that the nonlinearity fk : Xk × Xk+1 → Yk+1 fulfills fk (Xk , Xk+1 ) ⊆ im Bk+1 ,
fk (0, 0) = 0
for all k ∈ Z
(5.1e)
−1 fk < ∞, j = 1, 2. and that we have the Lipschitz estimates Lj := supk∈Z lipj Bk+1
In Sect. 4.3 we constructed asymptotic phases using the intersection of invariant fiber bundles Wi± and invariant foliations Vi∓ (ξ) (cf. Theorem 4.3.7). The following preparatory result investigates the intersection of invariant foliations:
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5 Linearization
˜ i ) for some Lemma 5.1.8. Suppose that Hypotheses 5.1.6, 5.1.7 are fulfilled with (G 1 ≤ i < N . If c ∈ Γ¯i , then there exists a unique continuous mapping Πi : X × X → X, geometrically given by Vi+ (κ, x1 ) ∩ Vi− (κ, x2 ) = {Πi (κ, x1 , x2 )}
for all (κ, x1 ), (κ, x2 ) ∈ X . (5.1f)
Furthermore, Πi is linearly bounded, i.e., for all (κ, x1 ), (κ, x2 ) ∈ X one has 1 Πi (κ, x1 , x2 ) ≤ 1 − ˜i (c)
+ − max Ki− + i (c), Ki i (c) 1+ (|x1 |i + |x2 |i ) , 1 − ˜i (c)
˜ where the constants ± i (c), i (c) ∈ [0, 1) are defined in Lemma 4.2.6, i (c) ∈ [0, 1) given in Theorem 4.3.7. Proof. Let κ ∈ Z, x1 , x2 ∈ Xκ and choose c ∈ Γ¯i . Since the mappings wi± (κ, ·) from Theorem 4.2.9 and vi± (κ, ·, ξ), (κ, ξ) ∈ X , from Proposition 4.3.5 have the same Lipschitz constants, we can derive analogously to (4.3t) that the relation lip2 vi+ · lip2 vi− < 1
(5.1g)
holds. The intersection Vi+ (κ, x1 ) ∩ Vi− (κ, x2 ) contains a point y ∈ Xκ , if and only if there exist p ∈ P1i (κ), q ∈ Qi1 (κ) so that y = q + vi+ (κ, q, x1 ), y = p + vi− (κ, p, x2 ), which, in turn, is equivalent to the fact that y allows the representation y = p + q, where p ∈ P1i (κ), q ∈ Qi1 (κ) solve the equations p = vi+ (κ, q, x1 ),
q = vi− (κ, p, x2 ).
(5.1h)
These equations are uniquely solvable by Corollary B.1.4 and we give the argument for later reference. Thanks to Proposition 4.3.5 and (5.1g), the two mappings Πκ− : P1i (κ) × Xκ2 → P1i (κ), (p, x1 , x2 ) → vi− (κ, vi+ (κ, p, x1 ), x2 ),
Πκ+ : Qi1 (κ) × Xκ2 → Qi1 (κ), (q, x1 , x2 ) → vi+ (κ, vi− (κ, q, x2 ), x1 )
are continuous and contractions in their first arguments (uniformly in the parameters κ, x1 , x2 ) with lip1 Πκ± ≤ ˜i (c) < 1 for all c ∈ Γ¯i (cf. (4.3t)). Hence, the contraction mapping principle from, e.g., [295, p. 361, Lemma 1.1] implies that there exist unique fixed point functions p∗κ : Xκ2 → X and qκ∗ : Xκ2 → X for Πκ− and Πκ+ , respectively. Thus, the sum Πi (κ, x1 , x2 ) := p∗κ (x1 , x2 ) + qκ∗ (x1 , x2 ) satisfies the geometric property (5.1f). Given x01 , x02 ∈ Xκ arbitrarily, we have
∗
pκ (x1 , x2 ) − p∗κ (x01 , x02 )
+ ∗ 0 0 1
Πκ (pκ (x1 , x2 ), x1 , x2 ) − Πκ+ (p∗κ (x01 , x02 ), x01 , x02 ) , ≤ 1 − ˜i (c)
5.1 Topological Conjugation and Decoupling
323
∗
qκ (κ, x1 , x2 ) − qκ∗ (κ, x01 , x02 )
− ∗ 0 0 1
Πκ (pκ (x1 , x2 ), x1 , x2 ) − Πκ− (p∗κ (x01 , x02 ), x01 , x02 ) ≤ 1 − ˜i (c) and these inequalities immediately imply the continuity of Πi inherited from the corresponding properties of vi+ and vi− . In order to prove that Πi (κ, ·) is linearly bounded, we proceed as follows. Using Proposition 4.3.5 we get the estimate
(5.1h) p∗κ (x1 , x2 ) = vi+ (κ, qκ∗ (x1 , x2 ), x1 ) (4.3o)
∗ i
≤ P1i (κ)x1 + ˜+ i (c) qκ (x1 , x2 ) − Q1 (κ)x1 (5.1h)
i
∗
˜+ −
≤ P1i (κ)x1 + ˜+ i (c) Q1 (κ)x1 + i (c) vi (κ, pκ (x1 , x2 ), x2 )
i
i ≤ P1 (κ)x1 + ˜+ i (c) Q1 (κ)x1
Qi1 (κ)x2 + ˜− (c) p∗κ (x1 , x2 ) − P1i (κ)x2 +˜+ i (c) i
i
Q (κ)x1 + Qi (κ)x2 ≤ P1i (κ)x1 + ˜i (c) P1i (κ)x2 + ˜+ 1 1 i (c) +˜i (c) p∗ (x1 , x2 ) κ
and similarly
qκ∗ (x1 , x2 ) ≤ Qi1 (κ)x1 + ˜i (c) Qi1 (κ)x2
+˜− (c) P i (κ)x1 + P i (κ)x2 + ˜i (c) q ∗ (x1 , x2 ) . i
1
1
κ
Thanks to the relation |Πi (κ, x1 , x2 )|i = max { p∗κ (x1 , x2 ) , qκ∗ (x1 , x2 ) } this finally implies the linear bound on Πi . We have established our claims.
Let us continue with a geometrical interpretation of the dynamics generated by the semilinear difference equation (S). Due to assumption (5.1e) the growth conditions (Γi± ) are trivially fulfilled. Under appropriate smallness conditions on the ˜ i ), Lipschitz constants L1 , L2 , formulated in terms of the spectral gap condition (G we infer from Theorems 4.2.9 and 4.3.7 that the pseudo-stable fiber bundle Wi+ possesses an asymptotic backward phase πi− and, dually, the pseudo-unstable fiber bundle Wi− admits an asymptotic forward phase πi+ . These asymptotic phases assign to any given solution φ : I → X of (S) two further solutions: + − • A solution φ− in Wi , which is given by φ− (k) := ϕ k; κ, πi (κ, φ(κ)) and − i can be identified with its projection φ− − , φ− (k) := Q1 (k)φ− (k); note that it + solves the Wi -reduced equation Bk+1 q = Ak q + Qi1 (k)fk (q + wi+ (k, q), q + wi+ (k + 1, q ))
(5.1i)
in the vector bundle Qi1 – for this we assume the general solution ϕ to (S) exists and Wi+ becomes an invariant fiber bundle (see Remark 4.2.10(2)).
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5 Linearization
− + • A solution φ+ in Wi given by φ+ (k) := ϕ k; κ, πi (κ, φ(κ)) and being + − i identified with its projection φ+ + , φ+ (k) := P1 (k)φ+ (k). It solves the Wi reduced equation Bk+1 p = Ak p + P i1 (k)fk (p + wi− (k, p), p + wi− (k + 1, p ))
(5.1j)
in the invariant vector bundle P1i (cf. (4.2s)). + We conclude that the assignment φ → (φ− − , φ+ ) leads to a decoupling of (S) into components in the invariant vector bundles P1i and Qi1 , respectively. The following result puts the above explanations into a precise framework:
Theorem 5.1.9 (topological decoupling). We suppose that Hypotheses 5.1.6, 5.1.7 ˜ i ) for some 1 ≤ i < N . Then there exists a topological conare fulfilled with (G ˜ jugation Tk : Xk → Xk , k ∈ Z, between the difference equations (S) and (S) with f˜k (x, x ) = Qi1 (k)fk (Qi1 (k)x + wi+ (k, x), Qi1 (k + 1)x + wi+ (k + 1, x )) + P i1 (k)fk (P1i (k)x + wi− (k, x), P1i (k + 1)x + wi− (k + 1, x )),
(5.1k)
which has the following properties: (a) For all κ ∈ Z one has the linear bounds ˜+ (c) ˜− (c) 1 + i i ≤ max Ki+ 1 + ˜+ 1+ , i (c) ˜ 1 − i (c) ˜+ (c) 1 + ˜− (c) i i |ξ|i , Ki− 1 + ˜− 1+ i (c) 1 − ˜i (c)
+ − max Ki− + 2 i (c), Ki i (c) 1+ |ξ|i . ≤ (5.1l) 1 − ˜i (c) 1 − ˜i (c)
|Tκ (ξ)|i
−1 T (ξ) κ
i
(b) The fiber bundles Wi+ and Wi− of (S) are mapped to the invariant vector bun˜ resp., i.e., for the corresponding fibers we have dles Qi1 and P1i of (S), Tκ (Wi+ (κ)) = Qi1 (κ),
Tκ (Wi− (κ)) = P1i (κ)
for all κ ∈ Z
˜± where the constants ± i (c), i (c) ∈ [0, 1) are defined in Lemma 4.2.6, i (c) ∈ [0, 1) ˜ from Theorem 4.2.9 and i (c) ∈ [0, 1) given in Theorem 4.3.7. Proof. Let (κ, ξ) ∈ X . Using the asymptotic phases πi− , πi+ from Theorem 4.3.7 and the mapping Πi from Lemma 5.1.8, we define mappings Tκ , T˜κ : Xκ → Xκ ,
5.1 Topological Conjugation and Decoupling
325
Tκ (ξ) := Qi1 (κ)πi− (κ, ξ) + P1i (κ)πi+ (κ, ξ), T˜κ (ξ) := Πi κ, P i (κ)ξ + w− (κ, ξ), Qi (κ)ξ + w+ (κ, ξ) 1
i
1
i
and use the notation from Definition 5.1.3. From Theorem 4.3.7 we know that Tκ is continuous, and thanks to Lemma 5.1.8 and Theorem 4.2.9 the same holds true for T˜κ . ˜ we In order to show that Tκ is a topological conjugation between (S) and (S), remark that πi+ (κ, ξ) ∈ Wi− (κ) and πi− (κ, ξ) ∈ Wi+ (κ) evidently imply P1i (κ)πi+ (κ, ξ) + wi− (κ, πi+ (κ, ξ)) = πi+ (κ, ξ), Qi1 (κ)πi− (κ, ξ) + wi+ (κ, πi− (κ, ξ)) = πi− (κ, ξ), respectively. This yields T˜κ (Tκ (ξ)) ≡ Πi (κ, πi+ (κ, ξ), πi− (κ, ξ)) ≡ ξ on Xκ , since we also have πi+ (κ, ξ) ∈ Vi+ (κ, ξ) and πi− (κ, ξ) ∈ Vi− (κ, ξ); similarly one shows the identity Tκ (T˜κ (ξ)) ≡ ξ on Xκ , and thus the continuous mappings Tκ , T˜κ are inverse to each other, i.e., Tκ−1 = T˜κ . Next we show the uniform limit relations required in Definition 5.1.3(iii), which immediately follow from (5.1l). These relations (5.1l), in turn, can be derived as follows. In case of Tκ , it is an easy consequence Theorem 4.3.7(a1) and (b1 ); concerning the inverse T˜κ , this follows from Proposition 4.3.5(c) and the Lipschitz ˜i) estimates for wi± (κ, ·) stated in Theorem 4.2.9. Note that Remark 4.3.8 and (G ± ˜ guarantee i (c) < 1. Therefore, it remains to establish the properties (i)–(ii) of Definition 5.1.3. For this, let the sequence φ : Z → X be a complete solution of (S). We define the sequence φ− (k) := ϕ k; κ, πi− (κ, φ(κ)) and obtain the inclusion φ− (κ) = πi− (κ, ξ) ∈ Wi+ (κ) ∩ Vi− (κ, φ(κ)). Due to the invariance of Vi− (φ(κ)) (cf. (4.3l)) and Wi+ (for this, cf. Remark 4.2.10(2)) one deduces φ− (k) ∈ Wi+ (k) ∩ Vi− (k, φ(k)) for all k ∈ Z and by Theorem 4.3.7(a) and Proposition 4.3.5(b), respectively, P1i (k)φ− (k) = wi+ (k, φ− (k)),
Qi1 (k)φ− (k) = Qi1 (k)πi− (k, φ(k))
for all k ∈ Z. Hence, Qi1 (·)πi− (·, φ(·)) is a solution of the Wi+ -reduced equation (5.1i) and analogously P1i (·)πi+ (·, φ(·)) solves the Wi− -reduced equation (5.1j). ˜ and whence This means, we have established that k → Tk (φ(k)) is a solution of (S) ˜ ˜ (i) holds. Conversely, let φ : Z → X be a solution of (S). We define a solution of (S) by ˜ ˜ ˜ ˜ ψ(k) := ϕ k; κ, Πi (κ, P1i (κ)φ(κ) + wi− (κ, φ(κ)), P1i (κ)φ(κ) + wi− κ, φ(κ))
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5 Linearization
and obtain from (4.3p) and Proposition 4.3.5(a) that ˜ ˜ ˜ ˜ φ(k) ∈ Vi+ k, P1i (k)φ(k) + wi− (k, φ(k)) ∩ Vi− k, Qi1 (k)φ(k) + wi+ (k, φ(k)) ˜ ˜ and hence k → T˜k (φ(k)) is a for all k ∈ Z, which implies ψ(k) = T˜k (φ(k)) solution of (S); we have shown (ii). Referring to the estimates (5.1l) we know that exponential boundedness of so˜ is preserved under the mapping sequence Tk (or T˜k , resp.). lutions for (S) (or (S)) Therefore, due to their dynamical characterization, the invariant fiber bundles Wi+ and Wi− of (S) are bijectively mapped onto the respective invariant vector bundles ˜ This was claim (b) and we have shown Theorem 5.1.9. Qi1 and P1i of (S).
˜ with nonlinearity f˜k given in (5.1k) The transformed difference equation (S) is structurally simpler than the original equation (S), since it is decoupled. In fact, ˜ the sequence Qi (·)ϕ(·; given the general solution ϕ˜ of (S), ˜ κ, Qi1 (κ)ξ) solves (5.1i), 1 i i while P1 (·)ϕ(·; ˜ κ, P1 (κ)ξ) is a solution to (5.1j). Conversely, the general solutions ˜ via the ϕ− to (5.1i) and ϕ+ to (5.1j), resp., yield the general solution ϕ˜ of (S) i i relation ϕ(k; ˜ κ, ξ) = ϕ+ (·; κ, P1 (κ)ξ) + ϕ− (·; κ, Q1 (κ)ξ). We close this section by discussing the periodic and autonomous situation. Corollary 5.1.10. Let p ∈ N. (a) If (S) is p-periodic, then Tκ+p = Tκ for all κ ∈ Z. (b) If (S) is autonomous, then Tκ does not depend on κ ∈ Z. Proof. Let (κ, ξ) ∈ X . Due to Tκ (ξ) = Qi1 (κ)πi− (κ, ξ) + P1i (κ)πi+ (κ, ξ) our claim instantly follows from Corollary 4.3.12.
5.2 Generalized Hartman–Grobman Theorem A fundamental problem in the theory of dynamical systems is to understand the relationship between the local behavior of a difference equation near a reference solution φ∗ , and the dynamics of its linearization along φ∗ . E.g., for the sake of stability properties or to relate the dynamics in the topological category, it is desirable that the two corresponding equations are topologically conjugated w.r.t. φ∗ . In fact, it is possible to establish such a topological conjugacy, provided φ∗ is a hyperbolic complete solution, and we will deduce a corresponding Hartman– Grobman theorem. Nonetheless, even simple scalar equations like x = x ± x3 illustrate that the stability of the trivial equilibrium differs from the stability of the corresponding linearization x = x. Thus, one cannot expect a complete linearization result in the nonhyperbolic situation. This indicates that also the behavior on the center manifold or fiber bundle must be taken into account. With these goals in mind, we are finally in the position to demonstrate how fruitful our preparations have been. In order to harvest a generalized topological
5.2 Generalized Hartman–Grobman Theorem
327
linearization theorem, it remains to prove only one further result. Again, this section is based on the assumption that I is the whole integer axis Z. Moreover, we suppose the general solution of (S) exists. We are dealing with semilinear difference equations Bk+1 x = Ak x + fk (x, x ),
(S)
whose linear part satisfies our Hypothesis 5.1.6. This means we have an exponential N -splitting (5.1d). Initially, our interest is focused on the nonhyperbolic situation in which there exists a 1 ≤ i∗ < N such that 1 ∈ (bi∗ +1 , ai∗ ).
(5.2a)
We denote the general solution of (S) by ϕ and additionally consider an equation Bk+1 x = Ak x + f˜k (x, x )
˜ (S)
as in Sect. 5.1, with corresponding general solution ϕ. ˜ Hypothesis 5.2.1. Let the general solution ϕ of (S) exist as a continuous mapping. Suppose that fk : Xk × Xk+1 → Yk+1 fulfills fk (Xk , Xk+1 ) ⊆ im Bk+1 , fk (0, 0) ≡ 0
on Z, M :=
sup
(k,x,x )∈X ×X
−1
B fk (x, x ) k+1 X
k+1
< ∞ (5.2b)
−1 fk < ∞, j = 1, 2. and that we have the Lipschitz estimates Lj := supk∈Z lipj Bk+1
Remark 5.2.2. In contrast to Hypothesis 4.2.3 or 5.1.7 we need to assume that the nonlinearity is globally bounded (cf. (5.2b)). This is just a technical assumption causing no trouble when deducing local results using cut-off techniques. We first formulate an important technical tool for our upcoming linearization result, which is based on the admissibility property from Theorem 3.5.10. Lemma 5.2.3. Let M1 , M2 ≥ 0. Suppose that Hypothesis 5.1.6 holds true with −1 (5.2a) and Bk+1 Ak ∈ GL(Xk , Xk+1 ), k ∈ Z, and fk , f˜k : Xk × Xk+1 → Yk+1 satisfy: −1 −1 ˜ fk (x, x ), Bk+1 fk (x, x ) ∈ Qi∗ (k). (i) Bk+1 (ii) The following mappings are homeomorphisms
IXk+1 IXk
−1 IXk+1 − Bk+1 fk (x, ·), −1 ˜ −B fk (x, ·) : Xk+1 → Xk+1 ,
k+1 −1 −1 IXk + (Bk+1 Ak )−1 Bk+1 fk (·, x ), −1 −1 ˜ + (Bk+1 Ak )−1 Bk+1 fk (·, x ) : Xk →
Xk .
