Almost Automorphic Type and Almost Periodic Type Functions in Abstract Spaces

Toka Diagana

Almost Automorphic Type and Almost Periodic Type Functions in Abstract Spaces

Almost Automorphic Type and Almost Periodic Type Functions in Abstract Spaces

Toka Diagana

Almost Automorphic Type and Almost Periodic Type Functions in Abstract Spaces

123

Toka Diagana Department of Mathematics Howard University Washington, DC, USA

ISBN 978-3-319-00848-6 ISBN 978-3-319-00849-3 (eBook) DOI 10.1007/978-3-319-00849-3 Springer Cham Heidelberg New York Dordrecht London Library of Congress Control Number: 2013942386 Mathematics Subject Classification (2010): 34K14, 35B15, 34C27, 35L10, 35L25, 43A60, 47A56, 47A62, 47-XX, 47D06, 58D25, 65J08, 58J35, 46B70, 46M35, 12H20, 35R20 © Springer International Publishing Switzerland 2013 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. Exempted from this legal reservation are brief excerpts in connection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Duplication of this publication or parts thereof is permitted only under the provisions of the Copyright Law of the Publisher’s location, in its current version, and permission for use must always be obtained from Springer. Permissions for use may be obtained through RightsLink at the Copyright Clearance Center. Violations are liable to prosecution under the respective Copyright Law. The use of general descriptive names, registered names, trademarks, service marks, 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. While the advice and information in this book are believed to be true and accurate at the date of publication, neither the authors nor the editors nor the publisher can accept any legal responsibility for any errors or omissions that may be made. The publisher makes no warranty, express or implied, with respect to the material contained herein. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)

In Memory of My Sister Coumba Diagana and My Brother Demba Diagana

Preface

Almost Automorphic Type and Almost Periodic Type Functions in Abstract Spaces presents a comprehensive and reader-friendly introduction to the concepts of almost periodicity, asymptotic almost periodicity, almost automorphy, asymptotic almost automorphy, pseudo-almost periodicity, and pseudo-almost automorphy as well as their recent generalizations. Further, it presents a wide range of sufficient conditions for the boundedness, existence, uniqueness, and stability of solutions to various classes of first-, second-, third-, and higher-order abstract differential equations, difference equations, and integrodifferential equations whose coefficients belong to the above-mentioned classes of functions. It also offers various applications to partial differential equations including the beam boundary-value problem, the heat boundary-value problem, and some partial neutral integrodifferential equations with delay arising in control systems. Some of the results presented are either new or else cannot be easily found in the mathematical literature. In order to establish the main results of the book, one makes extensive use of a myriad of tools ranging from functional analysis to operator theory. Noteworthy progress has been made during the past 15 years in studying almost periodic, almost automorphic, and pseudo-almost periodic functions as well as their applications to abstract differential equations. In recent years, various generalizations of the above classes of functions have been introduced in the mathematical literature including, but are not limited to, the concepts of Stepanov-like almost automorphy, weighted pseudo-almost periodicity, pseudo-almost automorphy, weighted pseudo-almost automorphy, Stepanov-like pseudo-almost periodicity, Stepanov-like pseudo-almost automorphy, and many more. Despite the noticeable and rapid progress made on these important topics, the only standard references that currently exist on those new classes of functions and their applications are still scattered research articles. One of the main objectives of this book consists of closing that gap. Namely, the book collects and presents in a unified manner most of the recent material introduced in the literature upon the above-mentioned classes of functions as well as their applications to differential equations, difference equations, integrodifferential equations, and partial differential equations. vii

viii

Preface

The book is divided into 12 main chapters and an appendix. It also offers some useful bibliographical notes at the end of each chapter. The first chapter collects the basic background on metric, Banach, and Hilbert spaces. Proofs of some of the most important results presented will be given including that of the Banach fixed-point theorem. Chapter 2 collects basic material in functional analysis and operator theory needed in the sequel including compact operators, self-adjoint operators, operators with compact resolvents, sectorial operators, analytic semigroups, abstract interpolation spaces, and various preliminary results on evolution families under Acquistapace–Terreni conditions and exponential dichotomy. Chapter 3 gathers basic properties on almost periodic, Stepanov-like almost periodic, and asymptotically almost periodic functions, and then introduces and studies differentiable almost periodic and Stepanov-like almost periodic functions. Chapter 4 collects basic results on almost automorphic and asymptotically almost automorphic functions, and then introduces and studies Stepanov-like almost automorphic, differentiable almost automorphic and Stepanov-like almost automorphic functions. The main objective of Chap. 5 consists of reviewing basic properties of pseudo-almost periodic functions including some composition theorems as well as some of their recent extensions such as weighted pseudo-almost periodic, differentiable pseudo-almost periodic and Stepanov-like pseudo-almost periodic functions. Chapter 6 reviews basic properties of pseudo-almost automorphic functions as well as some of their recent extensions such as weighted pseudo-almost automorphic, differentiable pseudo-almost automorphic, and Stepanov-like pseudoalmost automorphic functions. Chapter 7 studies the existence of almost periodic (respectively, almost automorphic) solutions to the classes of nonautonomous damped second-order differential equations d2u du + b(t)B + a(t)Au = h(t, u), dt 2 dt where A is a self-adjoint linear operator with compact resolvent, B is a closed linear operator satisfying some additional conditions, and the functions a, b, h are almost periodic (respectively, almost automorphic). Chapter 8 studies the existence of asymptotically almost automorphic mild solutions to the abstract partial neutral integrodifferential equations d D(t, ut ) = AD(t, ut ) + dt

 t 0

B(t − s)D(s, us )ds + g(t, ut ), t ∈ [σ , σ + a),

uσ = ϕ ∈ B, where A, B(t) are densely defined closed linear operators with a common domain D(A), which is independent of t, the history ut , is the function defined by ut (θ ) := u(t + θ ), which belongs to an abstract phase space B defined axiomatically, f , g are functions subject to some additional conditions, and

Preface

ix

D(t, ϕ ) := ϕ (0) + f (t, ϕ ). Chapter 9 studies the existence of differentiable pseudo-almost automorphic solutions to some general higher-order differential equations involving operator coefficients given by d n u n−1 d k u + ∑ Ak k + Au = f , dt n k=1 dt where A and Ak (k = 1, 2, . . . , n − 1) are densely defined closed linear operators on a Banach space X , and f : R → DA (α + m, p) with DA (α + m, p) being a real interpolation space. Our study concerns only the special case Ak = Aαk (if it exists) with the exponents 0 ≤ αk < 1 for k = 1, 2, . . . , n − 1. A nonautonomous version of the above higher-order differential equation will also be considered. In Chap. 10, using the Schauder and Banach fixed-point theorems, and exponential dichotomy techniques, we study and obtain, in various contexts, the existence of pseudo-almost periodic (respectively, weighted pseudo-almost periodic) solutions to the nonautonomous third-order differential equations   d d2u + g(t, Bu) = w(t)Au + f (t,Cu), dt dt 2

t ∈R

where A is a self-adjoint linear operator, B and C are linear operators such that their algebraic sum B +C is nontrivial, the function w given by w(t) = −ρ (t) is assumed to be almost periodic, there exist two constants ρ0 , ρ1 > 0 satisfying the following conditions ρ0 ≤ ρ (t) ≤ ρ1 , and the functions f , g are pseudo-almost periodic (respectively, weighted pseudo-almost periodic) in the first variable uniformly in the second one. Chapter 11 studies the existence of pseudo-almost automorphic solutions to the class of Sobolev-type differential equations given by d [u + f (t, u)] = A(t)u + g(t, u), dt where A(t) is a family of densely defined closed linear operators on a domain D, independent of t, and f , g are Stepanov-like pseudo-almost automorphic functions. In Chap. 12, one studies and obtains the existence of a globally asymptotically stable almost periodic (respectively, globally asymptotically stable almost automorphic) solutions to the higher-order difference equations x(t + n) +

n−1

∑ Am (t)x(t + m) + A0 (t)x(t) = f (t, x(t)),

m=1

x

Preface

where Am (t) for m = 0, 1, . . . , n − 1 are sequences of bounded linear operators on a Banach space X , and the forcing term f is almost periodic (respectively, almost automorphic) in the first variable uniformly in the second one, and satisfies some additional conditions. The prerequisite for the book is the basic introductory course in real analysis. It is suitable for beginning graduate and/or advanced undergraduate students. Moreover, it may be of interest to researchers in mathematics as well as in engineering, physics, and related areas. Further, some parts of the book may be used for various graduate and advanced undergraduate courses. In particular, it contains lots of examples which might be used in graduate and advanced undergraduate lectures. Chapter 1 provides some basic material for a course in functional analysis. Chapter 2 provides the reader with some needed background for a course in operator theory. Chapters 3–6 may be used for the study of periodic and almost periodic functions and their extensions. Chapters 7–11 may be used for a course on abstract differential equations and their applications to partial differential equations. I extend my deepest gratitude to both Prof. Martin Bohner and Prof. Terrence Mills for careful proofreading of all versions of the book. Their invaluable comments and suggestions have significantly improved this book. I am very grateful to Elizabeth Loew and Dahlia Fisch from Springer for their editorial assistance and useful comments. I owe a huge debt of gratitude to all the three reviewers for careful reading of the book and insightful comments. Last but certainly not least, I am incredibly grateful to my wife Sally and our wonderful children for their continued support and encouragements, and especially for their putting up with me during all those long hours I spent away from them, while writing this book. Washington, DC November 2012

Toka Diagana

Contents

1

Metric, Banach, and Hilbert Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Metric Spaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.1 Basic Definitions and Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.2 The Banach Fixed-Point Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.3 Compact Metric Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Banach Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 Basic Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.2 Examples of Banach Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.3 The Schauder Fixed-Point Theorem . . . . . . . . . . . . . . . . . . . . . . . 1.3 Hilbert Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.1 Basic Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.2 Examples of Hilbert Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.3 The Projection Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 2 17 20 22 22 26 32 33 33 34 35

2

Linear Operators on Banach Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Bounded Linear Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 Examples of Bounded Linear Operators. . . . . . . . . . . . . . . . . . . 2.1.2 Properties of Bounded Operators . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Unbounded Linear Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Examples of Unbounded Operators. . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 Closed Linear Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.3 Spectral Theory for Linear Operators . . . . . . . . . . . . . . . . . . . . . 2.2.4 Sectorial Linear Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.5 Semigroups of Linear Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.6 Intermediate Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Evolution Families . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Evolution Families and Their Estimates . . . . . . . . . . . . . . . . . . . 2.3.2 Acquistapace–Terreni Conditions. . . . . . . . . . . . . . . . . . . . . . . . . .

43 43 43 45 53 54 55 56 60 61 65 72 72 73

3

Almost Periodic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Almost Periodic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1 Definitions and Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

79 79 80 xi

xii

Contents

3.1.2 Fourier Series Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.3 Composition of Almost Periodic Functions . . . . . . . . . . . . . . . C(n) -Almost Periodic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Asymptotically Almost Periodic Functions . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 Basic Definitions and Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2 Composition Theorems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Almost Periodic Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.1 Basic Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.2 Properties of Almost Periodic Sequences . . . . . . . . . . . . . . . . . 3.4.3 Asymptotically Almost Periodic Sequences . . . . . . . . . . . . . . 3.4.4 Composition Theorems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S p -Almost Periodic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.1 Composition of S p -Almost Periodic Functions . . . . . . . . . . . (n) S p -Almost Periodic Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

87 91 93 96 96 98 100 100 101 104 105 106 108 109

4

Almost Automorphic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Almost Automorphic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.1 Basic Definitions and Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.2 Composition Theorems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 C(n) -Almost Automorphic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Asymptotically Almost Automorphic Functions . . . . . . . . . . . . . . . . . . . 4.3.1 Basic Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.2 Composition Theorems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Almost Automorphic Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.1 Basic Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.2 Composition Theorems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.3 Asymptotically Almost Automorphic Sequences . . . . . . . . . 4.5 S p -Almost Automorphic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.1 Composition Theorems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (n) 4.6 S p -Almost Automorphic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

111 111 111 117 120 121 121 124 128 128 129 131 132 134 138

5

Pseudo-Almost Periodic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Pseudo-Almost Periodic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.1 Basic Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.2 Composition Theorems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Weighted Pseudo-Almost Periodic Functions . . . . . . . . . . . . . . . . . . . . . . 5.2.1 Definitions and Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.2 Weighted Mean for Almost Periodic Functions . . . . . . . . . . . 5.2.3 A Composition Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 C(n) -Pseudo-Almost Periodic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.1 Definitions and Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.2 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 S p -Pseudo-Almost Periodic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

141 141 141 146 149 149 154 156 159 159 161 162

3.2 3.3

3.4

3.5 3.6

Contents

6

xiii

Pseudo-Almost Automorphic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Pseudo-Almost Automorphic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.1 Basic Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.2 Properties of Pseudo-Almost Automorphic Functions. . . . 6.2 Weighted Pseudo-Almost Automorphic Functions . . . . . . . . . . . . . . . . . 6.2.1 Basic Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.2 A Composition Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 C(n) -Pseudo-Almost Automorphic Functions . . . . . . . . . . . . . . . . . . . . . . . 6.3.1 Basic Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.2 Properties of C(n) -Pseudo-Almost Automorphic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 S p -Pseudo-Almost Automorphic Functions. . . . . . . . . . . . . . . . . . . . . . . . . 6.4.1 Basic Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.2 A Composition Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (n) 6.5 S p -Pseudo-Almost Automorphic Functions . . . . . . . . . . . . . . . . . . . . . . .

167 167 167 168 178 178 179 180 180

7

Existence Results for Some Second-Order Differential Equations . . . . 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Existence of Almost Periodic Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.1 First-Order Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.2 Sobolev-Type Second-Order Differential Equations . . . . . . 7.2.3 Damped Second-Order Evolution Equations . . . . . . . . . . . . . . 7.2.4 The Linear Beam Equation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Existence of Almost Automorphic Solutions . . . . . . . . . . . . . . . . . . . . . . . 7.3.1 First-Order Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.2 Sobolev-Type Second-Order Evolution Equations . . . . . . . . 7.3.3 Damped Second-Order Evolution Equations . . . . . . . . . . . . . . 7.3.4 The Linear Beam Equation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

189 189 191 191 198 202 203 204 204 206 206 207

8

Existence Results to Some Integrodifferential Equations . . . . . . . . . . . . . . 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Existence of Asymptotically Almost Automorphic Solutions . . . . . 8.3 Existence Result for Some Neutral Integrodifferential Equations . 8.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.2 Asymptotically Almost Automorphic Solutions . . . . . . . . . .

209 209 211 216 216 216

9

Existence of C(m) -Pseudo-Almost Automorphic Solutions . . . . . . . . . . . . . 9.1 Existence Results for Some Autonomous Higher-Order Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1.2 First-Order Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . 9.1.3 Higher-Order Differential Equations . . . . . . . . . . . . . . . . . . . . . . 9.1.4 Second-Order Differential Equations . . . . . . . . . . . . . . . . . . . . . . 9.1.5 An Autonomous Beam Equation . . . . . . . . . . . . . . . . . . . . . . . . . .

221

181 182 182 183 186

221 221 222 227 227 229

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9.2

Existence Results for Some Nonautonomous Higher-Order Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2.2 First-Order Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . 9.2.3 Higher-Order Differential Equations . . . . . . . . . . . . . . . . . . . . . . 9.2.4 Second-Order Damped Differential Equations . . . . . . . . . . . . 9.2.5 A Nonautonomous Beam Equation . . . . . . . . . . . . . . . . . . . . . . . .

231 231 233 238 238 241

Pseudo-Almost Periodic Solutions to Some Third-Order Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2 Existence of Pseudo-Almost Periodic Solutions . . . . . . . . . . . . . . . . . . . . 10.2.1 First-Order Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . 10.2.2 Third-Order Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . 10.2.3 Some General Third-Order Differential Equations . . . . . . . 10.3 Existence of Weighted Pseudo-Almost Periodic Solutions . . . . . . . . 10.3.1 First-Order Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . 10.3.2 Third-Order Differential Equations . . . . . . . . . . . . . . . . . . . . . . . .

245 245 247 247 251 255 255 255 260

Pseudo-Almost Automorphic Solutions to Some Sobolev-Type Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Existence of Pseudo-Almost Automorphic Solutions . . . . . . . . . . . . . . 11.2.1 Existence Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2.2 Existence Results for the Heat Equation . . . . . . . . . . . . . . . . . .

261 261 262 263 271

12

Stability Results for Some Higher-Order Difference Equations . . . . . . 12.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2 Almost Periodic Higher-Order Difference Equations . . . . . . . . . . . . . . 12.2.1 First-Order Difference Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2.2 Existence Results for the Beverton–Holt Model . . . . . . . . . . 12.2.3 Higher-Order Difference Equations . . . . . . . . . . . . . . . . . . . . . . . 12.3 Almost Automorphic Higher-Order Difference Equations . . . . . . . . . 12.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.3.2 First-Order Difference Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 12.3.3 Higher-Order Difference Equations . . . . . . . . . . . . . . . . . . . . . . .

275 275 276 276 281 282 283 283 283 288

A

Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.1 Spectral Decomposition of the Self-Adjoint Operator A . . . . . . . . . . . A.2 Proof of an Interpolation Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.3 Bi-Almost Automorphy and Positive Bi-Almost Automorphy . . . .

289 289 290 291

10

11

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301

Chapter 1

Metric, Banach, and Hilbert Spaces

In this introductory chapter, we collect the basic background on metric, Banach, and Hilbert spaces needed in the sequel. The exposition is self-contained, with a wealth of illustrative examples. Proofs of some of the most important results are given including that of the Banach fixed-point theorem. N – the set of nonnegative integers Z+ – the set of positive integers Z – the set of all integers R – the field of real numbers R+ – the set of nonnegative real numbers C – the field of complex numbers Q – the field of rational numbers X –a generic Banach space H – a generic Hilbert space Lp – Lebesgue spaces Wk,p –Sobolev spaces All the vector spaces considered in this book will be over a field F, where F = R or C is equipped with its natural absolute value | · |.

1.1 Metric Spaces Metric spaces play a crucial role in various areas of mathematics especially in functional analysis. In particular, one makes use of these spaces to deal with various mathematical issues (convergence, approximation, etc.). Classical examples of metric spaces include, but are not limited to, Banach and Hilbert spaces. Metric spaces were introduced in 1906 by the French mathematician M. Fr´echet [98] as a way of unifying the works of C. Arzel`a, G. Ascoli, G. Cantor, J. Hadamard, and V. Volterra on function spaces theory. In 1914, F. Hausdorff T. Diagana, Almost Automorphic Type and Almost Periodic Type Functions in Abstract Spaces, DOI 10.1007/978-3-319-00849-3 1, © Springer International Publishing Switzerland 2013

1

2

1 Metric, Banach, and Hilbert Spaces

generalized the work of M. Fr´echet and also conceptualized the notions of topological and metric spaces. This section is devoted to a self-contained review of the basic properties of metric spaces. Among other things, we will be discussing about the concepts of continuity, equi-continuity, uniform continuity, convergence, completeness, homeomorphism, separability, denseness, compactness, and the Banach fixed-point theorem in metric spaces. Moreover, various illustrative examples will be discussed.

1.1.1 Basic Definitions and Examples Definition 1.1. A metric space is a pair (X , d) consisting of a nonempty set X and a mapping d : X × X → R+ satisfying the following properties: (a) d(x, y) = 0 if and only if x = y. (b) d(x, y) = d(y, x) for all x, y ∈ X —(Symmetry). (c) d(x, z) ≤ d(x, y) + d(y, z) for all x, y, z ∈ X —(Triangle Inequality). Elements of the set X are referred to as points while the mapping d is called a distance (or a metric) on X × X . Further, the quantity d(x, y) is called the distance between the points x and y of the set X . In what follows, we provide and discuss a few examples of metric spaces. Example 1.2. Let X be an arbitrary nonempty set and consider the mapping d defined from X × X into R+ by d(x, y) =

⎧ ⎨0 ⎩

1

if x = y, if x = y.

The mapping d defined above is a distance. Indeed, first note that d maps X ×X into {0, 1} ⊂ R+ . Moreover, d(x, y) = 0 if and only if x = y. In addition, we also have, d(x, y) = d(y, x) for all x, y ∈ X . Now let x, y, z ∈ X . If x = z, then x may or may not be equal to y. If x = z = y, then d(x, z) = 0 ≤ 0 + 0 = d(x, y) + d(y, z). If x = z and x = y, then d(x, z) = 0 < 2 = 1 + 1 = d(x, y) + d(y, z). If x = y, then y may or may not be equal to z. The case x = y and y = z has already been treated. Now if x = y and y = z, then d(x, z) = 1 = 0 + 1 = d(x, y) + d(y, z). If y = z, then y may or may not be equal to x. The case y = z = x has already been treated. Now if y = z and y = x, then d(x, z) = 1 = 1 + 0 = d(x, y) + d(y, z). If x = y and y = z, then d(x, z) = 1 < 2 = 1 + 1 = d(x, y) + d(y, z). In view of the above, it follows that d is a distance commonly called the discrete distance. Example 1.3. Let p ≥ 1 and let X = Fn (n ≥ 2 being an integer). Define the mappings d1 , d p , and d∞ as follows:

1.1 Metric Spaces

3 n

∑ |x j − y j |,

d1 (x, y) =

j=1



1/p

n

∑ |x j − y j | p

d p (x, y) =

, and

j=1

d∞ (x, y) = max(|x1 − y1 |, |x2 − y2 |, . . . , |xn − yn |) for all x = (x1 , x2 , . . . , xn ) and y = (y1 , y2 , . . . , yn ) ∈ Fn . Each of the mappings d1 , d p , and d∞ defined above is a distance. (a) Let us prove that d1 is a distance. Indeed, d1 (x, y) = 0 if and only if |x j − y j | = 0 for j = 1, . . . , n, that is, x j = y j for j = 1, . . . , n, and hence x = y. Also, d1 (x, y) = d1 (y, x) for all x, y ∈ Fn . Now for all x = (x1 , x2 , . . . , xn ), y = (y1 , y2 , . . . , yn ), and z = (z1 , z1 , . . . , zn ) in Fn , using the triangle inequality for the absolute value, we obtain, d1 (x, z) = ≤

n

n

j=1

j=1

∑ |x j − z j | = ∑ |x j − y j + y j − z j | n

∑ (|x j − y j | + |y j − z j |)

j=1

=

n

n

j=1

j=1

∑ |x j − y j | + ∑ |y j − z j |

= d1 (x, y) + d1 (y, z), and hence d1 is a distance on Fn × Fn . (b) Let us prove that d p is a distance. Indeed, d p (x, y) = 0 if and only if |x j − y j | = 0 for j = 1, . . . , n, that is, x j = y j for j = 1, . . . , n, and hence x = y. Also, d p (x, y) = d p (y, x) for all x, y ∈ Fn . To prove the triangle inequality for d p , one essentially makes use of the so-called (discrete) Minkowski’s inequality in Fn given by: for all A j , B j ∈ F for j = 1, 2, . . . , n, then, 



1/p

n

∑ |A j + B j | p

≤

j=1



1/p

n

∑ |A j | p

j=1

+

n

1/p

∑ |B j | p

.

j=1

Indeed, for all x = (x1 , x2 , . . . , xn ), y = (y1 , y2 , . . . , yn ), and z = (z1 , z1 , . . . , zn ) in Fn , letting A j = x j − y j and B j = y j − z j , then we obtain  d p (x, z) =

n

∑ |x j − z j |

j=1



1/p p

=

n

∑ |(x j − y j ) + (y j − z j )|

j=1

1/p p

4

1 Metric, Banach, and Hilbert Spaces

 ≤



1/p

n

∑ |x j − y j |

n

∑ |y j − z j |

+

p

j=1

1/p p

j=1

= d p (x, y) + d p (y, z), and hence d p is a distance on Fn × Fn . (c) Let us prove that d∞ is a distance. Indeed, 0 = d∞ (x, y) ≥ |x j − y j | for j = 1, 2, . . . , n yields |x j − y j | = 0 for j = 1, . . . , n, that is, x j = y j for j = 1, . . . , n, and hence x = y. Similarly, d∞ (x, y) = d∞ (y, x) for all x, y ∈ X . Now for all x = (x1 , x2 , . . . , xn ), y = (y1 , y2 , . . . , yn ), and z = (z1 , z1 , . . . , zn ) in Fn , d∞ (x, z) = max |x j − z j | = max |x j − y j + y j − z j | j=1,..,n

j=1,..,n

≤ max (|x j − y j | + |y j − z j |) j=1,...,n

≤ max |x j − y j | + max |y j − z j | j=1,...,n

j=1,...,n

= d∞ (x, y) + d∞ (y, z), and hence d∞ is a distance on Fn × Fn . Example 1.4. Let (Xk , dk ) for k = 1, 2, . . . , n be a finite sequence of n metric spaces. Define their Cartesian product X = X1 × X2 × · · · × Xn as the set of all n-tuples x = (x1 , x2 , . . . , xn ) such that xk ∈ Xk for k = 1, 2, . . . , n. This (Cartesian) product space is sometimes denoted by n

X = ∏ Xk . k=1

Define for all x = (x1 , x2 , . . . , xn ) and y = (y1 , y2 , . . . , yn ) in X , the mapping d by setting n

d(x, y) :=

∑ dk (xk , yk ).

k=1

The mapping d defined above is a distance. Indeed, d(x, y) =

n

∑ dk (xk , yk ) = 0

k=1

yields dk (xk , yk ) = 0 for k = 1, 2, . . . , n. Since each dk is a distance it easily follows that xk = yk for k = 1, 2, . . . , n, that is, x = y. Similarly, d(x, y) =

n

n

k=1

k=1

∑ dk (xk , yk ) = ∑ dk (yk , xk ) = d(x, y).

1.1 Metric Spaces

5

Now, for all x = (x1 , x2 , . . . , xn ), y = (y1 , y2 , . . . , yn ), and z = (z1 , z2 , . . . , zn ) in X , we have d(x, z) = =

n

n

k=1

k=1

n

n

k=1

k=1

∑ dk (xk , zk ) ≤ ∑ [dk (xk , yk ) + dk (yk , zk )] ∑ dk (xk , yk ) + ∑ dk (yk , zk )

= d(x, y) + d(y, z), and hence d is a distance on X × X , which means that (X , d) is a metric space, which is referred to as the product metric space. Example 1.5. A normed vector space is a pair (X , · ) consisting of a vector space X over F and a mapping called norm · : X → R+ satisfying the following properties: (a) x = 0 if and only if x = 0. (b) λ x = |λ | . x for all λ ∈ F and x ∈ X . (c) x + y ≤ x + y for all x, y ∈ X —(Triangle Inequality). If (X , · ) is a normed vector space, then we set d(x, y) = x − y for all x, y∈X . The mapping d is a distance. Indeed, using (a) it follows d(x, y) = x−y = 0 if and only if x = y. Moreover, d(x, y) = d(y, x) for all x, y ∈ X . Now, using (c) it follows that for all x, y, z ∈ X , d(x, z) = x − z = (x − y) + (y − z) ≤ x − y + y − z = d(x, y) + d(y, z), and hence d is a distance on X × X . Proposition 1.6. If (X , d) is a metric space, then the following inequality holds, |d(x, y) − d(y, z)| ≤ d(x, z)

(1.1)

for all x, y, z ∈ X . Proof. Indeed, from the triangle inequality it follows that d(x, z) ≤ d(x, y) + d(y, z), which yields, d(x, z) − d(y, z) ≤ d(x, y). Interchanging x and y in the previous inequality and using the symmetry property, we obtain d(y, z) − d(x, z) ≤ d(y, x) = d(x, y) which yields, −(d(x, z) − d(y, z)) ≤ d(x, y). Combining, we obtain Eq. (1.1). Proposition 1.7. If (X , d) is a metric space, then the following (polygonal) inequality holds,

6

1 Metric, Banach, and Hilbert Spaces

d(x1 , xn ) ≤

n−1

∑ d(xk , xk+1 )

k=1

for all x1 , x2 , . . . , xn ∈ X . Let (X , d) be a (nonempty) metric space. If x ∈ X , then we define the open ball centered at x with radius R > 0 by B(x, R) := {y ∈ X : d(x, y) < R}. Similarly, the closed ball and the sphere, centered at x with radius R > 0, are defined respectively by B(x, R) := {y ∈ X : d(x, y) ≤ R} and S(x, R) := {y ∈ X : d(x, y) = R}. In addition to the above concepts, we also define the diameter of a nonempty subset A ⊂ X by diam A := sup [d(x, y)]. x,y∈A

Definition 1.8. Let (X , d) be a metric space. A subset A ⊂ X is said to be bounded if its diameter is finite. Obviously, if A ⊂ B, then diam A ≤ diam B. Definition 1.9. Let (X , d) be a metric space and let A ⊂ X be a subset. Let ε > 0. A subset B ⊂ X is said to be an ε -net for A if A⊂



B(x, ε ).

x∈B

Definition 1.10. A subset A is called totally bounded if, for every ε > 0, there is a finite ε -net for A. In other words, for every ε > 0 there is a finite set B such that A⊂



B(x, ε ).

x∈B

Obviously, every totally bounded set A is bounded with the converse being false. Definition 1.11. If A ⊂ X is a subset and if x ∈ X , then the distance between A and the point x is defined by d(x, A) := inf d(x, y). y∈A

1.1 Metric Spaces

1.1.1.1

7

Convergence

Definition 1.12. Let (X , d) be a metric space. A sequence (xn )n∈N ⊂ X is said to converge to a point x ∈ X , if d(xn , x) → 0 as n → ∞. Equivalently, for every ε > 0 there exists N0 ∈ N such that d(xn , x) < ε for all n ≥ N0 . In that case, we say that x is a limit point of the sequence (xn )n∈N and write lim xn = x. n→∞

Definition 1.13. Let (X , d) be a metric space. A sequence (xn )n∈N is said to be a Cauchy sequence, if d(xn , xm ) → 0 as n, m → ∞. Equivalently, for every ε > 0 there exists N0 ∈ N such that d(xn , xm ) < ε for all n, m ≥ N0 . Proposition 1.14. Let (X , d) be a metric space and let (xn )n∈N ⊂ X be a sequence. Then the following hold: (a) (b) (c) (d) (e)

If (xn )n∈N converges, then its limit point is unique. If (xn )n∈N converges, then (xn )n∈N is bounded. If (xn )n∈N is a Cauchy sequence, then (xn )n∈N is bounded. If (xn )n∈N converges, then (xn )n∈N is a Cauchy sequence. If (xn )n∈N converges to some x ∈ X and if (yn )n∈N is another sequence which converges to some y ∈ X , then d(xn , yn ) converges to d(x, y).

Proof. (a) Suppose (xn )n∈N converges to two limit points x1 and x2 . Then using the triangle inequality it follows that, 0 ≤ d(x1 , x2 ) ≤ d(x1 , xn ) + d(xn , x2 ) and hence letting n → ∞ in the previous inequality it follows that d(x1 , x2 ) = 0, that is, x1 = x2 . (b) Suppose (xn )n∈N converges to some x ∈ X . Thus for ε = 1, there exists N0 ∈ N such that d(xn , x) < 1 whenever n ≥ N0 . Let R > 1 be such that d(xn , x) < R for 1 ≤ n ≤ N0 − 1. Clearly, xn ∈ B(x, R) for all n and therefore xn is bounded. (c) The proof is very similar to that of (b). Take ε = 1. Using the fact that (xn )n∈N is a Cauchy sequence, there exists N0 ∈ N such that d(xn , xN0 ) < 1 for all n ≥ N0 . Let R > 1 be such that d(xn , xN0 ) < R for 1 ≤ n ≤ N0 − 1. Clearly, xn ∈ B(xN0 , R) for all n and therefore xn is bounded. (d) Suppose (xn )n∈N converges to some x ∈ X . Equivalently, for every ε > 0 there N0 ∈ N such that d(xn , x) < ε whenever n ≥ N0 . Using the triangle inequality it follows that, d(xn , xm ) ≤ d(xn , x) + d(x, xm ) ε ε < + 2 2 =ε whenever n, m ≥ N0 and hence (xn )n∈N is a Cauchy sequence. (e) It suffices to use both the triangle inequality for the distance δ defined on R × R by δ (x, y) = |x − y| for all x, y ∈ R and Eq. (1.1). Indeed,

8

1 Metric, Banach, and Hilbert Spaces

|d(xn , yn ) − d(x, y)| = |d(xn , yn ) − d(yn , x) + d(x, yn ) − d(x, y)| ≤ |d(xn , yn ) − d(yn , x)| + |d(x, yn ) − d(x, y)|, ≤ d(xn , x) + d(yn , y) and hence |d(xn , yn ) − d(x, y)| → 0 as n → ∞ and therefore d(xn , yn ) → d(x, y) as n → ∞.

1.1.1.2

Open and Closed Sets

Definition 1.15. Let (X , d) be a metric space and let A ⊂ X be a subset. A point x ∈ A is called an interior point of A if there exists R > 0 such that B(x, R) ⊂ A. The set of all interior points of A is called the interior of A and denoted by Int(A). If x ∈ X , then a subset B of X is said to be a neighborhood of x if x ∈ Int(B). Example 1.16. Let X = R be equipped with its natural distance and let A = Q ⊂ R, where Q is the field of rational numbers. Using the fact that between any two rational numbers, there is an irrational number, one can easily see that every open ball centered at a rational number contains irrational numbers, therefore Int(Q) = 0. / Definition 1.17. Let (X , d) be a metric space. A subset A ⊂ X is said to be open if A = Int(A). Similarly, a subset A ⊂ X is said to be closed if its complement Ac = X \A is open. Example 1.18 ([141]). Suppose X = [0, 1] ∪ [2, 3] is equipped with the distance defined by d(x, y) = |x − y| for all x, y ∈ X . One can easily check that the subset A = [0, 1] is both open and closed in X . Definition 1.19. Let (X , d) be a metric space and let A ⊂ X be a subset. A point x is called an adherent point of A if for every ε > 0, A ∩ B(x, ε ) = 0. / The collection of all adherent points of A is denoted by A. Proposition 1.20. Let (X , d) be a metric space and let A ⊂ X be a subset. Then, A is closed if and only if (xn )n∈N ⊂ A is a sequence which converges to some x ∈ X , then x ∈ A. Proof. The proof is left to the reader as an exercise. Definition 1.21. Let (X , d) be a metric space and let A ⊂ X be a subset. The boundary ∂ A of A is defined by ∂ A = A\Int(A). Definition 1.22. Let (X , d) be a metric space and let A ⊂ X be a (nonempty) subset. One says that x ∈ A is an accumulation point if and only if for all ε > 0, the subset A intersects B(x, ε ) at some point y with x = y. Equivalently, there exists a sequence (xn )n∈N ⊂ A which converges to x. It can be shown that the intersection of a finite number of open sets is also an open set. Indeed, let Ak for k = 1, 2, . . . , n be n open subsets of X . If

1.1 Metric Spaces

9

x∈

n 

Ak ,

k=1

then there exists Rk > 0 such that B(x, Rk ) ⊂ Ak for k = 1, 2, . . . , n. Setting R to be the smallest of those Rk one can easily see that B(x, R) ⊂

n 

Ak .

k=1

Similarly, it can be easily seen that the union of any collection of open sets is also an open set. Indeed, if x∈



Aα

α ∈Λ

where Aα ⊂ X is an open subset for all α ∈ Λ , then x belongs to one of these subsets, say Aβ for some β ∈ Λ . Since x ∈ Int(Aβ ), then 

x ∈ Int

.

Aα

α ∈Λ

Conversely, if

 x ∈ Int



Aα

,

α ∈Λ

then x belongs to one of these subsets, say Aβ for some β ∈ Λ . Since x ∈ Int(Aβ ), then  x ∈ Int



Aα

.

α ∈Λ

Using the above-mentioned facts, we obtain and prove the next proposition. Proposition 1.23. Let (X , d) be a metric space. Then the following properties hold: (a) (b) (c) (d) (e)

Both the metric space X and the empty space 0/ are closed (and open). Every open ball B(x, R) is an open set. Every closed ball B(x, R) is a closed set. The intersection of any collection of closed sets is a closed set. The union of any finite collection of closed sets is a closed set.

Proof. (a) We have X c = 0/ and 0/ c = X and hence both X and 0/ are closed (and open). (b) Let y ∈ B(x, R) and let R := R − d(x, y) > 0. Clearly, for all z ∈ B(y, R ), then using the triangle inequality we obtain,

10

1 Metric, Banach, and Hilbert Spaces

d(x, z) ≤ d(x, y) + d(y, z) < d(x, y) + R = d(x, y) + R − d(x, y) = R, which yields z ∈ B(x, R), that is, B(y, R ) ⊂ B(x, R), and hence B(x, R) is an open subset. (c) We need to show that B(x, R)c = X \B(x, R) is an open set. For that, let y be an arbitrary point in B(x, R)c . Using the fact that y ∈ B(x, R) it follows that d(x, y) > R. Choose R such that d(x, y) > R + R . Clearly, B(x, R) ∩ B(y, R ) = 0. / If not, there exists z ∈ B(x, R) ∩ B(y, R ). Using the triangle inequality it follows that d(x, y) < d(x, z) + d(z, y) < R + R , which contradicts the choice of R . In view of the above, every y ∈ B(x, R)c is the center of an open ball B(y, R ), which is contained in B(x, R)c . Therefore, B(x, R)c is an open set, that is, B(x, R) is a closed set. (d) Let {Bα : α ∈ Λ } be a collection of closed sets, indexed by Λ , and let 

B=

Bα .

α ∈Λ

Then we have  B = c



c =

Bα

α ∈Λ

α ∈Λ

Bcα

which yields Bc is open set as the union of a collection of open sets, and hence B is closed. (e) Let Bk for k = 1, . . . , n be a finite collection of closed sets. Let B=

n

Bk

k=1

be a finite union of closed sets. Then we have:  Bc =

n

k=1

c Bk

=

n 

Bck

k=1

which yields Bc is open set as a finite intersection of open sets, and hence B is closed.

1.1 Metric Spaces

1.1.1.3

11

Continuity

Definition 1.24. Let (X , d1 ) and (Y , d2 ) be two metric spaces. A function f : X → Y is said to be continuous at x0 ∈ X if for every ε > 0 there exists δ > 0 such that for all x ∈ X , d1 (x, x0 ) < δ yields d2 ( f (x), f (x0 )) < ε . The function f is said to be continuous if it is continuous at every point x of X . The collection of all continuous functions from X to Y is denoted by C(X , Y ). Notice that Definition 1.24 is also equivalent to the following: for every ε > 0 there exists δ > 0 such that f (B(x0 , δ )) ⊂ B ( f (x0 ), ε ) .