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5 Linearization
(iii) One has the global Lipschitz estimates ˜ j := sup lipj B −1 f˜k < ∞ L k+1
for j = 1, 2.
(5.2c)
k∈Z
(iv) One has the boundedness conditions
−1 −1 ˜ fk (x, x ) , Qii∗ (k + 1)Bk+1 max Qii∗ (k + 1)Bk+1 fk (x, x ) ≤ M1 ,
−1 −1 ˜ max Pii∗ −1 (k + 1)Bk+1 fk (x, x ) , Pii∗ −1 (k + 1)Bk+1 fk (x, x ) ≤ M2 for all triple (k, x, x ) ∈ X × X . If the condition ˜1 + L ˜2 < 1 2νi∗ L
(5.2d)
holds, then for each κ ∈ Z there exists a unique mapping Jκ : Xκ → Xκ such that Jκ (ξ) = Jκ (Qi∗ (κ)ξ) ∈ Qi∗ (κ) ϕ(·; ˜ κ, Jκ (ξ)) − ϕ(·; κ, ξ) ∈ X1
for all (κ, ξ) ∈ X ,
for all (κ, ξ) ∈ Qi∗ .
Moreover, the following holds: (a) Jκ : Xκ → Xκ is continuous with limξ→0 Jκ (ξ) = 0 uniformly in κ ∈ Z. (b) Jκ is “near identity” with Jκ (ξ) − ξ ≤
νi∗ (M1 + M2 ) ˜1 + L ˜ 2) 1 − νi∗ (L
for all (κ, ξ) ∈ Qi∗ .
(5.2e)
(c) For every solution φ : Z → X of (S) satisfying φ(κ) ∈ Qi∗ (κ) for some κ ∈ Z, ˜ ˜ the sequence φ(k) := Jk (φ(k)) solves (S), − Ki−1 Ki+ where we have abbreviated νi := 2 max 1−ai , bi−1 −1 . Remark 5.2.4. (1) Sufficient conditions for assumption (i) to hold have been given in Propositions 4.1.3 and 4.1.4. ˜ are p-periodic, then the relation Jκ+p = (2) Let p ∈ N. Provided both (S) and (S) Jκ holds for all κ ∈ Z. Proof. Let (κ, ξ) ∈ X be fixed and we abbreviate ϕξ (k) := ϕ(k; κ, ξ) for all k ∈ Z. Note here that the general solution ϕ of (S) exists due to assumption (ii). Central for our considerations is the difference equation Bk+1 x = Ak x + f˜k (x + ϕξ (k), x + ϕξ (k)) − fk (ϕξ (k));
(5.2f)
5.2 Generalized Hartman–Grobman Theorem
329
thanks to assumption (ii) its general solution φ¯ exists and the same holds for the ˜ Moreover, assumption (i) implies that Qi∗ is invariant general solution ϕ˜ of (S). ˜ and (5.2f). w.r.t. (S), (S) Given a sequence ψ ∈ X1 , we define the Lyapunov–Perron operator
Rκ (ψ; ξ) :=
·−1
−1 f˜n (ψ(n) + ϕξ (n))−fn (ϕξ (n)) Φ(·, n + 1)Qi1∗ (n + 1)Bn+1
n=−∞
+
∞ n=·
i∗ −1 −1 ˜n (ψ(n) + ϕξ (n))−fn (ϕξ (n)) . f Φ− (n+1)B i∗ −1 (·, n+1)P1 n+1 P 1
As in the proof of Theorem 3.5.10 or Lemma 4.3.2 we can establish that the mapping Rκ : X1 × Qi∗ (κ)→ X1 is well-defined and satisfies the uniform Lipschitz ˜1 + L ˜ 2 . Hence, due to (5.2d) we know that Rκ (·; ξ) is a condition lip1 Rκ ≤ νi∗ L contraction uniformly in its parameters. Its unique fixed point is exactly the uniquely determined bounded solution φ∗κ (ξ) : Z → X of (5.2f) with φ∗κ (κ, ξ) ∈ Qi∗ (κ), and φ∗κ (k, ξ) ≤
2νi∗ (M1 + M2 ) ˜1 + L ˜ 2) 1 − 2νi∗ (L
for all (κ, ξ) ∈ Qi∗ .
(5.2g)
(III) We define the mapping Jκ : Xκ → Xκ by Jκ (ξ) := Qi1∗ (κ) + P1i∗ −1 (κ) ξ + φ∗κ κ, Qi1∗ (κ) + P1i∗ −1 (κ) ξ ∈ Qi∗ (κ). ˜ and due to construction the difference Evidently, ϕ(·; ˜ κ, Jκ (ξ)) is a solution of (S), ϕ(·; ˜ κ, Jκ (ξ))−ϕ(·; κ, ξ) solves our initial equation (5.2f). By uniqueness, it follows (5.2g)
¯ κ, φ∗κ (κ, ξ)) ∈ X1 . ϕ(·; ˜ κ, Jκ (ξ)) − ϕ(·; κ, ξ) = ϕ(·; (a) The continuity assertion on the fixed point mapping φ∗κ : Xκ → X1 is not shown in Theorem 3.5.10. Nevertheless, this can be derived using very similar techniques as employed in the proof of Lemma 4.3.4. (b) Due to the definition of Jκ , the claimed estimate in (b) follows from (5.2g). (c) Let φ : Z → X be a solution of (S) with φ(κ0 ) ∈ Qi∗ (κ0 ) for κ0 ∈ Z. Then ˜ and, by construction, φ˜ := ϕ˜ ·; κ0 , Jκ0 (φ(κ0 )) solves the difference equation (S) ˜ the difference φ − φ isbounded. On the other hand, Jκ (φ(κ)) is the unique element of Qi∗ (κ) such that ϕ˜ ·; κ, Jκ (φ(κ)) − ϕ(·; κ, φ(κ)) ∈ X1 . Therefore, the identity ˜ ϕ(·; κ, φ(κ)) = φ implies ϕ˜ ·; κ, Jκ (φ(κ)) = φ˜ and also k → φ(k) = Jk (φ(k)), ˜ which in turn yields that k → Jk (φ(k)) solves (S).
The assumptions required in Hypotheses 5.1.6, 5.2.1 imply some consequences for the dynamics of (S) under the nonhyperbolicity condition (5.2a). For this, we retreat to a coarser splitting (P1i∗ −1 , Pi∗ , Qi1∗ ) of X , i.e., an exponential trichotomy
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5 Linearization
which emphasizes the crucial nonzero central projector Pi∗ . Indeed, by Proposition 4.2.17 and Corollary 4.2.19 we obtain the classical hierarchy of invariant fiber bundles as illustrated in Fig. 4.4. The nonautonomous sets Wji ⊆ X possesses the
representation Wji = (κ, η + wji (κ, η)) ∈ X : (κ, η) ∈ Pji as graphs of uniquely determined continuous mappings wji : X → X fulfilling wji (κ, x0 ) = wji (κ, Pji (κ)x0 ) ∈ Qij (κ) for all (κ, x0 ) ∈ X . Concerning the classical hierarchy of invariant fiber bundles as in Fig. 4.4, only the existence of the general forward solution to (S) was required. However, under the unique existence of backward solutions, we can prove our main result, the so-called generalized Hartman–Grobman theorem, which dates back to Palmer and ˇ sitaˇiˇsvili in the case of autonomous ordinary differential equations. Our version Soˇ is the following extension to nonautonomous implicit difference equations: ˇ sitaˇiˇsvili). Suppose that Hypotheses 5.1.6, 5.2.1 are fulTheorem 5.2.5 (Palmer–Soˇ −1 ˜ filled with (Gn ) for n ∈ {i∗ − 1, i∗ } and the condition Bk+1 Ak ∈ GL(Xk , Xk+1 ), k ∈ Z holds. If beyond the nonhyperbolicity condition (5.2a) one has 2νi∗ (L1 + L2 ) < 1, then there exists a topological conjugation Tk : Xk → Xk between (S) and the Wc -reduced difference equation Bk+1 x = Ak x + P i∗ (k)fk (Pi∗ (k)x + wc (k, x), Pi∗ (k)x + wc (k + 1, x )), (5.2h) where νi∗ is defined in Lemma 5.2.3. Proof. The proof relies on various of our previous results together and for the sake of a simple notation we introduce Pˆ1 := Qi1∗ , Pˆ2 := Pii∗∗ , Pˆ3 := P i∗ −1 , 1
+1 w1 := wii∗∗ +1 = ws ,
w2 := wii∗∗ = wc , w3 := w1i∗ −1 = wu ,
w23 := w1i∗ = wcu
j and Pˆij := n=i Pˆn . In this context, note that wji (κ, 0) ≡ 0 holds on Z and thanks ˜ n ), the mappings wi : X → X fulfill to the spectral gap condition (G j lip2 wji < 1 for (i, j) = (i∗ , i∗ )
(5.2i)
and thus, I + wji (k, ·) : Xk → Xk , k ∈ Z, are homeomorphisms (cf. Theorem B.3.1).
5.2 Generalized Hartman–Grobman Theorem
331
Our general approach to the above linearization problem is to proceed in essentially two steps. After a further preparation, we first decouple (S) and then linearize the resulting equations as far as possible. (0) By Hypothesis 5.2.1 the general (forward) solution ϕ of (S) exists as a continuous mapping and also ϕˆk := ϕ(k+1; k, ·) is continuous. So, for each x ∈ Xk there −1 −1 exists a unique solution x = ϕˆk (x) ∈ Xk+1 to x = Bk+1 Ak x + Bk+1 fk (x, x ). −1 This means that IXk+1 − Bk+1 fk (x, ·) : Xk+1 → Xk+1 is a homeomorphism. Analogously, using the existence of the continuous general backward solution to (S) −1 −1 ensures that IXk + (Bk+1 Ak )−1 Bk+1 fk (·, x ) : Xk → Xk is a homeomorphism. (I) Claim: There is a topological conjugation Uk : Xk → Xk between (S) and the decoupled equation Bk+1 x = Ak x +
Pˆi (k)fk (Pˆi (k)x + wi (k, x), Pˆi (k)x + wi (k + 1, x )).
i∈{1,2,3}
(5.2j) 3 ˆ ˆ We can apply Proposition 5.1.9 with the splitting (P1 , P2 ) to (S) and obtain a topological conjugation to the decoupled difference equation Bk+1 x = Ak x + Pˆ 1 (k)fk (Pˆ1 (k)x + w1 (k, x), Pˆ1 (k)x + w1 (k + 1, x ))
ˆ 3 (k + 1)fk (Pˆ23 (k)x + w23 (k, x), Pˆ23 (k + 1)x + w23 (k + 1, x )) +P 2 by virtue of a homeomorphism Uk1 : Xk → Xk . Next, we are able to again apply Proposition 5.1.9 with the splitting (Pˆ2 , Pˆ3 ) to the Wi−∗ -reduced equation Bk+1 x = Ak Pˆ23 (k)x ˆ 3 (k + 1)fk (Pˆ 3 (k)x + w23 (k, x), Pˆ 3 (k + 1)x + w23 (k + 1, x )), +P 2 2 2 which lives in P1i∗ . Referring to Corollary 4.2.13(a), its general solution exists on P1i∗ as a continuous mapping. Moreover, its nonlinearity fulfills global Lipschitz ˜− estimates with constants ˜− i∗ (c)L1 in the first, and i∗ (c)L2 in the second argument. ˜ i∗ ) guarantees ˜− (c) < 1 (cf. Remark Thus, since our spectral gap condition (G i∗ 4.3.8), Proposition 5.1.9 is applicable and we have a topological conjugation Uk2 to Bk+1 x = Ak Pˆ23 (k)x+
ˆ (k)fk (Pˆi (k)x+wi (k, x), Pˆ (k)x+wi (k+1, x )). P i i
i∈{2,3}
Therefore, the composition Uk (x) := Pˆ1 (k)Uk1 (x) + Uk2 (Pˆ23 (k)Uk1 (x)) provides a topological conjugation between the initial equation (S) and (5.2j). The inverse of ˜ 1 (Pˆ1 (k)x + U ˜ 2 (Pˆ23 (k)x)). Uk is given by Uk−1 (x) := U k k (II) Claim: There is a topological conjugation Vk : Xk → Xk between (5.2j) and the reduced difference equation (5.2h).
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5 Linearization
The basic tool in this step is Lemma 5.2.3, whose assumption (ii) will be verified by means of step (0) and (5.2i). Indeed, we will successively apply Lemma 5.2.3 to the following difference equations
Bk+1 x = Ak
Pˆi (k)x
i∈{1,3}
+
ˆ (k)fk (Pˆi (k)x + wi (k, x), Pˆ (k)x + wi (k + 1, x )) P i i
i∈{1,3}
(5.2k) and its linearization Bk+1 x = Ak
Pˆi (k)x,
(5.2l)
i∈{1,3}
both living in the vector bundle Qi∗ , with miscellaneous nonlinearities: ˆ ˆ ˆ • For fk (x, x ) := i∈{1,3} P i (k)fk (Pi (k)x + wi (k, x), Pi (k)x + wi (k + 1, x )) and f˜k (x, x ) :≡ 0 we obtain a unique continuous mapping Wk : Xk → Xk with Φ(·, κ)Wκ (ξ) − ϕ(·; κ, ξ) ∈ X1
for all (κ, ξ) ∈ Qi∗ .
The assumptions of Lemma 5.2.3 hold with M1 = Ki+∗ M and M2 = Ki−∗ −1 M . • For fk (x, x ) :≡ 0 and f˜k (x, x ) :=
Pˆ i (k)fk (Pˆi (k)x + wi (k, x), Pˆi (k)x + wi (k + 1, x ))
i∈{1,3}
˜ k : Xk → Xk with we obtain a unique continuous mapping W ˜ κ (Wκ (ξ)) − Φ(·, κ)Wκ (ξ) ∈ X1 ϕ ·; κ, W
for all (κ, ξ) ∈ Qi∗ .
The assumptions of Lemma 5.2.3 hold with constants M1 = (Ki+∗ + Ki−∗ −1 )M ˜ 1 = L1 and L ˜ 2 = L2 . and M2 = (Ki+∗ +Ki−∗ −1 )M ; besides from (5.2i) one has L • Finally, defining nonlinearities fk (x, x ) := f˜k (x, x ) Pˆ (k)fk (Pˆi (k)x + wi (k, x), Pˆi (k)x + wi (k + 1, x )) := i
i∈{1,3}
yields a unique continuous Jk : Xk → Xk with ϕ(·; κ, Jκ (ξ)) − ϕ(·; κ, ξ) ∈ X1
for all (κ, ξ) ∈ Qi∗
5.2 Generalized Hartman–Grobman Theorem
333
where obviously Jκ (ξ) = ξ. The assumptions of Lemma 5.2.3 hold with constants M1 = (Ki+∗ + Ki−∗ −1 )M and M2 = (Ki+∗ + Ki−∗ −1 )M ; again by (5.2i) one ˜ 1 = L1 and L ˜ 2 = L2 . deduces Lipschitz constants L Since the bounded sequences X1 form a linear space, this implies the inclusion ˜ κ (Wκ (ξ)) − ϕ(·; κ, ξ) ∈ X1 for all pairs (κ, ξ) ∈ Qi∗ and consequently, ϕ ·; κ, W due to the uniqueness assertion in Lemma 5.2.3, for all (κ, ξ) ∈ Qi∗ .
˜ κ (Wκ (ξ)) = Jκ (ξ) = ξ W
˜ κ (ξ)) = ξ and thus the mappings Wκ , Analogously, one shows the identity Wκ (W ˜ κ are inverse to each other. The remaining properties to show that Wk is a topoW logical conjugation between (5.2k) and (5.2l) directly follow from Lemma 5.2.3. Hence, the desired topological conjugation between (5.2j) and (5.2h) is given by Vk (x) = Pˆ22 (k)x + Wk ([Pˆ1 (k) + Pˆ3 (k)]x). Summarizing the steps (I) and (II), the composition Tk = Vk ◦ Uk is the claimed topological conjugation between (S) and (5.2j).
For hyperbolic linear parts we arrive at Corollary 5.2.6 (Hartman–Grobman). If Hypothesis 5.1.6 holds with 1 ∈ S(A, B; P ), then there exists a topological conjugation Tk : Xk → Xk between (S) and the linear difference equation (L0 ), which is “near identity” in the following sense:
max Tκ (ξ) − ξ Xκ , Tκ−1 (ξ) − ξ X ≤ νi∗ (Ki+∗ + Ki−∗ −1 )M κ
(5.2m)
for all (κ, ξ) ∈ X , where νi∗ is defined in Lemma 5.2.3. Proof. We apply Theorem 5.2.5 with Pi∗ = 0 and rely on the notation of its proof. Our present assumptions guarantee that Uk2 decouples (S) into (5.2j), which degenerates into (5.2k). Accordingly, the topological conjugation between (L0 ) and (S) is given by the composition Tk : Xk → Xk , Tk := Wk ◦ Uk1 . Thus, it remains to establish (5.2m). Referring to Lemma 5.2.3 we know by construction that Tk satisfies the estimate (5.2e). Then the particular choice of the functions fk and f˜k in the above step (II) together with (5.2i) implies one inequality in (5.2m). In addition, replacing ξ ∈ Xκ by Tk−1 (ξ) yields the remaining relation in (5.2m).