(1.2)

Proposition 1.25. Let (X , d1 ) and (Y , d2 ) be two metric spaces. The function f : X → Y is continuous if and only if for every sequence (xn )n∈N ⊂ X such that d1 (xn , x0 ) → 0 as n → ∞, then d2 ( f (xn ), f (x0 )) → 0 as n → ∞. Proof. Suppose f is continuous at x0 ∈ X . Thus for all ε > 0 there exists δ > 0 such that for all x ∈ X , d2 ( f (x), f (x0 )) < ε whenever d1 (x, x0 ) < δ . Suppose (xn )n∈N ⊂ X is a sequence such that d1 (xn , x0 ) → 0 as n → ∞. Equivalently, there exists N0 ∈ N such that d1 (xn , x0 ) < δ whenever n ≥ N0 . Using the continuity of f it follows that d2 ( f (xn ), f (x0 )) < ε whenever n ≥ N0 . Conversely, suppose that the function f is not continuous at x0 . Equivalently, there is an ε > 0 such that for all δ > 0 there is an x such that d1 (x, x0 ) < δ and d2 ( f (x), f (x0 )) ≥ ε . Letting xn represent x corresponding to δ = n−1 , then we obtain that d1 (xn , x0 ) → 0 as n → ∞ while d2 ( f (xn ), f (x0 )) ≥ ε for all n. Therefore, f (xn ) does not converge to f (x0 ). Proposition 1.26. Let (X , d1 ) and (Y , d2 ) be two metric spaces. The function f : X → Y is continuous if and only if for every open set B ⊂ Y , then the pre-image f −1 (B) of B is open in X , where the pre-image f −1 (B) is defined by f −1 (B) := {x ∈ X : f (x) ∈ B}. Proof. Suppose f is continuous and let B ⊂ Y be an open subset. Clearly, x ∈ f −1 (B) yields f (x) ∈ B. Using the fact that B is open in Y and f (x) ∈ B it follows that there exists ε > 0 such that B( f (x), ε ) ⊂ B. Using Eq. (1.2) it follows that there exists δ > 0 such f (B(x, δ )) ⊂ B( f (x), ε ). Consequently, B(x, δ ) ⊂ f −1 ( f (B(x, δ ))) ⊂ f −1 (B), which yields f −1 (B) is open in X . Conversely, let us suppose f −1 (B) is open in X for every open set B in Y . Let x ∈ X and let ε > 0. Using the fact that B( f (x), ε ) is open in Y it follows that f −1 (B( f (x), ε )) is open in X . Since x ∈ f −1 (B( f (x), ε )) it follows that there exists

12

1 Metric, Banach, and Hilbert Spaces

δ > 0 such that B(x, δ ) ⊂ f −1 (B( f (x), ε )), which yields f (B(x, δ )) ⊂ B( f (x), ε ); which, by Eq. (1.2), yields f is continuous. Definition 1.27. Let (X , d1 ) and (Y , d2 ) be two metric spaces. A function f : X → Y is said to be uniformly continuous if for every ε > 0 there exists δ > 0 such that d2 ( f (x), f (y)) < ε for all x, y ∈ X satisfying d1 (x, y) < δ . Definition 1.28. Let (X , d1 ) and (Y , d2 ) be two metric spaces. A sequence of functions fn : X → Y is said to converge pointwise to some function f : X → Y if for every x ∈ X and for every ε > 0 there exists N = N(x, ε ) ∈ N such that d2 ( fn (x), f (x)) < ε for all n ≥ N. Similarly, a sequence of functions fn : X → Y is said to converge uniformly to some function f : X → Y if for every ε > 0, there exists N = N(ε ) ∈ N such that d2 ( fn (x), f (x)) < ε for all n ≥ N and all x ∈ X . Equivalently, sup [d2 ( fn (x), f (x))] → 0 as n → ∞. x∈X

Proposition 1.29. Let (X , d1 ) and (Y , d2 ) be two metric spaces. If the sequence of continuous functions fn : X → Y converges uniformly to some function f : X → Y , then f is continuous. Proof. In order to prove that f is continuous at x0 ∈ X , we have to prove that for every ε > 0 there exists a δ > 0 such that if x ∈ X and d1 (x0 , x) < δ , then d2 ( f (x0 ), f (x)) < ε . Now since ( fn )n∈N converges uniformly to f , then there exists N ∈ N such that for all n ≥ N d2 ( f (x), fn (x)) <

ε 3

for all x ∈ X . Using the continuity of fN at x0 it follows that there exists δ > 0 such that if x ∈ X and d1 (x0 , x) < δ , then

ε d2 ( fN (x), fN (x0 )) < . 3 Let x ∈ X with d1 (x0 , x) < δ . Then, using the triangle inequality and in view of the above, we obtain d2 ( f (x0 ), f (x)) ≤ d2 ( f (x0 ), fN (x0 )) + d2 ( fN (x0 ), fN (x)) + d2 ( fN (x), f (x))

1.1 Metric Spaces

13

ε ε ε + + 3 3 3 = ε.

<

Definition 1.30. Let (X , d) be a metric space. A subset A of X is said to be dense if X = A. Equivalently, for each x ∈ X , there exists a sequence (xn )n∈N ⊂ A such that d(xn , x) → 0 as n → ∞.

1.1.1.4

Completeness

Definition 1.31. A metric space (X , d) is called complete if every Cauchy sequence in X converges. Proposition 1.32. If (X , d) is a complete metric space and Y ⊂ X is a closed subspace, then (Y , d) is complete. Proof. Let (xn )n∈N ⊂ Y be a Cauchy sequence. Clearly, (xn )n∈N ⊂ X is a Cauchy sequence, too. Now since (X , d) is complete, there exists x ∈ X such that d(xn , x) → 0 as n → ∞. Now since Y is closed it follows that x ∈ Y , that is, Y is complete. Example 1.33. Not all metric spaces are complete. A concrete example is Q when it is equipped with its natural metric, that is, d(x, y) = |x − y| for all x, y ∈ Q. To see that Q is not complete, we need to construct a Cauchy sequence in it, which does not converge in Q. Let α be an irrational number and let (xn )n∈N be the sequence given by, xn = 10−n E(10n α ), n ∈ N, where E(·) is the greatest integer function. Clearly, xn ∈ Q for all n ∈ N. Using the inequality 0 ≤ y − E(y) < 1 for all y ∈ R it follows that 0 ≤ α − xn < 10−n for all n ∈ N. Letting n → ∞ in the previous inequality it follows that (xn )n∈N converges to α ∈ Q. Therefore, (xn )n∈N is a Cauchy sequence which does not converge in Q. Example 1.34. The metric spaces (Fn , d1 ), (Fn , d p ), and (Fn , d∞ ) are complete. Example 1.35. Let X = C[−2, 2] be the collection of all continuous functions f : [−2, 2] → F equipped with the metric given by

δ ( f , g) =

 2 −2

| f (t) − g(t)|dt

for all f , g ∈ X . Proposition 1.36 ([147]). The metric space (C[−2, 2], δ ) is not complete. Proof. It suffices to construct a Cauchy sequence in X = C[−2, 2] which does not converge. Indeed, consider the sequence of functions fn in X defined by

14

1 Metric, Banach, and Hilbert Spaces

⎧ ⎪ 0 ⎪ ⎪ ⎪ ⎪ ⎨ fn (t) = nt + 1 − n ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 1

if − 2 ≤ t ≤ 1 − 1n , if 1 − 1n ≤ t ≤ 1, if 1 ≤ t ≤ 2.

Now

δ ( fn , fm ) =

1 2

1 1 − n m

 → 0 as n, m → ∞,

and hence fn is a Cauchy sequence. Let us notice however that the sequence fn does not converge in X . Indeed, assuming that there exists f ∈ X such that δ ( fn , f ) → 0 as n → ∞ it follows from the well-known Fatou’s lemma (see [29]) that  2 −2

lim inf | fn − f |dt ≤ lim inf n

n

 2 −2

| fn − f |dt = 0

which yields f = lim inf fn = lim fn a.e. on [−2, 2], where f is given by n

n→∞

f (t) =

⎧ ⎨0 ⎩

1

if t > 1, if t ≤ 1.

Now f ∈ X as f is not continuous at t = 1, and hence the space (X , δ ) is not complete. Example 1.37. Let (X , d1 ) and (Y , d2 ) be two metric spaces and let BC(X , Y ) denote the collection of all bounded continuous functions f : X → Y . Define the mapping d : BC(X , Y ) × BC(X , Y ) → R+ by d( f , g) := sup d2 ( f (x), g(x)) x∈X

for all f , g ∈ BC(X , Y ). It can be shown that d is a distance on BC(X , Y ) × BC(X , Y ). Theorem 1.38. If (Y , d2 ) is complete, then so is the metric space (BC(X , Y ), d). Proof. Let ( fn )n∈N ⊂ (BC(X , Y ), d) be a Cauchy sequence. Thus for every x ∈ X and ε > 0 there exists N such that d2 ( fn (x), fm (x)) ≤ d( fn , fm ) < ε

(1.3)

1.1 Metric Spaces

15

for all n, m ≥ N . Consequently, the sequence ( fn (x))n∈N ⊂ Y is a Cauchy sequence. Now since (Y , d2 ) is complete, it follows that there exists f (x) ∈ Y such that d2 ( fn (x), f (x)) → 0 as n → ∞. Clearly, the limit, f (x) = lim fn (x) n→∞

with respect to the distance d2 defines a function f : X → Y , which is well defined. To complete the proof, we have to prove three things: (a) That d( fn , f ) → 0 as n → ∞. (b) That f is continuous. (c) That f is bounded. Letting m → ∞ in Eq. (1.3) it follows d2 ( fn (x), f (x)) ≤ ε for all n ≥ N . Consequently, d( fn , f ) ≤ ε for all n ≥ N , and hence d( fn , f ) → 0 as n → ∞. The proof for the continuity of f has already been done (see Proposition 1.29). Using the fact that d( fn , f ) → 0 as n → ∞ and taking ε = 1, one can find N0 ∈ N such that for all x, y ∈ X , we have d2 ( fN0 (x), f (x)) < 1 and d2 ( fN0 (y), f (y)) < 1. Now d2 ( f (x), f (y)) ≤ d2 ( f (x), fN0 (x)) + d2 ( fN0 (x), fN0 (y)) + d2 ( fN0 (y), f (y)) < 2 + d2 ( fN0 (x), fN0 (y)) ≤ 2 + diam( fN0 (X )) <∞ and hence f is bounded.

1.1.1.5

Separable Spaces

Recall that a map f : X → Y is said to be a bijection if f is one-to-one (injective) and onto (surjective). Namely, a function f is said to be one-to-one if f (x) = f (y) yields x = y. Similarly, a function f is said to be onto if for each y ∈ Y there exists some x ∈ X such that y = f (x). For instance, the function f : R → R given by f (t) = t 3 is a bijection. Two sets A and B are said to have the same cardinality if there exists a bijection between A and B. In particular, a set C is said to be finite if C is either empty or there exists N ∈ N such that there is a bijection between C and the set {1, 2, 3, . . . , N}. If a set is not finite, it is said to be infinite. Definition 1.39. A set A is said to be countable if A is finite or has the same cardinality as N. A set B is said to be uncountable if it is infinite and not countable. Classical examples of countable sets include Z and Q. And classical examples of uncountable sets include R and R+ .

16

1 Metric, Banach, and Hilbert Spaces

Definition 1.40. A metric space (X , d) is said to be separable if it contains a countable subset A that is dense in X . Classical examples of separable metric spaces include Rn . Indeed, Qn is a countable dense subset in Rn .

1.1.1.6

Homeomorphisms

Definition 1.41. Let (X , d) and (Y , ρ ) be two metric spaces. A function f : (X , d) → (Y , ρ ) is said to be homeomorphic if: (a) f a bijection (one-to-one and onto). (b) f is continuous. (c) f −1 the inverse of f is continuous. If a function f : (X , d) → (Y , ρ ) is homeomorphic, then the metric spaces X and Y are said to be homeomorphic or topologically equivalent. Example 1.42. Consider the unit circle S1 given by S1 = {(x, y) ∈ R2 : x2 + y2 = 1}. Similarly, consider the square D = {(x, y) ∈ R2 : |x| + |y| = 1}. It is well known that both S1 and D are homeomorphic (topologically equivalent) as the map ϕ : S1 → D defined by

ϕ (x, y) =

x y , |x| + |y| |x| + |y|



is a homeomorphism whose inverse ϕ −1 : D → S1 is given by 

ϕ

−1

(x, y) =



x x2 + y2

,

y x2 + y2

.

Example 1.43. The interval (−1, 1) and R are homeomorphic (topologically equivalent) as the map ϕ : R → (−1, 1) defined by

ϕ (x) =

x |x| + 1

is a homeomorphism whose inverse ϕ −1 : (−1, 1) → R is given by

ϕ −1 (x) =

x . 1 − |x|

Theorem 1.44. Let (X , d1 ), (Y , d2 ), and (Z , d3 ) be metric spaces. If f : X → Y and g : Y → Z are homeomorphisms, then so is their composition g ◦ f .

1.1 Metric Spaces

17

Proof. Set h = g ◦ f . Clearly, h is continuous as the composition of two continuous functions. Since both f and g are homeomorphisms, then their inverses g−1 : Z → Y and f −1 : Y → X are continuous. Consequently, the composition f −1 ◦ g−1 is continuous. Now (g ◦ f ) ◦ ( f −1 ◦ g−1 ) = g ◦ ( f ◦ f −1 ) ◦ g−1 = g ◦ g−1 = IY where IY is the identity function of Y . Similarly, it can be easily shown that ( f −1 ◦ g−1 ) ◦ (g ◦ f ) = IX where IX is the identity function of X . In view of the above, it follows that g ◦ f is a homeomorphism.

1.1.2 The Banach Fixed-Point Theorem The Banach fixed-point theorem plays an important role in many fields including differential equations and partial differential equations. In this subsection, we discuss such an important theorem as well as its proof, and provide a concrete example. Let (X , d) be a metric space. A mapping T : X → X is said to have a fixedpoint if there exists x0 ∈ X such that T (x0 ) = x0 . The collection of all fixed-points for T will be denoted FT . Definition 1.45. Let (X , d) be a metric space. A mapping T : X → X is said to be Lipschitz if there exists a constant L ≥ 0 called Lipschitz constant such that d(T (x), T (y)) ≤ L · d(x, y)

(1.4)

for every x, y ∈ X . Example 1.46. Let √ X = R equipped with the distance d(x, y) = |x − y| for all x, y ∈ R and let T (x) = x 5 − 10. One can easily see that √ √ T is Lipschitz with Lipschitz constant L = 5. Indeed, we have |T (x) − T (y)| = 5 |x − y| for all x, y ∈ R. Definition 1.47. Let (X , d) be a metric space. A mapping T : X → X is said to be a strict contraction if it is Lipschitz with Lipschitz constant L ∈ [0, 1). Example 1.48. Let X = R be equipped with the distance d(x, y) = |x − y| for all x, y ∈ R and let T (x) = sin(dx) where 0 ≤ d < 1. One can easily see that T is a strict contraction as its Lipschitz constant d ∈ [0, 1). Indeed, we have | sin(dx) − sin(dy)| ≤ d |x − y| for all x, y ∈ R. Theorem 1.49 (Banach Fixed-Point Theorem). Let (X , d) be a complete metric space and let T : X → X be a strict contraction, that is, there exists L ∈ [0, 1) such that Eq. (1.4) holds. Then T has a unique fixed-point, that is, there exists a unique x¯ ∈ X such that T (x) ¯ = x. ¯ Equivalently, FT = {x}. ¯ Proof (Existence of a fixed-point). Choose any arbitrary point y ∈ X and consider the sequence (xn )n∈N ⊂ X defined by

18

1 Metric, Banach, and Hilbert Spaces

x0 = y, and xn+1 = T (xn ), n = 1, 2, 3, . . . . Using the fact that T is Lipschitz it follows that d(x2 , x1 ) ≤ Ld(x1 , y), d(x3 , x2 ) ≤ Ld(x2 , x1 ) ≤ L2 d(x1 , y), ... d(xn+1 , xn ) ≤ Ln d(x1 , y), and therefore d(xn+m , xn ) ≤ d(xn+m , xn+m−1 ) + d(xn+m−1 , xn+m−2 ) + · · · + d(xn+1 , xn ) ≤ (Lm−1 + Lm−2 + · · · + L + 1)Ln d(x1 , y) ≤

Ln d(x1 , y). 1−L

It is then clear that (xn )n∈N is a Cauchy sequence. Now since (X , d) is complete it ¯ → 0 as n → ∞. Also, since T is follows that there exists x¯ ∈ X such that d(xn , x) Lipschitz, it is (uniformly) continuous and hence passing to the limit in the equation xn+1 = T (xn ) it follows x¯ = T (x), ¯ that is, x¯ is a fixed-point of T . Uniqueness of a fixed-point—Suppose T has two fixed-points z and z with z = z . By assumption, z = T (z) and z = T (z ). Using the fact that T is a strict contraction it follows that d(z, z ) = d(T (z), T (z )) ≤ Ld(z, z ) < d(z, z ) and this is a contradiction. Therefore, one must have z = z .

1.1.2.1

Example

Let X = R be equipped with its natural distance, that is, d(x, y) = |x − y| for all x, y ∈ R. Consider the first-order autonomous differential equation given by u (t) = −Au(t) + f (t, u(t)),

t ∈R

(1.5)

where A > 0 is a constant and f : R × R → R is a continuous periodic function, that is, there exists ω > 0 such that f (t + ω , u) = f (t, u) for all t ∈ R and u ∈ R. Further, we assume that f is Lipschitz, that is, there exists L > 0 such that | f (t, u) − f (t, v)| ≤ L|u − v| for all u, v ∈ R and t ∈ R. Theorem 1.50. Under previous assumptions, then Eq. (1.5) has a unique continuous periodic solution whenever L is small enough, that is, L < A.

1.1 Metric Spaces

19

Proof. All bounded solutions to Eq. (1.5) are given by u(t) =

 t −∞

e−A(t−s) f (s, u(s))ds,

t ∈ R.

Consider the nonlinear integral operator G defined by (Gu)(t) :=

 t −∞

e−A(t−s) f (s, u(s))ds,

t ∈ R.

Let w be a continuous periodic function with period ω . It is clear that g(t) := f (t, w(t)) is a continuous periodic function with period ω . Further, it is easy to see that the function t → (Gw)(t) is continuous. Let us show that t → G(w)(t) is periodic. Indeed, (Gw)(t + ω ) = = = =

 t+ω −∞

 t

−∞

 t

−∞

 t

−∞

e−A(t+ω −s) f (s, w(s))ds

e−A(t−s) f (s + ω , w(s + ω ))ds e−A(t−s) f (s + ω , w(s))ds e−A(t−s) f (s, w(s))ds

for all t ∈ R. Let u, v be two arbitrary continuous periodic functions. It is then easy to see that |G(u)(t) − G(v)(t)| ≤ LA−1 sup |u(t) − v(t)| t∈R

for all t ∈ R, which yields sup |G(u)(t) − G(v)(t)| ≤ LA−1 sup |u(t) − v(t)|. t∈R

t∈R

Using the Banach fixed-point theorem (Theorem 1.96) in the (complete) metric space BC(R, R) equipped with sup-distance defined by d( f , g) = sup | f (t) − g(t)| t∈R

for all f , g ∈ BC(R, R), it follows that Eq. (1.5) has a unique continuous periodic solution whenever L is small enough, that is, L < A.

20

1 Metric, Banach, and Hilbert Spaces

1.1.3 Compact Metric Spaces The notion of compactness is one of the most powerful concepts in functional analysis. There are various equivalent definitions of the notion of compactness but the most used in applications is the so-called sequential compactness. Definition 1.51. A metric space (X , d) is said to be sequentially compact if every sequence in X contains a convergent subsequence. A subset S of X is said to be sequentially compact if (S, d) is sequentially compact. That is, every sequence in S contains a subsequence that converges to a point of S. Lemma 1.52 ([141]). Let (X , d) be a metric space. If S ⊂ X is sequentially compact, then S is closed. Proof. The proof is left to the reader as an exercise. Theorem 1.53 ([141]). Let (X , d) be a sequentially compact metric space. A set S ⊂ X is sequentially compact if and only if S is closed. Proof. Suppose A ⊂ X is sequentially compact and so according to Lemma 1.52, A is closed. Conversely, assume that A is closed and let (xn )n∈N ⊂ A ⊂ X be a sequence. Using the fact X is sequentially compact it follows that (xn )n∈N has a subsequence (xnk )k∈N such that xnk → x as k → ∞ where x ∈ X . From the closedness of A it follows that x ∈ A. Therefore, A is sequentially compact. In addition to the above-mentioned characterizations of sequentially compact metric spaces, we also have (we refer the reader to [141] for their proofs): Proposition 1.54. Let (X , d) be a metric space. Then the following properties hold: (a) If X is sequentially compact, then it is complete. (b) If X is sequentially compact, then it is totally bounded. (c) X is sequentially compact if and only if it is totally bounded and complete. Definition 1.55 ([141]). Let (X , d) be a metric space and let S be a subset of X .  A collection of sets {Oλ }λ ∈Λ in X is called a covering of S if S ⊂ λ ∈Λ Oλ . A  subcollection {Oγ }γ ∈Γ of the covering {Oλ }λ ∈Λ such that S ⊂ γ ∈Γ Oγ is said to be a subcovering of {Oλ }λ ∈Λ . Any covering or subcovering made up entirely of open sets is called an open covering or subcovering. We now have the following equivalent definition of the compactness using coverings and subcoverings. Definition 1.56. A metric space (X , d) is said to be compact if every open covering of X contains a finite open subcovering. Theorem 1.57 ([141]). A metric space (X , d) is sequentially compact if and only if it is compact.

1.1 Metric Spaces

21

Proof. The proof is left to the reader as an exercise. Theorem 1.58 ([141]). Let (X , d) be a complete metric space. A subset S ⊂ X is compact (sequentially compact) if and only if it is closed and totally bounded. Proof. The proof is left to the reader as an exercise. Theorem 1.59. Let (X , d1 ) and (Y , d2 ) be two metric spaces. If K ⊂ X is a compact metric space and if f : X → Y is continuous, then, f (K) is compact in Y . In particular, if K = X is compact, then f (X ) is compact. Proof. To show that f (K) is compact, let {Oλ }λ ∈Λ be an open covering of f (K), that is, f (K) ⊂



Oλ .

λ ∈Λ

Since f is continuous it follows that f −1 (Oλ ) is an open set for all λ ∈ Λ . Consequently, the collection { f −1 (Oλ ) : λ ∈ Λ } is a family of open sets whose union contains K. Now using the fact that K is compact it follows that there exists a finite number of open sets f −1 (Oλ1 ), f −1 (Oλ2 ), . . . , f −1 (Oλn ) such that K⊂

n

f −1 (Oλk )

k=1

and therefore, f (K) ⊂

n

Oλk ,

k=1

that is, f (K) is compact. Corollary 1.60. If (X , d) is a compact metric space and if f : X → F is a continuous function, then, max | f (x)| and min | f (x)| exist. x∈X

x∈X

Proof. Using Theorem 1.59 it follows that f (X ) is a compact subset of F. Consequently, f (X ) is bounded. Let M = maxx∈X | f (x)|. Now let xn ∈ X such that M − n−1 ≤ | f (xn )| ≤ M for all n = 1, 2, . . .. Since X is compact, then there exists a subsequence (xnk )k∈N ⊂ (xn )n∈N which converges to some x ∈ X . Now since | f | is continuous it follows | f (x)| = limk→∞ | f (xnk )| = M. The proof for the minimum makes use of similar ideas as above.

1.1.3.1

Arzel`a–Ascoli Theorem

One of the most important theorems in functional analysis is that of Arzel`a–Ascoli Theorem, which formulates a compactness criterion in the well-known metric space

22

1 Metric, Banach, and Hilbert Spaces

of continuous functions on a compact. To formulate such a theorem, we need to introduce the concept of equi-continuity whose definition is given below. Definition 1.61. Let (X , d) and (Y , ρ ) be two metric spaces. A subclass F ⊂ C(X , Y ) is said to be equi-continuous if for all x0 ∈ X and for all ε > 0 there exists δ = δ (x0 , ε ) > 0 such that d(x, x0 ) < δ yields ρ ( f (x), f (x0 )) < ε for all f ∈ F . Suppose (X , d) is compact and let Y = C. Let C(X ) denote the collection of all continuous functions from X to C. The space C(X ) is equipped with the supdistance defined by d( f , g) = maxx∈X | f (x) − g(x)| for all f , g ∈ C(X ). Recall that a subclass F ⊂ C(X ) is said to be uniformly bounded if there exists C > 0 such that | f (x)| ≤ C for every x ∈ X and f ∈ F . Theorem 1.62 (Arzel`a–Ascoli Theorem). A subclass F ⊂ C(X ) is relatively compact (i.e., the closure of F is compact) if and only if the following conditions hold: (a) F is equi-continuous; and (b) F is uniformly bounded. Proof. The proof is highly technical and so we refer the reader to the excellent book by Naylor and Sell [141]. Corollary 1.63. A subclass F of C(X ) is compact if and only if it is closed, uniformly bounded, and equi-continuous.

1.2 Banach Spaces A Banach space is a straightforward example of a metric space. The notion of Banach spaces plays a key role in several fields especially in functional analysis. Such a powerful concept was introduced in the literature by the Polish mathematician S. Banach in 1922. Banach introduced these spaces in his Ph.D. Thesis, which marked the birth of functional analysis. Classical examples of Banach spaces include, but are not limited to, Euclidean spaces, L p -spaces, l p -spaces, Sobolev spaces, the space of bounded continuous functions, the space of almost periodic functions, the space of almost automorphic functions, the space of pseudo-almost periodic functions, etc.

1.2.1 Basic Definitions Recall that the definition of a normed vector space (X , · ) is given in Example 1.5. The distance here is defined by d(x, y) = x − y for all x, y ∈ X . Prior to discussing about Banach spaces, let us establish a few results for normed vector spaces. First of all, note that if (X , · ) is a normed vector space and if (xn )n∈N ⊂ X is a sequence, then if xn → x as n → ∞ for some x ∈ X , that is,

1.2 Banach Spaces

23

lim d(xn , x) = lim xn − x = 0,

n→∞

n→∞

one says that (xn )n∈N converges strongly to x. This is sometimes denoted by s − limn→∞ xn = x. Proposition 1.64. Let (X , · ) be a normed vector space. If (xn )n∈N ⊂ X is a Cauchy sequence and if there is a subsequence (xnk )k∈N of (xn )n∈N such that xnk → x as k → ∞, then xn → x as n → ∞. Proof. For each ε > 0 there exists N ∈ N such that xn − xm < ε for all n, m ≥ N. Similarly, there exists k ∈ N such that x − xnk < ε for all nk ≥ N. Now x − xn = x − xnk + xnk − xn ≤ x − xnk + xnk − xn < 2ε for n > N and hence xn → x as n → ∞. Proposition 1.65. Let (X , · ) be a normed vector space. If (xn )n∈N ⊂ X is a Cauchy sequence, then there is a subsequence (xnk )k∈N of (xn )n∈N such that for all k ≥ 1, we have, xnk+1 − xnk ≤ 2−k . Proof. Since (xn )n∈N is a Cauchy sequence, there is n1 ∈ N such that xn − xm ≤ 2−1 for all n, m ≥ n1 . Similarly, there is n2 > n1 such that xn − xm ≤ 2−2 for all n, m ≥ n2 . Proceeding in this way, it follows that there exists a sequence n1 < n2 < n3 < · · · such that for all k ≥ 1, xn − xm ≤ 2−k for all n, m ≥ nk . Now, since nk+1 > nk , it follows that for all k ≥ 1, xnk+1 − xnk ≤ 2−k . Definition 1.66. A normed vector space (X , · ) is said to be a Banach space if it is complete. Proposition 1.67. Let (X , · ) be a Banach space. Then any closed subspace M of X is also a Banach space. Proof. Let (xn )n∈N ⊂ M be a Cauchy sequence. Using the facts that M ⊂ X and that (X , · ) is complete it follows that there exists x ∈ X such that d(xn , x) = xn − x → 0 as n → ∞. Now since M is closed it follows that x ∈ M, and therefore, (M, · ) is complete which yields it is a Banach space.

1.2.1.1

The Dual of a Banach Space

If (X , · ) is a vector normed space, then its (topological) dual denoted X ∗ consists of all continuous linear functionals ξ : X → F. That is, all ξ : X → F satisfying:

24

1 Metric, Banach, and Hilbert Spaces

(a) ξ (α x + β y) = αξ (x) + β ξ (y) for all α , β ∈ F and for all x, y ∈ X . (b) There exists C ≥ 0 such that |ξ (x)| ≤ C x for all x ∈ X . If ξ ∈ X ∗ and x ∈ X , then the quantity ξ (x) will be denoted by ξ , x. Definition 1.68. A sequence (xn )n∈N ⊂ X is said to converge weakly to some x ∈ X if ξ , xn  → ξ , xn  as n → ∞ in F for all ξ ∈ X ∗ . This is also denoted by xn  x or w − lim xn = x. n→∞

Definition 1.69. If (X , · ) is a normed vector space, then its (topological) dual X ∗ is equipped with the norm defined by ξ ∗ = sup |ξ , x| = sup |ξ , x| = sup x ≤1

0=x∈X

x =1

|ξ , x| . x

In view of the above, |ξ , x| ≤ ξ ∗ x for all ξ ∈ X ∗ and x ∈ X . Theorem 1.70. The normed vector space (X ∗ , · ∗ ) is a Banach space. Proof. Let (ξn )n∈N ⊂ X ∗ be a Cauchy sequence. Now for each x ∈ X , the sequence ξn , x is a Cauchy sequence in F as |ξn , x − ξm , x| ≤ ξn − ξm ∗ . x . Now since (F, | · |) is complete, then (ξn , x)n∈N converges to some ξ , x. Clearly, the functional ξ defined in this fashion is linear. Let y ∈ X . Then for all ε > 0 there exists N0 ∈ N such that ξn − ξm ∗ < ε2 for all n, m ≥ N0 . Now |ξn , y| = |ξn − ξm , y + ξm , y| ≤ |ξn − ξm , y| + |ξm , y| ε  + ξm ∗ ≤ y 2

 ε ≤ y + sup ξm ∗ . 2 m∈N Letting n → ∞ it follows that

 ε + sup ξm ∗ , |ξ , y| ≤ y 2 m∈N that is, ξ : X → F is continuous. (Note that sup ξm ∗ < ∞ as (ξn )n∈N ⊂ X ∗ is a m∈N

Cauchy sequence.) To complete the proof, we have to prove that ξn − ξ ∗ → 0 as n → ∞. Clearly, for each ε > 0, there exists N ∈ N such that ξn − ξm ∗ < ε2 for all n, m ≥ N. Fixing n > N, one can see that for any m > N and for any x ∈ X ,

ε |ξ − ξn , x| ≤ |ξ − ξm , x| + |ξm − ξn , x| ≤ |ξ − ξm , x| + x . 2

1.2 Banach Spaces

25

Clearly, we can choose m large enough so that |ξ − ξm , x| ≤ ε x 2 . Therefore, for n > N, we have |ξ − ξn , x| ≤ 2−1 ε x , which yields ξn − ξ ∗ → 0 as n → ∞. The proof is complete. Since the dual (X ∗ , · ∗ ) is a Banach space, then we can form the dual of X ∗ denoted by X ∗∗ and called the bidual of X . Clearly, X ∗∗ is the collection of all continuous linear functionals ϖ : X ∗ → F. Note that X can be identified with a subspace of X ∗∗ as follows: define the so-called canonical injection by J : X → X ∗∗ with J(x)(ξ ) = ξ (x) for all x ∈ X and ξ ∈ X ∗ . Clearly, J(x) ∈ X ∗∗ for each x∈X. Definition 1.71. Let (X , · ) be a Banach space and let J : X → X ∗∗ be the canonical injection from X into X ∗∗ . We say that X is reflexive, if J is surjective, that is, J(X ) = X ∗∗ . Example 1.72. (a) Any finite dimensional space X is reflexive as X =X ∗ =X ∗∗ . (b) Hilbert spaces are reflexive. (c) l p is reflexive for 1 < p < ∞. Theorem 1.73 (Kakutani [29]). A Banach space (X , · ) is reflexive if and only its unit closed ball B = {x ∈ X : x ≤ 1} is weakly compact. Theorem 1.74 ([29]). Let (X , · ) be a reflexive Banach space. Then, every bounded sequence (xn )n∈N ⊂ X has a subsequence (xnk )k∈N which converges weakly. The converse of Theorem 1.74 is also true. Theorem 1.75 (Eberlein–Smulian [29]). If (X , · ) is a Banach space in which every bounded sequence (xn )n∈N ⊂ X has a subsequence (xnk )k∈N which converges weakly, then X is reflexive. Definition 1.76. A Banach space (X , · ) is called uniformly convex if for each ε > 0 there exists δ > 0 such that   x+y  < 1−δ.  x, y ∈ X , x ≤ 1, y ≤ 1 and x − y ≥ ε yields  2  Example 1.77. Let X = Rn be equipped with the Euclidean norm, that is, the norm defined by x n = (x12 + x22 + · · · + xn2 )1/2 . It can be shown that (Rn , · n ) is uniformly convex. However, the space Rn equipped with either x 1 = |x1 | + |x2 | + · · · + |xn | or x ∞ = max(|x1 |, |x2 |, . . . , |xn |), is not uniformly convex. Theorem 1.78 (Milman–Pettis [29]). Every uniformly convex Banach space is reflexive.

26

1 Metric, Banach, and Hilbert Spaces

1.2.2 Examples of Banach Spaces 1.2.2.1

The L p -Spaces

Let Ω ⊂ Rn be a domain and let 1 ≤ p < ∞. If (X , · ) is a Banach space, then we define the space L p (Ω , X ) (or L p (Ω ), if there is no risk of confusion) to be the collection of all (Lebesgue) measurable functions u : Ω → X such that u p :=



1 u(x) dx p

Ω

p

< ∞.

(1.6)

This is understood to mean that in L p (Ω ), two functions u and v are equal whether they are equal almost everywhere on Ω . Obviously, · p is a norm on L p (Ω , X ). Similarly, one defines L∞ (Ω , X ) (or L∞ (Ω ), if there is no risk of confusion) as the collection of all (Lebesgue) measurable functions u : Ω → X such that there exists a constant C ≥ 0 such that f (x) ≤ C a.e. x ∈ Ω . The space L∞ (Ω , X ) is equipped with the following norm: u ∞ := inf{C : u(x) ≤ C a.e. x ∈ Ω }.

(1.7)

Theorem 1.79 (Monotone Convergence Theorem [29]). Let (un )n∈N be a sequence of (real-valued) functions that belongs to L1 (Ω ) and satisfies (a) u0 (x)  ≤ u1 (x) ≤ · · · ≤ un (x) ≤ un+1 (x) . . . a.e. x ∈ Ω . (b) sup

n∈N Ω

un (x)dx < ∞.

Then the sequence of functions (un (x))n∈N converges to some u(x) as n → ∞ for a.e. x ∈ Ω . Moreover, u ∈ L1 (Ω ) with un − u 1 → 0 as n → ∞. Theorem 1.80 (Lebesgue Dominated Convergence Theorem [29]). Let (un )n∈N ⊂ L p (Ω ) be a sequence that converges a.e. to some measurable function u. Suppose that there exists a positive real-valued function v ∈ L p (Ω ) such that un (x) ≤ v(x) a.e. for each n ∈ N. Then u ∈ L p (Ω ) and un − v p → 0 as n → ∞. Proposition 1.81 (H¨older’s Inequality). Let 1 ≤ p ≤ ∞ and let 1 < q < ∞ such that p−1 + q−1 = 1. If u ∈ L p (Ω ) and v ∈ L p (Ω ) where u is X -valued and v is F-valued, then their product uv ∈ L1 (Ω ). Moreover,  Ω

||uv(x)||dx ≤ u p · v q .

The proof of Proposition 1.81 requires the next technical lemma whose proof will be left to the reader as an exercise.

1.2 Banach Spaces

27

Lemma 1.82. Let 1 < p, q < ∞ such that p−1 + q−1 = 1. Then for any positive real numbers a and b, we have the following identity ab ≤ a p/q + bq/p . Proof (Proposition 1.81). The inequality holds for p = 1 or p = ∞ and so it is enough to suppose 1 < p < ∞. It is clear that uv is well defined and is measurable. It is enough to suppose that u p > 0 and v q > 0. Indeed, if either u or v = 0 a.e., then Proposition 1.81 obviously holds. Letting a = u ( u p )−1 and b = |v|( v q )−1 in Lemma 1.82 one obtains  −1  −1 u · |v| ≤ u p p u pp + |v|q q v qq . u p · v q By assumption the right side of the previous inequality is integrable and hence uv ∈ L1 (Ω ). Integrating both side of the inequality it follows that  −1  −1 uv 1 ≤ u pp p u pp + v qq q v qq =1 u p · v q and hence, uv 1 ≤ u p · v q . Using H¨older’s inequality (Proposition 1.81) we obtain the so-called Minkowski’s inequality. Proposition 1.83 (Minkowski’s Inequality). Let 1 ≤ p ≤ ∞. If u, v ∈ L p (Ω ), then u + v ∈ L p (Ω ). Moreover, u + v p ≤ u p + v p . Proof. Clearly, the proposition holds for p = 1 or p = ∞. Consequently, it is enough to assume 1 < p < ∞. Clearly u + v p ≤ ( u + v ) p + 2 p ( u p + v p ) which yields u + v ∈ L p (Ω ). Now u + v pp = = ≤

 Ω



Ω



Ω

u(x) + v(x) p dx u(x) + v(x) p−1 · u(x) − v(x) dx u(x) + v(x) p−1 · u(x) dx +

 Ω

u(x) + v(x) p−1 · v(x) dx.

28

1 Metric, Banach, and Hilbert Spaces

Since the function x → u(x) + v(x) p−1 belongs to Lq (Ω ) with p−1 + q−1 = 1, using H¨older’s inequality (Proposition 1.81) it follows that u + v pp ≤ u + v p−1 p ( u p + v p ), which yields u + v p ≤ u p + v p . Proposition 1.83 shows that L p (Ω ) for 1 ≤ p ≤ ∞ is a vector space and hence p ) is a normed vector space.

(L p (Ω ), ·

Theorem 1.84 (Fischer–Riesz). The space (Lr (Ω ), · r ) is a Banach space for each 1 ≤ r ≤ ∞. Proof. There are two cases we will be considering, that is, r = ∞ and 1 ≤ r < ∞. Case r = ∞—Let (u p ) p∈N ⊂ L∞ (Ω ) be a Cauchy sequence. Letting ε = k−1 it follows that there exists Nk ∈ N such that u p − uq ∞ < k−1 whenever p, q ≥ Nk . Consequently, there exists a set Sk whose Lebesgue measure is zero such that u p (x) − uq (x) < k−1 for all x ∈ Ω − Sk , p, q ≥ Nk .

(1.8)

Clearly, the Lebesgue measure of the set S = ∪k=1,2,... Sk is zero. Moreover, for all x ∈ Ω − S, the sequence (u p (x)) p∈N ⊂ X is a Cauchy sequence. Since X is complete it follows that there exists u(x) ∈ X such that u p (x) → u(x) as p → ∞ for all x ∈ Ω − S. Now letting q → ∞ in Eq. (1.8) it follows that u p (x) − u(x) < k−1 for all x ∈ Ω − S, p ≥ Nk . Therefore, u ∈ L∞ (Ω ) and u p − u ∞ < k−1 for p ≥ Nk . Case 1 ≤ r < ∞—Let (u p ) p∈N be a Cauchy sequence in Lr (Ω ). In view of Proposition 1.64, it is sufficient to show that (u p ) p∈N has a subsequence which converges in Lr (Ω ). In view of Proposition 1.65, one can find a subsequence (u pk )k∈N of (u p ) p∈N such that u pk+1 − u pk r ≤ 2−k for k ≥ 1.