Corollary 5.2.7. Let p ∈ N and Tk be the topological conjugation from Theorem 5.2.5 or Corollary 5.2.6. (a) If (S) is p-periodic, then Tκ+p = Tκ for all κ ∈ Z. (b) If (S) is autonomous, then Tκ does not depend on κ ∈ Z. Proof. The assertion follows from Corollary 5.1.10 and Remark 5.2.4(2).
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5 Linearization
Without an invertibility assumption, the classical Hartman–Grobman theorem for mappings is not true. This can be seen from the following simple Example 5.2.8. Consider the scalar autonomous equation x = f (x) with right-hand side f : F → F, f (x) := x2 . Its linearization in the fixed point 0 is Df (0) = 0. If we assume that there exists a homeomorphism T : F → F satisfying the conjugation relation T ◦ f = Df (0)T , then we arrive at T (x)2 = f (T (x)) = Df (0)T (x) = 0 and consequently T (x) ≡ 0. Such a mapping T , however, cannot be one-to-one.
5.3 Solution Conjugation In its classical autonomous formulation, the linearization theorem due to Hartman and Grobman (cf. Corollary 5.2.6) states that, near an equilibrium x∗ , there is a bicontinuous correspondence between the phase portraits of a C 1 -difference equation x = f (x) and its linearization x = Df (x∗ )x – provided Df (x∗ ) has no spectral points on the unit circle. In addition, as we have seen in Example 5.2.8 above, this result requires the additional assumption that f is a homeomorphism. In order to get rid of this invertibility condition, we introduce the weaker concept of a solution conjugation. Rather than finding homeomorphisms between all ˜ k as in Definition 5.1.3, we now search for a homecorresponding fibers Xk and X omorphism between solution spaces. For this purpose, as in Sect. 5.1 we consider general difference equations Hk+1 (x ) = Fk (x, x )
(D)
in X , alongside with equations of the form ˜ k+1 (x ) = F˜k (x, x ) H
˜ (D)
in the extended state space X˜ . Given a discrete interval J ⊆ I, we suppose that SJ and S˜J are spaces of solutions ˜ to (D) ˜ in X˜ equipped with an associated φ : J → X to (D) in X resp. φ˜ : J → X metric topology. Definition 5.3.1. A homeomorphism Ψ : SJ → S˜J is called (SJ , S˜J )-conjugation ˜ if following properties hold: between (D) and (D), ˜ (i) For every solution φ ∈ SJ of (D) the sequence φ˜ := Ψ (φ) solves (D). −1 ˜ ˜ ˜ ˜ (ii) For every solution φ ∈ SJ of (D) the sequence φ := Ψ (φ) solves (D). ˜ are called (SJ , S˜J )-conjugated. If such a homeomorphism Ψ exists, then (S) and (S) In order to investigate the relationship between topological and solution conjugacies, we have to constitute a metric topology on the solution spaces SJ := {φ : J → X| φ solves (D)} ,
˜ . S˜J := φ˜ : J → X| φ˜ solves (D)
(5.3a)
5.3 Solution Conjugation
335
For this, given a sequence (ωk )k∈J of positive reals ωk such that equip SJ with the metric
¯ := d(φ, φ)
ωk
k∈J
k∈J
ωk < ∞, we
¯ dXk (φ(k), φ(k)) ¯ 1 + dXk (φ(k), φ(k))
and similarly we introduce a metric d˜ on S˜J using the weight sequence (ωk )k∈J . ˜ are topologically conjugated along Corollary 5.3.2. Suppose that (D) and (D) ˜ k , k ∈ J. If the uniform limit φ∗ ∈ SJ by virtue of the mappings Tk : Xk → X ˜ ˜ are (SJ , S˜J )relations (5.1c) hold for all solutions in SJ resp. SJ , then (D) and (D) conjugated via the mapping Ψ (k, φ) := Tk (φ(k)) for all k ∈ J. ˜ = T −1 (φ(k)) ˜ Proof. First of all, the inverse of Ψ : SJ → S˜J is given by Ψ −1 (k, φ) k for every k ∈ J and thus Ψ is bijective. We next show that Ψ is continuous. Let φ0 ∈ SJ and suppose the sequence (φn )n∈N converges to φ0 in SJ . This implies pointwise convergence limn→∞ dXk (φn (k),φ0 (k)) = 0 for all k ∈ J and our assumptions on Tk guarantees limn→∞ dX˜k Tk (φn (k)), Tk (φ0 (k)) = 0 for all k ∈ J. Equivalently, for each ε > 0 there exists an N ∈ N such that dX˜ k Tk (φn (k)), Tk (φ0 (k)) < with ω :=
k∈J
ε ω
for all k ∈ J, n ≥ n
ωk . This yields dX˜k Tk (φn (k)), Tk (φ0 (k)) ωk 1 + dX˜ k Tk (φn (k)), Tk (φ0 (k)) k∈J ωk dX˜ k Tk (φn (k)), Tk (φ0 (k)) < ε ≤
˜ (φn ), Ψ (φ0 )) = d(Ψ
k∈J
for all n ≥ N and consequently limn→∞ Ψ (φn ) = Ψ (φ0 ) in S˜J . Thanks to (5.1c) the continuity of Ψ −1 can be shown along the same lines.
Now we retreat to semilinear difference equations (S). Here, differing from the above Hypothesis 5.1.7 or 5.2.1 it is not required that the general solution of (S) exists. Hypothesis 5.3.3. Suppose that the nonlinearity fk : Xk × Xk+1 → Yk+1 fulfills fk (Xk , Xk+1 ) ⊆ im Bk+1 ,
fk (0, 0) = 0 for all k ∈ I
(5.3b)
and that we have the global estimates for j = 1, 2,
−1
Ak L(X sup Bk+1
k∈I
k ,Xk+1 )
< ∞,
−1 Lj := sup lipj Bk+1 fk < ∞. k∈I
(5.3c)
336
5 Linearization
Now we can deduce a conjugation result relating forward and backward solutions of (L0 ) and (S). For bounded nonlinearities we can even estimate the difference of bounded or exponentially decaying solutions: Theorem 5.3.4 (of Cushing). Suppose that Hypotheses 5.1.6 and 5.3.3 are fulfilled. Under the spectral gap condition (Gi ) for one 1 ≤ i < N we equip the solution spaces
L Z± := φ ∈ X± κ,c : φ solves (L0 ) , κ ,c
SZ ± := φ ∈ X± κ,c : φ solves (S) κ ,c
±
for c ∈ Γ¯i with the · κ,c -norm and obtain the following assertions: , SZ + )-conjugated. (a) If I is unbounded above, then (L0 ) and (S) are (LZ+ κ ,c κ ,c − )-conjugated. (b) If I is unbounded below, then (L0 ) and (S) are (LZ− , S Zκ ,c κ ,c (c) The corresponding (LZ± , SZ ± )-conjugation Ψi is a Lipeomorphism with κ ,c κ ,c lip Ψi ≤ (1 − i (c))−1 ,
lip Ψi−1 ≤ 1 + i (c).
−1
fk (x, x ) < ∞ the (d) For nonlinearities with M := sup(k,x,x )∈X ×X Bk+1 mapping Ψi is “near identity” such that for all k ∈ Z± κ one has Ψi (k, φ) − φ(k) Xk
−1 ˜ − φ(k) ˜
Ψi (k, φ)
Xk
max Ki− , Ki+ M ec (k, κ), ≤ ς
− + max Ki , Ki M ec (k, κ), ≤ ς
(5.3d)
where the constant i (c) ∈ [0, 1) is defined in Lemma 4.2.6. Remark 5.3.5. (1) In the hyperbolic situation 1 ∈ S(A, B; P ) one can choose c = 1 and therefore the bounded forward (or backward) solutions to (L0 ) and (S) are connected via the correspondence Ψi . In this sense, Theorem 5.3.4 generalizes the classical Hartman–Grobman theorem from Corollary 5.2.6. (2) A local version of Theorem 5.3.4 relating solutions near the 0 can be deduced using [104, Lemma 1]. Here, one replaces the restrictive global conditions in form of (5.3c) by local Lipschitz conditions as in Hypothesis 4.6.1. Proof. Choose 1 ≤ i < N such that (Gi ) holds and c ∈ Γ¯i . The proof is based on Theorem B.3.6 with X = Z = X± κ,c and we split it into several steps: (a) Above all, thanks to (5.3c) it is easy to see that the linear operator + L : X+ κ,c → Xκ,c ,
−1 (Lφ)(k) := φ (k) − Bk+1 Ak φ(k)
i is bounded and we define S := {φ ∈ X+ κ,c : φ solves (L0 ) and φ(κ) ∈ P1 (κ)}, + which is obviously a linear subspace of Xκ,c.
5.3 Solution Conjugation
337
(I) Claim: The restriction LS := L|S : S → X+ κ,c is one-to-one and onto with
−1
L + S L(X
κ,c )
≤ Ci (c),
(5.3e)
where the constant Ci (c) > 0 is defined in Theorem 3.5.3. i Let φ ∈ X+ κ,c be a solution to (L0 ), i.e., φ ∈ ker L, with φ(κ) ∈ P1 (κ). Then Corollary 3.4.21(a) guarantees φ(κ) = 0 and LS is one-to-one. In order to show + that LS is onto X+ κ,c , we choose an arbitrary g ∈ Xκ,c,B . Setting ξ = 0 in Theorem 3.5.3(a) implies the existence of a solution φ∗ ∈ X+ κ,c to (Lg ) with Qi1 (κ)φ∗ (κ) = 0, which means φ∗ (κ) ∈ P1i (κ). Hence, LS is onto and (3.5b) implies the estimate (5.3e). + (II) Claim: The substitution operator F : X+ κ,c → Xκ,c , −1 fk (φ(k)) (F (φ))(k) := Bk+1
for all k ∈ I
(5.3f)
is well-defined with lip F ≤ (L1 + c L2 ). We first show the Lipschitz condition on F . For this, let φ, φ¯ ∈ X+ κ,c and we obtain ¯ = B −1 fk (φ(k)) − fk (φ(k)) ¯ F (k, φ) − F (k, φ) k+1 i
i
¯ + L2 φ (k) − φ¯ (k) ≤ L1 φ(k) − φ(k) i i
+ ≤ (L1 + c L2 ) φ − φ¯ κ,c ec (k, κ),
(5.3c)
which yields the claimed Lipschitz estimate on F after passing over to the supre+ ¯ mum over Z+ κ . If we set φ = 0 in the above estimate, one obtains F (φ) ∈ Xκ,c such that F is well-defined. (III) Now we are in a position to apply Theorem B.3.6. Thanks to (Gi ) we have (5.3e)
−1 (4.2i)
L lip F ≤ i (c) < 1 S
for all c ∈ Γ¯i
and therefore also the condition (B.3e) holds. The assertion (a) follows. (b) The proof is dual using Corollary 3.4.21(b) and Theorem 3.5.3(b). (c) and (d) immediately follow from Theorem B.3.6.
−1 Corollary 5.3.6. Let m ∈ N and c ∈ Γ¯i . For C m -mappings Bk+1 fk one has:
, SZ + )-conjugation Ψi is C m . (a) If I is unbounded above, c ≤ 1, then the (LZ+ κ ,c κ ,c (b) If I is unbounded below, c ≥ 1, then the (LZ− , SZ − )-conjugation Ψi is C m . κ ,c κ ,c Proof. Referring to Corollary B.3.7 the present proof boils down to find conditions ± m such that the substitution operator F : X± κ,c → Xκ,c from (5.3f) is of class C . −1 m In [26, Lemma 4.8] we have shown that this holds for C -nonlinearities Bk+1 fk , provided the corresponding assumptions above hold true.
338
5 Linearization
By identifying the space of c± -bounded solutions of the linear equation (L0 ) with the bundles P1i (κ) resp. Qi1 (κ) (cf. Corollary 3.1.6), we can formulate the next Corollary 5.3.7 (invariant fiber bundles). With the nonautonomous sets Wi± from Theorem 4.2.9 one has Wi+ (κ) = Ψi (Qi1 (κ)) and Wi− (κ) = Ψi (P1i (κ)), κ ∈ I. Proof. Let κ ∈ I. We have to show that the κ-fiber Wi+ (κ) is the image of Qi1 (κ).
i i + (I) Claim: The mapping qκ : Q1 (κ) → φ ∈ Xκ,c : φ solves (L0 ) given by qκi (ξ) := Φ(·, κ)ξ is a toplinear isomorphism. Obviously, qκi is linear and due to Proposition 3.4.7(a) also well-defined. The unique forward solvability of (L0 ) ensures that qκi is one-to-one. Moreover, qκi is onto, since for every solution φ ∈ X+ κ,c of (L0 ) the pre-image reads as φ(κ); in fact, as in the proof of Proposition 3.4.7(a) one sees φ(κ) ∈ Qi1 (κ). From the dichotomy estimate
i
(3.4g) one deduces qκ ≤ Ki+ and referring to [295, p. 388, Corollary 1.4] also the inverse of qκi is
bounded. + (II) The spaces φ ∈ X+ κ,c : φ solves (S) and Wi (κ) are related by the onto evaluation map evκ φ := φ(κ). Thus, by Theorem 5.3.4 one has Wi+ (κ) = evκ ◦ Ψi ◦ qκi (Q+ i (κ)), i.e., Wi+ (κ) is the Lipschitzian image of the linear space Q+ i (κ). In order to show the remaining backward time assertions relating the fibers P1i (κ) and Wi− (κ), we only remark the following: The backward solutions of (L0 ) and (S) in P1i and Wi− , resp., exist and are uniquely determined. Hence, an analogous statement to (I) can be shown using Proposition 3.4.16(b).
From the above Corollary 5.3.7 (and its proof) we obtained that the global fiber bundles Wi+ and Wi− are essentially the images of the invariant vector bundles Qi1 resp. P1i under the Lipschitzian mapping Ψi from Theorem 5.3.4. Therefore, we can conclude that the fibers Wi± (κ) are globally Lipschitzian graphs – as we have shown in Theorem 4.2.9 using different methods. With regard to the technically complex smoothness proof for the bundles Wi± in Theorem 4.4.6, it suggests itself to ask whether the solution conjugacy Ψi delivered from Theorem 5.3.4 is even a C m -mapping. We have answered this question in Corollary 5.3.6. It guarantees that invariant fiber bundles consisting of forward resp. backward bounded solutions – in particular global stable and unstable fiber bundles – inherit their smoothness from the corresponding difference equation.
5.4 Applications Given stepsize bounds 0 < τ ≤ T and a balancing parameter ∈ (0, 1] with τ = T , let us suppose that we have given a discretization mesh (tk )k∈I with hk := tk+1 − tk ∈ [τ, T ]
for all k ∈ I .
We briefly sketch two applications of our linearization concept from Sect. 5.3.
5.4 Applications
339
5.4.1 Time-Discretized Abstract Evolution Equations Given two Banach spaces X, Y with X ⊆ Y , we deal with an abstract equation (AE)
ut + B(t)u = f (t, u)
in Y under Hypothesis 1.5.4. Consequently, as discretization of (AE) we consider the explicit difference equation x = Ak x + fk (x)
(AΔE)
in X = I × X, like previously in Sect. 2.6.2. Our current goal is to relate the dynamical behavior of (AΔE) to the dynamics of its linear part x = Ak x.
(5.4a)
Note here that, differing from the (generalized) Hartman–Grobman situation discussed in Sect. 5.2, neither the operator Ak = U (tk+1 , tk ) ∈ L(X), k ∈ I , needs not to be invertible, nor does the general forward solution to (AΔE) necessarily extend to a 2-parameter group. Yet, the tools from Sect. 5.3 apply. We simplify our situation by assuming that (AE) admits the zero solution. Corollary 5.4.1. Suppose that Hypotheses 1.5.4, 3.7.4 and 4.9.10 hold with L := sup (ρ) ρ>0
1−r
and that max eωT , E1−r (μT ), bT1−r
≤
√ 2. If the spectral gap condition
μT ) max eω T , eωT T 1−r β−α max {K + , K − } K 2 E1−r (¯ L< α T αT 1−r min {T e ,Te } 2 is fulfilled and we choose a fixed ς∈
1−r
max K + , K − K 2 LE1−r (¯ μT ) max eω T , eωT T1−r , inf
eβhk −eαhk 2 k∈I
,
then the following holds for c ∈ [a1 + ς, b1 − ς]: , SZ + )-conjugated. (a) If I is unbounded above, then (5.4a) and (AΔE) are (LZ+ κ ,c κ ,c − )-conjugated (b) If I is unbounded below, then (5.4a) and (AΔE) are (LZ− , S Zκ ,c κ ,c , SZ ± )-conjugation Ψ : LZ± → SZ ± is a Lipeomorphism satand the (LZ± κ ,c κ ,c κ ,c κ ,c 1/(1−r)
isfying the properties from Theorem 5.3.4. Here, μ ¯ := (KLΓ (1 − r)) from Lemma 1.5.5 and the growth rates a1 , b1 are defined in Lemma 3.7.5.
, μ is
340
5 Linearization
Remark 5.4.2. In case Hypothesis 1.5.4(ii) holds with c = 0, then also Theorem Kb 1−r 5.3.4(d) can be applied with M = 1−r T (cf. Lemma 4.9.5). Proof. The assumptions of Theorem 5.3.4 have to be verified. This, however, can be done as in the proof of Theorem 4.9.9, where the growth conditions (Γi± ) become trivial due to Hypothesis 4.9.10.