(1.9)

Setting Sq (x) := ∑qk=1 u pk+1 (x) − u pk (x) it follows that Sq r ≤ 1. Using the Monotone Convergence Theorem (see Theorem 1.79) it follows that Sq (x) converges to some S(x) for a.e. x ∈ Ω . Moreover, S ∈ Lr (Ω ). Now for m ≥ n ≥ 2, u pm (x) − u pn (x) ≤ u pm (x) − u pm−1 (x) + · · · + u pn+1 (x) − u pn (x) ≤ S(x) − Sn−1 (x). Hence for a.e. x ∈ Ω , the subsequence (u pn (x))n∈N is a Cauchy sequence in X and thus converges to some u(x) ∈ X satisfying, for a.e. x ∈ Ω , u pn − u(x) ≤ S(x), and thus u ∈ Lr (Ω ).

n≥2

1.2 Banach Spaces

29

Now using the Lebesgue Dominated Convergence Theorem (see Theorem 1.80) it follows that, u pn − u r → 0 as n → ∞ since u pn (x) − u(x) → 0 as n → ∞ for a.e. x ∈ Ω and u pn − u r ≤ Sr where Sr ∈ L1 (Ω ). Classical examples of separable spaces include L p (Ω ) for 1 ≤ p < ∞. Theorem 1.85. Let 1 < p ≤ ∞ and let 1 ≤ q < ∞ be a real number such that p−1 + q−1 = 1, then the (topological) dual of L p (Ω ) is Lq (Ω ). Moreover, the space L p (Ω ) is reflexive if and only if 1 < p < ∞. In that case, the dual of L p (Ω ) is Lq (Ω ). Proof. The proof is left to the reader as an exercise. p Define Lloc (Ω ) for 1 ≤ p < ∞ to be the collection of all measurable functions f : Ω → X such that

 Ω

1 f (x) p dx

p

<∞

(1.10)

for any Ω ⊂ Ω compact subset. p In Lloc (Ω ) we define the notion of convergence as follows: a sequence of p p functions ( fn )n∈N ∈ Lloc (Ω ) is said to converge to some f ∈ Lloc (Ω ) whenever fn − f p → 0 as n → ∞ in L p (Ω ) for any Ω ⊂ Ω compact subset. Although p Lloc (Ω ) is a topological space, it is not a Banach space. 1.2.2.2

The Sobolev Spaces W k,p (Ω )

Let 1 ≤ p ≤ ∞ and let k ∈ N. Let us suppose that Ω ⊂ Rn is an open locally Jordan measurable subset. Moreover, we only consider functions, which will go from Ω to F. There are several equivalent definitions of Sobolev spaces. In this book, these spaces will be defined as follows: Definition 1.86. The Sobolev spaces W k,p (Ω ) are defined by W k,p (Ω ) := {u ∈ L p (Ω ) : Dα u ∈ L p (Ω ) for |α | ≤ k},

(1.11)

where Dα is the partial differential operator defined by Dα =

∂ x1α1

∂ |α | ∂ x2α2 . . . ∂ xnαn

with α = (α1 , α2 , . . . , αn ), αi ∈ N for i = 1, . . . , n, and |α | = α1 + α2 + · · · + αn . For p = 2, the corresponding Sobolev space W k,2 (Ω ) is denoted by H k (Ω ). One equips the vector space W k,p (Ω ) with the norm defined by

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1 Metric, Banach, and Hilbert Spaces

 u k,p =

1

∑

α

|α |≤k

D

p

,

u pp

if

1 ≤ p < ∞,

(1.12)

and u k,∞ = max |Dα u|∞

if

|α |≤k

p = ∞.

(1.13)

Let C0∞ (Ω ) denote the collection of all functions u : Ω → F of class C∞ with compact support in Ω . Definition 1.87. The Sobolev space W0k,p (Ω ) is defined to be the closure of C0∞ (Ω ) in the space W k,p (Ω ), that is, W k,p (Ω )

W0k,p (Ω ) = C0∞ (Ω )

.

If p = 2, then the space W0k,2 (Ω ) is denoted by H0k (Ω ). It should also be mentioned that in the case when Ω = Rn , then W0k,p (Rn ) = W k,p (Rn ). Theorem 1.88. The Sobolev space W k,p (Ω ) is a Banach space. The proof of Theorem 1.88 requires the following technical lemma whose proof will be left to the reader as an exercise. 1 (Ω ) such that u → u as n → ∞ in L1 (Ω ). Lemma 1.89. Let (un )n∈N ∈ Lloc n loc 1 (Ω ) and Dα u → v in L1 (Ω ). Then Dα u = v. Suppose Dα un ∈ Lloc n loc

Proof (Theorem 1.88). Let (un )n∈N be a Cauchy sequence in W k,p (Ω ). It is not hard to see that all the sequences (Dα un )n∈N for |α | ≤ k are Cauchy sequences in L p (Ω ). Using the fact L p (Ω ) is complete it follows that there exist two functions u and uα in L p (Ω ) such that un − u p → 0 and Dα un − uα p → 0 as n → ∞. Also, we have 1 (Ω ) as n → ∞. From Lemma 1.89, we conclude that un → u, Dα un → uα in Lloc α D u = uα . The definition of Sobolev spaces can be extended to the case of real orders. Definition 1.90. Let 1 ≤ p < ∞ and let s = k + σ where k ∈ N and σ ∈ (0, 1). We then define the Sobolev space W s,p (Ω ) as follows  W

s,p

(Ω ) :=

u∈W

k,p

(Ω ) :

|Dα u(x) − Dα u(y)| x − y σ + p n

whose norm is given by  u s,p =

u k,p +

∑



|α |=k Ω ×Ω

 ∈ L (Ω × Ω ), ∀α , |α | = k , p

1/p |Dα u(x) − Dα u(y)| p dxdy . x − y pσ +n

1.2 Banach Spaces

31

Note that if p = 2, then W s,2 (Ω ) is denoted H s (Ω ). If s ≥ 0, we then define to be the closure of the space C0∞ (Ω ) in the Sobolev space W s,p (Ω ). In particular, W0s,2 (Ω ) will be denoted by H0s (Ω ). Let s ≥ 0 and let 1 ≤ p < ∞. If q is given such that p−1 + q−1 = 1, we then define the (topological) dual of W0s,p (Ω ) to be the Sobolev space W −s,q (Ω ). In particular,

W0s,p (Ω )

W −s,2 (Ω ) = H −s (Ω ). For more on Sobolev spaces and related topics, we refer the interested reader to the landmark book by Adams [5]. 1.2.2.3

The Space BC(R, X )

Let (X , · ) be a Banach space and consider BC(R, X ) the space of all bounded continuous functions equipped with sup-norm defined by u ∞ := sup u(t) . t∈R

As we have previously seen, BC(R, X ) is a metric space when it is equipped with the metric defined by d( f , g) := f − g ∞ = sup f (t) − g(t) t∈R

for all f , g ∈ BC(R, X ). Theorem 1.91. The metric space (BC(R, X ), d) is complete and hence BC(R, X ) equipped with · ∞ is a Banach space. Proof. This is an immediate consequence of Theorem 1.38 and hence is omitted. An immediate consequence of Theorem 1.91 is the following: Proposition 1.92. If ( fn )n∈N ⊂ BC(R, X ) such that fn converges to some f with respect to the sup-norm, then f ∈ BC(R, X ). 1.2.2.4

The Space BCm (R, X )

Let (X , · ) be a Banach space and let C(n) (R, X ) be the collection of all functions f : R → X such that f (k) exists and belongs to C(R, X ) for k = 0, 1, 2, . . . , n. (The symbol f (k) stands for the k-derivative of f . Similarly, f (0) corresponds to the continuity of f .) Definition 1.93. The space BC(n) (R, X ) consists of all functions f ∈ C(n) (R, X ) such that n

f (n) := sup ∑ f (k) (t) < ∞. t∈R k=0

32

1 Metric, Banach, and Hilbert Spaces

It can be easily checked that f (n) is a norm on BC(n) (R, X ). Moreover, if f , g ∈ BC(n) (R, X ) and α , β ∈ F, then α f + β g ∈ BC(n) (R, X ). Thus BC(n) (R, X ) is a normed vector space. Theorem 1.94. The space (BC(m) (R, X ), · (m) ) is a Banach space. (k)

Proof. Let ( f p ) p≥0 ⊂ BC(m) (R, X ) be a Cauchy sequence. It is clear that ( f p ) p≥0 is a Cauchy sequence in BC(k) (R, X ) ⊂ BC(R, X ) for k = 0, 1, . . . , m. Now since BC(R, X ) equipped with the sup-norm · ∞ is a Banach space (Theorem 1.91) (k) it follows that ( f p ) p≥0 converges uniformly to some fk ∈ BC(R, X ) as p → ∞ for k = 0, 1, . . . , m. Setting f0 = g it is easily seen that fk = g(k) and that g(k) = fk ∈ BC(R, X ) for k = 0, 1, . . . , m, that is, g ∈ BC(m) (R, X ). Moreover, f p − g (m) → 0 as p → ∞, which yields (BC(m) (R, X ), · (m) ) is a Banach space.

1.2.3 The Schauder Fixed-Point Theorem The Schauder fixed-point theorem plays a crucial role in several fields especially when it comes to dealing with the existence of solutions to differential equations, integrodifferential equations, functional differential equations, difference equations, and partial differential equations. It extends the well-known Brouwer’s fixed-point theorem to the infinite dimensional setting. Definition 1.95. A nonempty set S is said to be convex if for all x, y ∈ S and λ ∈ [0, 1], then λ x + (1 − λ )y ∈ S. Theorem 1.96 (The Brouwer Fixed-Point Theorem [124]). Let S ⊂ Fn be a nonempty bounded closed convex subset. If the mapping T : S → S is continuous, then T has at least one fixed-point, that is, FT = 0. / Despite the usefulness and the beauty of Brouwer’s fixed-point theorem, it cannot be extended to infinite dimensional Banach spaces. To illustrate that, the following example was given in Khamsi and Kirk [124]. Example 1.97. Let c0 = {x = (xk )k∈N : xk ∈ F, ∀k ∈ N, and xk → 0, k → ∞}. Clearly, c0 is a Banach space when it is equipped with the sup-norm defined by x = sup |xk | for all x = (xk )k∈N . k∈N

Let S = {x ∈ c0 : x ≤ 1}. It is clear that S is a nonempty bounded closed convex set. Define the mapping T by setting T (x) = (1 − |x0 |, x0 , x1 , . . .) for all x = (xk )k∈N ∈ S. It is easy to check that T maps S into itself. Moreover, T is continuous. In fact, T (x)−T (y) = x−y for all x, y ∈ S. But assuming that T (x) = x yields x0 = x1 = · · · = 0, which, in turn, yields 1 − |x0 | = 0, and this is a contradiction. In summary, / although T satisfies all the assumptions of Brouwer’s fixed-point theorem, FT = 0.

1.3 Hilbert Spaces

33

Theorem 1.98 (The Schauder Fixed-Point Theorem [124]). Let X be a Banach space and let S ⊂ X be a nonempty compact convex subset. If the mapping T : S → S is continuous, then T has at least one fixed-point, that is, FT = 0. /

1.3 Hilbert Spaces Hilbert spaces play a crucial role in several fields such as partial differential equations, quantum mechanics, Fourier analysis, and physics. Classical examples of Hilbert spaces include, but are not limited to, the spaces Fn , L2 (Ω ), l 2 , H k (Ω ), etc. In this section we introduce and discuss basic properties of these very important spaces.

1.3.1 Basic Definitions Definition 1.99. Let H be a vector space over F. An inner product or scalar product on H is a mapping, ·, · : H × H → F, (x, y) → x, y satisfying, ∀x, y, z ∈ H , and λ ∈ F, (a) (b) (c) (d)

x + y, z = x, z + y, z. λ x, y = λ x, y and x, λ y = λ x, y. x, y = y, x. x, x ≥ 0; and x, x = 0 if and only if x = 0.

Definition 1.100. A vector space H over F endowed with an inner product is called a pre-Hilbert space. If (H , ·, ·) is a pre-Hilbert space, then we define a norm · on H by setting: x := x, x for all x ∈ H . In this event, we say that the norm · is deduced from the inner product ·, ·. Theorem 1.101 (Parallelogram Law). In a pre-Hilbert space H , we have x + y 2 + x − y 2 = 2( x 2 + y 2 ), ∀x, y ∈ H . Proof. Clearly, x + y 2 = x + y, x + y = x 2 + y 2 + 2ℜex, y. Similarly, replacing y by −y in the previous identity, we obtain x − y 2 = x|2 + y 2 − 2ℜex, y, and hence

34

1 Metric, Banach, and Hilbert Spaces

x + y 2 + x − y 2 = 2( x 2 + y 2 ) + 2ℜex, y − 2ℜex, y = 2( x 2 + y 2 ) for all x, y ∈ H . Theorem 1.102 (Cauchy–Bunyakowsky–Schwarz Inequality). In a pre-Hilbert space H , we have |x, y| ≤ x · y , ∀x, y ∈ H .

(1.14)

The equality holds if and only if x and y are linearly independent. Proof. Observe that 0 ≤ x− λ y, x− λ y = x, x−2ℜe[λ y, x]+|λ |2 y, y. Clearly, if x, y = 0, then there is nothing to prove. Now assume that x, y = 0. Taking x, x it easily follows that λ= y, x 0 ≤ −x, x2 +

x, x4 y, y2 , |y, x|2

which yields the Cauchy–Bunyakowsky–Schwarz inequality. Further, |x, y| = x, xy, y if and only if x = λ y. Theorem 1.103 (Polarization Identity). In a pre-Hilbert space H , 1 x, y = { x + y 2 − x − y 2 + i x + iy 2 − i x − iy 2 }, 4 for all x, y ∈ H . Proof. The proof is left to the reader as an exercise. Definition 1.104. A pre-Hilbert space H which is complete is called a Hilbert space.

1.3.2 Examples of Hilbert Spaces Example 1.105. Let Ω ⊂ Rn be an open subset and let H = L2 (Ω ), which we equip with its natural inner product given by u, v2 :=

 Ω

u(x1 , x2 , . . . , xn )v(x1 , x2 , . . . , xn )dx1 dx2 . . . dxn

for all u, v ∈ L2 (Ω ), whose associated norm is given by

1.3 Hilbert Spaces

35

u 2 =



1/2

Ω

|u(x1 , x2 , . . . , xn )|2 dx1 dx2 . . . dxn

.

The space (L2 (Ω ), ·, ·2 , · 2 ) is complete and hence is a Hilbert space. Example 1.106. Let Ω ⊂ Rn be an open subset and let H = H k (Ω ), which we equip with its natural inner product defined by u, vH k (Ω ) :=



∑

Ω |α |≤k

Dα u(x)Dα v(x)dx

for all u, v ∈ H k (Ω ), whose associated norm is u H k =



1/2

∑

α

Ω |α |≤k

|D u(x)| dx 2

.

The space (H k (Ω ), ·, ·H k , · H k ) is complete and hence is a Hilbert space. Example 1.107. Let a, b ∈ R with a < b and let H = C[a, b] be the collection of all complex-valued continuous functions on the interval [a, b]. Define  f , g2 :=

 b a

f (t)g(t)dt

for all f , g ∈ H . This is an inner product with associated norm given by f 2 =

 a

b

1/2 | f (t)|2 dt

for all f ∈ H . Consequently, the space (H , ·, ·2 , · 2 ) is a pre-Hilbert space. However, H is not a Hilbert space as it is not complete. The proof of the next theorem can be found in any good book in functional analysis and hence is omitted. Theorem 1.108 (Riesz Representation Theorem). For each ξ ∈ H ∗ there exists an element y ∈ H such that ξ (x) = x, y for all x ∈ H . Furthermore, ξ ∗ = y .

1.3.3 The Projection Theorem In this subsection, unless otherwise specified, H stands for a Hilbert space equipped with the inner product and norm given by, ·, ·, and · .

36

1.3.3.1

1 Metric, Banach, and Hilbert Spaces

Orthogonality and Orthonormal Systems

Two vectors u and v in H are said to be orthogonal and we write u ⊥ v if u, v = 0. Similarly, if L, M ⊂ H are subspaces, then we say that L and M are orthogonal and write L ⊥ M if u, v = 0 for all u ∈ L and v ∈ M. Let M ⊂ H be a subspace. The orthogonal complement M ⊥ of M is defined by M ⊥ = {u ∈ H : u, v = 0 for all v ∈ M}. The orthogonal complement M ⊥ of a closed subspace M of H is very often denoted by H  M. By definition, the subspaces M and M ⊥ are orthogonal. In particular, {0}⊥ = H and H ⊥ = {0}. Theorem 1.109 (Pythagorean Theorem). If u, v ∈ H and u ⊥ v, then u + v 2 = u 2 + v 2 . More generally, if ui ∈ H for i = 1, 2, . . . , n and ui ⊥ u j for i = j, then u1 + u2 + u3 + · · · + un 2 = u1 2 + u2 2 + · · · + un 2 . Proof. It suffices to notice that u + v 2 = u 2 + 2ℜeu, v + u 2 , which yields u + v 2 = u 2 + v 2 as u, v = 0. Similarly, 

n

i=1

j=1



∑ ui , ∑ u j

u1 + u2 + u3 + · · · + un 2 = =

n

n

n

∑ ∑ ui , u j 

i=1 j=1

=

n

∑ ui 2 +

i=1

=

n

∑

i, j=1, i= j

ui , u j 

n

∑ ui 2 .

i=1

Definition 1.110. A system of vectors (ek )k∈N in a Hilbert space H is said to be an orthonormal system whenever ei , e j  = δi j for all i, j ∈ N. Example 1.111. Consider the Hilbert space l 2 of all square-summable complexvalued infinite sequences, equipped with the inner product given by

1.3 Hilbert Spaces

37 ∞

x, y = ∑ xi yi i=0

for all x = (xi )i∈N and y = (yi )i∈N in l 2 . In the Hilbert space l 2 , we consider the sequence of vectors e0 = (1, 0, . . .), e1 = (0, 1, 0, . . .), . . . , e j = (0, 0, . . . , 1, 0, . . .), where the 1 appears in the ( j + 1)th position. It is then clear that (e j ) j∈N is an orthonormal sequence of l 2 . Example 1.112. In L2 [−1, 1] equipped with its natural inner product and norm given by u, v =

 1 −1

u(t)v(t)dt

and u 2 =



1

−1

1/2 |u(t)|2

,

one can easily check that the normalized Legendre polynomials  Pn (x) =

 2n + 1 1 d n  2 (x − 1)n , n = 1, 2, . . . n 2n n! dxn

form an orthonormal system. Definition 1.113. A system of vectors (ei )i∈N is said to be a basis of H if each u ∈ H there exists a unique sequence (ci )i∈N with ci ∈ F for each i ∈ N such that ∞

u = ∑ ci ei . i=0

If in addition (ei )i∈N is an orthonormal system of vectors, then it is said to be an orthonormal basis for H . Proposition 1.114 ([102]). Let (ek )k for k = 0, 1, 2, . . . , n be an orthonormal system of vectors in the Hilbert space H . Then for every u ∈ H , ∞

(a)

∑ |u, ek |2 ≤ u 2 —(Bessel’s Inequality).

k=1 ∞

(b)

∑ u, ek ek converges.

k=1 ∞

(c)

∑ ck ek converges if and only if (ck )k ∈ l 2 .

k=1

38

1 Metric, Banach, and Hilbert Spaces

(d) If v =

∞

∑ ck ek , then ck = v, ek .

k=1

Proof. (a) Write  0≤



n

n

k=1

k=1

u − ∑ u, ek ek , u − ∑ u, ek ek n

n

k=1

k=1

= u 2 − 2 ∑ |u, ek |2 + ∑ |u, ek |2 which yields n

∑ |u, ek |2 ≤ u 2 .

k=1

Letting n → ∞, one obtains the desired result. (b) Using the partial sum Sn =

n

∑ u, ek ek it follows from (a) that for n > m, we

k=1

have 

n

∑

Sn − Sm = 2

u, ek ek ,

k=m+1

=

n

∑

n

∑

 u, ek ek

k=m+1

|u, ek |2

k=m+1

→ 0, n, m → ∞. Consequently, the sequence (Sn )n is a Cauchy sequence and hence converges as H is complete. (c) Setting dn =

n

n

k=1

k=1

∑ ck ek and Dn = ∑ |ck |2 , one can see that for n > m, dn − dm 2 = Dn − Dm

which yields (dn )n is a Cauchy in H if and only if (Dn )n is a Cauchy sequence in R. In other words, (dn )n converges if and only (Dn )n converges. (d) We will prove (d) in two steps: Step 1—We first show that if un → u and vn → v in H , then un , vn  → u, v as n → ∞. Indeed,

1.3 Hilbert Spaces

39

|un , vn  − u, v| ≤ |un , vn − v| + |un − u, v| ≤ un vn − v + un − u v → 0, n → ∞ as un → u and vn → v as n → ∞. Step 2—Let v =

∞

∑ ck ek . Using Step 1 it follows that

k=1

 v, em  = lim

n→∞

∞



∑ ck ek , em ,

= cm .

k=1

Definition 1.115. A system (en )n∈N in a normed vector space X is said to be a complete system if the vector space spanned by (en )n∈N is a dense set in X . Theorem 1.116 ([102]). A Hilbert space H is separable if and only if it contains a complete orthonormal system (en )n∈N .

1.3.3.2

Projections

Definition 1.117. A projection P on a Hilbert space H is a linear operator satisfying P2 = P. If P : H → H is a projection, then its kernel and image are defined by N(P) = {x ∈ H : Px = 0} and R(P) = {Px : x ∈ H }. Proposition 1.118. If P is a projection on a Hilbert space H , then, (a) (b) (c) (d)

R(P) = N(I − P). N(P) = R(I − P). R(P) ∩ N(P) = {0}. H = N(P) + R(P). Moreover, if P is continuous, then H = N(P) ⊕ R(P).

Proof. The proof is left to the reader as an exercise. Theorem 1.119 ([18, 57, 70]). If C is a closed convex subset of H , and u is a point in H , then there exists a unique v ∈ C such that v − u = inf w − u . w∈C

Remark 1.120. The unique v ∈ C satisfying v − u = inf w − u is then called the projection of u onto C; let us denote it v = PC (u).

w∈C

40

1 Metric, Banach, and Hilbert Spaces

Proof (Theorem 1.119). Set γ = infz∈C x − z . Obviously, there exists a minimizing sequence (zn )n∈N ⊂ C such that zn → γ as n → ∞. Now since C is convex, it is clear that 12 (zn + zm ) ∈ C for all n, m ∈ N. And hence

γ≤

1 zn + zm 2

(1.15)

for all n, m ∈ N. Using the Parallelogram identity, one can easily see that  2   1  zn − zm = 2 x = zn + 2 x − zm − 4 x − (zn + zm )  2 2

2

2

(1.16)

for all n, m ∈ N. Since x − zn → γ and x − zm → γ as n, m → ∞, then (1.16) implies by using (1.15) that zn − zm 2 ≤ 2 x − zn 2 + 2 x − zm 2 − 4γ 2 → 0 as n, m → ∞. Consequently, (zn )n∈N is a Cauchy sequence and hence there exists y ∈ H such that zn → y. Now since C is closed, it is clear that y ∈ C. Obviously, γ = x − y . Now suppose that there exists another y˜ ∈ C such that γ = x − y . ˜ Clearly, 2    1  ≤ 0, y − y x − (y + y) ˜ ˜ 2 = 4γ 2 − 4    2 as 12 (y + y) ˜ ∈ C. Therefore, y = y, ˜ which shows that the element y such that γ = x − y is unique. Theorem 1.121 ([70]). Let M ⊂ H be a closed subspace, then x − PM x is orthogonal to M, that is, x − PM x, y = 0, ∀y ∈ M. Proof. Let z ∈ C and let y ∈ M. Since M is subspace of H and PM x ∈ M it follows that PM x + zy ∈ M. Now, let γ = x − PM x . From Theorem 1.119 it follows that

γ 2 ≤ x − (PM x + zy) 2 = x − PM x 2 + |z|2 y 2 − z¯x − PM x, y − zy, x − PM x, and hence 0 ≤ |z|2 y 2 − z¯x − PM x, y − zy, x − PM x. As z is arbitrary, then taking z = z x − PM x, y, where z ∈ R it easily follows that 0 ≤ z 2 |x − PM x, y|2 y 2 − 2z |x − PM x, y|2 .

1.3 Hilbert Spaces

41

Now since the last inequality holds for each z ∈ R it follows that |x−PM x, y|2 =0, that is, x − PM x, y = 0. Finally, since x − PM x, y = 0 for any y ∈ M, it then follows that x − PM x is orthogonal to M. Theorem 1.122 (Orthogonal Decomposition [70]). Let M be a closed subspace of a Hilbert space H . Every x ∈ H has a unique decomposition given by x = y + z, where y = PM x and z = (I − PM )x.

(1.17)

Proof. Suppose x = y + z where y ∈ M and z ∈ M ⊥ . Now one can write x = PM x + (x − PM x) where PM x ∈ M and x − PM x ∈ M ⊥ (see Theorem 1.121). Obviously, y − PM x = (x − PM x) − z and y − PM x ⊥ (x − PM x) − z. Hence, x − PM x = 0 = (x − PM x) − z, as a vector orthogonal to itself.

Bibliographical Notes The results on metric, Banach, and Hilbert spaces presented in this chapter are mainly taken from Diagana [57, 70], Bezandry and Diagana [18], Gohberg et al. [102], Naylor and Sell [141], Oden and Demkowicz [147], Gohberg et al. [102], Kato [123], Rudin [151], Weidmann [160], Khamsi and Kirk [124], and Yosida [169].

Chapter 2

Linear Operators on Banach Spaces

Let (X , · ) and (Y , · 1 ) be two Banach spaces over the same field F. A mapping A : D(A) ⊂ X → Y satisfying A(α x + β y) = α Ax + β Ay for all x, y ∈ D(A) and α , β ∈ F, is called a linear operator or a linear transformation. In this chapter we study various properties of bounded and unbounded linear operators needed in the sequel.

2.1 Bounded Linear Operators Definition 2.1. A linear operator A : X → Y satisfying the following property: there exists K ≥ 0 such that Ax 1 ≤ K x for all x ∈ X , is called a bounded (or continuous) linear operator. The collection of all bounded linear operators from X into Y will be denoted by B(X , Y ) with B(X , X ) := B(X ).

2.1.1 Examples of Bounded Linear Operators Example 2.2. Let X = Y = Fn equipped with its natural norm given by x = (x12 + x22 + · · · + xn2 )1/2 for all x = (x1 , x2 , . . . , xn ) ∈ Fn . T. Diagana, Almost Automorphic Type and Almost Periodic Type Functions in Abstract Spaces, DOI 10.1007/978-3-319-00849-3 2, © Springer International Publishing Switzerland 2013

43

44

2 Linear Operators on Banach Spaces

Consider the canonical base for Fn , that is, e1 = (1, 0, 0, . . . , 0), e2 = (0, 1, 0, . . . , 0), . . ., en = (0, 0, . . . , 1). Thus every x ∈ Fn can be written in the orthonormal basis (e j ) j=1,...,n as follows: x=

n

∑ x je j

for some x j ∈ F, j = 1, 2, . . . , n.

j=1

In particular, there exists ai j ∈ K for i, j = 1, 2, . . . , n such that n

Ae j = ∑ ai j ei . i=1

We will show that every linear operator from Fn into Fn is necessarily bounded. Indeed, let A : Fn → Fn be a linear operator. Then, for all x, y ∈ Fn with x=

n

∑ x je j

and y =

j=1

n

∑ y je j,

j=1

we have    n    Ax − Ay =  ∑ (x j − y j )Ae j   j=1  ≤ C max(|x1 − y1 |, |x2 − y2 |, . . . , |xn − yn |) ≤ C x − y , where C =

n

n

j=1

j=1



n

1/2

∑ Ae j = ∑ ∑ |ai j |2

< ∞. Therefore, A is continuous.

i=1

Example 2.3. Fix p, q ≥ 1 such that p−1 + q−1 = 1. Let X = L p (Rn ) and Y = L1 (Rn ) be equipped with their natural topologies. Consider the so-called multiplication operator defined by A f = Q f where Q ∈ Lq (Rn ). Using H¨older’s inequality (Proposition 1.81), it easily follows that A f 1 ≤ Q q f p for all f ∈ L p (R) and hence A ∈ B(L p (Rn ), L1 (Rn )). Example 2.4. Fix a, b ∈ R with a < b. Let X = Y = C[a, b], that is, the collection of all continuous functions from [a, b] into F, which we equip with the sup-norm, that is, · ∞ . Consider the so-called Volterra operator defined by (A f )(t) =

 t a

f (s)ds

for all t ∈ [a, b] and f ∈ C[a, b]. It can be easily seen that A f ∞ ≤ (b − a) f ∞ for all f ∈ C[a, b], which yields A ∈ B(C[a, b]).

2.1 Bounded Linear Operators

45

Example 2.5. Let Ω ⊂ RN be a bounded closed subset. Fix m ∈ Z+ and let C(m) (Ω ) be the collection of all functions f : Ω → F such that Dα f exists and belongs to C(Ω ) for |α | ≤ m. The space C(m) (Ω ) equipped with the norm f m,∞ := max sup |Dα f (x)| |α |≤m x∈Ω

is a Banach space. Now let X = C(Ω ) be equipped with its corresponding sup-norm and let Y = C(m) (Ω ) be equipped with its above-mentioned norm. Consider the differential operator given by Af =

∑

|α |≤m

aα Dα f (x),

where the coefficients aα ∈ F are constants. Obviously, A is a linear operator from C(m) (Ω ) to C(Ω ). Moreover, it is not hard to show that A is continuous.

2.1.2 Properties of Bounded Operators The identity and zero operators of X will be respectively denoted by I and O and are defined by I(x) = x and O(x) = 0 for all x ∈ X . Now, if A : X → Y is a bounded linear operator, then its kernel and range are respectively defined by N(A) = {x ∈ X : Ax = 0} and R(A) = {Ax : x ∈ X }. Proposition 2.6. If A, B ∈ B(X , Y ) and γ ∈ F, then the following properties hold, (a) A + B ∈ B(X , Y ). (b) γ A ∈ B(X , Y ). (c) If A, B ∈ B(X ), then AB, BA ∈ B(X ). Proof. (a) Using the linearity of both A and B it follows that (A + B)(λ x + μ y) = A(λ x + μ y) + B(λ x + μ y) = λ Ax + μ Ay + λ Bx + μ By = λ (A + B)x + μ (A + B)y, and hence A + B is linear. Similarly, using the triangle inequality and the continuity of both A and B, we obtain (A + B)x = Ax + Bx

46

2 Linear Operators on Banach Spaces

≤ Ax + Bx ≤ K x + K x = (K + K ) x , which yields A + B is continuous. (b) The proof is obvious and hence is omitted. (c) We will prove it for AB as the proof for BA is quite similar. Using the linearity of both A and B and (b) it follows that (AB)(λ x + μ y) = A[B(λ x + μ y)] = A[λ Bx + μ By] = λ (AB)x + μ (AB)y and hence AB is linear. Similarly, using the continuity of both A and B, we obtain that (AB)x = A(Bx) ≤ K Bx = (KK ) x , which yields AB is bounded. In view of Proposition 2.6, B(X , Y ) is a vector space. Moreover, if A ∈ B(X , Y ), then we define A = sup

0=x∈X

Ax 1 x

which turns out to be a norm on B(X , Y ) commonly called operator-norm. Further, it can be shown that A = sup

0=x∈X

Ax 1 x

= sup { Ax 1 : x ≤ 1} x∈X

= sup { Ax 1 : x = 1}. x∈X

By definition of the operator-norm, Ax 1 ≤ A · x for all x ∈ X . Moreover, if A, B ∈ B(X ), then AB ≤ A B . Theorem 2.7. The space B(X , Y ) is a Banach space when it is equipped with its operator norm. Proof. In view of Proposition 2.6 it follows that B(X , Y ) is a normed vector space. Now let (An )n∈N ⊂ B(X , Y ) be a Cauchy sequence, that is, An − Am → 0 as

2.1 Bounded Linear Operators

47

n, m → ∞. To complete the proof we have to show that there exists A ∈ B(X , Y ) such that An − A → 0 as n → ∞. Indeed, for all x ∈ X , we have, (An − Am )x 1 ≤ An − Am · x → 0 as n, m → ∞ and so (An x)n∈N is a Cauchy sequence in Y . Since Y is a Banach space, it follows that there exists ξ ∈ Y such that An x − ξ 1 → 0 as n → ∞. Thus Ax = ξ = lim An x n→∞

defines a linear operator A : X → Y . It remains to show that A is bounded. Using the fact that (An )n∈N is a Cauchy sequence it follows that supn∈N An < ∞. Consequently, Ax 1 = lim An x 1 ≤ K x n→∞

for all x ∈ X , that is, A ∈ B(X , Y ). Theorem 2.8. If A : X → Y is a linear operator, then the following statements are equivalent: (a) A is continuous. (b) A is continuous at 0. (c) There exists a constant C > 0 such that Au 1 ≤ C . u for each u ∈ X . Proof. It is not hard to see that (a) yields (b). Now if (b) holds, then there exists η > 0 such that Ax 1 ≤ 1 whenever x ≤ η . Thus for each 0 = x ∈ X , we have    ηx     x  = η . Now    ηx    = η Ax 1 , 1 ≥ A x 1 x and hence Ax 1 ≤ η −1 x , that is (c) holds. Now if (c) holds, then Ax − Ax0 1 = A(x − x0 ) 1 ≤ C x − x0 . Consequently, for each ε > 0 one can find an η = ε C−1 such that Ax − Ax0 1 < ε whenever x − x0 ≤ η . Therefore, A is continuous at x0 , which yields A is continuous on X as x0 is arbitrary.

48

2 Linear Operators on Banach Spaces

In the rest of this chapter, the spaces (H1 , ·, ·1 , · 1 ), (H2 , ·, ·2 , · 2 ), and (H3 , ·, ·3 , · 3 ) stand for Hilbert spaces over the same field F.

2.1.2.1

Adjoint for Bounded Operators

Let A ∈ B(H1 , H2 ). For each y ∈ H2 , the functional x → ξy (x) := Ax, y2 is linear and bounded. Therefore, from the Riesz representation theorem (Theorem 1.108), there exists a unique y∗ ∈ H1 such that Ax, y2 = ξy (x) = x, y∗ 1 for all x ∈ H1 , y ∈ H2 . The transformation A∗ : H2 → H1 defined by y → y∗ = A∗ y is called the adjoint of the linear operator A. In view of the above, Ax, y2 = x, A∗ y1 for all x ∈ H1 , y ∈ H2 . Proposition 2.9. If A : H1 → H2 is a bounded linear operator, then A∗ ∈ B(H2 , H1 ). Furthermore, A = A∗ . Proof. First of all, let us show that A∗ is a bounded linear operator. Indeed, given v, w ∈ H2 and α , β ∈ F, we have u, A∗ (α v + β w)2 = Au, α v + β w2 = α Au, v2 + β Au, w2 = α u, A∗ v1 + β u, A∗ w1 = u, α A∗ v1 + u, β A∗ w1 = u, α A∗ v + β A∗ w1 for all u ∈ H1 . Hence A∗ (α v + β w) = α A∗ v + β A∗ w, that is, A∗ is a linear operator from H2 into H1 . Now A∗ u 21 = A∗ u, A∗ u1 = AA∗ u, u2 ≤ AA∗ u 2 · u 2 ≤ A · A∗ u 1 · u 2 and hence A∗ u 1 ≤ A · u 2 , that is, A∗ ≤ A . Similarly, (A∗ )∗ ≤ A∗ . Now using the fact (A∗ )∗ = A it follows that A ≤ A∗ . Combining, we obtain the desired result.

2.1 Bounded Linear Operators

49

Proposition 2.10 ([18]). If A : H1 → H2 is a bounded linear operator, then A∗ A ∈ B(H1 ) and AA∗ ∈ B(H2 ). Moreover, AA∗ = A∗ A = A∗ 2 = A 2 . Proposition 2.11 ([18]). If A, B are bounded linear operators on H and if λ ∈ C, then, (a) (b) (c) (d) (e)

I ∗ = I. O∗ = O. (A + B)∗ = A∗ + B∗ . (λ A)∗ = λ A∗ . (AB)∗ = A∗ B∗ .

Definition 2.12. A bounded linear operator A : H → H is called self-adjoint or symmetric if A = A∗ . Proposition 2.13 ([70]). If A ∈ B(H1 , H2 ), then both AA∗ and A∗ A are selfadjoint. Proposition 2.14 ([18]). Let A : H → H be a bounded self-adjoint operator. Then the following properties hold: (a) Ax, x ∈ R for all x ∈ H . |Ax, x| (b) A = sup . 2 x=0 x (c) If B ∈ B(H ) is also self-adjoint and if AB = BA, then AB is also self-adjoint. Example 2.15. Let H1 = H2 = L2 [a, b], which we equip with the L2 -topology. Consider the integral operator defined by (A f )(t) =

 b a

K(t, s) f (s)ds

for all f ∈ L2 [a, b], where K ∈ L2 ([a, b] × [a, b]). It can be easily seen that A is a bounded linear operator on L2 [a, b]. Moreover, the adjoint of A is given by (A∗ f )(t) =

 b a

K(s,t) f (s)ds

for all f ∈ L2 [a, b]. In particular, A is self-adjoint if and only if K(t, s) = K(s,t) for all t, s ∈ [a, b].

50

2 Linear Operators on Banach Spaces

2.1.2.2

The Inverse Operator

Definition 2.16. An operator A ∈ B(X ) is called invertible if there exists B ∈ B(X ) such that AB = BA = I. In that event, the operator B is called the inverse operator of A and denoted by B = A−1 . Theorem 2.17 ([18]). If A ∈ B(X ) and if A < 1, then the linear operator I − A is invertible and its inverse is given by (I − A)−1 =

k

∑ Ak .

k=0

Proof. First of all, observe that (I − A)(I + A2 + · · · + An ) = I − An+1 and that An+1 ≤ A n+1 → 0 as n → ∞ as A < 1. Consequently, lim (I − A)(I + A2 + · · · + An ) = I in B(X ).

n→∞

Again from A < 1 and the fact that B(X ) is a Banach algebra, then the quantity S := lim (I + A2 + · · · + An ) n→∞

does exist and (I − A)S = I. In fact, we have   (I − A)S = I + (I − A) S − (I + A2 + · · · + An ) . On other hand,   (I − A) S − (I + A2 + · · · + An ) ≤ I − S · S − (I + A2 + · · · + An ) → 0 as n → ∞, hence (I − A)S = I. Thus (I − A) is invertible and (I − A)−1 = S, where S=

∞

∑ Ak

(A0 being I).

k=0

Observe that if A, B ∈ B(X ) are invertible, so is AB with (AB)−1 = B−1 A−1 .

2.1 Bounded Linear Operators

51

Proposition 2.18 ([102]). If A ∈ B(X ) is invertible and if B ∈ B(X ) is given such that A − B <

1 , A−1

then B is invertible and its inverse is given by B−1 =

∞

∑ [A−1 (A − B)]k A−1

k=0

and A−1 − B−1 ≤

A−1 2 · A − B . 1 − A−1 · A − B

Proof. It suffices to write B = A[I − A−1 (A − B)]. From A−1 (A − B) < 1 and Theorem 2.17 it follows that I − A−1 (A − B) is invertible. Using the fact that A is invertible it follows that A[I −A−1 (A−B)] is invertible. Further, using Theorem 2.17 it follows that B−1 = [I − A−1 (A − B)]−1 A−1 =

∞

∑ [A−1 (A − B)]k A−1 .

k=0

Moreover, A−1 − B−1 ≤

2.1.2.3

A−1 2 · A − B . 1 − A−1 · A − B

Spectrum for Bounded Linear Operators

Definition 2.19. If A : X → X is a bounded linear operator, then its spectrum σ (A) is defined by

σ (A) = {λ ∈ C : λ I − A is not invertible}. Similarly, the resolvent ρ (A) of A is defined by ρ (A) = C\σ (A), that is, the collection of all λ ∈ C such that the operator A − λ I is one-to-one (N(A − λ I) = 0), onto (R(A − λ I) = X ), and bounded. Example 2.20 ([18]). Let q : [α , β ] → C be a continuous function. Define the bounded linear operator Mq on X = L2 ([α , β ]) by (Mq φ )(s) = q(s)φ (s), ∀s ∈ [α , β ].