5.4.2 Time-Discretized Parabolic Evolution Equations Given a bounded domain Ω ⊆ Rd with Lipschitzian boundary, we return to parabolic evolution equations (SE)
ut + Bu = f (t, u)
under the assumptions stated in Sect. 4.9.3. This guarantees that (SE) can be seen as abstract sectorial equation in Y = L2 (Ω), where Dirichlet boundary conditions yield D(B) = H 2 (Ω) ∩ H01 (Ω) and an interpolation space X = Y 1/2 = H01 (Ω). Moreover, we assume f : R×X → Y satisfies Hypothesis 4.9.10 and is m-times, m ∈ N, continuously Frech´et differentiable in the second argument such that the derivatives map bounded subsets of X into bounded sets – uniformly in t ∈ R. The derivative A :≡ D2 f (t, 0) ∈ L(X, Y ) is supposed to be independent of t ∈ R and σ(B − A) = cl {λn }n∈N , where the eigenvalues λn ∈ R, n ∈ N, of B − A are ordered in a nondecreasing way. We again restrict to constant stepsizes hk ≡ τ on I and apply a linearly implicit Euler method to (SE), which yields a semi-implicit difference equation Bk x = Ak x + fk (x)
(S )
with Ak x := x, Bk := IY + τ (B − A) and fk (x) := τ [f (tk , x) − D2 f (tk , 0)x]. Having this at hand, we can relate the dynamics of the weakly nonlinear problem (S ) to its linearization Bk x = Ak x (L0 ) and obtain the following result on solution conjugation: Corollary 5.4.3. Let α < β be positive reals with (α, β) ∩ cl
1 1 + τ λn
=∅
(5.4b)
n∈N
and that the nonlinearity f : R × H01 (Ω) → L2 (Ω) satisfies Hypothesis 4.9.10 with L := sup lip[I + τ (B − A)]−1 [f (tk , ·) − D2 f (tk , 0)] < ∞. k∈I
5.5 Remarks
341
β−α + − Under the {K , K } L < 2τ and for every fixed spectral gap condition max , the following holds for c ∈ [α + ς, β − ς]: real ς ∈ τ max {K + , K − } L, β−α 2
, SZ + )-conjugated. (a) If I is unbounded above, then (L0 ) and (S ) are (LZ+ κ ,c κ ,c − )-conjugated , S (b) If I is unbounded below, then (L0 ) and (S ) are (LZ− Zκ ,c κ ,c , SZ ± )-conjugation Ψ : LZ± → SZ ± is a Lipeomorphism satisand the (LZ± κ ,c κ ,c κ ,c κ ,c − + fying the properties from Theorem 5.3.4. Here, K , K ≥ 1 are the constants from the exponential 2-splitting associated to assumption (5.4b). Proof. One applies Theorem 3.4.28 in order to get an exponential splitting for the −1 resolvent operator Bk+1 = [I + τ (B − A)]−1 . Then the remaining proof is a direct application of Theorem 5.3.4.
5.5 Remarks Topological conjugation and decoupling: Our notion of a topological conjugation from Definition 5.1.3 is a global version of [437, Definition 1]. In contrast to our usual philosophy in Chaps. 1–4, throughout the Sects. 5.1–5.2 we assumed the existence and uniqueness of backward solutions, i.e., the restrictive fact that (D) generates a 2-parameter group. Without this assumption, there are results on partial decoupling of discrete dynamical systems in Banach spaces due to [21, 22]. They provide a topological decoupling into triangular systems and consequently a linearization of the unstable part (see [397, Sect. 4]). Furthermore, in this context we also like to point out the work of [397, 398] valid in complete metric spaces. Decouplings into tridiagonal form are also considered in [357]. An alternative way to tackle the problem of nonexistent backward solutions is to restrict a difference equation to its (global) attractor on which the general (backward) solution exists (cf. Corollary 1.3.4). Generalized Hartman–Grobman theorem: The classical Hartman–Grobman theorem for homeomorphisms dates back to [199] and expanded into many textbooks on dynamical systems (cf., e.g., [230, 245, 348, 434, 447]). A modern and short proof can be found in [462], or already in [386]. The simple scalar Example 5.2.8 illustrating the failure of the classical Hartman– Grobman theorem in the noninvertible case is taken from [105, p. 159] (see also [133, p. 489, Example D.3]). A corresponding example in R2 can be found in [458, p. 114ff] or [34, Sect. 3]. Yet, topological conjugation results for noninvertible mappings can be found in [387] on the basis of inverse limit conjugates or [105, p. 160, Theorem A.2.3] using an operator formulation of difference equations (see also Sect. 5.3). While invariant foliations have been our basic tool to obtain linearization results, [325] shows that lineariziability of hyperbolic fixed points is actually equivalent to the existence of certain invariant foliations.
342
5 Linearization
For autonomous ordinary differential equations, the geometrically intuitive generalized Hartman–Grobman theorem is due to [443] and [349, 350]. A parallel treatment of the continuous and discrete case is contained in [253]. Generalizations to nonautonomous problems can be found in [458, pp. 116–117, Satz 2.7.7], [459], [206] and [34] under the assumption of a decoupled linear part. For further linearization results near hyperbolic solutions see [93]. In our discrete setting, smooth linearization results are less important than in the case of continuous dynamical systems – here one is interested that a strict solution of a differential equation is transformed into a strict solution of a new equation, and this requires differentiability. Yet, there are good reasons to support the interest of C m -linearization, m > 0: •
For example, two linear diffeomorphisms, both with a hyperbolic sink, one of which being a spiral and the other a focus, are topologically conjugated. Therefore, classification up to C 0 -conjugation can be quite rough. • Smoothness of invariant manifolds should be preserved under conjugation. Consequently, smooth conjugation results require non-resonance conditions on the eigenvalues of the linearization and we only refer to [431] in finite dimensions, or to [140, 402] in C m -Banach spaces. Another approach to smooth linearization is via normal form theory. For such techniques and the formulation of nonresonance conditions in terms of the dichotomy spectrum, see [437] dealing with general nonautonomous difference equations. Solution conjugation: Our final theoretical section can be seen as an outlook to a different approach when dealing with difference (or evolutionary differential) equations. Our main focus was to develop a “geometric theory”, i.e., to understand dynamical behavior in terms of the phase portrait as a subset of the extended state space X . A different proceeding is to formulate difference equations as abstract operator equations in ambient sequence spaces. In particular, this has the advantage that implicit or nonautonomous equations cause essentially no additional difficulties. Once a sufficient understanding of shift and substitution operators is obtained, one has various powerful tools from linear and nonlinear analysis at hand. Corresponding examples include: •
Well-known perturbation results for bounded linear operators yield information on exponential dichotomies (cf. [30,31]) and the dichotomy spectrum (see [378]) • Fixed point theorems to infer attractivity results (see [150, 377]) • The implicit function theorem (cf., e.g., [295, p. 364, Theorem 2.1]) for continuation results of bounded solutions (see [379, 380]) • Bifurcation results based on Fredholm theory (see [381]) As concrete example of such operator theoretical methods, we extended a linearization result of [106, p. 160, Theorem A.2.3], in form of Theorem 5.3.4. It has the additional advantage that no invertibility assumption on the difference equation (S) is required. Moreover, in form of (5.3d) it provides a precise estimate relating the bounded (forward or backward) solutions of (S) and of its linearization (L0 ).
5.5 Remarks
343
As we have seen in Corollary 5.3.7, the Hadamard–Perron Theorem 4.2.9 can also be approached using the linearization result from Theorem 5.3.4 – note that Wi± are Lipschitzian images of the respective vector bundles Qi1 and P1i . With a view to the technically involved proof of the differentiability result for Wi± in Theorem 4.4.6 one might ask for conditions that the solution conjugacy Ψi from Theorem 5.3.4 is of differentiability class C m . Referring to Corollary B.3.7 this essentially depends on the smoothness properties of the substitution operator −1 + F : X+ κ,c → Xκ,c defined in (5.3f). Provided the nonlinearities Bk+1 fk are of class m C , in Corollary 5.3.6 we made use of [26, Lemma 4.8] in order to show that F is a C m -mapping, if c ≤ 1. On the other hand, F needs not to be differentiable in case 1 c (see [26, Example 4.9] for a counterexample). In general, F becomes only differentiable when one continuously embeds its target space X+ κ,c into a larger + space Xκ,d with c d. Then, however, one has to use more subtle tools than the uniform C m -contraction principle from Theorem B.1.5 in the proof of our essential Theorem B.3.6 and Corollary B.3.7. Applications: The behavior of the classical Hartman–Grobman conjugation for ODEs under numerical discretization was investigated in [159] or [171, Corollary 2.3]. In addition, extensions to random dynamical systems are due to [13]. Linearization results for damped wave equations – whose solutions generate a flow – are due to [338]. Related investigations for semiflows generated by parabolic PDEs, namely Hartman–Grobman theorems for reaction-diffusion equations can be found in [43, 312]. The case of FDEs was investigated in [154] including applications to numerics around hyperbolic equilibria. A local version of Theorem 5.3.4 addressing the dynamical behavior of (S) near the trivial solution can be deduced. Thus, we claim that the concept of solution conjugation from Sect. 5.3 also applies to more general evolutionary PDEs by passing over to the time-h-map, instead of imposing global assumptions as in Corollaries 5.4.1 and 5.4.3.
Appendix A
Discrete Inequalities
Throughout, let I be a discrete interval.
A.1 Generalized Exponential Function We naturally equip the set of sequences a : I → F with the operations +, · yielding an algebra. For real-valued a, b : I → R we write a ≤ b, if a(k) ≤ b(k) is satisfied for all k ∈ I ; correspondingly one defines a < b and the uniform difference ab
:⇔
0 < b − a := inf (b(k) − a(k)), k∈I
as well as the supremum a := supk∈I a(k). In addition, intervals are defined as [a, b] := {c : I → R| a ≤ c ≤ b} ,
(a, b) := {c : I → R| a c b}
and similarly for half-open intervals. Merely for notational reasons, it is advantageous to introduce Definition A.1.1. Let k, κ∈ I and a : I → F. The generalized exponential function ea : (k, κ) ∈ I2 : κ ≤ k → F is given by the product ea (k, κ) :=
k−1
a(n) for all κ ≤ k.
n=κ
In case, a(n) = 0 for n ∈ {k, . . . , κ − 1} we extend ea to I2 by ea (k, κ) :=
κ−1
a(n)−1
for all k < κ.
n=k
The next result is elementary and merely an observation for later reference.
345
346
A Discrete Inequalities
Proposition A.1.2 (properties of ea ). Let k, l, κ ∈ I and a, b : I → F. Then one has e1 (k, l) = 1 and the following holds: (a) Semigroup property ea (k, l)ea (l, κ) = ea (k, κ)
for all κ ≤ l ≤ k
(A.1a)
and (A.1a) holds for all k, l, κ ∈ I, if a(n) = 0 for all n ∈ I . (b) Multiplication theorem ea (k, l)eb (k, l) = eab (k, l) for all l ≤ k
(A.1b)
and (A.1b) holds for all k, l ∈ I, if a(n) = 0 for all n ∈ I . (c) In case α ∈ Z and a(n) = 0 for n ∈ {l, . . . , k − 1} it is ea (k, l)α = eaα (k, l) for all l ≤ k
(A.1c)
and (A.1c) holds for all k, l ∈ I, if a(n) = 0 for all n ∈ I . (d) For F = R one has the monotonicity theorem 0
⇒
ea (k, l) ≤ eb (k, l) for all l ≤ k,
eb (k, l) ≤ ea (k, l) for all k ≤ l.
Proof. We omit the elementary proof.
Lemma A.1.3 (asymptotic behavior of ea ). Let κ ∈ I, γ > 0 and a : I → F. γ (a) If I is unbounded above and there exists a K > 1 with |a(k)| ≤ 1 − k1 for all k ≥ K, then limk→∞ ea (k, κ) = 0. (b) If I is unbounded γbelow, a(k) = 0 for k < κ and there exists a K < 1 with |a(k)| ≥ 1 − k1 for all k < K, then limk→−∞ ea (k, κ) = 0. Remark A.1.4. (1) Obviously, assertion (a) holds in case 0 ≤ a 1, whereas (b) is fulfilled for 1 a. In both situations one obtains even exponential convergence. (2) It is an easy consequence of Lemma A.1.3 that the following implications hold true for functions a, b : I → R with 0 < a b: lim sup b(k) k < b − a
⇒
lim sup − a(k) k < b − a
⇒
k→∞
k→−∞
lim e ab (k, κ) = 0,
k→∞
lim e b (k, κ) = 0.
k→−∞
a
(3) If for every ε > 0 there exists an N = N (ε) ≥ 0 such that |ea (k, κ)| ≤ ε
for all k, κ ∈ I, k − κ ≥ N,
then there exist C ≥ 1, α ∈ (0, 1) such that |ea (k, κ)| ≤ Cαk−κ for all κ ≤ k.
A.1 Generalized Exponential Function
347
Proof. In both cases we employ Proposition A.1.2(a). However, since assertion (a) can be shown analogously, we only prove (b). This follows from (A.1a)
|ea (κ, k)| = |ea (κ, K)| |ea (K, k)| ≥ |ea (κ, K)| = |ea (κ, K)|
#
k−1 K−1
$γ
K−1
n−1 γ n
n=k
−−−−−→ ∞
for all κ ∈ I,
k→−∞
−1
since we have limk→−∞ |ea (k, κ)| = limk→−∞ |ea (κ, k)|
= 0.
Lemma A.1.5. Let k1 , k2 , k, κ ∈ I with k1 ≤ k2 . If a, b : I → R are positive sequences, then the following holds: (a) In case a b one has k 2 −1
ea (k, n + 1)eb (n, κ) ≤
n=k1
4 ea (k, κ) 3 e b (k2 , κ) − e b (k1 , κ) . a b − a a
(A.1d)
4 ea (k, κ) 3 e b (k1 , κ) − e b (k2 , κ) . a a − b a
(A.1e)
(b) In case b a one has k 2 −1
ea (k, n + 1)eb (n, κ) ≤
n=k1
Proof. Let k1 , k2 , k, κ ∈ I with k1 ≤ k2 . We derive a preparatory identity, which yields from elementary properties of the exponential function in Proposition A.1.2: k 2 −1
(A.1a)
ea (k, n + 1)eb (n, κ) = ea (k, κ)
n=k1
k 2 −1
ea (κ, n + 1)eb (n, κ)
n=k1
(A.1f) (A.1b)
= ea (k, κ)
k 2 −1 n=k1
e b (n, κ) a
a(n)
= ea (k, κ)
k 2 −1 n=k1
e b (n + 1, κ) − e b (n, κ) a
a
b(n) − a(n)
.
(a) From (A.1f) one immediately gets by “telescopic summation” that k 2 −1
(A.1f)
ea (k, n + 1)eb (n, κ) ≤
n=k1
k2 −1 3 4 ea (k, κ) e b (n + 1, κ) − e b (n, κ) a a b − a n=k1
=
4 ea (k, κ) 3 e b (k2 , κ) − e b (k1 , κ) . a b − a a
(b) This yields analogously to (a) from (A.1f).
348
A Discrete Inequalities
A.2 Gronwall Inequalities We present a sufficiently general discrete version of the Gronwall lemma. Proposition A.2.1 (Gronwall inequality). Let κ ∈ I be given. (a) If the sequences a, b, u : I+ κ → R satisfy b(k) ≥ 0 and k−1
u(k) ≤ a(k) +
b(l)u(l) for all κ ≤ k,
(A.2a)
l=κ
then one has the explicit estimate for all κ ≤ k, k−1
u(k) ≤ e1+b (k, κ)a(κ) +
e1+b (k, l + 1) [a (l) − a(l)] .
(A.2b)
l=κ
(b) If the sequences a, b, u : I− κ → R satisfy b(k) ∈ [0, 1) and u(k) ≤ a(k) +
κ−1
b(l)u(l) for all k ≤ κ,
l=k
then one has the explicit estimate
u(k) ≤ e1−b (k, κ)a(κ) +
κ−1
e1−b (k, l + 1) [a(l) − a (l)]
for all k < κ.
l=k
Remark A.2.2. Inequality (A.2b) is the best possible, in the sense that equality in (A.2a) implies equality in (A.2b). Proof. Since the assertion (a) can be shown similarly we restrict to the proof of the slightly more involved assertion (b). We set for abbreviation
c(k) := a(k) +
κ−1
b(l)u(l) for all k ≤ κ
l=k
and according to our assumptions one has u(k) ≤ c(k), which yields c(k) − c (k) = a(k) − a (k) + b(k)u(k) ≤ a(k) − a (k) + b(k)c(k),
A.2 Gronwall Inequalities
349
hence, one has c(k) ≤ (1 − b(k))−1 (c (k) + a(k) − b (k)) for all k < κ. By mathematical induction in backward time we get c(k) ≤ e1−b (k, κ)c(κ) +
κ−1
e1−b (k, l + 1) [a(l) − a (l)]
for all k < κ
l=k
and due to c(κ) = a(κ), u(k) ≤ c(k) this leads to the assertion.
Proposition A.2.3 (uniform Gronwall inequality). Let κ ∈ I. If a, b, u : I+ κ → R are sequences satisfying u (k) ≤ a(k)u(k) + b(k) for all κ ≤ k,
(A.2c)
then the following holds: (a) In case a(k) ≥ 0 one has u(k) ≤ ea (k, κ)u(κ) +
k−1
ea (k, l + 1)b(l) for all κ ≤ k.
l=κ
(b) In case a(k) ≥ 1 and if there exist reals α1 , α2 , α3 ≥ 0 and N ∈ Z+ 0 with α1 := sup ea (k + N, k), k≥κ
α2 := sup
# then one has u(n) ≤ α1 α2 +
k≥κ
α3 N +1
$
k+N n=k
b(n) , a(n)
α3 := sup k≥κ
k+N
u(n),
n=k
for all κ ≤ k and n ∈ [k, k + N ]Z .
Proof. The proof of (a) is an easy induction and omitted. To deduce (b) we have
ea (κ, l + 1)u (l) − ea (κ, l)u(l) = ea (κ, l)
(A.2c) b(l) u (l) − u(l) ≤ ea (κ, l) a(l) a(l)
for all l ≥ κ and “telescopic” summation yields (note a(k) ≥ 1) ea (κ, n)u(n) − ea (κ, m)u(m) ≤
n−1
b(l) b(l) ≤ ea (κ, m) a(l) a(l) n−1
ea (κ, l)
l=m
l=m
for all n ≥ m ≥ κ which, in turn, implies u(n) ≤ ea (n, m) u(m) +
n−1 l=m
b(l) a(l)
for all n ≥ m ≥ κ.