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2 Linear Operators on Banach Spaces

It can be shown that λ I − Mq is invertible on L2 ([α , β ]) if and only if

λ − q(s) = 0, ∀s ∈ [α , β ].

(2.1)

The inverse (λ I − Mq )−1 of λ I − Mq is defined by   (λ I − Mq )−1 ψ (s) =



 1 ψ (s) λ − q(s)

with the following estimate:   (λ I − Mq )−1  ≤ max

s∈[α ,β ]



 1 . |λ − q(s)|

The spectrum σ (Mq ) of Mq is given by

σ (Mq ) = {q(s) : s ∈ [α , β ]}.

2.1.2.4

Compact Operators

Definition 2.21 ([102]). An operator A ∈ B(H1 , H2 ) is said to be compact if for each sequence (xn )n∈N in H1 with xn 1 = 1 for each n ∈ N, the sequence (Axn )n∈N has a subsequence which converges in H2 . The collection of all compact operators from H1 to H2 is denoted by K (H1 , H2 ). Theorem 2.22 ([102]). If K, L ∈ B(H1 , H2 ) are compact linear operators, then (a) α L is compact. (b) K + L is compact. (c) If A ∈ B(H3 , H1 ) and B ∈ B(H2 , H3 ), then KA and BK are compact. Proof. (a) This is straightforward. (b) Let (un ) ∈ H1 with xn 1 = 1. Since K is compact, (Kun )n∈N has a convergent subsequence (Kunk )k∈N . Similarly, (Lun )n∈N has a convergent subsequence (Lunk )k∈N . Therefore, ((K + L)unk )k∈N converges. (c) Let (vn )n∈N ⊂ H3 with vn 3 = 1 for each n ∈ N. Thus (Avn )n∈N is bounded. Now since K is compact, it is clear (KAvn )n∈N has a convergent subsequence and hence KA is a compact operator. Let (wn )n∈N ⊂ H1 with wn 1 = 1 for each n ∈ N. Now since K is compact, then (Kwn )n∈N has a convergent subsequence, say, (Kwnk )k∈N . Now by the continuity of B it follows that (BKwnk )k∈N converges and hence BK is a compact operator.

2.2 Unbounded Linear Operators

53

Example 2.23. In H = L2 [a, b], define the integral operator A by (A f )(t) :=

 b a

V (t, τ ) f (τ )d τ for each f ∈ L2 [a, b].

Assuming that V ∈ L2 ([a, b] × [a, b]), it can be shown that A is compact. Definition 2.24. A bounded linear operator is of finite rank if its image is a finitedimensional Banach space. The collection of all finite rank operators from H1 into H2 is denoted F (H1 , H2 ). Proposition 2.25 ([102]). Let A : H1 → H2 be an operator of finite rank n. Then there exist vectors x1 , x2 , . . . , xn in H1 and vectors y1 , y2 , . . . , yn in H2 such that Ax =

n

∑ x, xk yk .

k=1

The vectors y1 , y2 , . . . , yn can be chosen to be an orthonormal base for R(A). Proposition 2.26. F (H1 , H2 ) ⊂ K (H1 , H2 ). Proposition 2.27 ([102]). An operator A ∈ B(H1 , H2 ) is compact if and only if its adjoint A∗ is compact. Proof. If A is compact, then AA∗ is compact. Thus for any (xn )n∈N ⊂ H2 with xn 2 = 1 there exists a subsequence (xnk )k∈N of (xn )n∈N such that (AA∗ xnk )k∈N converges. Now using the fact A∗ xnk − A∗ xnl ≤ 2 AA∗ xnk − AA∗ xnl it follows that (A∗ xnk )k∈N is a Cauchy sequence, which converges as H1 is complete. Therefore, A∗ is compact. Similarly, if A∗ is compact, then using the above proved result, it follows that A = (A∗ )∗ is compact.

2.2 Unbounded Linear Operators Let A : D(A) ⊂ X → Y be a linear operator. If A is not continuous, then A is called an unbounded operator. The collection of all unbounded linear operators from X into Y will be denoted U (X , Y ) with U (X , X ) = U (X ). Because of the domain involved, elements of U (X ) need to be manipulated with a great care. For instance, if A, B ∈ U (X , Y ), then their (algebraic) sum A + B may or may be trivial depending on D(A) ∩ D(B). Note that the (algebraic) sum operator A + B is defined by D(A + B) = D(A) ∩ D(B) and (A + B)x = Ax + Bx for all x ∈ D(A) ∩ D(B).

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2 Linear Operators on Banach Spaces

Similarly, the product operator AB is defined by, D(AB) = {x ∈ D(B) : Bx ∈ D(A)} and ABx = A(Bx) for all x ∈ D(AB). As for bounded linear operators, if A : D(A) ⊂ X → Y is an unbounded linear operator, then we define its kernel N(A) and range R(A) as follows: N(A) := {x ∈ D(A) : Ax = 0} and R(A) = {Ax : x ∈ D(A)}. It is clear that N(A) ⊂ X while R(A) ⊂ Y . The operator A is said to be one-toone if N(A) = 0. The operator A is said to be onto if R(A) = Y . Of course, a linear operator A is bijective or invertible if it is one-to-one and onto. If A is invertible, then its inverse will be denoted by A−1 . If A is invertible, then D(A−1 ) = R(A) and R(A−1 ) = D(A).

2.2.1 Examples of Unbounded Operators Example 2.28 ([136]). Let X = Y = L2 (R) be equipped with its natural topology and consider the one-dimensional Laplace operator defined by D(A) = H 2 (R) and Au = −u for all u ∈ H 2 (R). Consider the sequence of functions defined by ψn (t) = e−n|t| , n = 1, 2, . . . . Clearly, for each n = 1, 2, . . ., ψn ∈ D(A) = H 2 (R). Furthermore, ψn 22 =

 +∞ −∞

e−2n|t| dt =

1 n

and Aψn 22 =

 +∞ −∞

n4 e−2n|t| dt = n3 .

Aψn 2 = n → ∞ as n goes to ∞, that is, A is an unbounded linear ψn 2 operator on L2 (R).

Therefore,

Example 2.29 ([136]). Let X = Y = L2 (0, 1) be equipped with its natural topology and consider the derivative operator defined by D(A) = C1 (0, 1) and Au = u

2.2 Unbounded Linear Operators

55

for all u ∈ C1 (0, 1), where C1 (0, 1) is the collection of continuously differentiable functions over (0, 1). Consider the sequence of functions defined by φn (t) = t n , n = 1, 2, . . .. Clearly, for each n = 1, 2, . . . , φn ∈ C1 (0, 1). Furthermore, φn 22 =

 1 0

t 2n dt =

1 , 2n + 1

and Aφn 22 =

 1 0

n2t 2n−2 dt =

n2 . 2n − 1

Here again,  Aφn 2 2n + 1 →∞ =n φn 2 2n − 1 as n goes to ∞, that is, A is an unbounded linear operator on L2 (R).

2.2.2 Closed Linear Operators Equip X × Y with its natural norm, that is, (x, y) = ( x 2 + y 2 )1/2 for all (x, y) ∈ X × Y . Definition 2.30. If A : D(A) ⊂ X → Y is a linear operator, then its graph is defined by G (A) = {(x, Ax) ∈ X × Y : x ∈ D(A)}. Definition 2.31. If A, B belong to U (X , Y ), then A is said to be an extension of B if D(B) ⊂ D(A) and Au = Bu for all u ∈ D(B). We then denote this by B ⊂ A. Moreover, B ⊂ A if and only if G (B) ⊂ G (A). Definition 2.32. A linear operator A : D(A) ⊂ X → X is called closed if its graph G (A) ⊂ X × X is closed. Namely, if un ∈ D(A) such that un → u and Aun → v in X as n → ∞, then u ∈ D(A) and Ax = v. The collection of all closed operators on X will be denoted cl(X ). It is clear that an operator A : D(A) ⊂ X → Y is closed if and only if G (A) ⊂ X × Y is closed.

56

2 Linear Operators on Banach Spaces

2.2.2.1

Examples of Closed Linear Operators

Example 2.33 ([70]). Every A ∈ B(X ) is closed. Indeed, let (un )n∈N ∈ D(A) such that un → u with Aun → v in X as n → ∞. Now since A is bounded, therefore D(A) = X . Again, from the continuity of A it is clear that u ∈ X and Au = v. Example 2.34. If A ∈ cl(X ) and B ∈ B(X ), then A + B ∈ cl(X ). Indeed, suppose (un )n∈N ∈ D(A + B) = D(A) such that un → u and (A + B)un → v in X as n → ∞. Now since B is bounded it follows that Aun → v − Bx in X as n → ∞. Since A is closed, then u ∈ D(A) and Au = v − Bu. Example 2.35. This is an illustration of Example 2.34. Let X = L2 (Rn ) and define A and B by D(A) = H 2 (Rn ), Au = −Δ u, ∀u ∈ H 2 (Rn ), and D(B) = {u ∈ L2 (Rn ) : γ (x)u ∈ L2 (Rn )}, Bu = γ (x)u, ∀u ∈ D(B), where Δ is the n-dimensional Laplace operator defined by

Δ=

n

∂2

∑ ∂ x2

k=1

k

and that γ ∈ L∞ (Rn ). It is then clear that B ∈ B(L2 (Rn )) and hence −Δ + γ ∈ cl(L2 (Rn )) with D(−Δ + γ ) = H 2 (Rn ). Example 2.36. Let X = Y = C[a, b] where a, b ∈ R with a < b. Define Cc1 (a, b] to be the set of all f ∈ C1 [a, b] with Supp( f ) ⊂ (a, b] where Supp( f ) = {x ∈ [a, b] : f (x) = 0}. Setting Au = u with D(A) = Cc1 (a, b] it can be easily shown that A ∈ cl(C[a, b]).

2.2.3 Spectral Theory for Linear Operators 2.2.3.1

Basic Definitions

Definition 2.37. If A ∈ cl(X ), then ρ (A) the resolvent set of A is defined by

ρ (A) = {λ ∈ C : λ I − A : D(A) → X is bijective, and (λ I − A)−1 ∈ B(X )}, and σ (A) the spectrum of A is the complement of the resolvent set ρ (A) in C.

2.2 Unbounded Linear Operators

57

Note that if λ ∈ ρ (A), then the operator-valued function R(λ , A) := (λ I − A)−1 : ρ (A) → B(X ) is called the resolvent of the operator A. It should be mentioned that ρ (A) = 0/ if A ∈ cl(X ). Example 2.38. Let X = Y = C[a, b] and let Au = u for all u ∈ D(A) = {u ∈ C1 [a, b] : u(b) = 0}. We want to compute ρ (A). For that, we have to find conditions on λ ∈ C such that (λ I − A)u = f

(2.2)

where λ ∈ C, u ∈ D(A), and f ∈ C[a, b], can be solved. This is equivalent to solving the differential equation u = λ u − f where u ∈ 1 C [a, b] and u(b) = 0, which is also equivalent to u(t) =

 b t

e(t−s)λ f (s)ds

for all t ∈ [a, b]. Thus for each λ ∈ C, Eq. (2.2) can be solved and its solution is given above. This means that ρ (A) = C, which yields σ (A) = 0. / Let us also mention that the operator (λ I − A)−1 is then defined by (λ I − A)−1 v(t) = R(λ , A)v(t) =

 b

e(t−s)λ v(s)ds

t

for each v ∈ C[a, b]. Proposition 2.39 ([18]). Let A, B ∈ cl(X ). (a) If λ , μ ∈ ρ (A), then R(λ , A) − R(μ , A) = (μ − λ )R(λ , A) R(μ , A). Furthermore, R(λ , A) and R(μ , A) commute. (b) If D(A) ⊂ D(B), then for all λ ∈ ρ (A) ∩ ρ (B) we have R(λ , A) − R(λ , B) = R(λ , A)(A − B)R(λ , B). (c) If D(A) = D(B), then for all λ ∈ ρ (A) ∩ ρ (B) we have R(λ , A) − R(λ , B) = R(λ , A)(A − B)R(λ , B) = R(λ , B)(A − B)R(λ , A). Proof. (a) Clearly R(λ , A) − R(μ , A) = R(λ , A)[(μ I − A) − (λ I − A)]R(μ , A) = (μ − λ )R(λ , A) R(μ , A).

58

2 Linear Operators on Banach Spaces

Similarly R(λ , A) R(μ , A) =

1 [R(λ , A) − R(μ , A)] μ −λ

=

1 [R(μ , A) − R(λ , A)] λ −μ

= R(μ , A) R(λ , A). (b) We have R(λ , A) − R(λ , B) = R(λ , A)[(λ I − B) − (λ I − A)]R(λ , B) = R(λ , A)(A − B)R(λ , B). (c) We have R(λ , A) − R(λ , B) = R(λ , A)[(λ I − B) − (λ I − A)]R(λ , B) = R(λ , A)(A − B)R(λ , B) = R(λ , B)(A − B)R(λ , A).

2.2.3.2

Self-Adjoint Linear Operators

Definition 2.40 ([57]). If A : D(A) ⊂ H1 → H2 is a densely defined linear operator, then its adjoint denoted by A∗ is defined in a unique fashion by D(A∗ ) = {v ∈ H2 : u → Au, v2 is H1 —continuous over D(A)}, and Au, v2 = u, A∗ v1 , ∀u ∈ D(A), v ∈ D(A∗ ). Proposition 2.41 ([102]). If A : D(A) ⊂ H1 → H2 is a densely defined (D(A) = H1 ) unbounded linear operator, then A∗ is closed. Proposition 2.42 ([160]). If A, B ∈ U (H ) are densely defined, then (a) A∗ B∗ ⊂ (BA)∗ . (b) If B ∈ B(H ), then A∗ B∗ = (BA)∗ . (c) If A + B is densely defined, we have (A + B)∗ ⊃ A∗ + B∗ . Proof. (a) Let us show that the operators A∗ B∗ and BA are adjoint to each other. Indeed, let u ∈ D(A∗ B∗ ) and v ∈ D(BA). Clearly, u ∈ D(B∗ ) such that B∗ u ∈ D(A∗ ). Similarly, v ∈ D(A) such that Av ∈ D(B). Using the definition of the adjoint it follows A∗ B∗ u, v = B∗ u, Av = u, BAv.

2.2 Unbounded Linear Operators

59

(b) Using (a) it is enough to show that D(BA)∗ ⊂ D(A∗ B∗ ). Indeed, let u ∈ D(BA)∗ . Now since B∗ is bounded it follows that for all v ∈ D(BA) = D(A), we obtain (BA)∗ u, v = u, BAv = B∗ u, v, and hence B∗ u ∈ D(A∗ ), that is, u ∈ D(A∗ B∗ ). (c) Let u ∈ D(A∗ + B∗ ) = D(A∗ ) ∩ D(B∗ ). Clearly, for all v ∈ D(A + B) = D(A) ∩ D(B), we have (A∗ + B∗ )u, v = A∗ u, v + B∗ u, v = u, Av + u, Bv = u, (A + B)v and hence u ∈ D((A + B)∗ ) and (A + B)∗ u = A∗ u + B∗ u. Definition 2.43. An operator A : D(A) ⊂ H → H is called self-adjoint if A = A∗ . Theorem 2.44 ([160]). Let A : D(A) ⊂ H → H be a closed linear operator. Then AA∗ and A∗ A are self-adjoint linear operators on H . Definition 2.45. A linear operator A is said to have a compact resolvent if ρ (A) = 0/ and R(λ , A) = (λ I − A)−1 is a compact operator for all λ ∈ ρ (A). In particular, A is said to have a compact inverse whether A−1 is compact. Theorem 2.46 ([102]). Suppose H is an infinite dimensional Hilbert space and A : D(A) ⊂ H → H is self-adjoint and has a compact inverse. Then, (a) There exists an orthonormal basis ( fn )n∈N of H consisting of eigenvectors of A. If the sequence (μn )n∈N are the corresponding eigenvalues, then μn is a real number and |μn | → ∞ as n → ∞. The numbers of repetitions of μn in the sequence(μn )n∈N is finite and equals the dimension of N(μn I − A).  (b) D(A) = (c) Ax =

∞

x∈H :

∞

∑ |μn |2 · |x, fn |2 < ∞

.

n=0

∑ μn x, fn  fn for all x ∈ D(A).

n=0

Example 2.47. Let λ = (λn )n∈N be a real-valued sequence. Define the diagonal operator Dλ on l 2 by D(Dλ ) = {x = (xk )k∈N ∈ l 2 : (λk xk )k∈N ∈ l 2 }, Dλ x = (λk xk )k∈N for all x ∈ D(Dλ ). It can be easily shown that Dλ is a self-adjoint linear operator. Moreover, for all μ ∈ ρ (Dλ ), the linear operator (μ I − Dλ )−1 is compact if and only if limn→∞ |λn | = ∞. Example 2.48. In L2 (a, b) consider the so-called Neumann Laplace operator defined by D(ΔN ) = {u ∈ H 2 (a, b) : u (a) = u (b) = 0} and ΔN u = u for all u ∈ D(ΔN ). It can be shown that the operator ΔN is self-adjoint and has compact resolvent.

60

2 Linear Operators on Banach Spaces

2.2.4 Sectorial Linear Operators 2.2.4.1

Basic Definitions

Definition 2.49 ([137]). A linear operator A : D(A) ⊂ X → X (not necessarily densely defined) is said to be sectorial if the following hold: there exist constants ζ ∈ R, θ ∈ ( π2 , π ), and M > 0 such that (a) ρ (A) ⊃ Sθ ,ζ := {λ ∈ C : λ = ζ , | arg(λ − ζ )| < θ }, and M for each λ ∈ Sθ ,ζ . (b) R(λ , A) ≤ |λ − ζ | Most of the unbounded linear operators encountered in the literature are sectorial. / Observe that if A is a sectorial operator, then A is necessarily closed as ρ (A) = 0. Thus the space (D(A), · A ), where x A = x + Ax for all x ∈ D(A), is a Banach space. The norm · A is called the graph norm of A. Proposition 2.50 ([137]). Let A be a linear operator on X such that ρ (A) contains the half-plane {λ ∈ C : ℜeλ ≥ ζ }, and λ R(λ , A) ≤ M for ℜ eλ ≥ ζ with ζ ∈ R and M > 0. Then A is sectorial.

2.2.4.2

Examples of Sectorial Operators

Example 2.51. Let Ω ⊂ Rn be a bounded open subset with C2 boundary ∂ Ω . Let X = L2 (Ω ) and define the second-order differential operator Au = Δ u for all u ∈ D(A) = W 2,p (Ω ) ∩W01,p (Ω ). It can be shown that the linear operator A is sectorial. Example 2.52 ([18]). Let Ω ⊂ RN be a bounded open subset whose boundary ∂ Ω is of class C2 . Let n(x) denote the outer normal to Ω for each x ∈ ∂ Ω . Consider the differential operator defined by A0 u(x) =

N

∑

ai j (x)

i, j=1

N ∂u ∂u + ∑ bi (x) + c(x)u(x), ∂ xi ∂ x j i=1 ∂ xi

where the coefficients ai j and bi and c are real, bounded, and continuous on Ω . Moreover, we suppose that for each x ∈ Ω , the matrix (ai j (x))i, j=1,...,N is symmetric and strictly positive definite, that is, N

∑

i, j=1

for all x ∈ Ω and ξ ∈ RN .

ai j (x)ξi ξ j ≥ ω |ξ |2

2.2 Unbounded Linear Operators

61

Theorem 2.53 ([137]). Let p > 1. (a) Let A p : W 2,p (RN ) → L p (RN ) be the linear operator defined by A p u = A0 u. Then the operator A p is sectorial in L p (RN ) and the domain D(A p ) is dense in L p (RN ). (b) Let A0 be defined as above and let A p be the linear operator defined by D(A p ) = W 2,p (Ω ) ∩W01,p (Ω ) and A p u = A0 u for all u ∈ D(A p ). Then the linear operator A p is sectorial in L p (Ω ). Moreover, D(A p ) is dense in L p (Ω ). (c) Let A0 be defined as above and let A p be the linear operator defined by D(A p ) = {u ∈ W 2,p (Ω ) : Bu|∂ Ω = 0}, A p u = A0 u, u ∈ D(A p ) where N

Bu(x) = b0 u(x) + ∑ bi (x) i=1

∂u ∂ xi

with the coefficients bi (i = 1, . . . , N) are in C1 (Ω ) and the condition N

∑ bi (x)ni (x) = 0

x ∈ ∂Ω

i=1

holds. Then A p is sectorial in L p (Ω ) and D(A p ) is dense in L p (Ω ).

2.2.5 Semigroups of Linear Operators Definition 2.54. A family of bounded linear operators (T (t))t∈R+ : X → X is said to be a semigroup if the following hold, (a) T (0) = I; and (b) T (t + s) = T (t)T (s) for all s,t ≥ 0. If in addition, (c) lim T (t) − I = 0, then the semigroup T (t) is said to be uniformly continuous. t0

Definition 2.55. A semigroup of bounded linear operators (T (t))t∈R+ : X → X is said to be a c0 -semigroup (or strongly continuous semigroup of bounded linear operators) if lim T (t)x − x = 0

t0

for each x ∈ X .

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2 Linear Operators on Banach Spaces

Observe that if (T (t))t∈R+ : X → X is a semigroup of bounded linear operators, one can associate with it an operator (D(A), A) called its infinitesimal generator defined by   T (t)u − u exists , D(A) := u ∈ X : lim t0 t

(2.3)

and Au := lim

t0

T (t)u − u t

(2.4)

for every u ∈ D(A). Remark 2.56. Note that an operator A is the infinitesimal generator of a uniformly continuous semigroup of bounded linear operators (T (t))t∈R+ if and only if A is bounded. In that event, it can be shown that T (t) = etA =

∞

∑

n=0

(tA)n . n!

Example 2.57 ([18]). Suppose that X = (BUC(R), · ∞ ) is the Banach space of bounded uniformly continuous functions on the real number line equipped with the sup norm. Define (S(t)ϕ ) (σ ) = ϕ (t + σ ) for all ϕ ∈ BUC(R). Then, (S(t))t∈R is a c0 -semigroup with S(t) ≤ 1 for each t ∈ [0, ∞). Moreover, its infinitesimal generator A is defined by D(A) = H 1 (R) and Aϕ = ϕ for all ϕ ∈ H 1 (R). Example 2.58 ([18]). Let 1 ≤ p < ∞ and let X = L p (R) equipped with its natural norm · p . Define (S(0))u(x) = u(x) for all x ∈ R, and (S(t))u(x) = √

1 4π t

 ∞ −∞

e

−|x−y|2 4t

u(y)dy, t > 0, x ∈ R.

Then S(t) is a c0 -semigroup satisfying S(t)u p ≤ u p and whose infinitesimal generator A p is defined by D(A p ) = W 2,p (R) and A p u = u for all u ∈ D(A p ).

2.2 Unbounded Linear Operators

63

Example 2.59 ([18]). This is a generalization of Example 2.58. Let 1 ≤ p < ∞ and let X = L p (RN ) (or BC(RN , C) equipped with the sup norm) equipped with its natural norm · p . Define (S(0))u(x) = u(x) for all x ∈ RN , and (S(t))u(x) =

1 (4π t)N/2

 ∞ −∞

e

− x−y 2 4t

u(y)dy, t > 0, x ∈ R.

Then S(t) is a c0 -semigroup satisfying S(t)u p ≤ u p whose infinitesimal generator A p is defined by D(A p ) = W 2,p (RN ) and A p u = Δ u for all u ∈ D(A p ).

2.2.5.1

Basic Properties of Semigroups

Theorem 2.60 ([148]). Let (T (t))t∈R+ : X → X be a semigroup of bounded linear operators, then (a) there are constants C, ζ such that T (t) ≤ C eζ t , t ∈ R+ ; (b) the infinitesimal generator A of the semigroup T (t) is a densely defined closed operator; (c) the map t → T (t)x which goes from R+ into X is continuous for every x ∈ X ; (d) the differential equation given by d T (t)x = AT (t)x = T (t)Ax, dt holds for every x ∈ D(A); (e) for every x ∈ X , then T (t)x = lim (exp(tAs ))x, with s0

As x :=

T (s)x − x , s

where the above convergence is uniform on every compact subset of R+ ; and (f) if λ ∈ C with ℜeλ > ζ , then the integral −1

R(λ , A)x := (λ I − A) x =

 ∞ 0

e−ζ t T (t)x dt,

defines a bounded linear operator R(λ , A) on X whose range is D(A) and (λ I − A) R(λ , A) = R(λ , A)(λ I − A) = I.

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2 Linear Operators on Banach Spaces

If ζ = 0 in (a) above, then the corresponding semigroup is uniformly bounded. Moreover, if C = 1, then (T (t))t∈R+ is said to be a c0 -semigroup of contractions. Theorem 2.61 (Hille–Yosida [148]). Let A : D(A) → X be an unbounded linear operator in a Banach space X . Then A is the infinitesimal generator of a c0 semigroup of contractions (T (t))t∈R+ if and only if: (a) A is a densely defined closed operator; and (b) the resolvent ρ (A) of A contains R+ and   (λ I − A)−1  ≤ 1 , ∀λ > 0. λ

(2.5)

Definition 2.62. Let X be a Banach space. The family of bounded operators (T (t))t∈R : X → X is said to be a c0 -group if the following statements hold: (a) T (0) = I. (b) T (t + s) = T (t)T (s) for every s,t ∈ R. (c) lim T (t)x − x = 0 for x ∈ X . t→0

Theorem 2.63 ([148]). Let A : D(A) → X be a linear operator on X . Then A is the infinitesimal generator of a c0 -group of bounded linear operators (T (t))t∈R satisfying T (t) ≤ C eζ |t| if and only if: (a) A is a densely defined closed operator; and (b) every λ ∈ R such that |λ | ≥ ζ is in ρ (A) and that for such a λ , the following holds: (λ I − A)−n ≤

C . (|λ | − ζ )n

(2.6)

Proposition 2.64 ([137]). Let A be a sectorial operator and let T (t) be the analytic semigroup associated with it. Then the following hold: (a) T (t)u ∈ D(Ak ) for all t > 0, u ∈ X , n ∈ N. If D(An ), then An T (t)u = T (t)An u, t ≥ 0; (b) there exist constants M0 , M1 , . . . such that T (t) ≤ M0 eζ t , t > 0, and t n (A − ζ I)n T (t) ≤ Mn eζ t , t > 0; and (c) the mapping t → T (t) belongs to C∞ ((0, ∞), B(X )) and dn T (t) = An T (t), t > 0, ∀n ∈ N. dt n

2.2 Unbounded Linear Operators

65

Proposition 2.65 ([137]). Let (T (t))t>0 be a family of bounded linear operators on X such that t → T (t) is differentiable, and (a) T (t + s) = T (t)T (s) for all t, s > 0; (b) there exist ζ ∈ R, M0 , M1 > 0 such that T (t) ≤ M0 eζ t , tT (t) ≤ M1 eζ t , ∀t > 0; (c) either (a) there exists t > 0 such that T (t) is one-to-one, or (b) for every x ∈ X , s − lim T (t)x = x. t→0

Then t → T (t) is analytic in (0, ∞) with values in B(X ), and there exists a unique sectorial operator A : D(A) ⊂ X → X such that (T (t))t≥0 is the semigroup associated with A.

2.2.6 Intermediate Spaces 2.2.6.1

Fractional Powers of Sectorial Operators

Let A be a sectorial linear operator on X whose associated analytic semigroup T (t) satisfies the following: for all t > 0, T (t) ≤ M0 e−ω t , tAT (t) ≤ M1 e−ω t , where M0 , M1 , ω > 0. For each α > 0 one defines the fractional powers of −A implicitly by

(−A)

−α

1 = Γ (α )

 +∞ 0

t α −1 T (t)dt,

where Γ is defined by

Γ (x) :=

 +∞

e−xt t x−1 dt

0

for each x > 0. Lemma 2.66 ([18]). For all α , β > 0, then the following properties hold: (a) (−A)−α (−A)−β = A−(α +β ) . (b) lim (−A)−α = I in the strong operator topology. α →0

(2.7)

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2 Linear Operators on Banach Spaces

Proof. (a) We have, (−A)−α (−A)−β =

1 Γ (α )Γ (β )

1 = Γ (α )Γ (β ) =

1 Γ (α )Γ (β )

=

1 Γ (α )Γ (β )

=

1 Γ (α + β )

 +∞  +∞ 0

0

 +∞

t

α −1

0

 +∞  r 0

0

 1 0

 +∞ t

(r − t)β −1 T (r)drdt

t α −1 (r − t)β −1 dtT (r)dr

τ α −1 (1 − τ )β −1 d τ

 +∞ 0

t α −1 sβ −1 T (t)T (s)dtds

 ∞ 0

rα +β −1 T (r)dr

rα +β −1 T (r)dr

= (−A)−α −β . (b) Since (−A)−α is one-to-one, if v ∈ D(A), there exists u ∈ X such that v = (−A)−α u. Thus (−A)−α v − v = (−A)−1−α u − (−A)−1 u → 0 as α → 0 by the fact that (−A)−α is continuous with respect to uniform operator norm. Remark 2.67 ([18]). (a) Let α ∈ (0, 1). Using the fact that (λ I − A)−1 =

 ∞ 0

e−λ t T (t)dt,

then Eq. (2.7) can be rewritten as (−A)−α =

sin(πα ) π

 +∞ 0

t −α (tI − A)−1 dt.

(2.8)

(b) The operator (−A)−α is one-to-one, and hence has an inverse, which obviously is (−A)α . The operator (−A)α is closed with domain D((−A)α ) = R((−A)−α ). The operators (−A)α are called fractional powers of −A. (c) If α > β , then D((−A)α ) ⊂ D((−A)β ). (d) D((−A)α ) is endowed with the norm u α = (−A)α u for each u ∈ D((−A)α ). (e) (−A)α commutes with T (t) on D(−A)α ) with T (t) B(D(−A)α )) ≤ M0 e−ω t , t > 0.

2.2 Unbounded Linear Operators

2.2.6.2

67

The Spaces DA (α ), DA (α , ∞), and DA (α + n, p)

Let A be a sectorial and let (T (t))t≥0 be the analytic semigroup associated with it. Clearly, (T (t))t≥0 maps (0, ∞) into B(X ) and there exist M0 , M1 > 0 [137] with T (t) ≤ M0 eω t ,

t > 0, ωt

t(A − ω )T (t) ≤ M1 e ,

(2.9) t > 0.

(2.10)

Definition 2.68. Let α ∈ (0, 1). A Banach space (Xα , · α ) is said to be an intermediate space between D(A) and X , or a space of class Jα , if D(A) ⊂ Xα ⊂ X and there is a constant c > 0 such that x α ≤ C x 1−α x αA ,

x ∈ D(A),

(2.11)

where · A is the graph norm of A. Concrete examples of the intermediate spaces Xα include D(Aα ) for α ∈ (0, 1), the domains of the fractional powers of A, the real interpolation spaces DA (α , ∞), α ∈ (0, 1), defined as follows:   ⎧ 1−α −ω t ⎪ ⎪ D ( α , ∞) := x ∈ X : [x] = sup t (A − ω )e T (t)x < ∞ α ⎨ A 0
⎪ ⎪ ⎩

x α = x + [x]α , · α

and the abstract H¨older spaces DA (α ) := D(A) . One should point out that DA (α , ∞) is characterized by the behavior of the quantity t → t 1−α AT (t)u near t = 0. Moreover, all the spaces DA (α , ∞) are subspaces of D(A). Namely, the following embedding hold with equivalent norms: D(A) ⊂ DA (β , ∞) ⊂ DA (α , ∞) ⊂ D(A) for all 0 < α < β < 1. If α ∈ (0, 1), it is not very hard to see that DA (α , ∞) can be characterized as being the subspace of all u ∈ X such that [[u]]α = sup t −α T (t)u − u < ∞. t∈(0,1]

Furthermore, the norm defined by u → u + [[u]]α is equivalent to the natural norm of DA (α , ∞). More generally, if α ∈ (0, 1) and p ∈ [1, ∞], we define the real interpolation space DA (α , p) as follows:

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2 Linear Operators on Banach Spaces

 ⎧ 1−α − 1p ⎪ AT (t)x ∈ L p (0, 1) ⎨ DA (α , p) := x ∈ X : t → v(t) = t ⎪ ⎩

x α ,p = x + v L p (0,1) ,

and for n ∈ N, DA (α + n, p) = {x ∈ D(An ) : An x ∈ DA (α , p)}, x α +n,p = x + An x α ,p . Proposition 2.69 ([137]). For α ∈ (0, 1) and 1 ≤ p ≤ ∞ and for (α , p) = (1, ∞), then DA (α , p) = (X , D(A))α ,p with equivalent norms. Moreover, for 0 < α < 1, then DA (α ) = (X , D(A))α . Proposition 2.70 ([137]). For α ∈ (0, 1), then DA (α , 1) ⊂ D((−A)α ) ⊂ (X , D(A))α ,p . Proof. First of all, note that D((−A)α ) belongs to the class Jα . Notice that for each u ∈ D(A), (−A)α u = (−A)−(1−α ) (−Au) and hence for each λ > 0,   λ  ∞     1 −α tA   + Ae udt t   Γ (1 − α ) 0 λ 

1 M0 M1 1−α −α ≤ Au λ + u λ . Γ (1 − α ) 1 − α α

(−A)α u =

Letting λ =

u it follows that Au (−A)α u ≤ c Au α u 1−α .

It remains to prove that D(−A)α is continuously embedded in DA (α , ∞). For that let u ∈ D((−A)α ) and let v = (−A)α u. So for 0 < ξ ≤ 1, we have ξ 1−α Aeξ A u = ξ 1−α Aeξ A (−A)α v  ∞   ξ 1−α  α −1 (ξ +t)A   t Ae vdt ≤   Γ (α ) 0

2.2 Unbounded Linear Operators

69

≤

M1 ξ 1−α Γ (α )

≤

M1 Γ (α )

 ∞ α −1 ξ

ξ +t

0

 ∞ α −1 s

1+s

0

dt v

ds (−A)α u

and hence u ∈ DA (α , ∞). Lemma 2.71 (Diagana and Nelson [75]). Suppose A is sectorial. If n < m, then, DA (α + m, p) → DA (α + n, p). Proof. Indeed, x ∈ DA (α + m, p) yields x ∈ D(Am ) ⊂ D(An ) and Am x ∈ DA (α , p). Now using the facts that A is invertible and n − m < 0 it follows that t 1−α − p AT (t)An x = t 1−α − p AT (t)An−m Am x 1

1

≤ An−m · t 1−α − p AT (t)Am x ∈ L p (0, 1) 1

and hence An x ∈ DA (α , p), that is, x ∈ DA (α + n, p). In this book, the bound of the injection DA (α + m, p) → DA (α + n, p) will be denoted by C . Lemma 2.72 (Diagana and Nelson [75]). Fix α ∈ (0, 1) and 1 ≤ p ≤ ∞. Let A : D(A) ⊂ X → X be a sectorial linear operator. Suppose that the analytic semigroup (T (t))t≥0 associated with A is exponentially stable, that is, there exists M, ω > 0 such that T (t) ≤ Me−δ t

for t ≥ 0.

(2.12)

Then for all m ∈ N, the function t → Am T (t) B(DA (α +m−1,p),X ) belongs to L1 (0, ∞), that is, there exists a measurable function H : (0, ∞) → (0, ∞) with

θ0 :=

 ∞ 0

H(t)dt < ∞

such that Am T (t) B(DA (α +m−1,p),X ) ≤ H(t),

t > 0.

Proof. Let x ∈ DA (α + m − 1, p) for α ∈ (0, 1), p ∈ [1, ∞], and n ∈ N. From [137, Corollary 2.2.3] one obtains that DA (α , p) → DA (α , ∞). Now using the definition of DA (α , ∞) it follows that Am T (t)x = t α −1t 1−α AT (t)Am−1 x ≤ t α −1 x α +m−1,∞ ≤ Kt α −1 x α +m−1,p for a.e. t ∈ (0, 1).

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2 Linear Operators on Banach Spaces

Since α ∈ (0, 1) it follows that t → Am T (t) ∈ L1 ((0, 1), B(DA (α +m−1, p), X )). Now using the exponential stability of T (t), it follows that there exists Cm > 0 such that for all x ∈ DA (α + m − 1, p) we have Am T (t)x ≤ Cmt −m e−δ t x ≤ Km t −m e−δ t x α +m−1,p ,

t > 0.

In particular, Am T (t)x ≤ Km t −m e−δ t x α +m−1,p ≤ Km e−δ t x α +m−1,p ,

t ≥1

and hence t → Am T (t) ∈ L1 ([1, ∞), B(DA (α + m − 1, p), X )).

2.2.6.3

Hyperbolic Semigroups

Definition 2.73 ([15]). Let A be a sectorial operator on X and let (T (t))t≥0 be the analytic semigroup associated with it. The semigroup (T (t))t≥0 is said to be hyperbolic if there exist a projection P and constants M, δ > 0 such that each T (t) commutes with P, N(P) is invariant with respect to T (t), T (t) : R(Q) → R(Q) is invertible, and T (t)Px ≤ Me−δ t x

for t ≥ 0,

(2.13)

T (t)Qx ≤ Meδ t x

for t ≤ 0,

(2.14)

where Q := I − P and T (t) := (T (−t))−1 for t < 0. Recall that an analytic semigroup (T (t))t≥0 is hyperbolic if and only if (see [90])

σ (A) ∩ iR = 0. /

(2.15)

For the hyperbolic analytic semigroup T (t), we can easily check that estimations similar to Eqs. (2.13) and (2.14) hold also with norms · α . In fact, as the part of A in R(Q) is bounded, it follows from the inequality Eq. (2.14) that AT (t)Qx ≤ c eδ t x

for t ≤ 0.

In view of the above, there exists a constant c(α ) > 0 such that T (t)Qx α ≤ c(α )eδ t x

for t ≤ 0.

Similarly, T (t)Px α ≤ T (1) B(X ,Xα ) T (t − 1)Px and then from Eq. (2.13), we obtain

for t ≥ 1,

(2.16)

2.2 Unbounded Linear Operators

71

T (t)Px α ≤ M e−δ t x ,

t ≥ 1,

where M depends on α . Clearly, T (t)Px α ≤ M t −α x , and hence there exist constants M(α ) > 0 and γ > 0 such that T (t)Px α ≤ M(α )t −α e−γ t x

for t > 0.

(2.17)

We need the next lemma, which will be very crucial for our computations. Lemma 2.74 (Diagana [57]). Let 0 < α , β < 1. Then AT (t)Qx α ≤ ceδ t x β

for t ≤ 0,

AT (t)Px α ≤ ct β −α −1 e−γ t x β

(2.18)

for t > 0.

(2.19)

Proof. As for Eq. (2.16), the fact that the part of A in R(Q) is bounded yields AT (t)Qx ≤ ceδ t x β ,

A2 T (t)Qx ≤ ceδ t x β

for t ≤ 0,

(2.20)

since Xβ → X . Hence, from Eq. (2.11) there is a constant c(α ) > 0 such that AT (t)Qx α ≤ c(α )eδ t x β

for t ≤ 0.