350
A Discrete Inequalities
Hence, we obtain u(n) ≤ α1 (u(m) + α2 ) for all m ≥ κ and n ∈ [m, m + N ]Z . The estimate leads to k+N k+N (N + 1)u(n) = u(n) ≤ α1 u(m) + (N + 1)α2 m=k
m=k
≤ α1 [α3 + (N + 1)α2 ] and division by N + 1 gives the result.
A.3 Remarks Generalized exponential function: Our use of the generalized exponential function is modeled after its counterpart from the calculus on time scales (cf. [204]). In order to avoid certain uniformity assumptions in discretization theory it turned out advantageous to have time-varying growth rates a(k), i.e., to work with ea (k, l) instead of αk−l ; this has been demonstrated, for instance, in [246]. Gronwall inequalities: Our Proposition A.2.1 extends previous work in [17], [20, Lemma 2.1] and more general versions of the discrete Gronwall inequality can be found in [3, pp. 184–192, Sect. 4.1] or [175, p. 1ff, Chap. 1]. Various versions of the Gronwall lemma and its application to discretizations of parabolic problems are presented in [143]. The uniform Gronwall inequality from Proposition A.2.3 plays a crucial role to study the behavior of dissipativity properties under discretization. The continuous version of Proposition A.2.3(b) is due to [453, Lemma 5.1] and similar variants can be found in [125, Lemma 8.2] or [142, Appendix 2]. The analysis of discretizations for Volterra integral equations with weakly singular kernels requires adapted discrete Gronwall inequalities, which might be considered as discrete counterparts to the Gronwall-Henry inequality (see [432, p. 625, Lemma D.4]). We refer to [45, 120] for corresponding results and to the monograph [67] for a survey.
Appendix B
Fixed Point and Inversion Theorems
In order to introduce measures of noncompactness, we follow an axiomatic path (cf. [35]) and focus on properties needed below. For this, let X be a complete metric space. A mapping χ : 2X → [0, ∞] is called measure of noncompactness on X, if the following conditions are met for A, B ⊆ X: (c0 ) A is bounded ⇔ χ(A) < ∞. (c1 ) (Regularity) A is relatively compact ⇔ χ(A) = 0. (c2 ) (Invariance under closure) χ(A) = χ(clX A) and, if X is a Banach space, then χ(A) = χ(coX A) (invariance under convex hull). (c3 ) (Semi-additivity) χ(A∪B) = max {χ(A), χ(B)} and, if X is a Banach space, then χ(A + B) ≤ χ(A) + χ(B) (algebraic semi-additivity). From these axioms, one deduces the properties (cf. [35, p. 19]): (c4 ) (Monotonicity) A ⊆ B ⇒ χ(A) ≤ χ(B). (c5 ) (Kuratowski property) If Ak ⊆ X, k ∈ N, are closed bounded sets with Ak ⊆ Al for l ≤ k and limk→∞ χ(Ak ) = 0, then for every sequence kn → ∞ in N and un ∈ Akn there is a convergent subsequence of (un )n∈N . Moreover, if X = X1 × X2 is the cartesian product of two metric spaces X1 , X2 with χ1 , χ2 denoting corresponding measures of noncompactness, then (c6 ) χ(A1 × A2 ) = ψ(χ1 (A1 ), χ2 (A2 )) for A1 ⊆ X1 , A2 ⊆ X2 defines a measure of noncompactness on X, where ψ : [0, ∞)2 → [0, ∞) is a convex function satisfying ψ(x1 , x2 ) = 0 if and only if x1 = x2 = 0 (cf. [40, p. 14, Theorem 3.3.2]). In functional analysis, the following three measures of noncompactness are commonly used: Example B.0.1. The Hausdorff measure of noncompactness is defined as α(A) := inf
< N ρ > 0 ∃N ∈ N : ∃x1 , . . . , xN ∈ X : A ⊆ Bρ (xn , X) , n=1
351
352
B Fixed Point and Inversion Theorems
the Kuratowski measure of noncompactness is given by ∃N ∈ N : ∃A1 , . . . , AN ⊆ X with N β(A) := inf ρ > 0 diamX An ≤ ρ : A ⊆ n=1 An and the separation measure of noncompactness reads as γ(A) := sup ρ > 0 ρ = inf dX (xm , xn ) with a sequence (xn ) in A . m=n
Since a ball of radius ρ has diameter at most 2ρ we have α(A) ≤ γ(A) ≤ β(A) ≤ 2α(A)
for all A ⊆ X
(B.0a)
(see [35, p. 26, Remark 3.2]). Moreover, it is shown in [35, p. 17ff, Chap. 2] that α, β, γ are measures of noncompactness satisfying (c0 )–(c5 ); in addition one has α(A) ≤ γ(A) ≤ β(A) ≤ diamX A for all A ⊆ X.
(B.0b)
B.1 Contractive Mappings This section centers around contractions depending on parameters, and the behavior of corresponding fixed points under varying parameters. As prototype result we get Theorem B.1.1 (uniform C 0 -contraction principle). Let X be a complete metric space and Y be a set. If a mapping T : X × Y → X satisfies lip1 T < 1, then there exists a unique function x∗ : Y → X with T (x∗ (y), y) ≡ x∗ (y) on Y and, if Y is a first countable topological space, one has: (a) If T (x, ·) is continuous for all x ∈ X, then x∗ ∈ C(X, Y ). lip2 T (b) If Y is metrizable and lip2 T < ∞, then lip x∗ ≤ 1−lip T. 1
Proof. In view of Banach’s fixed point theorem (see, for example, [295, p. 361, Lemma 1.1]) the existence of a unique function x∗ : Y → X is clear. (a) We only have to verify the continuity of x∗ . Since Y is first countable, it suffices to verify sequential continuity. To this end let (yn )n∈N be a sequence in Y converging to some arbitrarily given y0 ∈ Y . Then we have d(x∗ (yn ), x∗ (y0 )) = d(T (x∗ (yn ), yn ), T (x∗ (y0 ), y0 )) ≤ d(T (x∗ (yn ), yn ), T (x∗ (y0 ), yn )) + d(T (x∗ (y0 ), yn ), T (x∗ (y0 ), y0 )) ≤ lip1 T d(x∗ (yn ), x∗ (y0 )) + d(T (x∗ (y0 ), y), T (x∗ (y0 ), y0 ))
B.1 Contractive Mappings
353
for all n ∈ N and consequently d(x∗ (yn ), x∗ (y0 )) ≤ (1 − lip1 T )−1 d(T (x∗ (y0 ), yn ), T (x∗ (y0 ), y0 ). The continuity of T in the second argument implies our claim for n → ∞. (b) This immediately follows from (B.1a).
(B.1a)
Let X, Y be complete metric spaces equipped with respective measures of noncompactness χX , χY . A mapping f : X → Y is said to fulfill a Darbo condition, if there exists a real k ≥ 0 such that χY (f (B)) ≤ kχX (B)
for all B ⊆ X bounded;
(B.1b)
the smallest possible so-called Darbo constant k is denoted by dar f . In case of a compact mapping f one has dar f = 0. Furthermore, for the measures of noncompactness from Example B.0.1 it is dar f ≤ lip f and we will see in Remark C.2.2 that strict inequality can hold. Corollary B.1.2. Let Y be metrizable and complete. lip2 T (a) If lip2 T < ∞, then dar x∗ ≤ 1−lip . 1T (b) If every mapping T (x0 , ·) : Y → X, x0 ∈ x∗ (Y ), is bounded, then also the fixed point mapping x∗ : Y → X is bounded. Moreover, if T additionally fulfills a Darbo condition with dar T ≤ 1, then dar x∗ ≤ dar T .
Proof. (a) From Theorem B.1.1(b) we know that the fixed point function x∗ is bounded and therefore [35, p. 39, Example 5] implies our claim. (b) We initially show that x∗ maps bounded subsets of Y into bounded sets of X. For this, choose a fixed y0 ∈ Y and as in (B.1a) we deduce d(x∗ (y), x∗ (y0 )) ≤ (1 − lip1 T )−1 d(T (x∗ (y0 ), y), T (x∗ (y0 ), y0 )
for all y ∈ Y.
Hence, our assumptions guarantee supy∈B d(T (x∗ (y0 ), y), T (x∗ (y0 ), y0 )) < ∞ for every bounded subset B ⊆ Y , and consequently x∗ is bounded. Our property (c6 ) implies that χ(B1 × B2 ) := max {χX (B1 ), χY (B2 )}, where B1 ⊆ X, B2 ⊆ Y , defines a measure of noncompactness on the product space X × Y . Suppose B ⊆ Y is bounded. By assumption, there exists a k ∈ [0, 1] with χX (T (B1 , B)) ≤ kχ(B1 × B)
for all B1 ⊆ X bounded,
and since x∗ (B) ⊆ X is bounded, we can deduce (note k ≤ 1) that χX (x∗ (B)) = χX (T (x∗ (B), B)) ≤ kχ(x∗ (B) × B) = k max {χX (x∗ (B)), χY (B)} = kχY (B). This yields our assertion dar x∗ ≤ k.
354
B Fixed Point and Inversion Theorems
Lemma B.1.3. Let X, Y be metric spaces, Z be a first countable topological space and suppose a mapping T : X × Z → Y satisfies supn∈N lip T (·, zn ) < ∞ for all convergent sequences (zn )n∈N in Z. If T (x, ·) : Z → Y is continuous for all x ∈ X, then T is continuous itself. Proof. For sequences (xn , zn )n∈N in X × Z with limit (x0 , z0 ) we get d(T (xn , zn ), T (x0 , z0 )) ≤ d(T (xn , zn ), T (x0 , zn )) + d(T (x0 , zn ), T (x0 , z0 )) ≤ sup lip T (·, zn ) d(xn , x0 ) + d(T (x0 , zn ), T (x0 , z0 )) n∈N
for all n ∈ N. By assumption, both terms on the right-hand side tend to 0 in the limit n → ∞. Since Z is first countable, this implies our claim. Corollary B.1.4. Let X, Y be metric spaces, X be complete and Z be a set. If the mappings f : X × Z → Y , g : Y × Z → X satisfy lip1 f lip1 g < 1, then: (a) For each z ∈ Z there exist unique x∗ (z) ∈ X, y ∗ (z) ∈ Y satisfying the identities y ∗ (z) ≡ f (x∗ (z), z), x∗ (z) ≡ g(y ∗ (z), z) on Z. (b) If Z is a first countable topological space and f (x, ·) : Z → Y , g(y, ·) : Z → X are continuous for each y ∈ Y , x ∈ X, then x∗ : Z → X, y ∗ : Z → Y are continuous. Proof. We define the mapping T : X × Z → X by T (x, z) := g(f (x, z), z). (a) Due to lip1 f lip1 g < 1 we have lip1 T < 1 and Theorem B.1.1 implies the existence of a unique fixed point x∗ (z) ∈ X of T (·, z) for all z ∈ Z. The claim follows, if we set y ∗ (z) := f (x∗ (z), z). (b) Our Lemma B.1.3 guarantees that g is continuous, which implies that also T (x, ·), x ∈ X, is continuous. Theorem B.1.1(a) implies the continuity x∗ : Z → X and, thanks to Lemma B.1.3, also y ∗ : Z → Y is continuous. From now on, let X, Y be Banach spaces. Theorem B.1.5 (uniform Cm -contraction principle). Let m ∈ Z+ 0 and suppose both U ⊆ X, V ⊆ Y are open sets. If a mapping T : cl U × V → cl U satisfies lip1 T < 1,
T ∈ C m (U × V, X),
then there exists a unique x∗ ∈ C m (V, U ) with T (x∗ (y), y) ≡ x∗ (y) on V . Proof. See [87, p. 25, Theorem 2.2].
Corollary B.1.6. Let m ∈ Z+ 0 , Z be a Banach space and U ⊆ X, V ⊆ Z, W ⊆ Y be open subsets. If mappings f : cl U × V → W , g : W × V → cl U satisfy lip1 f lip1 g < 1 and f ∈ C m (U × V, Y ), g ∈ C m (W × V, X), then there exist unique C m -functions x∗ : V → X, y ∗ : V → Y so that y ∗ (z) ≡ f (x∗ (z), z), x∗ (z) ≡ g(y ∗ (z), z) holds on V .
B.2 Compact Mappings
355
Proof. We define the mapping T : cl U × V → cl U by T (x, z) := g(f (x, z), z). Then Theorem B.1.5 is applicable to T yielding a unique C m -function x∗ . By assumption, also y ∗ given by y ∗ (z) = f (x∗ (z), z) is of class C m .
B.2 Compact Mappings In this section, we present existence theorems for solutions of fixed point or further nonlinear problems involving a compact operator. This compactness can be weakened using the notion of a measure of noncompactness χ : 2X → [0, ∞] as introduced on p. 351. Then a mapping T : X → X on a Banach space X is called χ-set contraction, if it fulfills a Darbo condition with dar T ∈ [0, 1), and we denote T as χ-condensing, if χ(T (B)) < χ(B) for any bounded set B ⊆ X for which T (B) is bounded and χ(B) > 0. Examples for χ-set contractions (or χ-condensing maps) are compact or contracting mappings. Theorem B.2.1 (Darbo fixed point theorem). Let C be a nonempty bounded, closed and convex subset of a Banach space X. If T : C → C is a continuous χ-set contraction, then there exists a fixed point of T .
Proof. See [180, p. 133, (C.3)].
Theorem B.2.2 (Schauder fixed point theorem). Let C be a nonempty bounded, closed and convex subset of a normed space X. If a continuous mapping T : C → C has relatively compact image T (C) ⊆ X, then there exists a fixed point of T . Proof. For Schauder’s theorem see [345, p. 470], while the generalization to merely normed spaces can be done using [345, p. 472]. The following result is helpful to find zeros of nonlinear equations. Proposition B.2.3. Let B be a closed ball in an inner product space X. If a continuous map T : B → X has relatively compact image T (B) ⊆ X and T (x), x = 0
for all x ∈ bd B,
then there exists a x0 ∈ B such that T (x0 ) = 0. Proof. We assume the contrary, i.e., T (x) = 0 for all x ∈ B. The continuous mapping T (·), · : X → R does not change sign on bd B and we suppose it is r negative. Then the mapping T¯ : B → bd B, T¯(x) := T (x) T (x) is well-defined and continuous, where r > 0 denotes the radius of B. Moreover, it is easily seen that T¯(B) is relatively compact. Thus, by Theorem B.2.2 there exists a fixed point r ∗ x∗ = T (x ∗ ) T (x ) and we arrive at the contradiction 0 < r2 = x∗ , x∗ =
r r T (x∗ ), x∗ = T (x∗ ), x∗ < 0. T (x∗ ) T (x∗ )
356
B Fixed Point and Inversion Theorems
In the remaining case where T (·), · : X → R is positive, the same contradiction r can be derived using T¯ (x) := − T (x) T (x). With the use of asymptotic fixed point theorems one can get rid of the restriction to self-mappings of bounded sets into itself. Here we focus on the Kuratowski measure of noncompactness β: Theorem B.2.4. If a continuous map T : X → X on a Banach space X is (i) β-condensing and (ii) Compact dissipative, i.e., there is a bounded set B ⊆ X such that, for any n compact set K ⊆ X, there is an N = N (K) ∈ Z+ 0 such that T (K) ⊆ B for all n ≥ N , then there exists a fixed point of T .
Proof. See [344] or [196, Theorem 7].
B.3 Global Inverse Function Theorems Let X, Y, Z be sets. Given a mapping f : X × Y → Z, provided f (·, y), y ∈ Y , is bijective, we denote the mapping (z, y) → f (·, y)−1 (z) by f1−1 : Z × Y → X. The next result is applicable to contractive perturbations of the identity: Theorem B.3.1 (Lipschitz inverse function theorem). Let X be a complete metric space, Y be a set and Z be a metric linear space. If two mappings f, g : X ×Y → Z are such that f1−1 : Z × Y → X exists and one has lip1 f1−1 lip1 g < 1, then also the sum f (·, y) + g(·, y) : X → Z is invertible with lip1 (f + g)−1 1 ≤
lip1 f1−1 . 1 − lip1 f1−1 lip1 g
(B.3a)
Moreover, the mapping (f + g)−1 1 : Z × Y → X satisfies: (a) Provided Y is a first countable topological space and the mappings f1−1 , g(x, ·), x ∈ X, are continuous, then also (f + g)−1 1 is continuous. (b) Provided Y is metrizable with lip2 f1−1 , lip2 g < ∞, then lip2 (f + g)−1 1 ≤
lip1 f1−1 lip2 g + lip2 f1−1 . 1 − lip1 f1−1 lip1 g
(c) Provided X, Z are Banach spaces, Y an open subset of a Banach space and m f1−1 , g are of class C m , then (f + g)−1 1 ∈ C (Z × Y, X).