(2.21)

Furthermore, AT (t)Px α ≤ AT (1) B(X ,Xα ) T (t − 1)Px ≤ ce−δ t x β

for t ≥ 1.

(2.22) (2.23)

Now for t ∈ (0, 1], by Eq. (2.11), one has AT (t)Px α ≤ ct −α −1 x , and AT (t)Px α ≤ ct −α Ax , for each x ∈ D(A). Thus, by the Reiteration Theorem (see [137]), it follows that AT (t)Px α ≤ ct β −α −1 x β

72

2 Linear Operators on Banach Spaces

for every x ∈ Xβ and 0 < β < 1, and hence, there exist constants M(α ) > 0 and γ > 0 such that T (t)Px α ≤ M(α )t β −α −1 e−γ t x β

for t > 0.

2.3 Evolution Families 2.3.1 Evolution Families and Their Estimates In this section we study time-dependent linear operators of the form (A(t), D(A(t))t∈R as well as their associated evolution families. For more on evolution families and related topics, we refer to [137]. Definition 2.75. A family of bounded linear operators {U(t, s) : t, s ∈ R, t ≥ s} on X is called an evolution family whether the following hold: (a) U(t, s)U(s, r) = U(t, r) for t, s, r ∈ R such that t ≥ s ≥ r; (b) U(t,t) = I for t ∈ R; and (c) for each x ∈ X , the function (t, s) → U(t, s)x is continuous for t ≥ s. Evolution families (also called “evolution systems,” “evolution operators,” “evolution processes,” “propagators,” or “fundamental solutions”) play a crucial role especially when it comes to studying some partial differential equations. Classical examples of evolution families include U defined by U(t, s) = T (t − s) for all t, s ∈ R with t ≥ s, where (T (t))t≥0 is a strongly continuous semigroup. Another example of an evolution family consists of U(t, s) = f (t) f −1 (s) for all t, s ∈ R with t ≥ s, where f : R → R is a function satisfying, f (t) = 0 for all t ∈ R. Definition 2.76. An evolution family {U(t, s) : t, s ∈ R, t ≥ s} on X is said to be exponentially bounded if there exist M, ω > 0 such that U(t, s) ≤ Meω (t−s) , t, s ∈ R,

t ≥ s.

(2.24)

It can be easily seen that the evolution equation U(t, s) = T (t − s) for all t, s ∈ R with t ≥ s, where (T (t))t≥0 is a strongly continuous semigroup, is an example of an exponentially bounded evolution family. Definition 2.77. An evolution family {U(t, s) : t, s ∈ R, t ≥ s} on X is said to be exponentially stable if there exists M, ω > 0 such that U(t, s) ≤ Me−ω (t−s) , t, s ∈ R,

t ≥ s.

(2.25)

2.3 Evolution Families

73

It can be easily seen that the evolution equation U(t, s) = T (t − s) for all t, s ∈ R with t ≥ s, where (T (t))t≥0 is an exponentially stable C0 -semigroup, is an example of an exponentially stable evolution family.

2.3.2 Acquistapace–Terreni Conditions Definition 2.78. A family of linear operators (A(t))t∈R is said to satisfy the Acquistapace–Terreni conditions whether there exist λ0 ≥ 0 and the constants φ ∈ ( π2 , π ), L, K ≥ 0, and μ , ν ∈ (0, 1] with μ + ν > 1 such that

Σφ ∪ {0} ⊆ ρ (A(t) − λ0 )  λ ,

R(λ , A(t) − λ0 ) ≤

K 1 + |λ |

(2.26)

and (A(t) − λ0 )R(λ0 , A(t) − λ0 ) [R(λ0 , A(t)) − R(λ0 , A(s))] ≤ L |t − s|μ |λ |−ν (2.27) for t, s ∈ R, λ ∈ Σφ := {λ ∈ C \ {0} : | arg λ | ≤ φ } . For a given family of closed linear operators {A(t) : t ∈ R} on X , the existence of an evolution family associated with it is not always guaranteed. However, if the family A(t) satisfies Acquistapace–Terreni conditions, then the family of linear operators A(t) has an evolution family associated with it such that U(t, s)X ⊆ D(A(t)) for all t, s ∈ R. Moreover, the following hold: for t > s, the mapping (t, s) → U(t, s) ∈ B(X ) is continuously differentiable in t with ∂t U(t, s) = A(t)U(t, s). Furthermore, there exists a constant C > 0 which depends on constants in Eqs. (2.26) and (2.27) such that Ak (t)U(t, s) ≤ C (t − s)−k

(2.28)

for 0 < t − s ≤ 1 and k = 0, 1. Remark 2.79. (a) In the particular case of a constant domain D(A(t)), one can replace Eq. (2.27) (see for instance [148]) with the following: there exist constants L and 0 < μ ≤ 1 such that (A(t) − A(s)) R(λ0 , A(r)) ≤ L|t − s|μ , s,t, r ∈ R.

(2.29)

(b) The Acquistapace–Terreni conditions were introduced in the literature by Acquistapace and Terreni in [1, 2] for λ0 = 0. Definition 2.80. An evolution family {U(t, s) : t ≥ s with t, s ∈ R} ⊂ B(X ) is said to have an exponential dichotomy (or is hyperbolic) if there are projections P(t) (t ∈ R) that are uniformly bounded and strongly continuous in t (and we then let Q(t) = I − P(t)) and constants δ > 0 and N ≥ 1 such that

74

2 Linear Operators on Banach Spaces

(a) (i) U(t, s)P(s) = P(t)U(t, s) for all t ≥ s; (b) the restriction UQ (t, s) : Q(s)X → Q(t)X of U(t, s) is invertible for all t ≥ s (and we then set UQ (s,t) := UQ (t, s)−1 ); and (c) U(t, s)P(s) ≤ Ne−δ (t−s) and UQ (s,t)Q(t) ≤ Ne−δ (t−s) for t ≥ s and t, s ∈ R.

2.3.2.1

Estimates for the Evolution Family U(t, s)

This subsection is devoted to the study of some estimates related to U(t, s). For that, we introduce the corresponding interpolation spaces for A(t). We refer the reader to [90, 137] for proofs and further information on these spaces. Let A be a sectorial operator on X and let α ∈ (0, 1). Define the real interpolation space ! " XαA := x ∈ X : x Aα := supr>0 rα (A − ζ )R(r, A − ζ )x < ∞ , which, by the way, is a Banach space when endowed with the norm · Aα . For convenience we further write X0A := X , x A0 := x , X1A := D(A), and x A1 := (ζ − A)x . We will need the space Xˆ A := D(A). In particular, we will frequently be using the following continuous embedding: D(A) → XβA → D((ζ − A)α ) → XαA → Xˆ A ⊂ X ,

(2.30)

for all 0 < α < β < 1, where the fractional powers are defined in the usual way. In general, D(A) is not dense in the spaces XαA and X . However, we have the following continuous injection: · Aα

XβA → D(A)

(2.31)

for 0 < α < β < 1. Given the operators A(t) for t ∈ R, satisfying Acquistapace–Terreni conditions, we set A(t)

Xαt := Xα

,

Xˆ t := Xˆ A(t)

for 0 ≤ α ≤ 1 and t ∈ R, with the corresponding norms. Then the embedding in Eq. (2.30) holds with constants independent of t ∈ R. These interpolation spaces are of class Jα and hence there is a constant l(α ) such that y tα ≤ l(α ) y 1−α A(t)y α ,

y ∈ D(A(t)).

(2.32)

We will need the following estimates to establish some of the results of this book.

2.3 Evolution Families

75

Proposition 2.81 (Baroun et al. [15]). For x ∈ X , 0 ≤ α ≤ 1, and all t > s, the following hold: (a) There is a constant c(α ), such that δ

U(t, s)P(s)x tα ≤ c(α )e− 2 (t−s) (t − s)−α x .

(2.33)

(b) There is a constant m(α ), such that #Q (s,t)Q(t)x sα ≤ m(α )e−δ (t−s) x . U

(2.34)

Proof. (a) Using Eq. (2.32) we obtain U(t, s)P(s)x tα ≤ l(α ) U(t, s)P(s)x 1−α A(t)U(t, s)P(s)x α ≤ l(α ) U(t, s)P(s)x 1−α A(t)U(t,t − 1)U(t − 1, s)P(s)x α ≤ l(α ) U(t, s)P(s)x 1−α A(t)U(t,t − 1) α U(t − 1, s)P(s)x α ≤ l(α )N e−δ (t−s)(1−α ) e−δ (t−s−1)α x δ

δ

≤ c (α )(t − s)−α e− 2 (t−s) (t − s)α e− 2 (t−s) x for t − s ≥ 1 and x ∈ X . δ Since (t − s)α e− 2 (t−s) → 0 as t → ∞ it easily follows that δ

U(t, s)P(s)x tα ≤ c1 (α )(t − s)−α e− 2 (t−s) x . If 0 < t − s ≤ 1, we have U(t, s)P(s)xtα ≤ l(α )U(t, s)P(s)x1−α A(t)U(t, s)P(s)xα     t +s t +s U , s P(s)xα ≤ l(α )U(t, s)P(s)x1−α A(t)U t, 2 2     t +s t +s α 1−α , s P(s)xα ≤ l(α )U(t, s)P(s)x A(t)U t,  U 2 2 ≤ l(α )Ne−δ (t−s)(1−α ) 2α (t − s)−α e− ≤ l(α )Ne

− δ2 (t−s)(1−α )

2α (t − s)−α e

δ α (t−s) 2

x

− δ2α (t−s)

x

δ

≤ c2 (α )e− 2 (t−s) (t − s)−α x, and hence δ

U(t, s)P(s)x tα ≤ c(α )(t − s)−α e− 2 (t−s) x for t > s.

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2 Linear Operators on Banach Spaces

(ii) We have, U˜ Q (s,t)Q(t)x sα ≤ l(α ) U˜ Q (s,t)Q(t)x 1−α A(s)U˜ Q (s,t)Q(t)x α ≤ l(α ) U˜ Q (s,t)Q(t)x 1−α A(s)Q(s)U˜ Q (s,t)Q(t)x α ≤ l(α ) U˜ Q (s,t)Q(t)x 1−α A(s)Q(s) α U˜ Q (s,t)Q(t)x α ≤ l(α )Ne−δ (t−s)(1−α ) A(s)Q(s) α e−δ (t−s)α x ≤ m(α )e−δ (t−s) x . In the last inequality we made use of the fact that A(s)Q(s) ≤ c for some constant c ≥ 0 (see [153, Proposition 3.18]). Remark 2.82. It should be mentioned that if U(t, s) is exponentially stable, then P(t) = I and Q(t) = I − P(t) = 0 for all t ∈ R. In that case, Eq. (2.33) still holds and can be rewritten as follows: for all x ∈ X , δ

U(t, s)x tα ≤ c(α )e− 2 (t−s) (t − s)−α x .

(2.35)

We will need the following technical lemma later on. Lemma 2.83 (Diagana [76]). Let α ∈ (0, 1). Suppose that the evolution family (U(t, s))t≥s not only satisfies Acquistapace–Terreni conditions but also is exponentially stable. Then, there exists a function Hα ,δ : (0, ∞) → (0, ∞) with Hα ,δ ∈ L1 (0, ∞) such that A(t)U(t, s)x ≤ Hα ,δ (t − s) x α ,

t >s

(2.36)

for all x ∈ Xα . Proof. First of all, note that there exists a constant C > 0 such that A(t)U(t, s) B(Xα ,X ) ≤ C(t − s)α −1

(2.37)

for all t, s ∈ R such that 0 < t − s ≤ 1 (see [152]). Now for t − s ≥ 1, using Eq. (2.28) in the case when k = 1, we obtain that A(t)U(t, s)x = A(t)U(t,t − 1)U(t − 1, s)x ≤ A(t)U(t,t − 1) B(X ) U(t − 1, s)x ≤ C U(t − 1, s)x ≤ C Neδ e−δ (t−s) x ≤ M C Neδ e−δ (t−s) x α for all x ∈ Xα , where M is the bound of the continuous injection Xα → X .

2.3 Evolution Families

77

Choosing the function Hα ,δ : (0, ∞) → (0, ∞) as Hα ,δ (t) =

⎧ α −1 if t ∈ (0, 1] ⎨ Ct ⎩

N e−δ t if t ∈ (1, ∞)

where N = M C Neδ , one can easily check that Hα ,δ ∈ L1 (0, ∞) and that A(t)U(t, s)x ≤ Hα ,δ (t − s) x α ,

t >s

for all x ∈ Xα .

Bibliographical Notes The material of Sect. 2.1 is taken from the following sources: Diagana [57], and Bezandry and Diagana [18]. Additional relevant references include the following: Conway [40], Diagana [56], Naylar and Sell [141], Pazy [148], Rudin [151], Weidmann [160], Locker [136], Lunardi [137], and Engel and Nagel [90]. The material of Sects. 2.2 and 2.3 is taken from the following sources: Baroun et al. [15], Bezandry and Diagana [18], Diagana [76], Diagana and Nelson [75], and Lunardi [137]. Additional relevant references include Engel and Nagel [90], Schnaubelt [152, 153], and Chicone and Latushkin [38].

Chapter 3

Almost Periodic Functions

The theory of almost periodic functions was introduced in the literature around 1924–1926 with the pioneering work of the Danish mathematician Bohr [25]. A decade later, various significant contributions were then made to that theory mainly by Bochner [24], von Neumann [159], and van Kampen [155]. The notion of almost periodicity, which generalizes the concept of periodicity, plays a crucial role in various fields including harmonic analysis, physics, dynamical systems, etc. This chapter reviews basic properties of almost periodic functions as well as their discrete counterparts. Various concepts related to the notion of almost periodicity will be discussed including the notions of asymptotic almost periodicity, Stepanovlike almost periodicity, C(n) -almost periodicity, etc. The results presented in this chapter are either classical ones or else cannot be easily found in the literature. The main references include, but are not limited to, the books by Besicovitch [17], Bohr [26], Corduneanu [41, 42], Diagana [57], Bezandry and Diagana [18], Fink [97], Halanay and Rasvan [107], N’Gu´er´ekata [142, 143], Zhang [172], and papers such as Diagana et al. [58], and Fan [94], etc. In the rest of the book, the notations (X , · ) and (H , ·, ·, · ) stand respectively for a generic Banach space and a generic infinite dimensional separable Hilbert space over F. Similarly, the notations C(R, X ) and BC(R, X ) stand respectively for the collection of continuous functions from R to X , and the Banach space of bounded continuous functions from R to X equipped with the corresponding sup-norm. The notation l ∞ (Z) (respectively, l ∞ (Z+ )) stands for the Banach space of bounded X -valued sequences on Z (respectively, on Z+ ) equipped with the corresponding sup-norm.

3.1 Almost Periodic Functions Definition 3.1. If f : R → X is a function, we denote the translate of f by s ∈ R, the function Rs f defined by Rs f (t) := f (t + s) for all t ∈ R. T. Diagana, Almost Automorphic Type and Almost Periodic Type Functions in Abstract Spaces, DOI 10.1007/978-3-319-00849-3 3, © Springer International Publishing Switzerland 2013

79

80

3 Almost Periodic Functions

Definition 3.2. A subset S of BC(R, X ) is said to be translation-invariant if for any f ∈ S, then Rs f ∈ S for any s ∈ R.

3.1.1 Definitions and Properties Definition 3.3. A function f ∈ C(R, X ) is called (Bohr) almost periodic if for each ε > 0, there exists lε > 0 such that every interval of length lε contains at least a number τ with the following property: sup Rτ f (t) − f (t) < ε . t∈R

The number τ is then called an ε -translation or an ε -period of f . The collection of all almost periodic functions f : R → X will be denoted by AP(X ). Obviously, the space of almost periodic functions AP(X ) is translationinvariant. Prior to collecting properties of almost periodic functions, let us make the following remarks regarding τ , the ε -period of an almost periodic function f . If τ is an ε -period, then τ is also an ε -period for any ε > ε . In the same way, if τ is an ε -period and τ is an ε -period, then τ + τ and τ − τ are respectively (ε + ε )periods. Finally, if τ is an ε -period, then so is −τ . Thus denoting the collection of all the ε -periods for f by T (ε , f ), one can see that T (ε , f ) ⊂ T (ε , f ) for any ε > ε . Clearly, a function f ∈ C(R, X ) is (Bohr) almost periodic if for each ε > 0, the set T (ε , f ) is relatively dense, that is, one can find a number lε > 0 such that every interval of length lε contains at least a number τ ∈ T (ε , f ). The number lε is then called an interval inclusion for the set T (ε , f ) according to Besicovitch’s terminology. For more details on this and related issues we refer the reader to the excellent book by Besicovitch [17]. Classical examples of almost periodic functions include continuous periodic functions on R as well as trigonometric polynomials, that is, any function Pn : R → X of the form Pn (t) =

n

n

k=1

k=1

∑ ak eiλkt = ∑ ak (cos(λkt) + i sin(λkt))

(3.1)

where λk ∈ R and ak ∈ X for k = 1, . . . , n. A classical example of an almost periodic function which is not periodic is given by the following function √ f (t) = sint + sin 2t.

3.1 Almost Periodic Functions

81

Given two functions f : R → X and g : R → R, their convolution f ∗ g, if it exists, is defined by ( f ∗ g)(t) :=

 ∞ −∞

f (s)g(t − s)ds.

Given f ∈ AP(X ) one can generate various types of almost periodic functions using the convolution. Proposition 3.4. Fix f ∈ AP(X ). If g ∈ L1 (R), then f ∗ g ∈ AP(X ). Proof. Since f is continuous and that g ∈ L1 (R), it can be easily shown that the function t → ( f ∗ g)(t) is continuous. Furthermore, ( f ∗ g)(t) ≤ f ∞ g 1 for each t ∈ R, where g 1 is the L1 -norm of g, and so f ∗ g ∈ BC(R, X ). We now show that f ∗ g ∈ AP(X ). For that, it is enough to assume g 1 = 0. Indeed, if g 1 = 0 then g = 0 a.e. and hence f ∗ g = 0 a.e. and there is nothing to prove. Using the fact that f ∈ AP(X ) it follows that for every ε > 0 there exists lε > 0 such that for all δ ∈ R there exists τ ∈ [δ , δ + lε ] with Rτ f (σ ) − f (σ ) <

ε for each σ ∈ R. g 1

In particular, the following holds: Rτ f (t − s) − f (t − s) <

ε for each σ = t − s ∈ R. g 1

(3.2)

Now Rτ ( f ∗ g)(t) − ( f ∗ g)(t) =

 +∞ −∞

{ f (t − σ + τ ) − f (t − σ )}g(σ )d σ , t ∈ R.

From Eq. (3.2) and the fact that g ∈ L1 (R) it easily follows that Rτ ( f ∗ g) − ( f ∗ g) ∞ < ε , and hence f ∗ g ∈ AP(X ). Example 3.5. Let Tn be a trigonometric polynomial. Using Proposition 3.4 one can easily see that if g ∈ L1 (R), then the function defined by Cn (t) := is almost periodic.

 +∞ −∞

g(t − s)Tn (s)ds, (t ∈ R)

82

3 Almost Periodic Functions

The next definition of the almost periodicity was proposed by Bochner, which turns out to be equivalent to that of (Bohr) almost periodicity. Definition 3.6. A function f ∈ BC(R, X ) is called (Bochner) almost periodic if for any sequence (σn )n∈N of real numbers there exists a subsequence (σn )n∈N of (σn )n∈N such that ( f (t + σn ))n∈N converges uniformly in t ∈ R. It can be shown that Bohr and Bochner’s concepts of almost periodicity are in fact equivalent. Indeed, we have the following theorem whose proof can be found in the excellent book by Corduneanu [41]. Theorem 3.7. A function f ∈ BC(R, X ) is Bohr almost periodic if and only if it is Bochner almost periodic. A proof of Theorem 3.7 in the discrete setting will be discussed in Sect. 3.4 (see the proof of Theorem 3.56). Theorem 3.8. If f , g : R → X are almost periodic functions and if λ ∈ F, then the following hold: (a) (b) (c) (d) (e)

f + g ∈ AP(X ). λ f ∈ AP(X ). If g is F-valued, then f g ∈ AP(X ). The function t → f (t + α ) belongs to AP(X ). The function t → f (α t) belongs to AP(X ).

Proof. In order to prove all these properties, we will make extensive use of the Bochner’s almost periodicity (Definition 3.6). (a) Since f ∈ AP(X ), for any sequence (σn )n∈N of real numbers there exists a subsequence (σn )n∈N of (σn )n∈N such that the sequence of functions ( f (t + σn ))n∈N converges uniformly in t ∈ R. Since g ∈ AP(X ) it follows that there exists a subsequence (σn )n∈N of (σn )n∈N such that the sequence of functions (g(t + σn ))n∈N converges uniformly in t ∈ R. It is then clear that ( f (t + σn ))n∈N converges uniformly in t ∈ R as (σn )n∈N is also a subsequence of (σn )n∈N and therefore ( f (t + σn ) + g(t + σn ))n∈N converges uniformly in t ∈ R, that is, f + g ∈ AP(X ). (b) Since f ∈ AP(X ), for any sequence (σn )n∈N of real numbers there exists a subsequence (σn )n∈N of (σn )n∈N such that the sequence of functions ( f (t + σn ))n∈N converges uniformly in t ∈ R. It is then clear that (λ f (t + σn ))n∈N converges uniformly in t ∈ R and hence λ f ∈ AP(X ). (c) The proof can be done as in (a). Indeed, since f ∈ AP(X ), for any sequence (σn )n∈N of real numbers there exists a subsequence (σn )n∈N of (σn )n∈N such that the sequence of functions ( f (t + σn ))n∈N converges uniformly in t ∈ R. Since g ∈ AP(F) it follows that there exists a subsequence (σn )n∈N of (σn )n∈N such that the sequence of functions (g(t + σn ))n∈N converges uniformly in t ∈ R. It is then clear that ( f (t + σn ))n∈N converges uniformly in t ∈ R as (σn )n∈N is

3.1 Almost Periodic Functions

83

also a subsequence of (σn )n∈N and therefore ( f (t + σn ) · g(t + σn )n∈N converges uniformly in t ∈ R, that is, f g ∈ AP(X ). (d) Since f ∈ AP(X ), for any sequence (σn )n∈N of real numbers there exists a subsequence (σn )n∈N of (σn )n∈N such that the sequence of functions ( f (s + σn ))n∈N converges uniformly in s ∈ R. In particular, letting s = t + α it follows that ( f (t + α + σn ))n∈N converges uniformly in t ∈ R and hence θ → f (θ + α ) belongs to AP(X ). (e) Since f ∈ AP(X ), for any sequence (σn )n∈N of real numbers there exists a subsequence (σn )n∈N of (σn )n∈N such that the sequence of functions ( f (s + σn ))n∈N converges uniformly in s ∈ R. In particular, letting s = α t it follows that ( f (α t + σn ))n∈N converges uniformly in t ∈ R and hence θ → f (αθ ) belongs to AP(X ). Proposition 3.9 ([18]). Let f : R → X be an almost periodic function. Then, the following hold: (a) The function f is uniformly continuous in t ∈ R. (b) The range R( f ) = { f (t) : t ∈ R} is relatively compact in X . Proof. (a) Since f : R → X is almost periodic, therefore, for each ε > 0 there exists lε > 0 such that every interval of length lε contains a number τ with the property f (t + τ ) − f (t) <

ε 3

for all t ∈ R. From the uniform continuity of f on compact intervals, let δ ∈ (0, 1) such that f (t) − f (s) ≤ ε for 0 ≤ t, s ≤ lε + 1 and |t − s| ≤ δ . Let t, s ∈ R such that |t − s| ≤ δ . Choose τ such that 0 < t + τ , s + τ < lε + 1. Now f (t) − f (s) ≤ f (t) − f (t + τ ) + f (t + τ ) − f (s − τ ) + f (s + τ ) − f (s) < ε. (b) From the almost periodicity of f it follows that for each ε > 0 there exists lε > 0 such that every interval of length lε contains a number τ with the property f (t + τ ) − f (t) < for all t ∈ R.

ε 2

84

3 Almost Periodic Functions

Since f [0, lε ] is compact in X , let us choose a finite sequence t1 , . . . ,tn ∈ [0, lε ] such that f (t) ∈

 ε B f (ti ), . 2 i=1 n

Now let t ∈ R, τ = τ (t) such that 0 <t + τ < lε , t j an element of the sequence t1 , . . . ,tn such that f (t + τ ) ∈ B f (ti ), ε2 . Then f (t) − f (t j ) ≤ f (t) − f (t + τ ) + f (t + τ ) − f (t j ) < ε so that R( f ) ⊆

 ε . B f (ti ), 2 i=1 n

Since ε is arbitrary it follows that R( f ) is relatively compact. Proposition 3.10. If f ∈ AP(X ), then, sup f (t) < ∞. t∈R

Proof. This is a straightforward consequence of [Proposition 3.9, (b)]. Theorem 3.11 ([42]). Let M ⊂ AP(X ) be a subset. Then M is relatively compact in AP(X ) if and only if the following properties hold true: (a) M is equi-continuous, that is, for any ε > 0 there exists δε > 0 such that for any f ∈ M , one has f (t) − f (s) < ε if |t − s| < δε . (b) M is equi-almost-periodic, that is, for any ε > 0 there exists lε > 0 such that every interval of length lε contains an ε -translation τ for all f ∈ M . (c) For a fixed t ∈ R, the set { f (t) : f ∈ M } is relatively compact in the space X . Proposition 3.12. Fix N ∈ N. If f : R → F is almost periodic so is f N where f N (t) := f (t) f (t) . . . f (t) (N factors). Proof. It is enough to assume that f ∞ > 0. Indeed, f ∞ = 0 yields f (t) = 0 for all t ∈ R. In that event, there is nothing to prove as the proposition holds. Now using the fact that f ∈ AP(F), for each ε > 0, there exists lε > 0 such that every interval of length lε contains a number τ with the property | f (t + τ ) − f (t)| < ε C−1 for each t ∈ R, where C = N f N−1 ∞ . Using the identity X N −Y N = (X −Y )(X N−1 + X N−2Y + X N−3Y 2 + · · · + XY N−2 +Y n−1 )

3.1 Almost Periodic Functions

85

it follows that | f N (t + τ ) − f N (t)| ≤ C| f (t + τ ) − f (t)| < ε for each t ∈ R. An immediate consequence of [Theorem 3.8, (a)] and Proposition 3.12 is the following composition result. Corollary 3.13. If f : R → F is almost periodic and if P ∈ F[t] is a polynomial of degree N whose coefficients belong to F, then t → P( f (t)) belongs to AP(F). Proposition 3.14. If f : R → X and g : R → F are almost periodic such that 0 < M ≤ |g(t)| for each t ∈ R for some constant M, then the quotient function t → f g−1 (t) is almost periodic. Proof. Using the fact g ∈ AP(F) it follows that for each ε > 0 there exists lε > 0 such that every interval of length lε contains a number τ with the property |g(t + τ ) − g(t)| < M 2 ε for all t ∈ R. Now $ $ $ 1 1 $$ |g(t + τ ) − g(t)| $ $ g(t + τ ) − g(t) $ = |g(t + τ )g(t)| |g(t + τ ) − g(t)| M2 <ε ≤

for all t ∈ R. Now using [Theorem 3.8, (c)] it follows that the quotient function t → f g−1 (t) is almost periodic. The proof is complete. Corollary 3.15. If f , g : R → F are almost periodic and if P, Q ∈ F[t] are polynomials such that |Q(t)| ≥ M > 0 for all t ∈ R, then the function defined by t → R(t) =

P( f )(t) Q(g)(t)

belongs to AP(F). Proof. Set F(t) := P( f )(t) and G(t) := Q(g)(t). Using Corollary 3.13 it easily follows that both F and G belong to AP(F). Clearly, |G(t)| ≥ M > 0 for all t ∈ R. Thus using Proposition 3.14 it follows that R ∈ AP(F). Proposition 3.16. Let ( fn )n∈N be a sequence of almost periodic functions such that there exists f ∈ C(R, X ) with fn − f ∞ → 0 as n → ∞. Then f is almost periodic.

86

3 Almost Periodic Functions

Proof. Using the fact that fn − f ∞ → 0 as n → ∞ it follows that, for each ε > 0, there exists N(ε ) such that fn (t) − f (t) ≤

ε 3

for all t ∈ R and n ≥ N(ε ). Using the almost periodicity of the fN (t) it follows that there exists lε > 0 such that every interval of length lε contains a number τ with the property fN (t + τ ) − fN (t) <

ε 3

for all t ∈ R. Now f (t + τ ) − f (t) = f (t + τ ) − fN (t + τ ) + fN (t + τ ) − fN (t) + fN (t) − f (t) ≤ f (t + τ ) − fN (t + τ ) + fN (t + τ ) − fN (t) + fN (t) − f (t) ε ε ε < + + 3 3 3 =ε for all t ∈ R. The proof is complete. We now have the following theorem due to Bochner. Theorem 3.17 (Bochner). Let f : R → X be an almost periodic function such that f is uniformly continuous on R, then f is also almost periodic.     Proof. Let fn (t) = n f t + 1n − f (t) for each t ∈ R and for n ∈ Z+ . Clearly, fn : R → X is a sequence of almost periodic functions, which converges uniformly to f on R. To complete the proof, one makes use of Proposition 3.16. Theorem 3.18. Let X be a uniformly convex Banach space. If f : R → X is almost periodic, then, F(t) =

 t t0

f (σ )d σ

(3.3)

is almost periodic if and only if sup F(t) < ∞. t∈R

Proof. See the proof in Corduneanu [41, Proof of Theorem 6.20, pp. 179–180].

3.1 Almost Periodic Functions

87

3.1.2 Fourier Series Representation This subsection is devoted to the construction of a Fourier series theory for almost periodic functions. In order to achieve that, we need to prove the existence of a mean value for those almost periodic functions. Definition 3.19. If f : R → X is a bounded continuous function and if the limit 1 r→∞ r

 r

f (t)dt

lim

0

exists, we then call it the mean value of the function f and denote it M ( f ). Theorem 3.20 ([42]). If f : R → X is almost periodic, then the mean value of f , 1 r→∞ r

M ( f ) := lim

 r 0

f (t)dt

exists. Furthermore, 1 r→∞ r

 r

lim

0

1 r→∞ r

f (t)dt = lim

 a+r a

f (t)dt

uniformly in a ∈ R. Proof. For ε > 0, let lε > 0 be the length of the interval such that each (a, a + lε ), a ∈ R, contains an ε /2-translation number for f . For a fixed a ∈ R, let τ ∈ (a, a + lε ) be an ε /2-translation number for f . Now  a+r a

f (t)dt =

 τ a

f (t)dt +

 τ +r τ

f (t)dt +

 a+r τ +r

f (t)dt,

which yields  r     a+r  τ +r  1  r  1    f (t)dt − f (t)dt  ≤  f (t)dt − f (t)dt    r 0 r 0 τ a  τ  a+r    1  1    +  f (t)dt  +  f (t)dt   r a r τ +r ≤

1 r

+

1 r

 r 0

f (t) − f (t + τ ) dt +

 a+r τ +r

f (t) dt

ε f ∞ lε f ∞ lε + + 2 r r ε 2 f ∞ lε = + 2 r ≤

1 r

 τ a

f (t) dt

88

3 Almost Periodic Functions

Now choosing a = (k − 1)r for k = 1, 2, . . . , n it follows that  r   rk  ε 2 f ∞ lε 1  ≤ + f (t)dt − f (t)dt   2 r 0 r (k−1)r for k = 1, 2, . . . , n, which yields    r       1 1 1 nr 1 n kr  r    = f (t)dt − f (t)dt f (t)dt − f (t)dt   ∑ (k−1)r   r 0 n 0 r 0 n k=1     r kr  1 n   ≤ f (t)dt − f (t)dt  ∑   nr k=1 0 (k−1)r

 n ε 2 f ∞ lε < + n 2 r =

r1 r2

ε 2 f ∞ lε + . 2 r

In view of the above, if we choose two positive real numbers r1 , r2 > 0 such that ∈ Q, that is, mr1 = m r2 for some m, m ∈ N, then  r 

  1  1 1 r2   < ε + 2 f ∞ lε 1 + 1 . f (t)dt − f (t)dt  r1 0  r2 0 r1 r2  Clearly, the set (r1 , r2 ) ∈ (0, ∞) × (0, ∞) :

r1 r2

∈ Q is dense in (0, ∞) × (0, ∞).

It follows that Eq. (3.4) is true in general, that is, whether not. Choosing r1 , r2 > 0 such that r1 , r2 >

(3.4)

4 f ∞ lε , ε

r1 r2

is a rational number or

it follows, from Eq. (3.4), that

 r   1  1 1 r2   < 2ε . f (t)dt − f (t)dt  r1 0  r2 0 This yields the existence of M ( f ). Now using the fact that  a+r a

f (t)dt =

 r 0

f (t + a)dt,

a ∈ R,

it follows that M ( f (t)) exists if and only if M ( f (t + a)) exists. In that case, both means are obviously equal. Moreover, in view of the above  r   a+r  ε 2 f ∞ lε 1  ≤ + . f (t)dt − f (t)dt   2 r 0 r a

3.1 Almost Periodic Functions

Thus choosing r >

4 f ∞ lε ε

89

= r0 (ε ), one can see that

 r   a+r  1   < ε. f (t)dt − f (t)dt   r 0 a The proof is complete. The mean M ( f ) of an almost periodic function f can be formulated in a few different ways. Indeed, we have 1 r→∞ r

M ( f ) = lim

 r 0

1 r→∞ 2r

f (t)dt = lim

 r −r

1 r→∞ r

f (t)dt = lim

 0 −∞

f (t)dt.

Corollary 3.21 ([42]). If f : R → X is a continuous ω -periodic function ( f (t + ω ) = f (t) for all t ∈ R), then its mean value exists and is given by M(f) =

1 ω

 ω 0

f (σ )d σ .

Proof. Since f is almost periodic, its mean M ( f ) exists according to Theorem 3.20. Now 1 r→∞ r

M ( f ) = lim

1 r→∞ r

= lim

 r 0

f (t)dt



ω 0

f (t)dt +

 2ω ω

f (t)dt + · · · +

 r ωk

 f (t)dt



ω  r 1 = lim f (t)dt + f (t)dt k r→∞ r 0 ωk 

 ω  k ω +α k 1 = lim f (t)dt + f (t)dt k→∞ kω + α 0 kω + α kω for some α ∈ [0, ω ). In view of the above, M(f) =

1 ω

 ω 0

f (σ )d σ .

Proposition 3.22. Let f , g : R → X be almost periodic functions and let α ∈ C. Then (a) (b) (c) (d)

If f is C-valued, then M ( f ) = M ( f ). M (α f ) = α M ( f ). If f is R-valued, then M ( f ) ≥ 0 for all f ≥ 0. M ( f + g) = M ( f ) + M (g).

90

3 Almost Periodic Functions

(e) If ( fn )n∈N is a sequence of almost periodic functions, which converges uniformly to f , then lim M ( fn ) = M ( f ).

n→∞

Proposition 3.23 ([17]). If the mean value of a nonnegative almost periodic function f is zero, then f (t) = 0 for all t ∈ R. Proof. For each ε > 0 there exists lε > 0 such that every interval of length lε > 0 contains a τ such that | f (t + τ )− f (t)| < ε for all t ∈ R. Suppose that the proposition is not true. Thus suppose that there exists t0 such that f (t0 ) = a > 0. Since f is continuous, then there exists δ > 0 such that f (t) > 2a 3 for all t ∈ (t0 − δ ,t0 + δ ). Let l ε > 2δ . In view of the above, any interval of length l ε contains at least one 3 3   number of the form t0 + τ , where τ ∈ T 13 a, f and so at least one of the intervals (t0 + τ − δ ,t0 + τ ) and (t0 + τ ,t0 + τ + δ ). In each of these intervals, which are the intervals (t0 − δ ,t0 ),(t0 ,t0 + δ ), we have f (t) > a3 . Therefore in each interval of length l ε there exists a subinterval of length δ such 3 that f (t) > εa . Clearly,  b+l ε

3

b

f (t)dt >

aδ ε

for any b. Now 1 n→∞ nl ε

0 = M ( f ) = lim

3

 nl ε

3

0

f (t)dt >

aδ 3l ε

3

which is a contradiction. Let f : R → X be an almost periodic function and let λ ∈ R. Clearly, the function t → f (t)e−iλ t is almost periodic, too. We then define the so-called Fourier coefficients of f by 1 r→∞ 2r

a(λ , f ) := M ({ f (t)e−iλ t }) = lim

 r −r

f (t)e−iλ t dt.

Definition 3.24. If f : R → X is an almost periodic function, then the numbers λ1 , λ2 , . . . , λn , . . . for which a(λk , f ) = 0 are called the Fourier exponents of f . The set of Fourier exponents is called the Bohr spectrum and denoted by σb ( f ). Proposition 3.25 ([42]). If f ∈ AP(X ), then a(λ , f ) is nonzero at most at countably many points.

3.1 Almost Periodic Functions

91

Definition 3.26. If f ∈ AP(X ) and if σb ( f ) = {λk : k = 1, 2, . . .}, then the series fˆ(t) :=

∞

∑ a(λk , f )eiλkt

k=1

will be called the Fourier series associated with f . The previous definition is justified by the next approximation theorem. Theorem 3.27 ([126, 134]). If f ∈ AP(X ), then for every ε > 0 there exists a trigonometric polynomial Pε (t) =

n

∑ ak eiλkt

k=1

where ak ∈ X and λk ∈ σb ( f ) such that f (t) − Pε (t) < ε for all t ∈ R. Proposition 3.28 ([42]). If f , g ∈ AP(X ) and if fˆ = g, ˆ then f = g. For more on Fourier series theory for almost periodic functions and related issues we refer the interested reader to Besicovitch [17] and Corduneanu [41, 42].

3.1.3 Composition of Almost Periodic Functions The main objective of this subsection consists of studying a few composition theorems for almost periodic functions, which will be of a great interest especially when it comes to dealing with semilinear differential equations (respectively, difference equations) with almost periodic coefficients. Definition 3.29. A jointly continuous function F : (t, x) → F(t, x) is called almost periodic if t → F(t, x) is almost periodic uniformly in x ∈ B, where B is any bounded subset of X . That is, for each ε > 0 there exists lε > 0 such that every interval of length lε > 0 contains a number τ with the property F(t + τ , x) − F(t, x) < ε for all t ∈ R and x ∈ B. The collection of such functions will be denoted AP(R × X ). It should be noted that our definition of almost periodicity for functions of the form F : R × X → X , (t, x) → F(t, x) is stronger than some of the definitions proposed in the literature, which require that the almost periodicity be uniform in x ∈ K where K ⊂ X is an arbitrary compact subset. Our choice of considering bounded subsets rather than compact ones is mandated by the fact we want to have

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3 Almost Periodic Functions

some coherence in the definitions of the extensions of almost periodic functions, which we will be dealing with in this chapter as well as in Chaps. 4–6. Theorem 3.30. Let F : R × X → X , (t, x) → F(t, x) be an almost periodic function in t ∈ R uniformly in x ∈ B where B ⊂ X is an arbitrary bounded subset. Suppose that F is Lipschitz in x ∈ X uniformly in t ∈ R, i.e., there exists L > 0 such that F(t, x) − F(t, y) ≤ L x − y , for all x, y ∈ X ,

t ∈ R.