B.3 Global Inverse Function Theorems
357
Proof. In the first part we suppress the dependence of f, g on the fixed parameter y ∈ Y . We define a mapping T : X × Z → X by T (x, z) := f −1 (z − g(x)) and obtain lip1 T ≤ lip f −1 lip g < 1. Thus, by Theorem B.1.1 there exists a unique function x∗ : Z → X such that x∗ (z) is the unique fixed point of f −1 (z − g(·)) for all z ∈ Z. Because of the equivalence x = x∗ (z)
⇔
x = f −1 (z − g(x))
⇔
z = f (x) + g(x)
the fixed point mapping x∗ is the promised inverse of f + g. In addition, due to the estimate lip2 T ≤ lip f −1 one also gets (B.3a) from the triangle inequality. Now define T : X × Z × Y → X by T (x, z, y) := f1−1 (z − g(x, y), y): (a) The mapping T fulfills the assumptions of Theorem B.1.1(a), yielding that the fixed point mapping x∗ : Y × Z → X is continuous. (b) Follows from Theorem B.1.1(b) since lip(2,3) T ≤ lip1 f1−1 lip2 g + lip2 f1−1 . (c) Is finally a direct consequence of Theorem B.1.5. Theorem B.3.2. Let X be a normed space. If a mapping T : X → X is completely continuous and there exists a γ > 0 such that γ x − x ¯ ≤ x − T (x) − x ¯ + T (¯ x)
for all x, x ¯ ∈ X,
(B.3b)
then IX − T : X → X is bijective with lip(IX − T )−1 ≤ γ −1 . Proof. See [180, p. 130, Corollary (8.6)], where the Lipschitz condition for the inverse of I − T follows immediately from (B.3b). Our next inverse function theorem guarantees that a strongly monotone Lipschitz mapping is bijective with Lipschitzian inverse. Theorem B.3.3. Let X be a Hilbert space and Y be a set. Suppose that a mapping T : X × Y → X fulfills lip1 T < ∞ and there exists a γ > 0 such that 2
x, y), x − x ¯ γ x − x¯ ≤ T (x, y) − T (¯
for all x, x ¯∈X
(B.3c)
and y ∈ Y , then T (·, y) is a Lipeomorphism with lip1 T1−1 ≤ γ −1 and the inverse function T1−1 : X × Y → X satisfies: (a) Provided Y is a first countable topological space and T (x, ·), x ∈ X, is continuous, then also T1−1 is continuous. 2 T) (b) Provided Y is metrizable with lip2 T < ∞, then lip2 T1−1 ≤ 2(1+lip . γ (c) Provided X is a Banach space, Y an open subset of a Banach space and T is of class C m , then T1−1 ∈ C m (X × Y, X). Remark B.3.4. Given a smooth mapping T (·, y) ∈ C 1 (X, X) it is a consequence of the mean value theorem (see [295, p. 342, Corollary 4.3]) that the coercivity 2 condition (B.3c) is implied by γ x ≤ D1 T (ξ, y)x, x for x, ξ ∈ X, y ∈ Y .
358
B Fixed Point and Inversion Theorems
Proof. In the beginning, we suppress the dependence of T on the fixed parameter y ∈ Y . Above all we have lip T > 0, since otherwise T is constant and (B.3c) cannot hold. We begin the proof by showing that T : X → X is bijective, i.e., we show that for each ξ ∈ X there exists a unique x ∈ X with T (x) = ξ. For an arbitrary ε > 0, this in turn is equivalent to the existence of a unique fixed point of the mapping Φξ (x) := εξ − εT (x) + x. We obtain from (B.3c) that 2
2
2
x) = x − x ¯ + ε2 T (x) − T (¯ x) − 2ε T (x) − T (¯ x), x − x ¯ Φξ (x) − Φξ (¯ ¯2 − 2ε T (x) − T (¯ x), x − x ¯ ≤ 1 + ε2 (lip T )2 x − x 2 ¯ for all x, x ¯∈X ≤ 1 + ε(ε(lip T )2 − 2γ) x − x and it is easy to see that the function !(ε) := 1+ε(ε(lip T )2 −2γ) achieves its global 2 minimum 1 − (lipγ T )2 at ε0 := (lipγT )2 . The above estimate guarantees !(ε) ≥ 0 ' 2 for all ε > 0 and in case ε = ε0 one has lip Φξ ≤ !(ε0 ) = 1 − (lipγ T )2 ∈ [0, 1). Hence, Banach’s fixed point theorem (see [295, p. 361, Lemma 1.1]) yields the existence of a unique fixed point x∗ (ξ) ∈ X of Φξ and thus x∗ = T −1 . To establish a Lipschitz condition on this inverse, we pick x, x ¯ ∈ X and ξ, ξ¯ ∈ X with ¯ ξ = T (x), ξ = T (¯ x) and get from the Cauchy-Schwarz inequality (B.3c) % −1 % 2 ¯ %2 = x − x %T (ξ) − T −1 (ξ) ¯ ≤ γ −1 T (x) − T (¯ x), x − x ¯ % % . ¯ x−x ¯ ¯ ≤ γ −1 %ξ − ξ¯% x − x ≤ γ −1 ξ − ξ, % % % % ¯ %, = γ −1 %ξ − ξ¯% %T −1 (ξ) − T −1 (ξ)
which implies lip T −1 ≤ γ −1 and we are done. Now the mapping Φ : X × X × Y → X is given by Φ(x, ξ, y) := εξ − εT (x, y)+ x: (a) By assumption, Φ(·, ξ, y) is a uniform contraction in (ξ, y) ∈ X × Y and Φ fulfills the continuity conditions required in Theorem B.1.1(a). (b) We set ε = γ/(lip1 T )2 and the above estimate for Φξ yields = lip1 Φ ≤
1−
γ2 , (lip T )2
lip(2,3) Φ ≤
γ (1 + lip2 T ). (lip T )2
Having this at hand, we apply Theorem B.1.1(b) to Φ and obtain assertion (b). (c) Follows from Theorem B.1.5.
Theorem B.3.5. Let m ∈ N, X be a Hilbert space and Z be an open subset of a Banach space. If a C m -mapping T : X × Z → X satisfies that for every z ∈ Z ∞ there exists a continuous function ωz : [0, ∞) → (0, ∞) with 0 ωzds(s) = ∞ and
B.3 Global Inverse Function Theorems
359 2
| D1 T (x, z)y, y| ≥ ωz (x) y
for all x, y ∈ X,
(B.3d)
then T (·, z) : X → X, z ∈ Z, is a global C m -diffeomorphism and moreover the inverse T1−1 : X × Z → X is of class C m . Proof. The mapping g : X 2 × Z → X, g(x, y, z) = T (x, z) − y is of class C m . Thanks to D1 g(x, y, z) = D1 T (x, z) we can deduce from [390, Corollary 3.7] that Tz := T (·, z) : X → X is a global C 1 -diffeomorphism for all z ∈ Z. Moreover, from differentiating the identity x ≡ Tz (Tz−1 (x)) on X one obtains DTz (x) ∈ GL(X) for all x ∈ X and the local inverse function theorem (cf., for example, [295, p. 361, Theorem 1.2]) ensures that Tz is a C m -diffeomorphism. On the other hand, the inverse T1−1 : X × Z → X satisfies the identity g(T1−1 (y, z), y, z) ≡ 0 and by the implicit function theorem (see [295, p. 364, Theorem 2.1, p. 361, Theorem 1.2]) we know that T1−1 is a C m -mapping. Our final result is a global version of a tool due to [104, Lemma 1]. Theorem B.3.6. Let X, Z denote Banach spaces and suppose that the mappings L ∈ L(X, Z), F : X → Z satisfy: (i) For a subspace S ⊆ X the map LS := L|S is one-to-one and im LS is closed. (ii) F (0) = 0, F (X) ⊆ im LS and lip F < ∞. If the nonlinearity F fulfills the estimate % −1 % %L % lip F < 1, S
(B.3e)
then the following holds true: (a) There exists a Lipeomorphism Ψ : ker L → {x ∈ X : Lx = F (x)} with % % % lip f −1 , lip Ψ ≤ 1 − %L−1 S
% % % lip f. lip Ψ −1 ≤ 1 + %L−1 S
(B.3f)
(b) Under the assumption M := supx∈X F (x) < ∞ the Lipeomorphism Ψ is “near identity” in the sense of % % %M Ψ (y) − yX ≤ %L−1 S % −1 % % −1 % %Ψ (x) − x% ≤ %L % M S Z
for all y ∈ ker L, for all x ∈ Ψ (ker L).
Proof. Define Y := ker L and T : X × Y → X by T (x, y) := L−1 S F (x) + y. (a) Thanks to our assumptions we have % % (B.3e) % lip f < 1, lip1 T ≤ %L−1 S
lip2 T = 1
and by Theorem B.1.1 there exists a unique mapping Ψ : Y → X satisfying the identity T (Ψ (y), y) ≡ Ψ (y) on Y and the left estimate in (B.3f). Moreover, Ψ (y) solves the equation Lx = F (x). On the other hand, it is straight forward to see that
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B Fixed Point and Inversion Theorems
Ψ is one-to-one, while Ψ being onto can be seen as in [104, p. 435, Remark (b)]. Given Ψ (yi ) = xi , i = 1, 2, its inverse function satisfies % % % % −1 %Ψ (x1 ) − Ψ −1 (x2 )% = y1 − y2 = %x1 − L−1 f (x1 ) − x2 + L−1 T (x2 )% S S % % % lip f x1 − x2 for all x1 , x2 ∈ X, ≤ 1 + %L−1 S which implies the remaining estimate in (B.3f). (b) Let x ∈ ker L and y ∈ Ψ (ker L). The first estimate is an immediate consequence of the identity Ψ (y) − y ≡ L−1 S F (Ψ (y)) on ker L and in order to deduce the second relation, simply set y = Ψ −1 (x) into the first one. Corollary B.3.7. Let m ∈ N. If F ∈ C m (X, Z) holds, then Ψ : ker L → X is m-times continuously differentiable. Proof. Use Theorem B.1.5 instead of Theorem B.1.1 in the above proof.
B.4 Remarks Comprehensive introductions to measures of noncompactness can be found in various texts on nonlinear analysis, and in particular in [35, 40], [280, p. 187] or [5, p. 9ff]. Connections to the essential spectrum of an operator can be found in [192, p. 14, Lemma 2.3.3] or [465, p. 497, Proposition 11.9]. Contractive mappings: The uniform contraction principle from Theorem B.1.1 or B.1.5 appears in a variety of references (e.g., [87, p. 25, Theorem 2.2] or [180, 201, 465]) and turned out to be a very convenient tool in many applications. For instance, Theorem B.1.5 forms the basis to derive the implicit function theorem. However, in various cases the differentiability assumption in Theorem B.1.5 is too strong – in particular in applications to show smoothness of center and pseudostable/-unstable manifolds. Here, substitution operators become only differentiable, if one continuously embeds their ranges in a larger space. Such generalizations of Theorem B.1.5 to scales of Banach spaces can be found in [205, 409, 410, 457]. Compact mappings: The Schauder fixed point theorem is fundamental in nonlinear analysis (cf., e.g., [180]). Our more general version Theorem B.2.2 is valid in normed spaces and taken from [345, p. 472]. In Banach spaces, Theorem B.2.1 clearly implies Theorem B.2.2. The following Proposition B.2.3 generalizes a classical result due to Poincar´e (cf. [95, p. 58, Lemma 7.2]) to the infinite-dimensional setting. Conditions guaranteeing uniqueness in Schauder’s theorem are due to [249] and a generalization to set contractions comes from [441]. The asymptotic fixed point result in Theorem B.2.4 was discovered independently in [196, 344]. Global inverse function theorems: We refer to [1, p. 138, Exercise 2.5K] for a local version of the Lipschitz inverse function theorem given in Theorem B.3.1.
B.4 Remarks
361
A prototype global inversion theorem is due to Banach-Mazur (cf. [465, p. 174, Theorem 4.G]) and states that proper mappings are globally invertible, if and only if they are local homeomorphisms. The result from Theorem B.3.3 can be interpreted as a nonlinear version of the Lax–Milgram theorem (see, e.g., [432, p. 86, Lemma 36.5]), which also inspired its proof. Our Theorem B.3.5 is a corollary of a more general inverse function result. It belongs in the framework of the Hadamard– Levy theorem (cf. [1, p. 131, 2.5.17]) and a survey on such results can be found in [102,111,389,390]. On the other hand, both (B.3c) and (B.3d) can be interpreted as strong monotonicity conditions and an overview of surjectivity results under such assumptions is given in [161, 339]. We remark that coercivity conditions like given in (B.3c) or (B.3d) are sufficient for injectivity (cf. [180, p. 75, (B.1)]). On the other hand, there exists an array of conditions for a mapping to be surjective; among them we mention Minty’s theorem (cf. [180, p. 76, (B.7)]), quasi-boundedness conditions (cf. [180, p. 125, Theorem (5.6)]) or Schauder domain invariance (cf. [180, p. 130, Corollary (8.6)]). Finally, our Theorem B.3.6 is a global and quantitative version of a result due to [104, Lemma 1].
Appendix C
Smooth Mappings and Extensions
C.1 Differentiability Let X, Y, Z be Banach spaces and U ⊆ X, V ⊆ Y be nonempty open subsets. It is an elementary consequence of the mean value inequality (cf. [295, p. 342, Corollary 4.3]) that C 1 -functions satisfy a global Lipschitz condition on each convex compact set. Now we consider a somehow inverse situation. Proposition C.1.1. If f : U → Y is differentiable and lip f < ∞, then one has Df (x)L(X,Y ) ≤ lip f for all x ∈ U . Proof. Let ε > 0 and x ∈ U . Due to the differentiability of f : U → Y there exists a δ = δ(ε) > 0 with f (x + h) − f (x) − Df (x)hY ≤ ε hX for all h ∈ B˙ δ (0), and % % Df (x)hY ≤ f (x + h) − f (x)Y + %− f (x + h) − f (x) − Df (x)h %Y ≤ (lip f + ε) h for all h ∈ B˙ δ (0). X
Setting z :=
h hX
implies Df (x)zY ≤ lip f + ε for all z ∈ bd B1 (0) and thus
Df (x)L(X,Y ) = sup Df (x)zY ≤ lip f + ε for all x ∈ U. zX =1
Since ε > 0 was arbitrary, this gives the assertion.
The following result implies that completely continuous and differentiable mappings have compact derivatives. Proposition C.1.2. If f : U → Y is differentiable in x0 ∈ U and dar f < ∞, then one has dar Df (x0 ) ≤ dar f . Proof. Let ε > 0, x0 ∈ U and suppose χX , χY denote measures of noncompactness on X, Y , respectively. Due to the differentiability of f there exists a δ = δ(ε) > 0 with f (x + h) − f (x) − Df (x)hY ≤ ε hX for all h ∈ Bδ (0). Now let us
363
364
C Smooth Mappings and Extensions
assume that C ⊆ X is a nonempty bounded set. We can find a ρ > 0 such that ¯ρ (0) and so λC ⊆ B ¯δ (0) for λ = δ/ρ. Therefore, from (c3 ) we have C⊆B χY (Df (x0 )(λC)) ≤ χY (f (x0 + λC)) + 2εδ ≤ dar f χX (λC) + 2εδ, which implies χY (Df (x0 )C) ≤ dar f χX (C) + 2ερ and passing to the one-sided limit ε " 0 yields our claim. Theorem C.1.3 (chain rule). Let m ∈ N. If f : U → Z, g : V → Y are of class C m with g(V ) ⊆ U , then also the composition f ◦ g : V → Z is m-times continuously differentiable and for l ∈ {1, . . . , m}, x ∈ V the derivatives possess representations as a so-called partially unfolded derivative tree Dl (f ◦ g)(x) =
l−1 l−1 j
j=0
Dj [Df (g(x))] · Dl−j g(x)
(C.1a)
and as a so-called totally unfolded derivative tree Dl (f ◦ g)(x)x1 · · · xl =
l
(C.1b)
Dj f (g(x))D#N1 g(x)xN1 · · · D#Nj g(x)xNj
j=1 (N1 ,...,Nj )∈P < (l) j
for any x1 , . . . , xl ∈ X. Proof. A proof of (C.1a) follows by an easy induction argument (cf. [435, p. 266, Satz B.3]), while (C.1b) is shown in [408, Theorem 2].
C.2 Smooth Norms and Extensions Let X, Y be normed spaces. The existence of retractions or smooth bump functions on X is of crucial importance to deduce local results from global ones via extending functions. The next tool is sufficient in a Lipschitz setting. ¯1 (0, X), Lemma C.2.1. The so-called radial retraction rX : X → B rX (x) :=
x
for x ≤ 1,
1 x x
for x > 1
satisfies lip rX ∈ [1, 2] and for an inner product space X one has lip rX = 1. Remark C.2.2. Although [321, p. 83, Proposition 5.7] works with the Kuratowski measure of noncompactness β, only the properties (c2 ) and (c4 ) are needed to deduce
C.2 Smooth Norms and Extensions
365
dar rX =
0
for dim X < ∞,
1
otherwise.
Proof. In a general normed space X it is clear that lip rX ≥ 1 holds. On the other side, lip rX ≤ 2 is shown for example in [9, p. 262, Lemma (19.8)]. The assertion for an inner product space is due to [123]. Example C.2.3. (1) It is lip rC(Ω) = 2, where Ω is a compact subset of a locally compact space (see [243, Corollary 4]). (2) For every p ∈ (1, ∞) one has lip r p (N) < 2, whereas lip r p (N) = 2 for p ∈ {1, ∞} (see [243, Corollary 4]). Moreover, it is shown in [242] that (1 + tp(p−1) )1/p (1 + tq )1/q , t∈[0,1] 1 + tp
lip r p (N,R2 ) = max
where
1 1 + = 1. p q
(3) Let k ∈ N, p ∈ [1, ∞]. On the Lebesgue space Lp (Ω) we get lip rLp (Ω) < 2 for p ∈ (1, ∞) and lip rLp (Ω) = 2 in the limit cases p ∈ {1, ∞}. The same holds in the Sobolev spaces W k,p (Ω) (cf. [243, Corollary 4]) and results for such spaces of functions with values in Banach spaces can also be found in [243]. The next proposition easily follows from the Kirszbraun–Valentine extension theorem (cf. [455, Corollary]). For an alternative proof in the case of nonexpanding mappings see also [396, Theorem 5]. Proposition C.2.4. Let X, Y be Hilbert spaces. If S ⊆ X a nonempty set, then for any function F : S → Y with lip F < ∞ there exists an extension F¯ : X → Y with F (x) ≡ F¯ (x) on S and lip F¯ = lip F . Proof. Define the mapping T : S × Y → X × Y by T (x, y) := (0, F (x)), which satisfies lip T ≤ lip F . Then [455, Corollary] yields a global extension T¯ : X × Y → X × Y of T with T¯ (x, y) = T (x, y) for all (x, y) ∈ S × Y and lip T¯ ≤ lip T . If Π2 : X × Y → Y is the projection Π2 (x, y) := y, then F¯ (x) := Π2 T¯ (x, 0) satisfies the assertion. A simpler, yet more quantitative and explicit situation occurs when extending functions defined on balls in general linear spaces X, Y . For this, given a neighborhood Ω ⊆ X × Y of 0, it is convenient to introduce the condition ¯ρ (0, X) × B ¯ρ (0, Y ) ⊆ Ω B
(BΩ (ρ))
needed in the following Proposition C.2.5 (Lipschitzian extension). Let (Z, d) be a metric space and let Ω ⊆ X × Y denote a neighborhood of 0. If F : Ω → Z is a locally Lipschitzian mapping, i.e., for each r > 0 fulfilling (BΩ (r)) one has !i (r) := lipi F |Br (0,X)×Br (0,Y ) < ∞
for all i = 1, 2,
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C Smooth Mappings and Extensions
then for ρ > 0 satisfying (BΩ (ρ)) there exists a mapping F ρ : X × Y → Z with the following properties: (a) F ρ (x, y) = F (x, y) for all x ∈ Bρ (0, X), y ∈ Bρ (0, Y ). (b) One has the global Lipschitz estimates lip1 F ρ ≤ !1 (ρ) lip rX ,
lip2 F ρ ≤ !2 (ρ) lip rY .
Remark C.2.6. If the mapping F satisfies a Darbo condition, then Remark C.2.2 and [35, p. 39, Proposition 5.3(b)] implies dar F ρ ≤
0
for dim X × Y < ∞,
dar F
otherwise.
Proof. Choose a ρ > 0 such that (BΩ (ρ)) holds and define the extension F ρ : X × Y → Z,
F ρ (x, y) := F ρrX xρ , ρrY yρ .
By definition of the radial retractions rX , rY , the arguments of F are contained in the ball Bρ (0) × Bρ (0) ⊆ Ω and F ρ is defined on X × Y . Moreover, assertion (a) is valid. In order to establish assertion (b), we choose x, x¯ ∈ X and obtain # $ d(F ρ (x, y), F ρ (¯ x, y)) = d F ρrX xρ , ρrY yρ , F ρrX xρ¯ , ρrU yρ % % % % ¯ ≤ !1 (ρ)ρ %rX ( xρ ) − rX ( xρ¯ )% ≤ !1 (ρ) lip rX x − x for all y ∈ Y , which yields the first claimed global Lipschitz-estimate for F ρ . Since the second one follows along the same lines, the proof is finished. Our next aim is to deduce a counterpart to the previous Proposition C.2.5 for differentiable functions. Thus, from now on suppose X, Y are Banach spaces. In a first step it is important to have information about differentiability properties of norms on Banach spaces. Here, the following notion is appropriate: Definition C.2.7. We say (X, ·) is a C m -Banach space, m ∈ N ∪ {∞}, provided ∗ there exists a norm · equivalent to · such that the norm mapping nX : X \ {0} → [0, ∞),
nX (x) := x∗
is m-times continuously differentiable. Example C.2.8. Let I be an unbounded discrete interval. Given a nonempty infinite set Ω we consider spaces of real-valued functions or sequences. (1) !1 (I) canonically equipped with x := k∈I |xk | is not a C 1 -Banach space (see [151, p. 314, Fact 10.5] and [284, p. 158, 14.11(1)]).
C.2 Smooth Norms and Extensions
367
(2) !∞ (I) with the natural norm x := supk∈I |xk | is not a C 1 -Banach space (see [151, p. 347, 10.4]). (3) c0 (I) is the space of sequences x : I → R such that {t ∈ I : |x(t)| > ε} is finite for every ε > 0. The norm x := supt∈I |x(t)| makes it a C ∞ -Banach space (cf. [116, p. 189, Theorem 1.5]). However, the derivative Dnc0 (Ω) of the norm mapping is not globally Lipschitz [284, p. 164, Corollary 15.8]). (4) C[0, 1] equipped with norm x := supt∈[0,1] |x(t)| is not a C 1 -Banach space (see [151, p. 314, Fact 10.5] and [284, p. 158, 14.11(1)]). (5) We equip the Lebesgue space Lp (Ω), p ≥ 1, with its canonical norm and obtain (see [116, p. 184, Theorem 1.1 and p. 222, Corollary 4.11]): If p ∈ N is even, then Lp (Ω) is a C ∞ -Banach space. More precisely, the norm mapping nLp (Ω) is C ∞ off 0, Dp−1 npLp (Ω) is continuous, Dp npLp (Ω) is constant and Dp+1 npLp (Ω) vanishes identically. • If p ∈ N is odd, then Lp (Ω) is a C p−1 -Banach space. More precisely, nLp (Ω) is C p−1 off 0 and Dp−1 npLp (Ω) is locally Lipschitz; in addition, every infinitedimensional Lp (Ω) is not a C p -Banach space. • If p ∈ N, then Lp (Ω) is a C [p] -Banach space. More precisely, nLp (Ω) is C [p] and D[p] npLp (Ω) is (p − [p])-H¨older; moreover, for every infinite-dimensional Lp (Ω) there exists no norm equivalent to nLp (Ω) whose [p]-th order derivative is locally α-H¨older with α > p − [p]. •
For norms it is sufficient to show differentiability on spheres only. Proposition C.2.9. If there exists an r > 0 such that the norm mapping nX is differentiable in every x ∈ bd Br (0), then X is a C 1 -Banach space with DnX (x)L(X,R) ≤ 1 for all x ∈ X \ {0}.
(C.2a)
Proof. Due to its positive homogeneity, the norm mapping nX is differentiable off {0} (cf. [151, p. 242]) and [151, p. 244, Corollary 8.5] implies that nX is continuously differentiable in every x ∈ X \ {0}. By the lower triangle inequality one has lip nX ≤ 1, thus, Proposition C.1.1 implies (C.2a). Proposition C.2.10. Hilbert spaces are C ∞ -Banach spaces. Proof. The norm on a Hilbert space X is a composition√of C ∞ -functions, namely q : X → R, q(x) := x, x and the square root function · : (0, ∞) → R. Proposition C.2.11. Finite-dimensional spaces are C ∞ -Banach spaces. Proof. Let X be a linear space over F with basis {e1 , . . . , ed }, d ∈ N. For vectors d d x, y ∈ X with representations x = i=1 ξi ei , y = i=1 ηi ei and unique coefd ficients ξi , ηi ∈ F we observe that x, y := i=1 ξi ηi defines an inner product on X. Since all norms on finite-dimensional linear spaces are equivalent (cf. [295, p. 38, Corollary 3.14]), Proposition C.2.10 yields the assertion.
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C Smooth Mappings and Extensions
Next we provide criteria for the existence of differentiable norms. Proposition C.2.12. Let m ∈ N and S be a set. (a) If X is separable, then X is a C 1 -Banach space, and the converse holds on separable spaces X. (b) If X is a C 1 -Banach space, then X is reflexive. (c) If c0 (S) → X → Y is a short exact sequence1 and Y admits a C m -norm, then X is a C m -Banach space. (d) If there exists a continuous injection X → Y with closed range, and if Y admits a C m -norm, then X is a C m -Banach space. Proof. (a) See [116, p. 50, Theorem 3.1(i) and p. 51, Corollary 3.3]. (b) See [151, p. 244, Theorem 8.6]. (c) See [284, p. 143, Proposition 13.17]. (d) Suppose T ∈ L(X, Y ) is one-to-one, i.e., ker T = {0}. Then the mapping T ·Y is a C m -norm on X. It remains to establish that ·∗ := T ·Y is equivalent ∗ to the given norm on X. We obtain x ≤ T L(X,Y ) xX , x ∈ X, and in order to establish the converse inequality, we observe that T : X → im T is bijective with T −1 ∈ L(im%T, X)% (cf. [244, p. 167, 5.21]) and thus we eventually obtain to estimate xX ≤ %T −1 %L(im T,X) T xY . Example C.2.13. Let Ω ⊆ RN be a domain satisfying the cone condition. If the reals p, q ∈ [1, ∞) and k ∈ N fulfill one of the conditions q kq ≤ N and q ≤ p ≤ NN−kq , or • kq = N and q ≤ p < ∞,
•
then the Sobolev space W k,q (Ω) inherits its smoothness properties from Lp (Ω) (cf. Example C.2.8(5)); this is a consequence of Proposition C.2.12(d) and the Sobolev embedding theorem (cf. [432, p. 608, Theorem B.2]). Using the following result one obtains that smoothness properties for spaces of real-valued functions can be lifted to spaces of RN -valued functions. m Proposition C.2.14. Let m ∈ Z+ 0 . The product ' X × Y of C -Banach spaces X, Y
is also C m w.r.t. the norm (x, y)X×Y :=
2
2
xX + yY .
Proof. See [284, p. 143, Proposition 13.17(1)].
The following result prepares smooth extensions. It addresses bump functions and provides a kind of optimality in their Lipschitz constant (cf. Fig. C.1). Lemma C.2.15 (bump functions). For every real s > 1 there exists a function ϑ ∈ C ∞ (R) such that ϑ(t) ≡ 1 on (−∞, 1], ϑ(t) ∈ [0, 1] for t ∈ [1, 2], ϑ(t) ≡ 0 on [2, ∞) and Dϑ(t) ∈ [−s, 0] for t ∈ R, as well as tϑ(t) ∈ [0, s] for all t ≥ 0. The space c0 (S) is the closure of all functions on S with finite support in the Banach space of all bounded functions on S equipped with the supremum norm.
1
C.2 Smooth Norms and Extensions
369
ϑ(t)
1
2 − 1s
2
t
Fig. C.1 The bump function from Lemma C.2.15
Proof. For reals r > 0 consider the bump function ωr : R → R, r for |t| < 12 , exp − 1−4t 2 ωr (t) := 0 for |t| ≥ 12 t ∞ of class C ∞ (cf. [1, p. 94]). Then ϑr : R → R, ϑr (t) := −∞ ωr / −∞ ωr is an 1 1 increasing C ∞ -function with ϑr (t) ∞= 0 for t ≤ − 2 , ϑr (t) = 1 for t ≥ 2 and the derivative Dϑr (t) = ωr (t)/ −∞ ωr . From the properties of ωr we see that ∞ mint∈R Dϑr (t) = 0 and m(r) := maxt∈R Dϑr (t) = exp(−r)/ −∞ ωr . It is not difficult to prove that m : (0, ∞) → R is a strictly increasing continuous function with limr0 m(r) = 1. Thus, for every s > 1 there exists a r∗ > 0 such that m(r∗ ) ≤ s, and therefore Dϑr∗ (t) ∈ [0, s] for all t ∈ R. In conclusion, the function ϑ given by ϑ(t) := ϑr∗ ( 32 − t) satisfies the assertions. Yet, it remains the establish the final estimate. By construction, the minimal slope of ϑ is greater − 2) or1 equal than −s. Due to ϑ(2) = 0, this yields ϑ(t) ≤ −s(t 1 for all t ∈ 2 − , 2 (see Fig. C.1). Thus, it is tϑ(t) = st(2 − t) ≤ 2 − for all s s t ∈ 2 − 1s , 2 and since also tϑ(t) ≤ t ≤ 2 − 1s holds for t ∈ 0, 2 − 1s , we have deduced the desired inequality tϑ(t) ≤ 2 − 1s for all t ≥ 0. Having this at hand, the elementary estimate 2 − 1s ≤ s, s > 1, yields our claim. Proposition C.2.16 (cut-off functions). If X is a C m -Banach space, then for all reals ρ > 0 and s > 1 there exists a C m -function χρ : X → R such that χρ (x) ≡ 1 for x ≤ ρ, χρ (x) ∈ (0, 1) for x ∈ (ρ, 2ρ), χρ (x) ≡ 0 for x ≥ 2 and ¯sρ (0, X) for all x ∈ X. Dχρ (x) ≤ ρs , as well as xχρ (x) ∈ B with a bump Proof. Given ρ > 0 we define χρ : X → R by χρ (x) := ϑ x ρ function ϑ from Lemma C.2.15. In a neighborhood of 0 we have χρ (x) ≡ 1. Outside this set, by assumption, χρ is the composition of C m -mappings and therefore of class C m . The bound for the derivative follows from the chain rule in Theorem C.1.3 yielding Dχρ (x) ≤
1 ρ
% % (C.2a) s x % x % Dϑ ρ %Dn ρ % ≤ ρ
for all x ∈ X
370
C Smooth Mappings and Extensions
using Lemma C.2.15. It is a consequence of the final estimate in Lemma C.2.15 that $ # x x χρ (x)x = ρϑ ρ ρ ≤ ρs for all x ∈ X holds and we are done. After these preparations we finally arrive at a smooth version of Proposition C.2.5. Proposition C.2.17 (Cm -extension). Let Z be a Banach space, suppose that X, Y are C m -Banach spaces and let Ω ⊆ X ×Y be an open neighborhood of 0. Provided F : Ω → Z is a C m -mapping so that for each r > 0 with (BΩ (r)) one has !i (r) := lipi F |Br (0,X)×Br (0,Y ) < ∞
for all i = 1, 2,
then for reals s > 1 and ρ > 0 satisfying (BΩ (sρ)) there exists a C m -mapping F ρ : X × Y → Z with the following properties: (a) F ρ (x, y) = F (x, y) for all x ∈ Bρ (0, X), y ∈ Bρ (0, Y ). (b) One has the global Lipschitz estimates lip1 F ρ ≤ (1 + 2s)!1 (ρ),
lip2 F ρ ≤ (1 + 2s)!2 (ρ).
(c) If the derivatives Dn F , n ∈ {0, . . . , m}, are bounded on Bsρ (0, X) × Bsρ (0, Y ), then the same holds for F ρ : X × Y → Z. Remark C.2.18. In case F satisfies a Darbo condition, then Remark C.2.2 in combination with [35, p. 39, Proposition 5.3(b)] implies dar F ≤ (1 + 2s) ρ
0
for dim X × Y < ∞,
dar F
otherwise.
In particular, this means that compactness of F persists when passing over to the modified mapping F ρ ; this observation also applies to Remark C.2.6 above. Proof. For a given s > 1 choose a ρ > 0 such that the inclusion (BΩ (sρ)) holds. ¯ sρ (0) Thanks to the final inclusion in Proposition C.2.16 this guarantees χρ (x)x ∈ B and also (χρ (x)x, χρ (y)y) ∈ Ω for all x ∈ X, y ∈ Y . Therefore, we can define the extension F ρ : X × Y → Z by F ρ (x, y) := F (χρ (x)x, χρ (y)y), which is of class C m and satisfies assertions (a) and (c). In order to establish the remaining claim (b), we consider the C m -function θρ : X → X, θρ (x) := χρ (x)x. By the product rule (cf. [295, p. 336]), Proposition C.2.16 and Lemma C.2.15 one has the estimate Dθρ (x) ≤ Dχρ (x)x + |χρ (x)| ≤
s x + |χρ (x)| ≤ 2s + 1 ρ
for all x ∈ X
and consequently lip θρ ≤ 2s + 1 by the mean value inequality (cf. [295, p. 342, Corollary 4.3]). This yields the Lipschitz estimate F ρ (x, y) − F ρ (¯ x, y) ≤ !1 (sρ) θρ (x) − θρ (¯ x) ≤ (1 + 2s)!1 (sρ) x − x ¯
C.3 Remarks
371
for all x, x¯ ∈ X, y ∈ Y and the claimed condition for lip1 F ρ . Similarly, one shows the second Lipschitz estimate yielding lip2 F ρ .
C.3 Remarks Differentiability: Due to the lack of a reference for Proposition C.1.1, we gave a direct proof. Yet, Proposition C.1.2 is from [5, p. 25, Theorem 1.5.9] and somewhat illustrates parallel proof strategies for assertions on Lipschitz and Darbo conditions. The higher order chain rule from Theorem C.1.3 is an important technical tool needed in various contexts, e.g., to prove smoothness properties of invariant manifolds. On the one hand it is necessary to rigorously compute the higher order derivatives of compositions of maps, the so-called “derivative tree”. It turned out to be advantageous to use two different representations of the derivative tree, namely a “totally unfolded derivative tree” (in Taylor approximations or to compute explicit bounds of higher order derivatives) and besides a “partially unfolded derivative tree” to elaborate certain induction arguments in a recursive way. On the other hand, one needs explicit versions of the chain rule to get Taylor approximations of invariant manifolds (see [383]). Unfortunately, but due to its complexity, certain versions of the chain rule found in the literature need a little care. Nonetheless, a variety of different formulations for the higher order chain rule from Theorem C.1.3 can be found in [407, 408]. Smooth norms and extensions: Estimates for the Lipschitz constants of the radial retraction (also called radial projection) appeared for the first time in [123]. Indeed, the minimal Lipschitz constant 1 turned out to be a characteristic quantity to characterize inner product spaces (cf. [112]). The problem of finding Banach spaces which admit a smooth norm found a certain attention over the last decades. Thus, a comprehensive approach to this topic can be found in the monographs [116], [151, Chaps. 8–10], [284] or the book [451]. Here, also deeper questions on the geometry of Banach spaces are addressed. Referring to Example C.2.8 it is particularly interesting, whether such smoothness results also hold for norms in spaces of vector-valued (say Bochner-integrable) functions. This issue is discussed in [451, pp. 25–26, Theorem 2.2.9]. For a brief outline on smooth norms see [1, pp. 385–388, 5.5B]. Finally, classes of Orlicz spaces with a C m -norm can be found in [317]. It is shown in [167, Applications (ii)] that Λ-spaces (closures of C ∞ in the H¨oldertopology) admit a norm that is C ∞ away from the origin.