(3.5)

If g : R → X is almost periodic, then the function Γ (t) = F(t, g(t)) : R → X is also almost periodic. Proof. Since g ∈ AP(X ), for all ε > 0 there exists lε > 0 such that every interval of length lε > 0 contains a τ with the property that g(t + τ ) − g(t) <

ε 2L

(3.6)

for all t ∈ R. Now Γ (t + τ ) − Γ (t) = F(t + τ , g(t + τ )) − F(t, g(t)) ≤ F(t + τ , g(t + τ )) − F(t + τ , g(t)) + F(t + τ , g(t)) − F(t, g(t)) ≤ L g(t + τ ) − g(t) + F(t + τ , g(t)) − F(t, g(t)) . Using the almost periodicity of F it follows that

ε sup F(t + τ , g(t)) − F(t, g(t)) < . 2 t∈R

(3.7)

Combining Eqs. (3.6) and (3.7) one obtains that sup Γ (t + τ ) − Γ (t) = sup F(t + τ , g(t + τ )) − F(t, g(t)) < ε , t∈R

t∈R

and hence the function Γ : t → F(t, g(t)) is almost periodic. Theorem 3.31. Let F : R × X → X be an almost periodic function. Suppose u → F(t, u) is uniformly continuous on every bounded subset B ⊂ X uniformly for t ∈ R. If g ∈ AP(X ), then Γ : R → X defined by Γ (·) := F(·, g(·)) belongs to AP(X ). Proof. Since g ∈ AP(X ), for all ε > 0 there exists lε > 0 such that every interval of length lε > 0 contains a τ with the property that

3.2 C(n) -Almost Periodic Functions

93

g(t + τ ) − g(t) < ε

(3.8)

for all t ∈ R. Since g ∈ AP(X ) it follows that g is uniformly bounded. Now let B ⊂ X be a bounded subset such that g(t) ∈ B for all t ∈ R. Now F(t + τ , g(t + τ )) − F(t, g(t)) ≤ F(t + τ , g(t + τ )) − F(t + τ , g(t)) + F(t + τ , g(t)) − F(t, g(t)) . Taking into account Eq. (3.8) and using the uniform continuity of F on bounded subsets of X it follows that

ε sup F(t + τ , g(t + τ )) − F(t + τ , g(t)) < . 2 t∈R

(3.9)

Similarly, using the almost periodicity of F it follows that

ε sup F(t + τ , g(t)) − F(t, g(t)) < . 2 t∈R

(3.10)

Combining Eqs. (3.9) and (3.10) one obtains that sup F(t + τ , g(t + τ )) − F(t, g(t)) < ε , t∈R

and hence the function t → F(t, g(t)) is almost periodic. It should be mentioned that the Lipschitz condition in Theorem 3.30 and the uniform continuity in Theorem 3.31, as shown in the next theorem, can be relaxed. Theorem 3.32. If F : R × X → X , (t, x) → F(t, x) is a jointly continuous function such that t → F(t, x) is almost periodic uniformly in x ∈ K where K ⊂ X is an arbitrary compact subset and if g ∈ AP(X ), then Γ : R → X defined by Γ (·) := F(·, g(·)) belongs to AP(X ). For the proof of Theorem 3.32, we refer the interested reader to Fink [97] or Zhang [172] for instance.

3.2 C(n) -Almost Periodic Functions An important subclass of almost periodic functions is that of n-differentiable almost periodic functions (also called C(n) -almost periodic functions), that is, functions f : R → X such that f (k) ∈ AP(X ) for k = 0, 1, 2, . . . , n. These functions were introduced in the literature by Adamczak [3] for real-valued functions. Since then

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3 Almost Periodic Functions

such a notion has generated several developments, in particular Bugajewski and N’Gu´er´ekata [31] not only extended it to Banach spaces but also introduced the concept of asymptotic C(n) -almost periodicity, which is a natural generalization of the C(n) -almost periodicity. For more on these functions and related topics, we refer the reader to [4, 14, 31, 87, 92, 130]. Definition 3.33. A function f ∈ C(n) (R, X ) is said to be C(n) -almost periodic if f (k) ∈ AP(X ) for k = 0, 1, . . . , n. The collection of C(n) -almost periodic functions is denoted AP(n) (X ), which turns out to be a Banach space when it is equipped with the norm · (n) , that is, n

f (n) := sup ∑ f (k) (t) < ∞ for each f ∈ AP(n) (X ). t∈R k=0

We have the following inclusions . . . → AP(n+1) (X ) → AP(n) (X ) → . . . → AP(1) (X ) → AP(X ). Definition 3.34. A jointly continuous function F : (t, x) → F(t, x), R × X → X is said to be C(n) -almost periodic in t ∈ R uniformly in x ∈ B (B ⊂ X being an arbitrary bounded subset) if (k)

Dt f (t, x) :=

∂k f (t, x) ∂ tk

is almost periodic in t ∈ R uniformly in x ∈ B, where k = 0, 1, 2, . . . , n with (0)

Dt f (t, x) :=

∂0 f (t, x) = f (t, x). ∂ t0

The collection of such functions will be denoted by AP(n) (R × X ). Theorem 3.35. If ( fk )k∈N ⊂ AP(n) (X ) is a sequence such that fk − f (n) → 0 as k → ∞, then f ∈ AP(n) (X ). (m)

Proof. Clearly, fk (m) fk

− f (m) ∞ → 0 as k → ∞ for m = 0, 1, . . . , n. And since each

∈ AP(X ) for m = 0, 1, . . . , n, then one must have f (m) ∈ AP(X ) for m = 0, 1, . . . , n by using Proposition 3.16. Therefore, f ∈ AP(n) (X ). Theorem 3.36. The space AP(n) (X ) equipped with the norm · (n) is a Banach space. (m)

Proof. Let ( fk )k∈N ⊂ AP(n) (X ) be a Cauchy sequence. It is clear that ( fk )k∈N is a Cauchy sequence in AP(X ) for m = 0, 1, . . . , n. Now since (AP(X ), · ∞ ) is a (m) Banach space it follows that ( fk )k∈N converges uniformly to some fm ∈ AP(X )

3.2 C(n) -Almost Periodic Functions

95

as k → ∞ for m = 0, 1, . . . , n. Setting f0 = g it is easily seen that fm = g(m) and that g(m) = fm ∈ AP(X ) for m = 0, 1, . . . , n, that is, g ∈ AP(n) (X ). Moreover, fk − g (n) → 0 as k → ∞, which yields (AP(n) (X ), · (n) ) is a Banach space. Theorem 3.37. If f ∈ AP(n) (X ) and if g ∈ L1 (R), then their convolution f ∗ g ∈ AP(n) (X ). Proof. Let f ∈ AP(n) (X ) and let g ∈ L1 (R). Using the fact ( f ∗ g)(k) = f (k) ∗ g it follows that ( f ∗ g)(k) ∈ AP(k) (X ) for all k = 0, 1, 2, . . . , n as f (k) ∈ AP(X ) for all k = 0, 1, 2, . . . , n and g ∈ L1 (R) (see Proposition 3.4 for details). Theorem 3.38. If f ∈ AP(n) (X ) such that f (n+1) is uniformly continuous, then f ∈ AP(n+1) (X ). Proof. Let f˜ := f (n) . Since f˜ is uniformly continuous, then for every ε > 0 there exists δ > 0 such that for all t, s ∈ R with |t − s| < δ , then f˜ (t) − f˜ (s) < ε . For an arbitrary r ∈ R and δ > 1n , we have

   1 n 1 ( f˜ (r + τ ) − f˜ (r))d τ . n f˜ r + − f˜(r) − f˜ (r) = n n 0 Therefore, from Proposition 3.16 it follows that f˜ ∈ AP(X ), that is, f ∈ AP(n+1) (X ). Example 3.39. Let us give an example of a function f ∈ AP(2) (R) such that f ∈ AP(3) (R). Indeed, consider the function f defined by f (t) =

∞

sin(kt) . k4 k=1

∑

Clearly, f is a continuous 2π -periodic function and hence is almost periodic. Similarly, the functions f (t) =

∞

∞ cos(kt) sin(kt) , f (t) = − ∑ k3 ∑ k2 k=1 k=1

are continuous 2π -periodic functions and hence are almost periodic. Now ∞

cos(kt) k k=1

f (t) = − ∑

is not continuous for all real numbers. Namely, f is not defined at t = 0. Hence f is not almost periodic. Therefore, f ∈ AP(2) (R) while f ∈ AP(3) (R).

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3 Almost Periodic Functions

3.3 Asymptotically Almost Periodic Functions The concept of asymptotic almost periodicity was introduced in the literature by the French mathematician Fr´echet [99–101]. Such a notion is a natural generalization of the concept of almost periodicity. To introduce the notion of asymptotic almost periodicity, we need the space C0 (R+ , X ), that is, the collection of all continuous functions ϕ : R+ → X such that lim ϕ (t) = 0.

t→∞

3.3.1 Basic Definitions and Properties Definition 3.40. A continuous function f : R+ →  X is said to be asymptotically almost periodic if there exist h ∈ AP(X ) and ϕ ∈ C0 (R+ , X ) such that f (t) = h(t) + ϕ (t),

t ∈ R+ .

The collection of all asymptotically almost periodic functions will be denoted by AAP(X ). The functions h and ϕ are respectively called the principal and corrective terms of the function f . We have the following characterization of asymptotically almost periodic functions whose proof is left to the reader as an exercise. Proposition 3.41. A continuous function f : R+ → X is asymptotically almost periodic if and only if for any sequence (τn )n∈N with τn → ∞ as n → ∞ there exists a subsequence (τnk )k∈N for which f (t + τnk ) converges uniformly in t ∈ R+ . Lemma 3.42. If f ∈ AP(X ) such that lim f (t) = 0, then f = 0. t→∞

Proof. Since f ∈ AP(X ), for the sequence (n)n∈N there exists a subsequence (nk )k∈N ⊂ (n)n∈N such that lim f (t + nk ) = g(t)

k→∞

uniformly in t ∈ R. From lim f (t) = 0 and Eq. (3.11) it follows that g(t) = 0 for all t ∈ R. t→∞ Now 0 = lim g(t − nk ) = f (t) k→∞

uniformly in t ∈ R, and hence f = 0.

(3.11)

3.3 Asymptotically Almost Periodic Functions

97

The proof of Lemma 3.43 is left to the reader as an exercise. A proof of this lemma in a broader context will be given in the next chapter (see Lemma 4.28). Lemma 3.43. If f ∈ AAP(X ), that is, f = h + ϕ where h ∈ AP(X ) and ϕ ∈ C0 (R+ , X ), then {h(t) : t ∈ R} ⊂ { f (t) : t ∈ R+ }. Proposition 3.44. The decomposition of asymptotically almost periodic functions is unique, that is, AAP(X ) = AP(X ) ⊕C0 (R+ , X ). Proof. This is a straightforward consequence of Lemma 3.42. Indeed, if f = h1 + ϕ1 ∈ AAP(X ), where h ∈ AP(X ) and ϕ ∈ C0 (R+ , X ) and if f can also be represented by f = h2 + ϕ2 ∈ AAP(X ) where h2 ∈ AP(X ) and ϕ2 ∈ C0 (R+ , X ), then we obtain h1 − h2 = ϕ2 − ϕ1 and hence h1 − h2 ∈ AP(X ) and lim h1 − h2 = 0.

t→∞

Using Lemma 3.42 it follows that h1 = h2 , which yields ϕ1 = ϕ2 = 0. Proposition 3.45. If f , g ∈ AP(X ) and if there exists t0 ∈ R+ such that f (t) = g(t) for all t ≥ t0 , then f = g. Proof. Let h = f − g. Clearly, h ∈ AP(X ) and lim h(t) = 0. Therefore, h = 0 by t→∞ using Lemma 3.42. If f ∈ AAP(X ), that is, f = h + ϕ where h ∈ AP(X ) and ϕ ∈ C0 (R+ , X ), we define f AAP := sup h(t) + sup ϕ (t) . t∈R

t∈R+

Clearly, · AAP is a norm on AAP(X ) and the following holds: Theorem 3.46. The space AAP(X ) equipped with the norm · AAP is a Banach space. Proof. It is enough to show that the norm defined by f = supt∈R+ f (t) for f ∈ AAP(X ) is equivalent to f AAP . For that, we use ideas from Lemma 4.28. Indeed, if f ∈ AAP(X ), that is, f = h + ϕ where h ∈ AP(X ) and ϕ ∈ C0 (R+ , X ), then supt∈R h(t) ≤ f by using Lemma 3.43. Now f ≤ f AAP = sup h(t) + sup f (t) − h(t) t∈R

t∈R+

98

3 Almost Periodic Functions

≤ 3 sup f (t) t∈R+

= 3 f . The proof of the next proposition is clear and hence is omitted. Proposition 3.47. If ( fn )n∈N ⊂ AAP(X ) such that fn converges uniformly to f on R+ , that is, lim fn − f AAP = 0.

n→∞

Then f ∈ AAP(X ).

3.3.2 Composition Theorems This subsection is devoted to the composition of asymptotically almost periodic functions. For that, we need to introduce the space C0 (R+ × X ), which consists of all jointly continuous functions Ψ : R+ × X → X such that lim Ψ (t, x) = 0

t→∞

uniformly in x ∈ K where K ⊂ X is an arbitrary bounded subset. Definition 3.48. A jointly continuous function F : R+ × X → X , (t, x) → F(t, x) is called asymptotically almost periodic if t → F(t, x) is asymptotically almost periodic uniformly in x ∈ K, where K is an arbitrary bounded subset of X . That is, F = H + Φ where H ∈ AP(R × X ) and Φ ∈ C0 (R+ × X ). The collection of all such asymptotically almost periodic functions will be denoted by AAP(R+ × X ). Theorem 3.49. Let F : R+ ×X → X , (t, x) → F(t, x) be an asymptotically almost periodic function in t ∈ R+ uniformly in x ∈ B, where B is arbitrary bounded subset. Letting F = H + Φ where H ∈ AP(R × X ) and Φ ∈ C0 (R+ × X ), we suppose that both H and Φ are Lipschitz in x ∈ X uniformly in t, i.e., there exists L1 , L2 > 0 such that H(t, x) − H(t, y) ≤ L1 x − y

(3.12)

for all x, y ∈ X and t ∈ R, and Φ (t, x) − Φ (t, y) ≤ L2 x − y for all x, y ∈ X and t ∈ R+

(3.13)

3.3 Asymptotically Almost Periodic Functions

99

If g : R+ → X is asymptotically almost periodic, then the function Γ (t) = F(t, g(t)) : R+ → X is also asymptotically almost periodic. Proof. Let F = H + Φ where H ∈ AP(R × X ) and Φ ∈ C0 (R+ × X ) and let g = h + ϕ where h ∈ AP(X ) and ϕ ∈ C0 (R+ , X ). Now F(t, g(t)) = H(t, h(t)) + [F(t, g(t)) − H(t, h(t))] = H(t, h(t)) + [H(t, g(t)) − H(t, h(t)) + Φ (t, g(t))]. Using Theorem 3.30 it follows that t → H(t, h(t)) is almost periodic. Also, since g is bounded, it is clear that t → Φ (t, g(t)) belongs to C0 (R+ , X ). To complete the proof, we have to show that t → H(t, g(t)) − H(t, h(t)) belongs to C0 (R+ , X ). Indeed, using the fact that H(t, g(t)) − H(t, h(t)) ≤ L2 g(t) − h(t) = L2 ϕ (t) it follows that t → H(t, g(t)) − H(t, h(t)) belongs to C0 (R+ , X ), and hence the function t → F(t, g(t)) is asymptotically almost periodic. We also have the following theorem whose proof is omitted. Theorem 3.50. Let F : R+ × X → X , (t, x) → F(t, x) be asymptotically almost periodic in t ∈ R+ uniformly in x ∈ B where B is arbitrary bounded subset. Letting F = H + Φ where H ∈ AP(R × X ) and Φ ∈ C0 (R+ × X ), we suppose that u → H(t, u) is uniformly continuous on every bounded subset B ⊂ X uniformly for t ∈ R and that Φ is Lipschitz in x ∈ X uniformly in t ∈ R+ , i.e., there exists L > 0 such that Φ (t, x) − Φ (t, y) ≤ L x − y , for all x, y ∈ X . t ∈ R+ .

(3.14)

If g : R+ → X is asymptotically almost periodic, then the function Γ (t) = F(t, g(t)) : R+ → X is also asymptotically almost periodic. Theorem 3.51. Let F : R+ × X → X be an asymptotically almost periodic function. Suppose u → F(t, u) is uniformly continuous on every bounded subset B ⊂ X uniformly for t ∈ R+ . If g ∈ AAP(X ), then Γ : R+ → X defined by Γ (·) := F(·, g(·)) belongs to AAP(X ). Proof. Let F = H + Φ where H ∈ AP(R × X ) and Φ ∈ C0 (R+ × X ) and let g = h + ϕ where h ∈ AP(X ) and ϕ ∈ C0 (R+ , X ). Now F(t, g(t)) = H(t, h(t)) + [F(t, g(t)) − H(t, h(t))] = H(t, h(t)) + [H(t, g(t)) − H(t, h(t)) + Φ (t, g(t))].

100

3 Almost Periodic Functions

Using Theorem 3.31 it follows that t → H(t, h(t)) is almost periodic. Also, it is clear that t → Φ (t, g(t)) belongs to C0 (R+ , X ). To complete the proof, we have to show that t → H(t, g(t)) − H(t, h(t)) belongs to C0 (R+ , X ). First of all, note that t → g(t) − h(t) = ϕ (t) ∈ C0 (R+ , X ). Thus for each δ > 0 there exists A > 0 such that g(t) − h(t) ≤ δ for all |t| > A. Thus for ε > 0, H(t, g(t)) − H(t, h(t)) < ε for all |t| > A, that is, t → H(t, g(t)) − H(t, h(t)) belongs to C0 (R+ , X ), and hence the function t → F(t, g(t)) is asymptotically almost periodic. It should be mentioned that assumptions on Theorems 3.49–3.51, as shown in the next theorem, can be relaxed. Theorem 3.52. If F : R+ × X → X , (t, x) → F(t, x) such that t → F(t, x) is asymptotically almost periodic uniformly in x ∈ K where K ⊂ X is an arbitrary compact subset and if g ∈ AAP(X ), then Γ : R → X defined by Γ (·) := F(·, g(·)) belongs to AAP(X ). Proof. We refer the interested reader to the proof given in [172, Theorem 2.8].

3.4 Almost Periodic Sequences This section is mainly based on the work by Diagana et al. [58] in which a theory for almost periodic sequences was constructed. All the proofs presented here follow the work of Diagana et al. [58], which will be extended to the Banach spaces setting. Additional relevant references to this topic include Corduneanu [41, 42], Ding et al. [83], Halanay and Rasvan [107], and Zhang [172].

3.4.1 Basic Definitions Recall that l ∞ (Z+ ) is the Banach space of all bounded X -valued sequences equipped with the sup-norm defined for each x = {x(t)}t∈Z+ ∈ l ∞ (Z+ ), by x ∞ = sup x(t) . t∈Z+

Define  N(Z+ ) := x = (x(t))t∈Z+ ∈ l ∞ (Z+ ) : lim x(t) = 0 . t→∞

Definition 3.53 ([58]). An X -valued sequence x = {x(t)}t∈Z+ is called (Bohr) almost periodic if for each ε > 0, there exists a positive integer N0 (ε ) such that among any N0 (ε ) consecutive integers, there exists at least one integer τ with the following property:

3.4 Almost Periodic Sequences

101

x(t + τ ) − x(t) < ε , ∀t ∈ Z+ . As in the continuous setting, the integer τ is called an ε -period of the sequence x = {x(t)}t∈Z+ . The collection of all almost periodic X -valued sequences on Z+ will be denoted by AP(Z+ ). It is a Banach space when equipped with the sup-norm defined above. Definition 3.54. An X -valued sequence x = {x(t)}t∈Z+ is called Bochner almost periodic if for every sequence {h(t)}t∈Z+ ⊂ Z+ there exists a subsequence {h(Ks )}s∈Z+ such that {x(t + h(Ks ))}s∈Z+ converges uniformly in t ∈ Z+ .

3.4.2 Properties of Almost Periodic Sequences Proposition 3.55 ([58]). Let xm = {xm (t)}t∈Z+ be a Bohr almost periodic sequence converging uniformly in m ∈ Z+ to x, then the sequence x is Bohr almost periodic. Proof. We refer the interested reader to Halanay and Rasvan [107, Proposition 4.7, p. 229]. Theorem 3.56 ([58]). A sequence x = {x(t)}t∈Z+ is Bochner almost periodic if and only if it is Bohr almost periodic. Proof. First, we show that if x = {x(t)}t∈Z+ is Bochner almost periodic, then it is Bohr almost periodic. To achieve this, we show that if x = {x(t)}t∈Z+ is not Bohr almost periodic, then it is not Bochner almost periodic. Suppose that x = {x(t)}t∈Z+ is not Bohr almost periodic. Then there exists at least one ε > 0 such that for any positive integer N0 , there exist N0 consecutive positive integers which contain no ε -period related to the sequence {x(t)}t∈Z+ . Now, let h(1) ∈ Z+ and let 2α1 + 1, 2α1 + 2, 2α1 + 3, . . . , 2β1 − 2, 2β1 − 1 be (2β1 − 2α1 − 1)positive integers (α1 , β1 ∈ Z+ ) such that 2β1 − 2α1 − 2 > 2h(1) or β1 − α1 − 1 > h(1) and the sequence 2α1 + 1, 2α1 + 2, 2α1 + 3, . . . , 2β1 − 2, 2β1 − 1 does not contain any ε -period related to {x(t)}t∈Z+ . Next, let 1 h(2) = (2α1 + 2β1 ) = α1 + β1 . 2 Clearly, h(2) − h(1) is a (positive) integer such that 2α1 + 1 < h(2) − h(1) < 2β1 − 1, and hence h(2) − h(1) cannot be an ε -period. Thus, there exist 2α2 + 1, 2α2 + 2, 2α2 + 3, . . . , 2β2 − 2, 2β2 − 1 such that 2β2 − 2α2 − 2 > 2(h(1) + h(2)), which does not contain any ε -period related to {x(t)}t∈Z+ . Setting h(3) = 12 (2α2 + 2β2 ) = α2 + β2 , it follows that h(3) − h(2), h(3) − h(1) are respectively one of the terms

102

3 Almost Periodic Functions

2α2 + 1, 2α2 + 2, 2α2 + 3, . . . , 2β2 − 2, 2β2 − 1, and hence h(3) − h(2), h(3) − h(1) are not ε -period related to {x(t)}t∈Z+ . Proceeding as previously, one defines the numbers h(4), h(5), . . ., such that none of the expressions h(i) − h( j) for i > j is an ε -period for the sequence {x(t)}t∈Z+ . Consequently, for all i, j ∈ Z+ , sup x(t + h(i)) − x(t + h( j)) ≥ sup x(t + h(i)) − x(t + h( j)) i, j

i> j

= sup x(t + h(i) − h( j)) − x(t) i> j

≥ ε. This proves that {x(t + h(i))}i∈Z+ cannot contain any uniformly convergent sequence, and hence {x(t)}t∈Z+ is not Bochner almost periodic. Conversely, suppose that the sequence {x(t)}t∈Z+ is Bohr almost periodic and ! " t j j∈Z+ is a sequence of positive integers. Here, we adapt our proof to the one given in [107, Proof of Theorem 4.9, pp. 230–231]. For each ε > 0 there exists an integer N0 > 0 such that between t j and N0 + t j there exist an ε -period τ j with 0 ≤ τ j −t j ≤ N0 . Setting s j = τ j −t j , one can see that s j can take only a finite number (at most N0 + 1) values, and hence there is some s, 0 ≤ s ≤ N0 such that s j = s for an infinite numbers of j s. Let these indexes be numbered as jk , then we have x(t + t j ) − x(t + s j ) = x(t + τ j + s j ) − x(t + s j ) < ε . Hence, x(t + t j ) − x(t + s j ) < ε for all t ∈ Z+ . One may complete the proof by proceeding exactly as in [107, Proof of Theorem 4.9, pp. 230–231] and using [107, Proposition 4.7] relative to Z+ rather than Z. 1 Now let {εr }r∈Z+ be a sequence such that ε → 0 as r → ∞, say εr = . Now, r+1 ! " from the sequence x(n + t j ) j∈Z+ , consider a subsequence chosen so that x(n + t j1 ) − x(n + s1 ) ≤ ε1 . i

Next, from the previous sequence, we take a new subsequence such that x(n + t j2 ) − x(n + s2 ) ≤ ε2 . i

3.4 Almost Periodic Sequences

103

Repeating this procedure and for each r ∈ Z+ we obtain a subsequence  x(n + t jir ) such that i∈Z+

x(n + t jir ) − x(n + sr ) ≤ εr .  Now, for the diagonal sequence, x(n + t ji ) i

ε 2,

i∈Z+

, for each ε > 0 take k(ε ) ∈ Z+

such that εk(ε ) < where εr belongs to the previous sequence {εr }r∈Z+ . ! " ! " r Using the fact that the sequences t and t jss are both subsequences of j r   t

k(ε )

ji

, for r ≥ k(ε ) we have x(n + t jrr ) − x(n + t jss ) ≤ x(n + t jrr ) − x(n + sk ) + x(n + sk ) − x(n + t jss ) ≤ εk(ε ) + εk(ε ) ≤ ε.

 Thus, the sequence x(n + t ji ) i

i∈Z+

is a Cauchy sequence. The proof is complete.

Let x = {x(t)}t∈Z+ and α = {α (s)}s∈Z+ be X -valued sequences. Define  Tα x := y = (y(s))s∈Z+ : y(s) = lim x(s + α (t)) . t→∞

Theorem 3.57 ([58]). Let x be a sequence. Suppose that for every pair α , β of sequences in Z+ there exist common subsequences α , β where α is a subsequence of α and β that of β , such that Tα Tβ x = Tα +β x pointwise in Z+ . Then x is almost periodic. Lemma 3.58 ([58]). If {x(t)}t∈Z+ is almost periodic, then it is bounded. Proof. Assume that {x(t)}t∈Z+ is not bounded. Then for some subsequence x(ti ) → ∞ as i → ∞. Let ε = 1. Then there exists N(ε ) ∈ Z+ − {0} that satisfies the almost periodicity definition. There exists ti = s1 such that ti = s1 > N(ε ). Then among the integers {s1 − N(ε ) + 1, s1 − N(ε ) + 2, . . . , s1 } there exists s%1 such that x(t + s%1 ) − x(t) < 1.

104

3 Almost Periodic Functions

Next, choose t j = s2 such that t j = s2 > N(ε ) + s1 . Then among the integers {s2 − N(ε ) + 1, s2 − N(ε ) + 2, . . . , s2 } there exists s%2 such that x(t + s%2 ) − x(t) < 1. Repeating this process, we obtain a sequence {% si } → ∞ as i → ∞ such that x(t + s%i ) − x(t) < 1 for r = 1, 2, 3, . . . , and a subsequence {si } of {ti } with {si } → ∞ as i → ∞. Moreover, si = s%i + ui where 0 ≤ ui < N(ε ). Since {ui } is finite, there exists ui0 that is repeated infinitely many times and sir = s%ir + ui0 , where ir → ∞ as i → ∞. Therefore, x(t + s%ir ) − x(ui0 ) < 1. Moreover,   x(t + si ) − x(ui ) < 1. r 0 Hence, {x(sir )} is bounded and this is a contradiction

3.4.3 Asymptotically Almost Periodic Sequences Definition 3.59 ([94]). An X -valued sequence x = {x(t)}t∈Z+ is said to be asymptotically almost periodic if it can be decomposed as x(t) = u(t) + υ (t), where u = {u(t)}t∈Z+ ∈ AP(Z+ ) and {υ (t)}t∈Z+ ∈ N(Z+ ). The collection of all asymptotically almost periodic X -valued sequences will be denoted AAP(Z+ ). Lemma 3.60 ([58]). If x = {x(t)}t∈Z+ ∈ AP(Z+ ) and lim x(t) = 0, then x(t) = 0 t→∞ for all t ∈ Z+ .

3.4 Almost Periodic Sequences

105

1 for each t ∈ {0, 1, . . .}. Then there exists N(εt ) such that t +1 among t,t + 1, . . . ,t + N(εt ) − 1, there exists st such that

Proof. Let εt =

x(t + st ) − x(t) ≤ εt for all t ∈ Z+ . As t → ∞, st → ∞, x(t + st ) → 0, and εt → 0. Hence, x(t) ≤ 0. This implies that x(t) = 0 for all t ∈ Z+ . Lemma 3.61 ([58]). The decomposition of an asymptotically almost periodic sequence is unique. That is, AP(Z+ )∩ N(Z+ ) = {0}. Proof. Suppose that x = {x(t)}t∈Z+ can be decomposed as x(t) = u(t) + υ (t) and x(t) = v(t) + δ (t), where u = {u(t)}t∈Z+ , ν = {ν (t)}t∈Z+ ∈ AP(Z+ ) and {υ (t)}t∈Z+ , {δ (t)}t∈Z+ ∈ N(Z+ ). Clearly, u(t) − ν (t) = δ (t) − υ (t) ∈ AP(Z+ )∩ N(Z+ ). By Lemma 3.60, u(t) = ν (t) and υ (t) = δ (t) for all t ∈ Z+ .

3.4.4 Composition Theorems Definition 3.62. A sequence F : Z+ × X → X , (t, u) → F(t, u) is called almost periodic in t ∈ Z+ uniformly in u ∈ B where B ⊂ X is an arbitrary bounded subset, if for each ε > 0 there exists a positive integer N0 (ε ) such that among any N0 (ε ) consecutive integers, there exists at least one integer s with the property: F(t + s, u) − F(t, u) < ε for all t ∈ Z+ and u ∈ B. Theorem 3.63 ([58]). Suppose that F : Z+ × X → X , (t, u) → F(t, u) is almost periodic in t ∈ Z+ uniformly in u ∈ B, where B ⊂ X is a bounded subset of X . If in addition, F is Lipschitz in u ∈ X uniformly in t ∈ Z+ (that is, there exists L > 0 such that F(t, u) − F(t, v) ≤ L u − v

∀u, v ∈ X , t ∈ Z+ ),

then for every X -valued almost periodic sequence x = {x(t)}t∈Z+ , the X -valued sequence y(t) = F(t, x(t)) is almost periodic.

106

3 Almost Periodic Functions

Proof. Let x = {x(t)}t∈Z+ be an almost periodic sequence and let y(t) = F(t, x(t)). Then, for each ε > 0 there exists a positive integer N0 (ε ) such that among any N0 (ε ) consecutive integers, there exists at least one integer s with the following property: x(t + s) − x(t) <

ε L

∀t ∈ Z+ .

Moreover, y(t + s) − y(t) = F(t + s, x(t + s)) − F(t, x(t)) ≤ F(t + s, x(t + s)) − F(t + s, x(t)) + F(t + s, x(t)) − F(t, x(t)) ε ≤ L x(t + s) − x(t)) + 2 ε ε < + = ε. 2 2 As in the continuous setting, the Lipschitz condition in Theorem 3.63, as shown in the next theorem, can be relaxed. Theorem 3.64. If F : Z+ ×X → X , (t, x) → F(t, x) such that t → F(t, x) is almost periodic uniformly in x ∈ K where K ⊂ X is an arbitrary compact subset and if t → z(t) is almost periodic, then the sequence y : Z+ → X defined by y(·) := F(·, z(·)) is almost periodic.

3.5 S p -Almost Periodic Functions This section is devoted to the concept of S p -almost periodicity (or Stepanovlike almost periodicity), which is a natural generalization of the notion of almost periodicity. Various properties of these functions will be investigated. Definition 3.65. The Bochner transform f b (t, s), t ∈ R, s ∈ [0, 1], of a function f : R → X is defined by f b (t, s) := f (t + s). A function ϕ (t, s), t ∈ R, s ∈ [0, 1], is the Bochner transform of a certain function f : R → X , ϕ (t, s) = f b (t, s) , if and only if ϕ (t + τ , s − τ ) = ϕ (s,t) for all t ∈ R, s ∈ [0, 1] and τ ∈ [s − 1, s]. Definition 3.66. The Bochner transform F b : R × [0, 1] × X → X of a function F : R × X → X is defined by F b (t, s, u) := F(t + s, u) for each t ∈ R, s ∈ [0, 1], and u ∈ X . Definition 3.67. Let p ∈ [1, ∞). The space BS p (X ) of all Stepanov-like bounded functions, with the of all measurable functions f : R → X  exponent p, consists  such that f b ∈ L∞ R, L p ((0, 1), X ) . This is a Banach space with the norm

3.5 S p -Almost Periodic Functions

107

f S p = f b L∞ (R,L p ) = sup t∈R



t+1

t

f (τ ) p d τ

1/p .

We have the following inclusions: p L p (R, X ) ⊂ BS p (X ) ⊂ Lloc (R, X )

and BS p (X ) ⊂ BSq (X ) for p ≥ q ≥ 1. Similarly, for p ≥ 1, we have the following continuous injection: (BC(X ), · ∞ ) → (BS p (X ), · S p ).

(3.15)

p Definition 3.68. A function f ∈ BS p (X  )p is called S -almost periodic (or b Stepanov-like almost periodic) if f ∈ AP L ((0, 1), X ) . That is, for each ε > 0 there exists lε > 0 such that every interval of length lε > 0 contains a τ such that

 sup t∈R

t

t+1

f (s + τ ) − f (s) ds

1/p

p

< ε.

p The collection of such functions will be denoted by Sap (X ).

In other words, a function f ∈ L p (R, X ) is said to be S p -almost periodic if its Bochner transform f b : R → L p ((0, 1), X ) is almost periodic in the sense that f b ∈ AP(L p ((0, 1), X )). Example 3.69. Let α , β ∈ R be given such that αβ −1 is an irrational number. Consider the functions defined by

fα ,β (t) = sin

1 2 + cos α t + cos β t



and

gα ,β (t) = cos

1 2 + cos α t + cos β t



for all t ∈ R. Both fα ,β and gα ,β are examples of S p -almost periodic functions, which are not almost periodic (see [13, 127]).

108

3 Almost Periodic Functions

Proposition 3.70. If f ∈ AP(X ), then f is S p -almost periodic for any 1 ≤ p < ∞. Proof. Let f ∈ AP(X ). Using Eq. (3.15) one can easily see that not only f ∈ p BS p (X ) but also f ∈ Sap (X ). Definition 3.71. A function F : R × X → X , (t, u) → F(t, u) with F(·, u) ∈ BS p (X ) for each u ∈ X , is said to be S p -almost periodic in t ∈ R uniformly in u ∈ X if for each ε > 0 and each compact subset K ⊂ X , there exists lε > 0 such that every interval of length lε contains a τ with the property,  sup t∈R

0

1

F(t + s + τ , u) − F(t + s, u) ds

1/p

p

<ε

(3.16)

uniformly in u ∈ K. p The collection of these functions will be denoted by Sap (R × X ). p For any compact subset K ⊂ X , we denote by Sap,K (R × X ) the set of all p functions F ∈ Sap (R × X ) such that Eq. (3.16) is replaced with

 1 sup t∈R

0

 p 1/p < ε. sup F(t + s + τ , u) − F(t + s, u) ds

(3.17)

u∈K

3.5.1 Composition of S p -Almost Periodic Functions The following composition result is due to Long and Ding [135] whose proof will be omitted. p Theorem 3.72 ([135]). Let F ∈ Sap (R×X ) for p > 1 and suppose that F satisfies:  p there exists L ∈ BSr (R) with r ≥ max p, p−1 such that

F(t, x) − F(t, y) ≤ L(t) x − y for all x, y ∈ X and t ∈ R. p (X ) and if there exists a set E ⊂ R of measurable zero such that K = If g ∈ Sap {g(t) : t ∈ R\E} ⊂ X is compact, then there exists q ∈ [1, p) such that the function q Γ : R → X defined by Γ (t) = F(t, g(t)) belongs to Sap (X ). Other additional relevant references upon the composition of S p -almost periodic functions include, but are not limited to, Andres and Pennequin [13], Danilov [46], and Ding et al. [84].

(n)

3.6 S p -Almost Periodic Functions

109

(n)

3.6 S p -Almost Periodic Functions (n)

The concept of S p -almost periodicity (n-differentiable Stepanov-like almost periodicity) is new and was introduced in the literature by Diagana et al. [73]. This subsection discusses a few of the properties of these new functions. (n)

Definition 3.73. Let p ∈ [1, ∞) and let n ∈ N. The space BS p (X ) consists of all functions f : R → X such that f (k) ∈ BS p (X ) for k = 0, 1, . . . , n. We equip the (n) space BS p (X ) with the norm defined by n

f p,(n) := sup ∑

t∈R k=0



t+1

t

f (k) (τ ) p d τ

1/p .

(n)

Proposition 3.74 ([73]). The space BS p (X ) equipped with the norm f p,(n) is a Banach space. (n)

(k)

Proof. Let ( fm )m∈N ⊂ BS p (X ) be a Cauchy sequence. It is clear that ( fm )m∈N is a Cauchy sequence in BS p (X ) for k = 0, 1, . . . , n. Now since (BS p (X ), · S p ) is a Banach space it follows that there exists a function g ∈ BS p (X ), which is n-times (k) differentiable such that ( fm )m≥0 converges to g(k) with respect to the norm · S p as m → ∞ for k = 0, 1, . . . , n. Clearly, fm − g p,(n) → 0 as m → ∞, which yields (n)

BS p (X ) equipped with the norm · p,(n) is a Banach space. (n),p

Definition 3.75. The space Sap (X ) of Stepanov-like C(n) -almost periodic func(n) (n) tions (or S p -almost periodic) consists of all f ∈ BS p (X ) such that   ( f (k) )b ∈ AP L p ((0, 1), X ) for k = 0, 1, . . . , n. (n)

(n)

In other words, a function f ∈ BS p (X ) is said to be S p -almost periodic if ( f (k) )b is almost periodic for k = 0, 1, . . . , n in the sense that for every sequence ε > 0 there exists lε > 0 such that every interval of length lε contains a τ with the property 

t+1

sup t∈R

t

f

(k)

(s + τ ) − f

(k)

1/p (s) ds p

→0

for k = 0, 1, . . . , n, as m → ∞. Similarly, Definition 3.76. A function F : R × X →  X , (t, u) → F(t, u) with F(·, u) ∈ (n),p Sap (X ) for each u ∈ B where B ⊂ X is a bounded subset, is said to

110

3 Almost Periodic Functions (n)

(n)

be S p -almost periodic. The collection of those S p -almost periodic functions (n),p F : R × X → X will be denoted by Sap (R × X ). (n),p

Theorem 3.77. If f ∈ Sap (X ) and if g ∈ L1 (R), then their convolution f ∗ g ∈ (n),p Sap (X ). (n),p

1 Proof. Let f ∈ Sap (X  p) and let g ∈L (R). To complete the proof we have to show (k) b that [( f ∗ g) ] ∈ AP L ((0, 1), X ) for all k = 0, 1, 2, . . . , n. Indeed, using the fact

[( f ∗ g)(k) ]b = [ f (k) ∗ g]b = ( f (k) )b ∗ g it follows that   [( f ∗ g)(k) ]b ∈ AP L p ((0, 1), X ) for all k = 0, 1, 2, . . . , n. (n),p

Proposition 3.78. The space Sap (X ) equipped with the norm · p,(n) is a Banach space. (n),p

(n)

Proof. The proof is based on the fact Sap (X ) is a closed subspace of BS p (X ). (n),p

Proposition 3.79. If f ∈ AP(n) (X ), then f ∈ Sap (X ). That is, AP(n) (X ) ⊂ (n),p Sap (X ).

Bibliographical Notes The results on almost periodic, C(n) -almost periodic, asymptotically almost peri(n) odic, S p -almost periodic, and S p -almost periodic functions discussed in this chapter are mainly taken from the following books: Besicovitch [17], Bohr [26], Corduneanu [41,42], Diagana [57], Bezandry and Diagana [18], Fink [97], N’Gu´er´ekata [142, 143], and Zhang [172], and the papers by Diagana and Nelson [71], Diagana et al. [73], and Long and Ding [135]. The part of the chapter on almost periodic and asymptotically almost periodic sequences is based on Diagana et al. [58] and Fan [94].

Chapter 4

Almost Automorphic Functions

The concept of almost automorphy was introduced in the literature by S. Bochner in 1955 in the context of differential geometry [21] (see also Bochner [22, 23]). Since then, this concept has been extended in various directions. Veech [156] extended this concept to groups and then obtained various properties of these functions including the existence of their corresponding Fourier series (see also [157, 158]). Almost automorphic functions coincide with the so-called Levitan N-almost periodic functions [126]. Although every almost periodic function is almost automorphic, the converse, however, is not true. Johnson [122] constructed an example of an almost periodic system whose projective flow has two minimal subsets, one of which is almost automorphic but not almost periodic. Other important contributions to the theory of almost automorphic functions include those from Zaki [170, 171], N’Gu´er´ekata [142–145], and Shen and Yi [154]. Our main task in this chapter consists of introducing basic properties of almost automorphic functions as well as their discrete counterparts. In addition, we will introduce and study the notion of asymptotic almost automorphy [144,145] not only in the continuous setting but also in the discrete realm [60] as well as the notion of Stepanov-like almost automorphy [34, 146].

4.1 Almost Automorphic Functions 4.1.1 Basic Definitions and Examples Definition 4.1. A continuous function f : R → X is said to be almost automorphic if for every sequence of real numbers (s n )n∈N there exists a subsequence (sn )n∈N such that lim lim f (t + sn − sm ) = f (t)

m→∞ n→∞

for each t ∈ R. T. Diagana, Almost Automorphic Type and Almost Periodic Type Functions in Abstract Spaces, DOI 10.1007/978-3-319-00849-3 4, © Springer International Publishing Switzerland 2013

111

112

4 Almost Automorphic Functions

In fact Definition 4.1 can also be reformulated as follows: Definition 4.2 ([142, 143]). A continuous function f : R → X is almost automorphic if for every sequence of real numbers (s n )n∈N there exists a subsequence (sn )n∈N such that g(t) := lim f (t + sn ) n→∞

is well defined for each t ∈ R, and lim g(t − sn ) = f (t)

n→∞

for each t ∈ R. The collection of all almost automorphic functions which go from R to X is denoted by AA(X ). It should be mentioned that if the convergence in Definition 4.2 is uniform in t ∈ R, then we obtain the almost periodicity of f . Consequently, every almost periodic function is almost automorphic, with the converse being obviously false. The following example of an almost automorphic function, which is not almost periodic, was constructed by Veech [158], f (t) =

√ 2 √ |2 + eit + eit 2 |

2 + eit + eit

for all t ∈ R. Definition 4.3. A continuous function f : R → X is said to be compact almost automorphic if for every sequence of real numbers (s n )n∈N there exists a subsequence (sn )n∈N such that g(t) := lim f (t + sn ) n→∞

and lim g(t − sn ) = f (t)

n→∞

uniformly on compact subsets of R. The collection of all compact almost automorphic functions from R into X is denoted by KAA(X ). If f , g : R → X are almost automorphic functions and if λ ∈ F, then: (a) f + g ∈ AA(X ). (b) λ f ∈ AA(X ). (c) fλ ∈ AA(X ) where fλ (t) := f (t + λ ) for λ ∈ R.

4.1 Almost Automorphic Functions

113

(d) f# ∈ AA(X ) where f#(t) = f (−t). (e) The range of f , that is, R( f ) := { f (t) : t ∈ R} is relatively compact. Moreover, if f and g are the functions appearing in Definition 4.2, then not only f ∞ < ∞ and g ∞ = f ∞ but also R(g) ⊆ R( f ). Proofs of all these basic properties of almost automorphic functions can be found in [142, 143]. Classical examples of almost automorphic functions include not only compact almost automorphic functions but also almost periodic functions. Namely, we have the following inclusions: AP(X ) → KAA(X ) → AA(X ). Example 4.4. Other examples of almost automorphic functions include for instance  f (t) = cos

1 p(t)



where p(t) = 2 + sin α t + sin β t with α , β being real numbers such that αβ −1 is an irrational number. In fact, f is an another example of an almost automorphic function which is not almost periodic. Another way of constructing almost automorphic functions from a fixed almost automorphic function is through the convolution, that is, the convolution of that (fixed) almost automorphic function with any L1 -function. Theorem 4.5 (Bugajewski–Diagana [32,33]). Fix f ∈ AA(X ). If h ∈ L1 (R), then their convolution defined by ( f ∗ h)(t) := belongs to AA(X ).

 ∞ −∞

f (σ )h(t − σ )d σ

114

4 Almost Automorphic Functions

Proof. First of all, let us mention that since f is continuous and that h ∈ L1 (R) it is not hard to see that the function t → ( f ∗ h)(t) is continuous. Let (s n )n∈N be an arbitrary sequence of real numbers. Now since f is almost automorphic, there exists (sn )n∈N ⊂ (s n )n∈N a subsequence such that g(t − σ ) := lim f (t − σ + sn ) n→∞

and f (t − σ ) = lim g(t − σ − sn ) n→∞

for all t, σ ∈ R. Consider  +∞

( f ∗ h)(t + sn ) :=

f (t − σ + sn )h(σ )d σ

−∞

for all t ∈ R. Clearly, f (t − σ + sn )h(σ ) ≤ f ∞ |h(σ )| for all t, σ ∈ R. Using the fact h ∈ L1 (R) and in view of the above, we then deduce from the Lebesgue Dominated Convergence Theorem that lim ( f ∗ h)(t + sn ) =

n→∞

=

 +∞

lim f (t − σ + sn ) h(σ ) d σ

−∞ n→∞  ∞ −∞

g(t − σ ) h(σ ) d σ

:= (g ∗ h)(t) for each t ∈ R. Similarly, consider (g ∗ h)(t − sn ) :=

 +∞ −∞

g(t − σ − sn )h(σ )ds

for all t ∈ R. Clearly, g(t − σ − sn )h(σ ) ≤ g ∞ |h(σ )| for all t, σ ∈ R. Again, by the Lebesgue Dominated Convergence Theorem it follows that lim (g ∗ h)(t − sn ) =

n→∞

 +∞

lim g(t − σ − sn ) h(σ ) ds

−∞ n→∞

4.1 Almost Automorphic Functions

115

=

 +∞ −∞

f (t − σ ) h(σ ) d σ

:= ( f ∗ h)(t) for each t ∈ R, hence f ∗ h is almost automorphic. Theorem 4.6 ([142, Theorem 2.1.10]). Let ( fn )n∈N ⊂ AA(X ) be a sequence such that there exists f ∈ C(R, X ) with fn − f ∞ → 0 as n → ∞. Then f ∈ AA(X ). Theorem 4.7. The space (AA(X ), · ∞ ) is a Banach space. Proof. Clearly, AA(X ) ⊂ BC(R, X ). Now Theorem 4.6 yields AA(X ) is a closed subspace of BC(R, X ). Therefore, (AA(X ), · ∞ ) is itself a Banach space. Lemma 4.8 ([129, 164]). Let f ∈ AA(X ). Fix t0 ∈ R and for ε > 0, define the set  Bε := τ ∈ R : f (t0 + τ ) − f (t0 ) ≤ ε . Then there exist s1 , s2 , . . . , sm ∈ R such that m 

 sk + Bε = R.

k=1

Definition 4.9. A function f : R → X is said to be weakly continuous if for any ξ ∈ X ∗ (X ∗ being the topological dual of X ), the function f ∗ : R → F defined by f ∗ (t) := ξ ( f (t)) = ξ , f (t) is continuous. Definition 4.10. A weakly continuous function f : R → X is weakly almost automorphic if for every ξ ∈ X ∗ and every sequence of real numbers (s n )n∈N there exists a subsequence (sn )n∈N such that ξ , g(t) := lim ξ , f (t + sn ) n→∞

is well defined for each t ∈ R, and lim ξ , g(t − sn ) = ξ , f (t)

n→∞

for each t ∈ R. The collection of all weakly almost automorphic functions which go from R into X is denoted by WAA(X ). Obviously, AA(X ) → WAA(X ). Moreover, if f , g : R → X are weakly almost automorphic functions and if λ ∈ F, then:

116

(a) (b) (c) (d)

4 Almost Automorphic Functions

f + g ∈ WAA(X ). λ f ∈ WAA(X ). fλ ∈ WAA(X ) where λ ∈ R and fλ (t) := f (t + λ ). f# ∈ WAA(X ) where f#(t) = f (−t).

In addition, if f and g are the functions appearing in Definition 4.10, then f ∞ < ∞. Further, g ∞ = f ∞ . Definition 4.11. A weakly continuous function f : R → X is weakly compact almost automorphic if for every ξ ∈ X ∗ and every sequence of real numbers (s n )n∈N there exists a subsequence (sn )n∈N such that ξ , g(t) := lim ξ , f (t + sn ) n→∞

is well defined for each t ∈ R, and lim ξ , g(t − sn ) = ξ , f (t)

n→∞

uniformly on compact subsets of R. The collection of all weakly compact almost automorphic functions which go from R into X is denoted by WKAA(X ). Obviously the following inclusion holds: KAA(X ) → WKAA(X ). Theorem 4.12 ([170]). Let f ∈ WAA(X ) and suppose that the Banach space X is reflexive. Then  t

F(t) := t0

f (s)ds

belongs to WAA(X ) if and only if F is bounded. Theorem 4.13 ([170]). If f ∈ AA(X ), then  t

F(t) := t0

f (s)ds

belongs to AA(X ) if and only if the range of F is relatively compact in X . The next theorem generalizes Theorem 3.18. Theorem 4.14 ([170]). Let f ∈ AA(X ) and suppose that the Banach space X is uniformly convex. Then

4.1 Almost Automorphic Functions

117

 t

F(t) := t0

f (s)ds

belongs to AA(X ) if and only if F is bounded.

4.1.2 Composition Theorems Definition 4.15. A jointly continuous function F : R × X → X is called almost automorphic in t ∈ R uniformly in x ∈ B where B ⊂ X is any bounded subset, if for every sequence of real numbers (s n )n∈N there exists a subsequence (sn )n∈N such that G(t, x) := lim F(t + sn , x) n→∞

is well defined for each t ∈ R, and lim G(t − sn , x) = F(t, x)

n→∞

for each t ∈ R, uniformly in x ∈ B. The collection of those almost automorphic functions is denoted by AA(R×X ). The next theorem is a generalization of Theorem 3.30. Theorem 4.16 ([142, Theorem 2.2.6]). Let f : R × X → X be an almost automorphic function. Suppose that u → (t, u) is Lipschitzian uniformly in t ∈ R, that is, there exists L > 0 such f (t, x) − f (t, y) ≤ L x − y

(4.1)

for all (x, y) ∈ X × X and t ∈ R. If ϕ ∈ AA(X ), then F : R → X defined by F(·) := f (·, ϕ (·)) belongs to AA(X ). Proof. Let (sn )n∈N be an arbitrary sequence of real numbers. By definition of almost automorphic functions, we can extract a subsequence (τn )n∈N such that (i) lim f (t + τn , x) = g(t, x), for each t ∈ R and x ∈ B; n→∞

(ii) lim g(t − τn , x) = f (t, x), for each t ∈ R and x ∈ B (B ⊂ X being an arbitrary n→∞ bounded subset); (iii) lim ϕ (t + τn ) = ψ (t), for each t ∈ R; and n→∞

(iv) lim ψ (t − τn ) = ϕ (t), for each t ∈ R. n→∞

Letting G(t) := g(t, ψ (t)), we need to show that for each t ∈ R,

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4 Almost Automorphic Functions

lim F(t + τn ) = G(t),

n→∞

and lim G(t − τn ) = F(t).

n→∞

Now F(t + τn ) − G(t) ≤ f (t + τn , ϕ (t + τn )) − f (t + τn , ψ (t)) + f (t + τn , ψ (t)) − g(t, ψ (t)) . ≤ L ϕ (t + τn ) − ψ (t) + f (t + τn , ψ (t)) − g(t, ψ (t)) . From (iii) it follows that lim ϕ (t + τn ) − ψ (t) = 0.

n→∞

Similarly, from identity (i) we obtain lim f (t + τn , ψ (t)) − g(t, ψ (t)) = 0

n→∞

and hence lim F(t + τn ) = G(t).

n→∞

Using similar arguments as above we obtain that lim G(t − τn ) = F(t),

n→∞

which yields F is almost automorphic. The next theorem is a generalization of Theorem 3.31. Theorem 4.17 ([132, Lemma 2.2]). Let f : R × X → X be an almost automorphic function. Suppose u → F(t, u) is uniformly continuous on every bounded subset K ⊂ X uniformly for t ∈ R (i.e., for each ε > 0, there exists δ > 0 such that x, y ∈ K and x − y < δ yields f (t, x) − f (t, y) < ε for all t ∈ R). If ϕ ∈ AA(X ), then F : R → X defined by F(·) := f (·, ϕ (·)) belongs to AA(X ). Proof. Let (sn )n∈N be an arbitrary sequence of real numbers. By definition of almost automorphic functions, we can extract a subsequence (τn )n∈N such that

4.1 Almost Automorphic Functions

119

(i) lim f (t + τn , x) = g(t, x), for each t ∈ R and x ∈ X ; n→∞

(ii) lim g(t − τn , x) = f (t, x), for each t ∈ R and x ∈ B (B ⊂ X being an arbitrary n→∞ bounded subset); (iii) lim ϕ (t + τn ) = ψ (t), for each t ∈ R; and n→∞

(iv) lim ψ (t − τn ) = ϕ (t), for each t ∈ R. n→∞

Letting G(t) := g(t, ψ (t)), one can easily show that for each t ∈ R, lim F(t + τn ) = G(t),

n→∞

and lim G(t − τn ) = F(t).

n→∞

Now |F(t + τn ) − G(t) ≤ f (t + τn , ϕ (t + τn )) − f (t + τn , ψ (t)) + f (t + τn , ψ (t)) − g(t, ψ (t)) . Since ϕ is almost automorphic, then both ϕ and ψ are bounded. Thus, we can choose a bounded subset K of X such that ϕ (t) ∈ K and ψ (t) ∈ K for all t ∈ R. By identity (iii) and the uniform continuity of x → f (t, x) in x ∈ K, we obtain that lim f (t + τn , ϕ (t + τn )) − f (t + τn , ψ (t)) = 0.

n→∞

Similarly, by identity (i) we obtain lim f (t + τn , ψ (t + τn )) − g(t + τn , ψ (t)) = 0

n→∞

and hence lim F(t + τn ) = G(t).

n→∞

Using similar arguments as above we obtain that lim G(t − τn ) = F(t),

n→∞

which yields F is almost automorphic.

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4 Almost Automorphic Functions

4.2 C(n) -Almost Automorphic Functions An important subclass of almost automorphic functions is that of C(n) -almost automorphic functions (n-differentiable almost automorphic functions), that is, functions f : R → X such that f (k) ∈ AA(X ) for k = 0, 1, 2, . . . , n. These functions were introduced in the literature by Ezzinbi et al. [93]. Recent developments on these functions and related issues can be found for instance in [71, 73, 87, 91]. Definition 4.18. A function f ∈ C(n) (R, X ) is said to be C(n) -almost automorphic (respectively, C(n) -compact almost automorphic) if f (k) ∈ AA(X ) for k = 0, 1, . . . , n (respectively, f (k) ∈ KAA(X ) for k = 0, 1, . . . , n). The collection of C(n) -almost automorphic (respectively, C(n) -compact almost automorphic) functions is denoted by AA(n) (X ) (respectively, denoted by KAA(n) (X )), which turns out to be a Banach space when equipped with the n-sup-norm · (n) . Clearly, the following inclusions hold: · · · → AA(n+1) (X ) → AA(n) (X ) → · · · → AA(1) (X ) → AA(X ) and · · · → KAA(n+1) (X ) → KAA(n) (X ) → · · · → KAA(1) (X ) → KAA(X ). Definition 4.19. A jointly continuous function F : R × X →  X is said to be C(n) almost automorphic in t ∈ R uniformly in x ∈ B where B ⊂ X is an arbitrary bounded subset, if (k)

Dt f (t, x) :=

∂k f (t, x) ∂ tk

is almost automorphic in t ∈ R uniformly in x ∈ B for k = 0, 1, 2, . . . , n with (0)

Dt f (t, x) :=

∂0 f (t, x) = f (t, x). ∂ t0

The collection of such functions is denoted by AA(n) (R × X ).

  Theorem 4.20. The space AA(n) (X ) equipped with the n-sup-norm  · (n) is a Banach space. (k)

Proof. Let ( f p ) p∈N ⊂ AA(n) (X ) be a Cauchy sequence. It is clear that ( f p ) p∈N is a Cauchy sequence in AA(X ) for k = 0, 1, . . . , n. Now since (AA(X ), · ∞ ) is a (k) Banach space it follows that ( f p ) p∈N converges uniformly to some fk ∈ AA(X ) as p → ∞ for k = 0, 1, . . . , n. Setting f0 = g it is easily seen that fk = g(k) and that g(k) =

4.3 Asymptotically Almost Automorphic Functions

121

fk ∈ AA(X ) for k = 0, 1, . . . , n, that is, g ∈ AA(n) (X ). Moreover, f p − g (n) → 0 as p → ∞, which yields AA(n) (X ) equipped with the norm · (n) is a Banach space. Theorem 4.21. If f ∈ AA(n) (X ) and if g ∈ L1 (R), then f ∗ g ∈ AA(n) (R). Proof. Let f ∈ AA(n) (X ) and let g ∈ L1 (R). To complete we have to show that f ∗ g ∈ AA(n) (X ). Using the fact ( f ∗ g)(k) = f (k) ∗ g and Theorem 4.5 it follows that ( f ∗ g)(k) ∈ AA(X ) for all k = 0, 1, 2, . . . , n as f (k) ∈ AA(X ) for all k = 0, 1, 2, . . . , n and g ∈ L1 (R). Theorem 4.22. If f ∈ AA(n) (X ) such that f (n+1) is uniformly continuous, then f ∈ AA(n+1) (X ).

4.3 Asymptotically Almost Automorphic Functions The concept of asymptotic almost automorphy was introduced in the literature by N’Gu´er´ekata [144, 145]. This obviously is a natural generalization of the notions of asymptotic almost periodicity and that of almost automorphy. In this section we review and study basic properties of these functions.

4.3.1 Basic Definitions Definition 4.23 ([144]). A continuous function f : R+ → X is said to be asymptotically almost automorphic if there exist h ∈ AA(X ) and ϕ ∈ C0 (R+ , X ) such that f (t) = h(t) + ϕ (t),

t ∈ R+ .

The collection of all asymptotically almost automorphic functions is denoted by AAA(X ). Like in the asymptotically almost periodic case, the functions h and ϕ are respectively called the principal and corrective terms of the function f . Definition 4.24. A continuous function f : R+ → X is said to be compact asymptotically almost automorphic if there exist h ∈ KAA(X ) and ϕ ∈ C0 (R+ , X ) such that f (t) = h(t) + ϕ (t),

t ∈ R+ .

122

4 Almost Automorphic Functions

The collection of all compact asymptotically almost automorphic functions is denoted by KAAA(X ). We have the inclusions KAA(X ) → KAAA(X ) → AAA(X ). We now show that the decomposition of asymptotically almost automorphic functions appearing in Definition 4.23 is unique. Lemma 4.25. If f ∈ AA(X ) such that lim f (t) = 0, then f = 0. t→∞

Proof. Since f ∈ AA(X ), for the sequence (n)n∈N there exists a subsequence (nk )k∈N ⊂ (n)n∈N such that lim f (t + nk ) = g(t)

(4.2)

lim g(t − nk ) = f (t)

(4.3)

k→∞

exists for each t ∈ R, and k→∞

for each t ∈ R. From lim f (t) = 0 and Eq. (4.2) it follows that g(t) = 0 for each t ∈ R. t→∞ Now using Eq. (4.3) it follows that 0 = lim g(t − nk ) = f (t) k→∞

and hence f = 0. Proposition 4.26. The decomposition of asymptotically almost automorphic functions is unique, that is, AAA(X ) = AA(X ) ⊕C0 (R+ , X ). Proof. If f = h1 + ϕ1 ∈ AAA(X ) where h ∈ AA(X ) and ϕ ∈ C0 (R+ , X ) and if f can also be represented by f = h2 + ϕ2 ∈ AAA(X ), where h ∈ AA(X ) and ϕ ∈ C0 (R+ , X ), then we obtain h1 − h2 = ϕ2 − ϕ1 . Thus h1 − h2 ∈ AA(X ) and lim h1 − h2 = 0.

t→∞

Using Lemma 4.25 it follows that h1 = h2 which yields ϕ1 = ϕ2 . Corollary 4.27. The decomposition of compact asymptotically almost automorphic functions is unique, that is, KAAA(X ) = KAA(X ) ⊕C0 (R+ , X ). Lemma 4.28 ([80]). If f ∈ AAA(X ), that is, f = h + ϕ where h ∈ AA(X ) and ϕ ∈ C0 (R+ , X ), then {h(t) : t ∈ R} ⊂ { f (t) : t ∈ R+ }.

4.3 Asymptotically Almost Automorphic Functions

123

Proof. Since h is almost periodic, then there exists a sequence (τn )n∈N with τn → ∞ such that g(t) = lim h(t + τn ) n→∞

(4.4)

is well defined for each t ∈ R and h(t) = lim g(t − τn ) n→∞

(4.5)

for each t ∈ R. Now for any fixed t0 , we have t0 + τn → ∞, which by Eq. (4.4) yields f (t0 + τn ) = h(t0 + τn ) + ϕ (t0 + τn ) → g(t0 ) as n → ∞. Consequently, g(t0 ) ∈ { f (t) : t ∈ R+ }, which yields {g(t) : t ∈ R} ⊂ { f (t) : t ∈ R+ }. Now, using Eqs. (4.4) and (4.5) it follows that {g(t) : t ∈ R} = {h(t) : t ∈ R}. Therefore, {h(t) : t ∈ R} ⊂ { f (t) : t ∈ R+ }. If f ∈ AAA(X ), that is, f = h + ϕ where h ∈ AA(X ) and ϕ ∈ C0 (R+ , X ), then we define f AAA := sup h(t) + sup ϕ (t) . t∈R

t∈R+

Clearly, · AAA is a norm on AAA(X ) and the following holds: Theorem 4.29. The space AAA(X ) equipped with the norm · AAA is a Banach space. Proof. The proof is similar to that of Theorem 3.46, which, for the sake of clarity, is reproduced here. It is enough to show that the norm defined by f = supt∈R+ f (t) for f ∈ AAA(X ) is equivalent to f AAP . Indeed, if f ∈ AAA(X ), that is, f = h+ ϕ where h ∈ AA(X ) and ϕ ∈ C0 (R+ , X ), then supt∈R h(t) ≤ f by using Lemma 4.28. Now f ≤ f AAA = sup h(t) + sup f (t) − h(t) t∈R

t∈R+

≤ 3 sup f (t) t∈R+

= 3 f .

124

4 Almost Automorphic Functions

Proposition 4.30. If ( fn )n∈N ⊂ AAA(X ) such that lim fn − f AAA = 0, then f ∈ n→∞

AAA(X ).

Proposition 4.31. If f , g ∈ AA(X ) and if there exists t0 ∈ R+ such that f (t) = g(t) for all t ≥ t0 , then f = g.

4.3.2 Composition Theorems Definition 4.32. A jointly continuous function F : R+ × X → X , (t, x) → F(t, x) is called asymptotically almost automorphic if t → F(t, x) is asymptotically almost automorphic, uniformly in x ∈ B, where B ⊂ X is an arbitrary bounded subset. In other words, F = H + Φ, where H ∈ AA(R × X ) and Φ ∈ C0 (R+ × X ). The collection of all such asymptotically almost automorphic functions will be denoted by AAA(R+ × X ). Definition 4.33. A jointly continuous function F : R+ × X → X , (t, x) → F(t, x) is called compact asymptotically almost automorphic if t → F(t, x) is compact asymptotically almost automorphic, uniformly in x ∈ B, where B ⊂ X is an arbitrary bounded subset. In other words, F = H + Φ, where H ∈ KAA(R×X ) and Φ ∈ C0 (R+ ×X ). The collection of all such compact asymptotically almost automorphic functions will be denoted by KAAA(R+ × X ). Theorem 4.34. Let F ∈ AAA(R+ × X ) and write F = H + Φ where H ∈ AA(R × X ) and Φ ∈ C0 (R+ × X ). Suppose that both H and Φ are Lipschitz in x ∈ X uniformly in t, i.e., there exists L1 , L2 > 0 such that H(t, x) − H(t, y) ≤ L1 x − y , for all x, y ∈ X , t ∈ R,

(4.6)

Φ (t, x) − Φ (t, y) ≤ L2 x − y , for all x, y ∈ X . t ∈ R+ .

(4.7)

and

If g : R+ → X is asymptotically almost automorphic, then the function Γ (t) = F(t, g(t)) : R+ → X is also asymptotically almost automorphic. Proof. Let F = H + Φ where H ∈ AA(R × X ) and Φ ∈ C0 (R+ × X ) and let g = h + ϕ where h ∈ AA(X ) and ϕ ∈ C0 (R+ , X ).

4.3 Asymptotically Almost Automorphic Functions

125

Now F(t, g(t)) = H(t, h(t)) + [F(t, g(t)) − H(t, h(t))] = H(t, h(t)) + [H(t, g(t)) − H(t, h(t)) + Φ (t, g(t))]. Clearly, t → Φ (t, g(t)) belongs to C0 (R+ , X ). Also, the fact t → H(t, g(t)) − H(t, h(t)) belongs to C0 (R+ , X ) is clear (see the proof of Theorem 3.49). Therefore, the function t → F(t, g(t)) is asymptotically almost automorphic. Theorem 4.35 ([132]). Suppose that F = G + Φ is an asymptotically almost automorphic function where G ∈ AA(R × X ) and Φ ∈ C0 (R+ × X ), and x → G(t, x) is uniformly continuous on any bounded subset B ⊂ X uniformly in t ∈ R+ . If g : R+ → X is asymptotically almost automorphic, then the function Γ (·) = F(·, g(·)) : R+ → X is asymptotically almost automorphic. Proof. Write g = α + β , where α ∈ AA(X ) and β ∈ C0 (R+ , X ). Now F(t, g(t)) = G(t, α (t)) + F(t, g(t)) − G(t, α (t)) = G(t, α (t)) + G(t, g(t)) − G(t, α (t)) + Φ (t, g(t)). Using Theorem 4.17 it follows that t → G(t, α (t)) is almost automorphic. Also, using the facts that g is bounded and Φ ∈ C0 (R+ × X ) it follows that t → Φ (t, g(t)) belongs to C0 (R+ , X ). Since g(t), α (t) are bounded, there is a bounded set K ⊂ X such that g(t), α (t) ∈ K for all t ∈ R+ . Using the fact that β ∈ C0 (R+ , X ) it follows that for any fixed δ > 0, there exists A > 0 such g(t) − α (t) = β (t) < δ for |t| > A. Consequently, for a given ε > 0 G(t, g(t)) − G(t, α (t)) < ε for |t| > A. Therefore, the function t → F(t, g(t)) is asymptotically almost automorphic. The proof is complete. Let K ⊂ X be a subset and I ⊆ R be an interval. Let CK (I × X ) denote the collection of functions f : I × X → X such that f (t, ·) is uniformly continuous on K for every t ∈ I. Similarly, Cc (R+ × X ) stands for the space of continuous functions f : R+ × X → X such that lim f (t, z) = 0

t→∞

uniformly for z ∈ C where C ⊂ X is any compact subset. Lemma 4.36 (Diagana et al. [48]). If u ∈ KAA(X ) and f ∈ KAA(R+ × X ) ∩ CR(u) (R × X ), then the function Φ : R → X defined by Φ (t) = f (t, u(t)) belongs to KAA(X ). Proof. Since u ∈ KAA(X ), then for every sequence of real numbers (σn )n∈N there exists a subsequence (sn )n∈N ⊂ (σn )n∈N such that

126

4 Almost Automorphic Functions

v(t) = lim u(t + sn ) and u(t) = lim v(t − sn ) n→∞

n→∞

uniformly on compact subsets of R. Similarly, g(t, z) = lim f (t + sn , z) and f (t, z) = lim g(t − sn , z) in X n→∞

n→∞

uniformly on compact subsets of R, for each z ∈ X . Let ε > 0 and K ⊂ R be an arbitrary compact. Since lim u(t + sn ) = v(t)

n→∞

uniformly for all t ∈ K and R(v) ⊂ R(u), there exists there exist N0 ∈ N such that f (s, u(s + sn )) − f (s, v(s)) ≤ ε , for all s ∈ K and every n ≥ N0 . On the other hand, using the fact that R(v) is relatively compact, for all δ > 0, there exist points xi ∈ X (i = 1, . . . , n) such that for each t ∈ R one can find i(t) ∈ {1, . . . , n} with v(t) − xi(t) X ≤ δ . Since f , g ∈ CR(u) (R × X ), we infer that f (s + sn , xi ) − f (s + sn , v(s)) ≤ ε , g(s, xi ) − g(s, v(s)) ≤ ε , for all s ∈ R. From f ∈ KAA(R+ × X ), there exist N1 ∈ N such that f (s + sn , xi ) − g(s, xi ) ≤ ε , for all s ∈ K and for all i = 1, . . . , n, whenever n ≥ N1 . In view of the above, for each s ∈ K, and n ≥ Nε = max{N0 , N1 }, f (t + sn , u(t + sn )) − g(t, v(t)) ≤ f (t + sn , u(t + sn )) − f (t + sn , v(t)) + f (t + sn , v(t)) − f (t + sn , xi(t) ) + f (t + sn , xi(t) ) − g(t, xi(t) ) + g(t, xi(t) ) − g(t, v(t)) ≤ 4ε ,

4.3 Asymptotically Almost Automorphic Functions

127

which proves that the convergence is uniform on K. Since K is arbitrary, it follows that the convergence is uniform on any compact subset of R. Using similar arguments as previously it follows that the sequence of functions t → g(t − sn , u(t − sn )) − f (t, u(t)) converges uniformly to 0 on compact sets of R. The proof is complete. Lemma 4.37 is a straightforward consequence of [80, Lemma 2.4] and hence its proof is omitted. Lemma 4.37. Let K ⊂ X be a compact subset and f ∈ KAA(R+ ×X )∩CK (R+ × X ), then f ∈ CK (R × X ). Lemma 4.38 (Diagana et al. [48]). Let u ∈ KAAA(X ) and f ∈ KAAA(R×X )∩ CR (R × X ), where R = {u(t) : t ∈ R}. Then the function Φ : R → X defined by Φ (t) = f (t, u(t)) belongs to KAAA(X ). Proof. Suppose that the functions f and u have the following decompositions: f = k + h and u = y + z, where k ∈ KAA(R+ × X ), h ∈ Cc (R+ × X ), y ∈ KAA(X ) and z ∈ Cc (R+ × X ). Now f (t, u(t)) = f (t, u(t)) − f (t, y(t)) + k(t, y(t)) + h(t, y(t)) = q1 (t) + q2 (t) + q3 (t), where q1 (t) = f (t, u(t)) − f (t, y(t)), q2 (t) = k(t, y(t)), q3 (t) = h(t, y(t)). Since R = {u(t) : t ∈ R} is compact and y(t) ∈ R for t ∈ R, using lim [u(t) − y(t)] = 0

t→∞

and f ∈ CR (R × X ), we obtain lim q1 (t) = lim [ f (t, u(t)) − f (t, y(t))] = 0.

t→∞

t→∞

From the definition of Cc (R+ × X ), it is easy to see that h ∈ CR (R+ × X ) and hence k ∈ CR (R+ × X ). Let R1 = {y(t) : t ∈ R}, then R1 ⊆ R is compact and k ∈ CR1 (R+ × X ). Now, from Lemma 4.37 we obtain that k ∈ CR1 (R × X ) and by Lemma 4.36 it follows that q2 (·) ∈ KAA(X ). From h ∈ Cc (R+ ×X ) it follows that lim q3 (t) = 0. In view of above, f (·, u(·)) ∈ KAAA(X ). The proof is complete.

t→∞

128

4 Almost Automorphic Functions

4.4 Almost Automorphic Sequences 4.4.1 Basic Definitions In this section we study almost automorphic sequences which were introduced in the literature several decades ago, see, e.g., the work of Veech [156]. These sequences generalize in a natural fashion almost periodic sequences. The results presented in this section are based on a recent work by Diagana [60]. Definition 4.39. An X -valued sequence x = {x(t)}t∈Z is called almost automorphic if for every sequence {h (n)}n∈N ⊂ Z there exists a subsequence {h(n)}n∈N ⊂ Z such that lim x(t + h(n)) = y(t)

n→∞

is well defined for each t ∈ Z, and lim y(t − h(n)) = x(t)

n→∞

for each t ∈ Z. The collection of all almost automorphic X -valued sequences on Z will be denoted by AA(Z). It is a Banach space when it is equipped with the sup-norm. Example 4.40 (Veech [156]). Let θ be an irrational real number. Note that for each t ∈ Z, we have cos 2π t θ = 0. Consider the sequence given by ⎧ if cos 2π t θ > 0, ⎨1 z(t) = sgn cos 2π t θ = ⎩ −1 if cos 2π t θ < 0. This is an example of an almost automorphic sequence, which is not almost periodic. Definition 4.41. A sequence of functions F : Z × X → X , (t, u) → F(t, u) is called almost automorphic if for every sequence {h (n)}n∈N ⊂ Z there exists a subsequence {h(n)}n∈N ⊂ Z such that lim F(t + h(n), x) = G(t, x)

n→∞

is well defined for each t ∈ Z and lim G(t − h(n), x) = F(t, x)

n→∞

for each t ∈ Z and x ∈ B where B ⊂ X is an arbitrary bounded subset.

4.4 Almost Automorphic Sequences

129

4.4.2 Composition Theorems Theorem 4.42. Suppose that f : Z × X → X , (t, u) → f (t, u) is almost automorphic in t ∈ Z uniformly in u ∈ B where B ⊂ X is an arbitrary bounded subset. If in addition, f is Lipschitz in x ∈ X uniformly in t ∈ Z, that is, there exists, L > 0 such that f (t, u) − f (t, v) ≤ L u − v

∀u, v ∈ X , t ∈ Z,

then for every X -valued almost automorphic sequence x = {x(t)}t∈Z , the X valued sequence F(t) = f (t, x(t)) is almost automorphic. Proof. The proof can be done as in the continuous case but for the sake of clarity we reproduce it here. Let (sn )n∈N ⊂ Z be an arbitrary sequence. By definition of an almost automorphic sequence, we can extract a subsequence (τn )n∈N ⊂ Z such that (i) lim f (t + τn , u) = g(t, u), for each t ∈ Z and u ∈ B; n→∞

(ii) lim g(t − τn , u) = f (t, u), for each t ∈ Z and u ∈ B, where B ⊂ X is an arbitrary n→∞ bounded subset; (iii) lim x(t + τn ) = y(t), for each t ∈ R; and n→∞

(iv) lim y(t − τn ) = x(t), for each t ∈ Z. n→∞

Letting G(t) := g(t, y(t)), we need to show that for each t ∈ Z, lim f (t + τn ) = G(t),

n→∞

and lim G(t − τn ) = F(t).

n→∞

Now F(t + τn ) − G(t) ≤ f (t + τn , x(t + τn )) − f (t + τn , y(t)) + f (t + τn , y(t)) − g(t, y(t)) . ≤ L x(t + τn ) − y(t) + f (t + τn , y(t)) − g(t, y(t)) . From identity (iii) it follows that lim x(t + τn ) − y(t) = 0.

n→∞

130

4 Almost Automorphic Functions

Similarly, from identity (i) we obtain lim f (t + τn , y(t)) − g(t, y(t)) = 0

n→∞

and hence lim F(t + τn ) = G(t).

n→∞

Using similar arguments as above we obtain that lim G(t − τn ) = F(t),

n→∞

which yields F is almost automorphic. Theorem 4.43. Let f : Z × X → X be an almost automoprhic sequence. Suppose u → f (t, u) is uniformly continuous on every bounded subset K ⊂ X uniformly for t ∈ Z (i.e., for each ε > 0, there exists δ > 0 such that x, y ∈ K and x − y < δ yields f (t, x) − f (t, y) < ε for all t ∈ Z). If x ∈ AA(Z), then F : Z → X defined by F(·) := f (·, x(·)) belongs to AA(Z). Proof. The proof can be done as in the continuous case but for the sake of clarity we reproduce it here. Let (sn )n∈N be an arbitrary sequence of real numbers. Clearly, we can extract a subsequence (τn )n∈N such that (i) lim f (t + τn , u) = g(t, u), for each t ∈ Z and x ∈ B; n→∞

(ii) lim g(t − τn , u) = f (t, u), for each t ∈ Z and x ∈ B, where B ⊂ X is an arbitrary n→∞ bounded subset; (iii) lim x(t + τn ) = y(t), for each t ∈ R; and n→∞

(iv) lim y(t − τn ) = x(t), for each t ∈ R. n→∞

Letting G(t) := g(t, y(t)), one needs to show that for each t ∈ Z, lim F(t + τn ) = G(t),

n→∞

and lim G(t − τn ) = F(t).

n→∞

Now |F(t + τn ) − G(t) ≤ f (t + τn , x(t + τn )) − f (t + τn , y(t)) + f (t + τn , y(t)) − g(t, y(t)) .

4.4 Almost Automorphic Sequences

131

Since x is almost automorphic, both the sequences x and y are bounded. Thus we can choose a bounded subset K of X such that x(t) ∈ K and y(t) ∈ K for all t ∈ Z. By identity (iii) and the uniform continuity of x → f (t, x) in x ∈ K, we obtain that lim f (t + τn , x(t + τn )) − f (t + τn , y(t)) = 0.

n→∞

Similarly, by identity (i) we obtain lim f (t + τn , y(t + τn )) − g(t + τn , y(t)) = 0

n→∞

and hence lim F(t + τn ) = G(t).

n→∞

Using similar arguments as above we obtain that lim G(t − τn ) = F(t),

n→∞

which yields F is almost automorphic.