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Index
2-parameter group, 4, 5, 18, 22, 33, 99, 319,
339 continuous, 4 2-parameter semiflow, 26, 27, 32, 36, 74, 79, 299, 300 discretized, 24ff, 25, 30 2-parameter semigroup, 1, 2ff, 3 asymptotically compact, 12ff, 12 asymptotically smooth, 34 autonomous, 21ff, 21 backward, 4, 52 bounded, 4 compact, 12 continuous, 4 contracting, 15ff, 15 dissipative, 18, 58ff bounded, 18 compact, 18 uniformly bounded, 18 eventually bounded, 13 eventually compact, 12 generator, 5 linear, 4, 27, 173 parameter-dependent, 4, 53, 56, 57 periodic, 21ff, 21 uniformly contracting, 15 φ-neighborhood, 61 θ -method, 38, 40, 48, 58, 60, 71, 73, 78, 79, 92, 94, 171, 190, 295 2-stage, 38, 41, 51, 71, 78, 190 a-stable backward, 62, 103, 131, 135, 145, 148, 156 forward, 62, 64, 67, 102–104, 107, 124, 131, 135, 145, 147, 156
admissible, 96, 151ff, 184 on semiaxes, 151, 155 on the whole axis, 153, 157
asymptotic phase, 188, 214ff, 214, 224, 312ff, 312 backward, 225 existence, 225 forward, 225, 229, 314 periodic, 230 attractive, 62, 63 exponentially, 275 globally, 62, 104 pullback, 38, 63, 64 uniformly, 63 uniformly, 62 attractor, x, 16ff, 17 cocycle, 35 exponential, 35, 314 global, 1, 16ff, 16–19, 23, 25, 26, 30, 32, 34ff, 34, 38, 64, 65, 72, 79, 87, 89, 92, 106, 107, 187–189, 193, 287, 289, 290, 296, 298, 299, 307, 310, 314 and inertial bundles, 278 connectedness, 20 dynamical characterization, 17 of linear equations, 106 local, 17, 64 properties, 19 pullback, 35 uniform, 34 upper-semicontinuity, 20 ball, xviii Banach space, C m -, 258, 366 base flow, 33 base space, ix, 33, 35, 187 Bohl exponent, lower forward, 182 exponent, upper forward, 182 spectrum, 182 spectrum, forward, 182
393
394 bounded growth, 103, 131, 135 backward, 103 forward, 103, 131 Butcher tableau, 40, 41, 50
chain rule, 249, 272, 364 characteristic exponent, 38, 62, 107, 125 backward lower, 61 backward upper, 61 forward lower, 61 forward upper, 61 cocycle, 33 compactification time, 12 companion matrix, 185 complete, asymptotically, 188, 275, 314 complexification, of a real space, xx, 111 condition dissipativity, 25, 46–50 growth, 196, 204, 215, 261, 279, 298, 309, 311 invariance, 115–117 regularity, 115, 116, 118, 119, 122, 136, 170, 182 strong, 115, 118, 119, 269 spectral gap, 195, 196, 199, 202, 209, 223, 224, 226ff, 232, 233, 250, 261, 280, 281, 291, 292, 298, 309, 311, 313, 314, 339, 341 strengthened, 207, 208, 212, 225, 248, 251, 256, 261, 275, 278, 330 spectral ratio, 282 summability, 58, 59, 192 conjugation linear, 108, 135, 182, 184, 320 solution, 334ff, 334 topological, x, 317, 318ff, 320, 341 local, 320 periodic, 326, 333 contraction principle uniform C 0 -, 352 uniform C m -, 354 Crank–Nicholson method, 41
Darbo condition, 15, 34, 54, 74, 192, 353, 355, 366, 370, 371 constant, 15, 353 DDE, see differential equation, delay decoupling partial, 341 topological, 312, 317, 318ff, 324 delay endomorphism, 90
Index derivative tree partially unfolded, 231, 240, 364 totally unfolded, 231, 240, 364 dichotomy exponential, 128ff, 134, 135, 149, 154, 159, 165, 173, 181–184, 195, 250, 252 forward, 96, 130, 131, 133, 134, 154, 169 ordinary, 134 difference equation, 37ff d-th order, 41 and 2-parameter semigroups, 51ff autonomous, 68ff, 68 linear, 143ff delay, 37, 42, 43, 71, 72, 89, 90, 92, 172, 288, 314 discrete Krisztin–Walther, 288 driven, ix, 44, 90 explicit, 39 first order, 39 implicit, 39 integro, 90 linear, 95ff, 96, 171 homogeneous, 96 inhomogeneous, 96 scaled, 130 linearly implicit, 189 nonautonomous, 37ff of perturbed motion, 38, 67, 68, 102, 257, 321 and stability, 67 parametrically perturbed, 44, 91 partial, 43 evolutionary, 43 periodic, 68ff, 68 linear, 108ff, 143ff quasilinear, 310 random, ix, 91 reduced, 205, 206, 267, 330 semi-implicit, 39, 41, 44, 46, 52, 53, 55, 56, 74, 78, 96, 195, 251, 269, 275, 294, 340 semilinear, 151, 155, 180, 183, 187, 189ff, 189, 190, 192–195, 198, 207, 231, 279, 287, 290, 310, 318, 321, 323, 327, 335 Volterra, 37, 43, 90 with infinite delay, 43 well-posed, 51 differentiability, 363ff differential equation abstract evolution, 26, 28, 38 time-discrete, 73ff
Index time-discretized, 173ff, 289ff, 293ff, 339, 340ff abstract sectorial, 28ff time-discretized, 75ff Chafee–Infante, 31, 80, 189, 294, 301 delay, 25, 36, 72, 90, 92, 94, 172, 315 diffusion fully discretized, 175 doubly nonlinear, 31ff time-discretized, 88ff functional, viii, 1, 2, 12, 24ff, 24, 36, 71, 94, 170, 314, 315 discretized, 71ff, 170ff, 287ff Ginzburg–Landau, 31, 38, 80, 94, 189, 294, 304 finite difference, 80ff, 81, 87, 304ff, 307 Krisztin–Walther, 26, 288 Kuramoto–Sivashinsky, 36, 94, 316 ordinary, 2, 24, 25, 33, 36, 75, 76, 79–81, 90, 92, 94, 175, 181, 182, 184, 300, 311–315, 330, 342, 343 parabolic, 2, 31, 75, 91, 92, 94, 96, 185, 189, 290, 295, 315, 318, 340, 343 time-discretized, 174ff partial, viii, 1, 2, 24, 38, 90–92, 94, 299, 311, 313–315, 343 porous medium, 32 reaction-diffusion, 1, 2, 29ff, 29, 30, 32, 36, 76, 94, 294, 343 discretized, 75ff, 93, 295ff with piecewise constant argument, 36 dimension, fractal, 35 direct sum, xix distance, xviii driving system, 44 dual space, xix dynamical consistency, 90 dynamical system, 22, 34 metric, 33, 91 nonautonomous, ix, 1ff, 33, 35 discrete, 33, 91 random, 33, 35, 44, 91, 92, 343 set-valued, 35
embedding, 26, 173 compact, xviii, 12, 89 continuous, xviii, 87, 127, 179, 343, 360 equation characteristic, 171 homological, 273, 314 invariance, 200, 201, 221–223, 251, 253, 255, 262, 269, 270, 272, 275, 311 local, 259
395 Euler method backward, 41 explicit, 41, 79, 288 forward, 41 implicit, 38, 41, 51, 78, 81, 88, 92, 94, 174 linearly implicit, 94, 294, 299, 300, 340 evolution family, 26, 27, 74, 173, 185, 289 exponential function, generalized, 345ff, 345 multiplication theorem, 346 asymptotic behavior, 346 monotonicity, 346 exponential tracking, 312 exponentially decaying in backward time, 125 in forward time, 125 extension, 364ff C m -, 370 Lipschitzian, 365 FDE, see differential equation, functional fiber backward, through a point, 221 forward, through a point, 221 fiber bundle, 187 C m -, 259 approximation, 279ff center, 266 center-stable, 260, 266 center-unstable, 266 classical, 266 forward invariant, 187 global, 259 locally, 258 inertial, 188, 195, 196, 274ff, 275, 278, 279, 287, 314ff, 314 existence, 275 smoothness, 278 invariant, xi, 96, 187ff, 187, 194ff, 301, 312 existence, 194ff, 200 global, 259 hyperbolic, 312 intersection, 208 intersection, smoothness, 248 locally, xi, 195, 259 nonuniqueness, 263 periodic, 213 smoothness, 230ff, 233 Taylor approximation, 269ff, 273 Taylor expansion, 270 pseudo-center, 265 existence, 265 pseudo-stable, 256ff, 259 existence, 259 of a solution, 261
396 pseudo-unstable, 256ff, 260 existence, 259 of a solution, 261 stable, 260, 266 strongly stable, 260 strongly unstable, 260 Taylor approximation, 269ff unstable, 260, 266 Floquet multiplier, 109, 110, 112 representation, 111, 181 spectrum, 109 theory, xi, 95, 111ff, 147, 181, 274 flow, 33 foliation, 228, 312ff invariant, 188, 214ff, 214, 250 smoothness, 230ff pseudo-stable, 229 pseudo-unstable, 229 fractional power space, 29, 174, 291, 293 function bump, 368 cut-off, 258, 314, 327, 369ff Gamma-, 27 Lambert W -, 83 Mittag-Leffler, 27 step, 40, 46, 47, 57, 79, 82
Galerkin method, 30, 76 spectral, 38, 77, 299, 300, 303 generator of a 2-parameter semigroup, 5, 6, 15, 37, 52, 320 of a linear 2-parameter semigroup, 99 of a periodic 2-parameter semigroup, 21 generic, 149 graph transform, 310, 311 Green’s function, 115, 117, 118, 151, 194, 215 grid, 43 group property, 4, 51 linear, 99 growth exponential, 114ff rate, 130, 131, 134, 135
Hadamard method, 194, 310 Hausdorff separation, xviii hierarchy of invariant fiber bundles classical, 188, 266, 330 extended, 188, 208, 212 pseudo-stable, 204, 294
Index pseudo-unstable, 204, 294 stable, 260 unstable, 260 of invariant vector bundles, 123 classical, 139 extended, 123, 139 pseudo-center, 139 pseudo-stable, 138 pseudo-unstable, 138 stable, 138 unstable, 138 hyperbolic, 134, 146, 148, 184, 204, 230 forward, 130
image, xix inequality, 345ff Gagliardo–Nirenberg, 85, 179 Gronwall, 50, 51, 83, 85, 93, 191, 206, 207, 254, 256, 348ff, 348 Henry, 28, 350 uniform, 89, 94, 349 interpolation, 84, 179 inertial form, 275 inhomogeneity, 96 initial condition, 39 initial state, 39 initial time, 39 initial value problem, 39 inner product, xx, 77, 179 Dirichlet, 179 2s , 179 on HN 2 on LN , 75 interval, discrete, xxi, 279 invariant fibers existence, 221 smoothness, 249
kernel, xix kinematically similar, 108
lattice, 43 Lebesgue space, xxi discrete, 75, 96, 178, 296 limit set, 5ff α-, 8 ω -, 8 properties, 10, 14 limit superior, of a sequence of sets, 8 linearization, 312, 317ff, 317 along a solution, 55
Index Lipeomorphism, xviii, 277, 336, 339, 341, 357, 359 Lyapunov exponent, 181 lower forward, 107 upper forward, 107 spectrum, 96, 107, 182 forward, 107, 108 transformation, 108, 112, 181, 318 Lyapunov–Perron equation, 281, 282, 312, 313 truncated, 281, 282, 285, 286 method, 96, 188, 194, 279, 310–312, 316 operator, 189, 196, 198, 202, 209, 214, 224, 234, 280, 310, 329 truncated, 189, 280, 282, 285 sum, 106, 151, 153, 160, 164, 196, 273
manifold center, 187, 266, 313, 314 center-stable, 266 center-unstable, 187, 188, 266 inertial, 187–189, 274, 275, 299–301, 303, 311, 313, 314 invariant, viii, x–xii, 187, 194, 213, 266, 310–315, 317, 342 pseudo-stable, 261, 311–313 pseudo-unstable, 261, 311–313 stable, 187, 188, 266, 310, 312, 313 unstable, 187, 266, 312, 313 mapping bounded, xviii compact, xviii, 355ff compact dissipative, 356 completely continuous, xviii condensing, 355 contractive, 352ff dual, xix iterates, xvii near identity, 328, 333, 336 restriction, xvii smooth, 363ff uniformly bounded, 258 mass matrix, 75–77, 176 measure of noncompactness, 15, 16, 93, 142, 291, 351–353, 355, 360, 363 Hausdorff, 15, 351 invariance under closure, 351 invariance under convex hull, 351 Kuratowski, 15, 23, 34, 352, 356, 364 Kuratowski property, 351 monotonicity, 351 regularity, 351
397 semi-additivity, 351 algebraic, 351 separation, 15, 352 mesh, discretization, 24, 41, 70, 170, 287, 338 method collocation, 77 deformation, 311 finite difference, 38, 41, 76ff, 76, 80, 81, 90, 92–94, 175ff, 176, 177ff, 177 finite element, 38, 77, 180ff multistep, 90, 92, 314 linear, 42 one-step, 37, 38, 40, 46, 48, 57–59, 81, 92, 304, 314 parametrization, 311 Morse decomposition, 35, 92, 94 set, 35 motion, 33, 35, 52 complete, 2, 5, 6, 8, 16, 17, 69 periodic, 2, 22, 23, 317
nonautonomous set, 2ff, 2 ε-neighborhood, 3 absorbing, 2, 16, 18–20, 38, 58–60, 74, 79, 83, 86, 88, 94, 274, 279, 297, 300, 314 pullback, 35 uniformly, 18, 23, 58–60, 192, 274 attracting, 10 exponentially, 229 attractive, 1 pullback, 35 boundary, 3 bounded, 3 uniformly, 3 cartesian product, 2 closed, 2 closure, 2 compact, 2 fiber, 2 interior, 2 invariant, 5ff, 5 backward, 5 forward, 5 neighborhood, 3 open, 2 stable, of φ∗ , 257 unstable, of φ∗ , 257 norm, 364ff adapted, 144, 172 normal bundle of Wi− , 254
398 normal hyperbolicity, 188, 256 normally hyperbolic, 250ff, 250, 252, 256 nullspace, xix
ODE, see differential equation, ordinary operator sectorial, 28 complexification, xx difference, 176, 177, 179, 184 forward, 40, 89 elliptic differential, 174, 175, 310 embedding, xviii, 43, 127 evolution, 95, 98 backward, 98 forward, 98 Laplace, 77, 94, 175, 180, 295 finite difference, 76, 175 root, 111 shift, 91, 183 solution, 25, 26, 36 substitution, 230, 312, 337, 342, 343, 360 orbit, 6 truncated, 6
parameter space, 4 partition, xvii length, xvii ordered, xvii PDE, see differential equation, partial period map, 2, 22, 68, 113 linear equation, 109 properties, 22 perturbation parametric, ix principle of linearized asymptotic stability, 156 converse, 180 process, 33 projector bounded, 115 complenentary, 115 invariant, 115 regular, 115 spectral, 122 pseudo-hyperbolic, 134, 162, 187, 188, 313 pullback convergence, x, 60
radial projection, 371 radial retraction, 261, 298, 308, 364, 366, 371 range, xix
Index RDE, see differential equation, reactiondiffusion reducibility, 181 reduction principle, 187, 188, 313, 314 Pliss, 187, 188, 267, 279 root, of an operator, 111 rule midpoint, 40 trapezoidal, 41 Runge–Kutta method, 37, 40, 50, 51, 60, 75, 90, 92, 94 explicit, 40 stage, 40
semidynamical system, 22 semiflow, 33, 35, 36, 94, 343 semigroup property, 3, 51 linear, 99 of ea , 346 sequence c-bounded, 124 c, d-bounded, 124 c+ -bounded, 124 c− -bounded, 124 quasibounded, 181 shifted, 3 set boundary, xviii closure, xviii diameter, xviii interior, xviii invariantly connected, 34 set contraction, 355 side left-hand, 39 right-hand, 39 skew product, ix, 33, 34, 184, 185 smoothing property, 12, 13, 30, 32, 34, 73–75, 89, 94 Sobolev space, xxi discrete, 38, 96, 179 solution backward, 39 boundedness, 44ff complete, 39 existence, 44ff forward, 39 general, 51 general backward, 52, 100 general forward, 51, 100 existence, 53, 57 smoothness, 54, 57
Index hyperbolic, 134, 187, 188, 317, 326, 341–343 mild, 27, 29, 38, 74, 289 of a difference equation, 39 of an initial value problem, 39 solution identity, 39, 158, 203, 205, 223, 253–255, 257 solution space, 97, 318, 334, 336 spectral dichotomy, 184 spectral interval, 130, 134 spectral radius, xix spectral set, 122 spectrum, xix dichotomy, 96, 134, 135, 182, 287, 342 forward, 130–132 point, xix splitting, 114ff, 120, 128ff bounded, 120 exponential, 134ff, 135 for autonomous equations, 145 for compact equations, 146 for periodic equations, 147 roughness, 161ff exponential forward, 131, 134ff invariant, 114ff, 122 minimal, 120 spectral, 122, 144 stable, 60ff, 65, 102ff, 104, 267 asymptotically, 66, 104 pullback, 104, 267 uniformly, 66 exponentially, 62, 267 globally asymptotically, 66 pullback, 38, 66, 104, 267 asymptotically, 66 uniformly, 66 uniformly asymptotically, 66 structurally, x uniformly, 66, 104, 267 asymptotically, 104 pullback, 104, 267 state space, 3 extended, 3 stiffness matrix, 75–77, 176, 178, 180 superposition principle, 97 tangent bundle of Vi+ , 250 of Wi− , 250, 252 tempered, backward, 59 test space, 76
399 Theorem fixed point, 351ff global inverse function, 356ff Hadamard–Perron, 292 inversion, 351ff Lipschitz inverse function, 356 of Cushing, 336 of Darbo, 46, 355 of Floquet, 111 of Hadamard–Perron, 200, 230, 343 of Hartman–Grobman, 317, 326ff, 333, 336, 341, 343 generalized, xi, 318, 326ff, 330 ˇ sitaˇiˇsvili, 317, 330 of Palmer–Soˇ of Schauder, 355 roughness, 96, 161, 165, 170, 202 time scale, 33, 40, 350 trichotomy exponential, 135, 181, 183, 312, 329 forward, 131
universe absorption, 18, 58–60, 192 attraction, 10, 17, 229 unstable, 66, 267
variation of constants, 38, 95, 100, 113, 180, 190, 206, 207, 254, 255, 286 nonlinear, 180 variational equation, 55, 95, 98, 134, 252–255, 257, 267, 312 n-th order, 55 first order, 55 vector bundle, 3 center, 139 center-stable, 138, 139 center-unstable, 138, 139 classical, 139 pseudo-stable, 138 pseudo-unstable, 138 stable, 138, 139 strongly stable, 138 strongly unstable, 138 super-stable, 99, 115, 138 trivial, 115 unstable, 138, 139
Whitney sum, 3, 100, 115, 117, 121
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