4.4.3 Asymptotically Almost Automorphic Sequences In this subsection we introduce a new theory for asymptotically almost automorphic sequences on Z+ which is a natural generalization of the concept of asymptotic almost periodicity for sequences. Definition 4.44. An X -valued sequence x = {x(t)}t∈Z+ is said to be asymptotically almost automorphic if it can be decomposed as x(t) = u(t) + v(t),

t ∈ Z+ ,

where u = {u(t)}t∈Z ∈ AA(Z) and {v(t)}t∈Z+ ∈ N(Z+ ) with  N(Z+ ) = all bounded X − valued sequences y = (y(t))t∈Z+ : lim y(t) = 0 . t→∞

The collection of all asymptotically almost automorphic X -valued sequences will be denoted by AAA(Z). Lemma 4.45. If x = {x(t)}t∈Z ∈ AA(Z) and lim x(t) = 0, then x(t) = 0 for all t→∞ t ∈ Z. Proof. For the sequence (n)n∈N there exists a subsequence {n(k)}k∈N such that

132

4 Almost Automorphic Functions

lim x(t + n(k)) = y(t)

k→∞

is well defined for each t ∈ Z and lim y(t − n(k)) = x(t)

k→∞

for each t ∈ Z. From lim x(t) = 0, it follows that y(t) = 0 and hence x(t) = 0 for all t ∈ Z. t→∞

Lemma 4.46. The decomposition of an asymptotically almost automorphic sequence is unique. That is, AA(Z)∩ N(Z+ ) = {0}. Proof. Suppose that x = {x(t)}t∈Z+ ∈ AAA(Z) can be decomposed as x(t) = u(t) + v(t) and x(t) = u1 (t) + v1 (t), where u = {u(t)}t∈Z , ν = {ν (t)}t∈Z ∈ AA(Z) and {v(t)}t∈Z+ , {v1 (t)}t∈Z+ ∈ N(Z+ ). Clearly, u(t) − u1 (t) = v1 (t) − v(t) ∈ AA(Z)∩ N(Z+ ). By Lemma 4.45, u(t) = u1 (t) for all t ∈ Z and v(t) = v1 (t) for all t ∈ Z+ .

4.5 S p -Almost Automorphic Functions This section is devoted to S p -almost automorphic functions (or Stepanov-like almost automorphic functions). These functions, which generalize almost automorphic functions, were introduced in the literature by N’Gu´er´ekata and Pankov [146]. This section is entirely based on the study of basic properties of those functions. p Definition 4.47 ([146]). The space Saa (X ) of S p -almost automorphic (or Stepanov-like almost automorphic) functions consists of all f ∈ BS p (X ) such that   f b ∈ AA L p ((0, 1), X ) . p (R, X ) is said to be S p -almost automorphic if its In other words, a function f ∈ Lloc b p Bochner transform f : R → L ((0, 1), X ) is almost automorphic in the sense that for every sequence of real numbers (s n )n∈N , there exists a subsequence (sn )n∈N and p a function g ∈ Lloc (R, X ) such that



t+1

t

 t

as n → ∞ pointwise on R.

1/p f (sn + s) − g(s) p ds

t+1

→ 0, and 1/p

g(s − sn ) − f (s) ds p

→0

4.5 S p -Almost Automorphic Functions

133

It is clear that if 1 ≤ p < q < ∞ and f ∈ Lq (R, X ) is Sq -almost automorphic, then f is S p -almost automorphic. Also if f ∈ AA(X ), then f is S p -almost automorphic for any 1 ≤ p < ∞. It is easily seen that f ∈ KAA(X ) if and only if f b ∈ ∞ (X ). AA(L∞ ((0, 1), X )). Thus, KAA(X ) can be considered as Saa Example 4.48 ([146]). Let x = (xn )n∈Z ∈ l ∞ (X ) be an almost automorphic sequence and let ε0 ∈ (0, 12 ). Consider the function defined by

f (t) =

⎧ ⎨ xn ⎩

0

if t ∈ (n − ε0 , n + ε0 ) if t ∈ R\(n − ε0 , n + ε0 )

p (X ) for p ≥ 1 and that f ∈ AA(X ) as f is It can be shown that f ∈ Saa discontinuous.

Theorem 4.49 ([146]). The following statements are equivalent: p (a) f ∈ Saa (X ). (b) f b ∈ KAA(L p ((0, 1), X )). (c) For every sequence (s n )n∈N of real numbers there exists a subsequence (sn )n∈N such that

g(t) := lim f (t + sn )

(4.8)

n→∞

p (R, X ) and exists in the space Lloc

f (t) = lim g(t − sn )

(4.9)

n→∞

p (R, X ). in the sense of Lloc

Definition 4.50. A function F : R × X → X , (t, u) → F(t, u) with F(·, u) ∈ L p (R, X ) for each u ∈ B where B ⊂ X is an arbitrary bounded subset, is said to be S p -almost automorphic in t ∈ R uniformly in u ∈ B if t → F(t, u) is S p -almost automorphic for each u ∈ B. Namely, for every sequence of real numbers (s n )n∈N , p there exists a subsequence (sn )n∈N and a function G(·, u) ∈ Lloc (R, X ) such that



1 0

F(t + sn + s, u) − G(t + s, u) ds p



1 0

 1p

G(t − sn + s, u) − F(t + s, u) ds

as n → ∞ pointwise on R for each u ∈ B.

p

→ 0, and

 1p

→0

134

4 Almost Automorphic Functions

The collection of those S p -almost automorphic functions F : R × X → X will be p denoted by Saa (R × X ). p For each compact subset K ⊂ X , we let Saa,K (R × X ) be the collection of all p functions F ∈ Saa (R × X ) such that for every sequence of real numbers (s n )n∈N , p there exists a subsequence (sn )n∈N and a function G(·, u) ∈ Lloc (R, X ) such that  1  lim

n→∞

0

 p  1p sup F(t + sn + s, u) − G(t + s, u) ds = 0, and

u∈K

 1  lim

n→∞

0

 p  1p sup G(t − sn + s, u) − F(t + s, u) ds = 0

u∈K

for each t ∈ R. Obviously p p Saa,K (R × X ) ⊂ Saa (R × X ) ⊂ AA(R × X ).

4.5.1 Composition Theorems The next composition theorem, which is very useful for applications, is due to Fan et al. [95]. p Theorem 4.51 (Fan et al. [95]). Let f ∈ Saa (R × X ) such that there exists a p constant L > 0 such that for all x, y ∈ Lloc (R, X )

& 0

1

f (t+s, x(s))− f (t+s, y(s)) p ds

'1/p

≤L

&

1 0

x(s)−y(s) p ds

'1/p (4.10)

for all t ∈ R. p p If x ∈ Saa (X ) and K = {x(t) : t ∈ R} is compact, then f (·, x(·)) ∈ Saa (X ). The proof of Theorem 4.51 requires Lemma 4.52. p Lemma 4.52 (Fan et al. [95]). If f ∈ Saa (R×X ) satisfies Eq. (4.10), then for each p compact subset K ⊂ X , the function f belongs to Saa,K (R × X ).

Proof. By assumption, for every sequence of real numbers (s n ), one can find a p subsequence (sn )n∈N and a function g : R × X → X with g(·, u) ∈ Lloc (R, X ) such that lim f (t +sn +·, u)−g(t +·, u) p = lim g(t −sn +·, u)− f (t +·, u) p = 0 (4.11)

n→∞

n→∞

for each t ∈ R and each u ∈ X . For all ε > 0, u, v ∈ X , one can find an N ∈ N such that

4.5 S p -Almost Automorphic Functions

135

f (t + sn + ·, u) − g(t + ·, u) p < ε /2

∀ n > N,

(4.12)

f (t + sn + ·, v) − g(t + ·, v) p < ε /2

∀ n > N.

(4.13)

Using Eqs. (4.10), (4.12), and (4.13), it follows that g(t + ·, u) − g(t + ·, v) p ≤ g(t + ·, u) − f (t + sn + ·, u) p + f (t + sn + ·, u) − f (t + sn + ·, v) p + f (t + sn + ·, v) − g(t + ·, v) p

(4.14)

≤L u − v + ε . whenever n > N. Since ε is arbitrary, then g(t + ·, u) − g(t + ·, v) p ≤ L u − v .

(4.15)

Using the fact K ⊂ X is a compact subset, for each ε > 0, one can find x1 , x2 , . . . , xk ∈ K such that K⊆

k

B(xi , ε ).

i=1

Using Eq. (4.11), for ε > 0 above, there exists integer N such that f (t + sn + ·, xi ) − g(t + ·, xi ) p < ε

(4.16)

for all n > N and i = 1, 2, . . . , k. If u ∈ K, one can find x j ∈ {x1 , x2 , . . . , xk } such that u − x j ≤ ε .

(4.17)

Using Eqs. (4.15)–(4.17) and (4.10), it follows that f (t + sn + ·, u) − g(t + ·, u) p ≤ f (t + sn + ·, u) − f (t + sn + ·, x j ) p + f (t + sn + ·, x j ) − g(t + ·, x j ) p + g(t + ·, x j ) − g(t + ·, u) p ≤2L u − x j + ε ≤(2L + 1)ε , whenever n > N , which yields,

136

4 Almost Automorphic Functions

lim f (t + sn + ·, u) − g(t + ·, u) p = 0

n→∞

uniformly in u ∈ K. Similarly, it can be shown that lim g(t − sn + ·, u) − f (t + ·, u) p = 0

n→∞

(4.18)

uniformly in u ∈ K. p (R × X ). Therefore, f ∈ Saa,K Proof (Theorem 4.51). Our first task consists of showing that t → f (t, x(t)) ∈ p Lloc (R, X ). For that, choose an arbitrary compact subset E ⊂ R. Clearly, there exists T ∈ N such that E ⊆ [−T, T ]. Using Eq. (4.10) it follows that for every ε ∈ (0, 1), one can find a δ > 0 such that  E

f (s, u) − f (s, v) p ds ≤

2T −1  −T +i+1

∑

i=0

−T +i

f (s, u) − f (s, v) p ds ≤ ε p

(4.19)

for all u, v ∈ X satisfying u − v ≤ δ . Using the fact that K = {x(t) : t ∈ R} is compact, one can find a finite number of open balls Ok (k = 1, 2, . . . , m) centered at xk ∈ K with radius δ given above and such that {x(t) : t ∈ R} ⊂

m

Ok .

k=1

Define Bk such that Bk = {s ∈ R : x(s) ∈ Ok },

R=

m

Bk ,

k=1

and set E1 = B1 ,

Ek = Bk \

k−1

B j (2 ≤ k ≤ m).

j=1

Now Ei ∩ E j = 0, / for i = j, 1 ≤ i, j ≤ m. Consider the step function x : R → X defined by x(s) = xk ,

s ∈ Ek , k = 1, 2, . . . , m.

4.5 S p -Almost Automorphic Functions

137

One can easily see that x(s) − x(s) ≤ δ for all s ∈ R. Thus using Eq. (4.19), one obtains 

1/p f (s, x(s)) ds p

E



≤

1/p f (s, x(s)) − f (s, x(s)) ds

+

p

E

 ≤ε +

m

∑



k=1 E∩Ek



1/p f (s, x(s)) ds p

E

1/p f (s, xk ) ds p

.

p (R, X ), for every k = 1, 2, . . . , m, In view of the above and the fact that f (·, xk ) ∈ Lloc p it follows that t → f (t, x(t)) ∈ Lloc (R, X ). Using our assumptions and Lemma 4.52, for every sequence of real numbers (s n )n ∈N , there exist a subsequence (sn )n∈N and a function g : R × X → X with p g(·, u) ∈ Lloc (R, X ) such that

lim f (t +sn +·, u)−g(t +·, u) p = lim g(t −sn +·, u)− f (t +·, u) p = 0 (4.20)

n→∞

n→∞

p (X ) and Lemma 4.49, for each t ∈ R, uniformly in u ∈ K. Further, since x ∈ Saa b p it follows that x ∈ KAA(L ((0, 1), X )), which yields the existence of a function y : R → X such that

lim x(t + sn + ·) − y(t + ·) p = lim y(t − sn + ·) − x(t + ·) p = 0

n→∞

n→∞

uniformly in t on any compact interval. Using Minkowski’s inequality, it follows that f (t + sn + ·, x(t + sn + ·)) − g(t + ·, y(t + ·)) p ≤ f (t + sn + ·, x(t + sn + ·)) − f (t + sn + ·, y(t + ·)) p + f (t + sn + ·, y(t + ·)) − g(t + ·, y(t + ·)) p :=In + Jn . Again, using Eqs. (4.10) and (4.21), it follows that lim In = 0

n→∞

for each t ∈ R.

(4.21)

138

4 Almost Automorphic Functions

Now, using Eq. (4.21) and the fact that K = {x(t) : t ∈ R}, it follows that y(t + s) ∈ K for almost every s ∈ [0, 1]. Using Eq. (4.20) and the compactness of K, it follows that for each t ∈ R, lim Jn = lim f (t + sn + ·, y(t + ·)) − g(t + ·, y(t + ·)) p = 0,

n→∞

n→∞

and hence lim f (t + sn + ·, x(t + sn + ·)) − g(t + ·, y(t + ·)) p = 0

n→∞

for each t ∈ R. Using similar arguments as above, it can be shown that lim g(t − sn + ·, y(t − sn + ·)) − f (t + ·, x(t + ·)) p = 0

n→∞

for each t ∈ R. This completes the proof. p Theorem 4.53 ([85]). Let F ∈ Saa (R × X ) for p > 1 and suppose that F satisfies: p there exists L ∈ BSr (R) with r ≥ max(p, p−1 ) such that

F(t, x) − F(t, y) ≤ L(t) x − y for all x, y ∈ X and t ∈ R. p If g ∈ Saa (X ) and if K = {g(t) : t ∈ R} ⊂ X is compact, then there exists q ∈ [1, p) such that the function Γ : R → X defined by Γ (t) = F(t, g(t)) belongs to q Saa (X ).

(n)

4.6 S p -Almost Automorphic Functions Stepanov-like C(n) -almost automorphic functions (or Stepanov-like n-differentiable almost automorphic functions), which are natural generalizations of C(n) -almost automorphic functions, were recently introduced in the literature by Diagana et al. [73]. This section is entirely devoted to the study of basic properties of these new functions. (n),p

Definition 4.54 (Diagana et al. [73]). The space Saa (X ) of Stepanov-like C(n) (n) almost automorphic functions (or S p -almost automorphic) consists of all f ∈   (n) BS p (X ) such that ( f (k) )b ∈ AA L p ((0, 1), X ) for k = 0, 1, . . . , n. (n)

(n)

In other words, a function f ∈ BS p (X ) is said to be S p -almost automorphic if ( f (k) )b is almost automorphic for k = 0, 1, . . . , n in the sense that for every sequence of real numbers (s m )m∈N , there exists a subsequence (sm )m∈N and a function g ∈ (n) BS p (X ) such that

(n)

4.6 S p -Almost Automorphic Functions



t+1

t

 t

139

f (k) (sm + s) − g(k) (s) p ds

t+1

(k)

g (s − sm ) − f

(k)

1/p → 0, and 1/p

(s) ds p

→0

for k = 0, 1, . . . , n, as m → ∞ pointwise on R. Definition 4.55. A function F : R × X → X , (t, u) → F(t, u) with F(·, u) ∈ (n),p (n) Saa (X ) for each u ∈ X , is said to be S p -almost automorphic in t ∈ R uniformly (n) in u ∈ B where B ⊂ X is an arbitrary bounded subset. The collection of those S p (n),p almost automorphic functions F : R × X → X will be denoted by Saa (R × X ). (n),p

Theorem 4.56 ([73]). If f ∈ Saa (X ) and if g ∈ L1 (R), then their convolution (n),p f ∗ g ∈ Saa (X ). (n),p

Proof. Let f ∈ Saa (X ) and let g ∈ L1 (R). To complete the proof we have to show that   [( f ∗ g)(k) ]b ∈ AA L p (0, 1; X ) for all k = 0, 1, 2, . . . , n. Indeed, using the fact [( f ∗ g)(k) ]b = [ f (k) ∗ g]b = ( f (k) )b ∗ g   it follows that [( f ∗ g)(k) ]b ∈ AA L p ((0, 1), X ) for all k = 0, 1, 2, . . . , n. (n),p

Proposition 4.57 ([73]). The space Saa (X ) equipped with the norm · p,(n) is a Banach space. (n),p

Proof. The proof is based on the fact that Saa (X ) is a closed subspace of (n) BS p (X ). (n),p

Proposition 4.58 ([73]). If f ∈ AA(n) (X ), then f ∈ Saa (X ). That is, (n),p AA(n) (X ) ⊂ Saa (X ). (n)

Proof. Clearly, f (k) ∈ BS p (X ) for k = 0, 1, . . . , n and hence f ∈ BS p (X ). To complete the proof we have to show that ( f (k) )b ∈ AA(L p (((0, 1), X )) for k = 0, 1, 2, . . . , n. Using the fact that AA(X ) ⊂ AS p (X ) for p ∈ [1, ∞) (see [146]) it follows that ( f (k) )b ∈ AA(L p ((0, 1), X )) for k = 0, 1, 2, . . . , n. Example 4.59. This example was constructed by Diagana et al. [73] . Indeed, let p = 1 and let ε ∈ AA(3) (R) such that inf ε (t) = δ0 > 0.

t∈R

140

4 Almost Automorphic Functions (2),1

(3),1

We give an example of a function f ∈ Saa (R) such that f ∈ Saa (R). Indeed, consider the function f defined by f (t) =

∞

ε (kt) . 4 k=1 k

∑

(2),1

(3),1

We claim that f ∈ Saa (R) while f ∈ Saa (R). Indeed, f (t) =

∞

∞ ε (kt) ε (kt) and f (t) = ∑ 3 2 k=1 k k=1 k

∑

for all t ∈ R. (2) Clearly, f ∈ BS1 (R) as it can be easily shown that f , f , f ∈ BS1 (R). Moreover, (2),1 f , f , f ∈ AA(R). Consequently, f ∈ Saa (R). Now f (t) =

∞

ε (kt) ∑ k . k=1 (2),1

(3),1

Clearly, f ∈ BS1 (R). Therefore, f ∈ Saa (R) while f ∈ Saa (R).

Bibliographical Notes The results presented in this chapter are either classical ones or else cannot be easily found in the literature. The main references include, but are not limited to, the books by N’Gu´er´ekata [142, 143], and the following papers [18, 22, 23, 71, 73, 80, 81, 83– 85, 87, 91, 95, 132, 144–146, 156–158, 170, 171] etc. There are certainly several unanswered questions on almost automorphic sequences. Among other things it would be interesting to see how the concepts of (n) S p -almost automorphy and S p -almost automorphy look like in the discrete setting.

Chapter 5

Pseudo-Almost Periodic Functions

The concept of pseudo-almost periodicity, which is a natural generalization of the notion of almost periodicity, was introduced in the literature more than a decade ago by C. Zhang [172–174]. Since its introduction in the literature, the notion of pseudo-almost periodicity has generated several developments and extensions, see, e.g., [63,64,66–69]. Among other things, it has been utilized to study the qualitative behavior to various differential and partial differential equations involving pseudoalmost periodic coefficients, see, e.g., Diagana et al. [53, 54, 57, 59, 62–69, 71, 73, 74], Cuevas et al. [43–45], Ding et al. [79, 81, 82], Zhang [172–174], Ji and Zhang [121], Liang et al. [129], Fan et al. [95, 96], Agarwal et al. [6, 7], Ait et al. [9, 10], Pinto [150], Al-Islam et al. [11], Amir and Maniar [12], Cuevas et al. [8, 43, 44], Bugajewski et al. [33], Boukli-Hacenea and Ezzinbi [27, 28], Ezzinbi et al. [88, 89], Li et al. [128], Liang et al.[130], and Zhang and Li [175–177]. Our main task in this chapter consists of studying basic properties of pseudoalmost periodic functions including some composition theorems as well as some of their extensions that have recently been introduced in the literature. In particular, we will introduce and study the notions of weighted pseudo-almost periodicity due to Diagana [64], C(n) -pseudo-almost periodicity due to Diagana and Nelson [71], and that of Stepanov-like pseudo-almost periodicity due to Diagana [54].

5.1 Pseudo-Almost Periodic Functions 5.1.1 Basic Definitions Definition 5.1 (Zhang [172]). A continuous function f : R → X is called pseudoalmost periodic if it can be written as f = h + ϕ,

T. Diagana, Almost Automorphic Type and Almost Periodic Type Functions in Abstract Spaces, DOI 10.1007/978-3-319-00849-3 5, © Springer International Publishing Switzerland 2013

141

142

5 Pseudo-Almost Periodic Functions

with h ∈ AP(X ) and ϕ ∈ PAP0 (X ) where the space PAP0 (X ) is defined by   1 r PAP0 (X ) = g ∈ BC(R, X ) : lim g(t) dt = 0 . r→∞ 2r −r

The functions h and ϕ are respectively called the almost periodic and the ergodic components of f . The collection of all pseudo-almost periodic functions which go from R to X will be denoted by PAP(X ). Example 5.2. Consider the function defined by √ f (t) = sint + sin 2t + (1 + t 2 )−1 for all t ∈ R. It can be easily checked√that the function f is pseudo-almost periodic. Indeed, the function t → sint + sin 2t belongs to AP(R) while the function t → (1 + t 2 )−1 is in PAP0 (R). For a fixed pseudo-almost periodic function, one way of generating new pseudoalmost periodic functions is through the convolution of that function with any L1 function. Proposition 5.3. Fix f ∈ PAP(X ). If g ∈ L1 (R), then f ∗ g, the convolution of f with g on R, belongs to PAP(X ). Proof. Let f = h + ϕ where h ∈ AP(X ) and ϕ ∈ PAP0 (X ). Clearly f ∗ g = h ∗ g + ϕ ∗ g. It is enough to show that ϕ ∗ g ∈ PAP0 (X ) as we have already shown that h ∗ g ∈ AP(X ) (see Proposition 3.4). Now from ϕ ∈ PAP0 (X ) and g ∈ L1 (R) it is clear that ϕ ∗ g ∈ BC(R, X ). By assumption, 1 r→∞ 2r

 r

lim

−r

ϕ (t) dt = 0.

Now setting 1 L(r) := 2r

 r  +∞ −r −∞

ϕ (t − s) |g(s)| ds dt

it follows that 1 2r

 r −r

(ϕ ∗ g)(t) dt ≤ L(r) =

1 2r

 r  +∞ −r −∞

ϕ (t − s) |g(s)| ds dt

5.1 Pseudo-Almost Periodic Functions

143

  1 r = |g(s)| ϕ (t − s) dt ds 2r −r −∞

 r−s   +∞ 1 |g(s)| ϕ (r) dr ds = 2r −r−s −∞  +∞

=

 +∞ −∞

|g(s)| φr (s) ds,

where

ϕr (s) =

1 2r

 r−s −r−s

ϕ (σ ) d σ .

Obviously ϕr (s) → 0 as r → ∞. Using the fact that ϕr is bounded, g ∈ L1 (R), and the Lebesgue Dominated Convergence Theorem, it follows that lim L(r) = 0,

r→∞

and hence ϕ ∗ g ∈ PAP0 (X ). n+7

Example 5.4. Let f (t) = Pn (t) + t| sin π t|t where for n ∈ N and Pn is a trigonometric polynomial. Clearly, f ∈ PAP(R). And thus for any g ∈ L1 (R), the function defined by Qn (t) :=

 +∞ & −∞

Pn (t − s) + (t − s)| sin π (t − s)|(t−s)

n+7

' g(s)ds

for all t ∈ R, is a sequence of pseudo-almost periodic functions. In order to prove the next lemma we make use of properties of the mean value of an almost periodic function. Our proof is new and differs from the one given by Zhang [172]. Lemma 5.5. The decomposition of a pseudo-almost periodic appearing in Definition 5.1 is unique, that is, PAP(X ) = AP(X ) ⊕ PAP0 (X ). Proof. Let f ∈ AP(X ) ∩ PAP0 (X ). First of all, let us mention that the function g defined by g(t) := f (t) ≥ 0 for all t ∈ R, belongs to AP(R) ∩ PAP0 (R). Since M (g) = 0 it follows from Proposition 3.23 that g(t) = 0 for all t ∈ R, which yields f = 0. Now if f˜ = h0 + ϕ0 and f˜ = h1 + ϕ1 where h0 , h1 ∈ AP(X ) and ϕ0 , ϕ1 ∈ PAP0 (X ), then h0 − h1 = ϕ1 − ϕ0 . In view of the above, h0 − h1 = ϕ1 − ϕ0 = 0, that is, h0 = h1 and ϕ1 = ϕ0 . The proof is complete.

144

5 Pseudo-Almost Periodic Functions

Proposition 5.6 ([172]). If f = g + ϕ ∈ PAP(X ), then g(R) ⊂ f (R) and f ∞ ≥ g ∞ ≥ inf g(t) ≥ inf f (t) . t∈R

t∈R

Proof. Clearly, if we suppose that g(R) ⊆ f (R), then there exists a t0 such that inf g(t0 ) − f (s) > ε .

s∈R

Using the continuity of the function g, there exists δ > 0 such that |t| < δ yields inf Rt0 (t) − Rt0 (s) > ε .

s∈R

Using the fact that AP(X ) is translation-invariant it follows that Rt0 g ∈ AP(X ). Consequently, for every ε > 0 there exists lε /2 > 0 such that every interval of length lε /2 contains a τ with the property that sup Rτ Rt0 g(t) − Rt0 g(t) ≤ ε /2. t∈R

Now if t ∈ (−δ , δ ), then t + τ ∈ (τ − δ , τ + δ ). Now Rt0 ϕ (t + τ ) = Rt0 f (t + τ ) − Rt0 g(t + τ ) ≥ Rt0 f (t + τ ) − Rt0 g(t) − Rt0 g(t) − Rt0 g(t + τ ) ≥ inf Rt0 f (s) − Rt0 g(t) − Rt0 g(t) − Rτ Rt0 g(t) s∈R

> ε /2, which yields M ( Rt0 ϕ ) ≥ δ ε /lε and this a contradiction. We have Proposition 5.7. The space PAP(X ) is translation-invariant. Proof. The proof is not difficult and hence is left for the reader as an exercise. If f , g ∈ PAP(X ) and λ ∈ F, we can then show that f + g and λ f belong to PAP(X ). If g is F-valued for instance, it can be easily shown that f g ∈ PAP(X ). It is also clear that (PAP(X ), · ∞ ) is a normed space. In the next theorem, we show that the normed vector space (PAP(X ), · ∞ ) is in fact complete. Lemma 5.8. If ( fn )n∈N ∈ PAP(X ) such that fn − f ∞ → 0 as n → ∞, then f ∈ PAP(X ).

5.1 Pseudo-Almost Periodic Functions

145

Proof. Let ( fn )n∈N ∈ PAP(X ) such that fn − f ∞ → 0 as n → ∞. Let us show that f ∈ PAP(X ). Since ( fn )n∈N ∈ PAP(X ) write fn = hn + ϕn where (hn )n∈N ∈ AP(X ) and (ϕn )n∈N ∈ PAP0 (X ). According to Proposition 5.6, we have hn ∞ ≤ fn ∞ , which yields that hn − hm ∞ ≤ fn − fm ∞ . Since fn − f ∞ → 0 as n → ∞ it follows that ( fn )n∈N is a Cauchy sequence and hence hn − hm ∞ → 0 as n, m → ∞, too, that is, (hn )n∈N ∈ AP(X ) is a Cauchy sequence. Now since (AP(X ), · ∞ ) is a Banach space it follows that there exists h ∈ AP(X ) such that hn − h ∞ → 0 as n → ∞. Setting ϕ = f − h one can easily see that ϕn − ϕ ∞ → 0 as n → ∞, which yields ϕ ∈ BC(R, X ). Now 1 2r

 r

1 ϕ (t) dt = 2r −r ≤

1 2r

 r −r

 r

−r

ϕ (t) − ϕn (t) + ϕ (t) dt ϕ (t) − ϕn (t) dt +

≤ ϕ − ϕn ∞ +

1 2r

 r −r

1 2r

 r −r

ϕn (t) dt

ϕn (t) dt

Letting r → ∞ it follows that 1 r→∞ 2r

 r

lim

−r

ϕ (t) dt ≤ ϕ − ϕn ∞ .

And letting n → ∞ in the previous inequality it follows that 1 r→∞ 2r lim

 r −r

ϕ (t) dt = 0.

The proof is complete. Theorem 5.9. The space of pseudo-almost periodic functions PAP(X ) equipped with the sup-norm is a Banach space. Proof. Clearly, PAP(X ) ⊂ BC(R, X ) and by Lemma 5.8, the space PAP(X ) is closed. Therefore, (PAP(X ), · ∞ ) is a Banach space. Definition 5.10. A jointly continuous function f : R × X → X is called pseudoalmost periodic if F = G + Φ,

146

5 Pseudo-Almost Periodic Functions

where G ∈ AP(X × R) and Φ ∈ PAP0 (R × X ) where PAP0 (R × X ) is the collection of all bounded continuous functions Ψ : R × X → X satisfying 1 r→∞ 2r lim

 r −r

Ψ (t, x) dt = 0

uniformly in x ∈ B where B ⊂ X is an arbitrary bounded subset. The collection of such functions will be denoted by PAP(R × X ). Example 5.11. It can be easily checked that the function h : R × R → R defined by & ' h(t, x) = cos x sint + sin π t + (1 + t 2 )−1 is pseudo-almost periodic in t ∈ R uniformly in x ∈ B, where B ⊂ R is any arbitrary bounded subset. Theorem 5.12. Suppose that the Banach space X is uniformly convex. Let f = h + ϕ ∈ PAP(X ) with h ∈ AP(X ) and ϕ ∈ PAP0 (X ). Define the functions  t

H(t) := 0

h(s)ds and Φ (t) :=

 t 0

ϕ (s)ds,

t ∈R

and suppose (a) H is bounded; and (b) there exists ξ ∈ X such that Φ − ξ ∈ PAP0 (X ). Then the function defined by t → F(t) = H(t) + Φ (t) belongs to PAP(X ). Proof. From Theorem 3.18 it follows that H ∈ AP(X ). Similarly, from Zhang [174, Theorem 7] it follows that Φ ∈ PAP0 (X ). Therefore, F = H + Φ ∈ PAP(X ).

5.1.2 Composition Theorems Theorem 5.13 ([12, 57]). Let f ∈ PAP(R × X ) be given such that f satisfies the Lipschitz condition, that is, there exists L > 0 such that f (t, u) − f (t, v) ≤ L u − v for all u, v ∈ X and t ∈ R. If ϕ ∈ PAP(X ), then the function h(t) = f (t, ϕ (t)) ∈ PAP(X ). Proof. Let f ∈ PAP(R × X ) and write f = G + H where G ∈ AP(R × X ) and H ∈ PAP0 (R×X ). Moreover, write ϕ = ϕ1 + ϕ2 with ϕ1 ∈ AP(X ) and ϕ2 ∈ PAP0 (X ).

5.1 Pseudo-Almost Periodic Functions

147

Now f (t, ϕ (t)) ≤ L ϕ ∞ + f (t, 0) ≤ L ϕ ∞ + G(t, 0) + H(t, 0) ≤ L ϕ ∞ + G(·, 0) + H(·, 0) , and hence f (·, ϕ (·)) ∈ BC(R × X ), and f (·, ϕ (·)) = G(·, ϕ1 (·)) + f (·, ϕ (·)) − G(·, ϕ1 (·)) = G(·, ϕ1 (·)) + f (·, ϕ (·)) − f (·, ϕ1 (·)) + H(·, ϕ1 (·)). In view of Theorem 3.30 it easily follows that G(·, ϕ1 (·)) ∈ AP(X ). Now, using the fact that f is Lipschitzian and ϕ2 ∈ PAP0 (X ) it follows that f (·, ϕ (·)) − f (·, ϕ1 (·)) ∈ PAP0 (X ). In order to show that f (·, ϕ (·)) ∈ PAP(R × X ) we must show that 1 r→∞ 2r lim

 r −r

H(t, ϕ1 (t)) dt = 0.

Since ϕ1 (R) is relatively compact in X , for each ε > 0 there exists a finite number of open balls Ok , centered at xk ∈ ϕ1 (R) with radius less than ε /3L, and

ϕ1 (R) ⊂

N

Ok .

k=1

For k (k = 1, 2, . . . , N), the set Bk = {t ∈ R : ϕ1 (t) ∈ Ok } is open and R=

N

Bk .

k=1

Let Ek = Bk −

k−1

Oi

i=1

and E1 = B1 . It can be checked that Ei



E j = {0} /

148

5 Pseudo-Almost Periodic Functions

for all i = j. Using the argument that H ∈ PAP0 (R × X ), one can find r0 such that 1 2r

 r −r

H(t, xk ) dt <

ε , r ≥ r0 , k ∈ {1, 2, . . . , N}. 3N

(5.1)

Furthermore, since G ∈ AP(R × X ) is also uniformly continuous in R × ϕ1 (R) it follows that

ε G(t, xk ) − g(t, x) < , ∀x ∈ Ok , k ∈ {1, 2, . . . , N}. 3

(5.2)

From H(·, ϕ1 (·)) = f (·, ϕ1 (·)) − G(·, ϕ1 (·)) and H(t, xk ) = f (t, xk ) − G(t, xk ) it follows that 1 2r

 r −r

H(t, ϕ1 (t)) dt =

1 N ∑ 2r k=1

≤

1 N ∑ 2r k=1

 Ek ∩[−r,r]

 Ek ∩[−r,r]

1 N + ∑ 2r k=1 ≤

≤

1 N ∑ 2r k=1

+

1 N ∑ 2r k=1

1 N ∑ 2r k=1 +

N

Ek ∩[−r,r]

 Ek ∩[−r,r]

Ek ∩[−r,r]

1 k=1 2r

Ek ∩[−r,r]

−r

H(t, xk ) dt H(t, ϕ1 (t)) − H(t, xk ) dt

L ϕ1 (t) − xk dt



 r

H(t, xk ) dt

f (t, ϕ1 (t)) − f (t, xk ) dt





1 N ∑ 2r k=1

+∑

Ek ∩[−r,r]

Ek ∩[−r,r]

1 N ∑ 2r k=1

H(t, xk ) − H(t, xk ) dt





+

H(t, xk ) dt

G(t, xk ) − G(t, xk ) dt

H(t, xk ) dt.

(

For each t ∈ Ek [−r, r], ϕ1 (t) ∈ Ok . Thus it follows from (5.1) and (5.2) that 1 2r

 r −r

H(t, ϕ1 (t)) dt ≤ ε , r ≥ r0 ,

(5.3)

5.2 Weighted Pseudo-Almost Periodic Functions

149

and hence 1 r→∞ 2r lim

 r −r

H(t, ϕ1 (t)) dt = 0.

Theorem 5.14 ([128]). Let F : R×X → X be a pseudo-almost periodic function. Suppose u → F(t, u) is uniformly continuous on every bounded subset K ⊂ X uniformly for t ∈ R. If g ∈ PAP(X ), then Γ : R → X defined by Γ (·) := F(·, g(·)) belongs to PAP(X ).

5.2 Weighted Pseudo-Almost Periodic Functions This section is devoted to the concept of weighted pseudo-almost periodicity introduced in the literature in 2006 by Diagana [64]. Because of the weights involved, the concept of weighted pseudo-almost periodicity is more general and richer than the concept of pseudo-almost periodicity, which we have introduced and studied in Sect. 5.1. Since its introduction in the literature, the notion of weighted pseudo-almost periodicity has generated several developments, see for instance the following references [6, 7, 19, 27, 28, 57, 66, 67, 129]. One should mention that the theory of weighted pseudo-almost periodicity was recently extended by Blot et al. [20] using measure theory.

5.2.1 Definitions and Properties Let U denote the collection of functions (weights) μ : R → (0, ∞), which are locally integrable over R such that μ > 0 almost everywhere. If μ ∈ U and for r > 0, we then set Qr := [−r, r] and

μ (Qr ) :=

 Qr

μ (x)dx.

As in the particular case when μ (x) = 1 for each x ∈ R, in this setting, we are exclusively interested in those weights, μ , for which, lim μ (Qr ) = ∞. Consequently, r→∞ we define the space of weights U∞ by  U∞ := μ ∈ U : inf μ (x) = μ0 > 0 and lim μ (Qr ) = ∞ . r→∞

x∈R

In addition to the above, we define the set of weights UB by  UB := μ ∈ U∞ : sup μ (x) = μ1 < ∞ . x∈R

150

5 Pseudo-Almost Periodic Functions

Definition 5.15. Let μ , ν ∈ U∞ . One says that μ is equivalent to ν and denote it μ μ ≺ ν , if ∈ UB . ν Let μ , ν , γ ∈ U∞ . It is clear that μ ≺ μ (reflexivity); if μ ≺ ν , then ν ≺ μ (symmetry); and if μ ≺ ν and ν ≺ γ , then μ ≺ γ (transitivity). Therefore, ≺ is a binary equivalence relation on U∞ . Definition 5.16 (Diagana [64]). Fix μ ∈ U∞ . A continuous function f : R → X is called weighted pseudo-almost periodic if it can be written as f = h + ϕ, with h ∈ AP(X ) and ϕ ∈ PAP0 (X , μ ) where the space PAP0 (X , μ ) is defined by  PAP0 (X , μ ) = g ∈ BC(R, X ) : lim

1 r→∞ μ (Qr )

 Qr

g(t) μ (t)dt = 0 .

The function h and ϕ are respectively called the almost periodic and the weighted ergodic components of f . The collection of all weighted pseudo-almost periodic functions from R into X will be denoted by PAP(X , μ ). Definition 5.17. Fix μ ∈ U∞ . A jointly continuous function F : R × X → X is called pseudo-almost periodic if F = G+Φ with G ∈ AP(X × R) and Φ ∈ PAP0 (R × X , μ ) where the space PAP0 (R × X , μ ) is the collection of all bounded continuous functions Ψ : R × X → X satisfying 1 r→∞ μ (Qr )



lim

Qr

Ψ (t, x) μ (t)dt = 0

uniformly in x ∈ B where B ⊂ X is an arbitrary bounded subset. The collection of such functions will be denoted by PAP(R × X , μ ). We have seen in Sect. 5.1 that the decomposition of a pseudo-almost periodic is unique (Lemma 5.5). This is not always the case for weighted pseudo-almost periodic functions (see Example 5.18). In fact, the uniqueness of such a decomposition depends upon the translation-invariance of the space PAP0 (X , μ ). Example 5.18 (Liang et al. [129]). Consider the weight μ = 1 + μ˜ ∈ U∞ with

μ˜ (t) =

⎧    ⎪ ⎪ ⎨−|k| t − 2kπ t − 2kπ − π ,

if 2kπ ≤ t ≤ 2kπ + π , k ∈ Z,

⎪ ⎪ ⎩0,

if 2kπ + π ≤ t ≤ 2kπ + 2π , k ∈ Z.

5.2 Weighted Pseudo-Almost Periodic Functions

151

Letting

f (t) = max(− sint, 0) =

⎧ ⎪ ⎪0, ⎨

if 2kπ ≤ t ≤ 2kπ + π , k ∈ Z,

⎪ ⎪ ⎩sint,

if 2kπ + π ≤ t ≤ 2kπ + 2π , k ∈ Z,

one can show that f ∈ AP(R) ∩ PAP0 (R, μ ), which yields the decomposition for PAP(R, μ ) is not unique for the weighted μ given above. In fact, f ∈ PAP0 (R, μ ) while Rπ f ∈ PAP0 (R, μ ). In other words, PAP0 (μ , R) is not translation-invariant. Proposition 5.19 (Ji and Zhang [121]). Fix μ ∈ U∞ . Suppose for any s ∈ R,

μ (t + s) < ∞. |t|→∞ μ (t) lim

Then PAP0 (X , μ ) is translation-invariant. The proof of Proposition 5.19 is highly technical; we therefore refer the interested reader to [121]. In view of Proposition 5.19, we introduce the following new set of weights, which makes the spaces of weighted pseudo-almost periodic functions translationinvariant:  μ (t + s) U∞Inv := μ ∈ U∞ : for all s ∈ R, lim <∞ . |t|→∞ μ (t) It can be easily seen that if μ ∈ U∞Inv , then the corresponding space of weighted pseudo-almost periodic functions PAP(X , μ ) is translation-invariant. In particular, since UB ⊂ U∞Inv , it follows that for each μ ∈ UB , then PAP(X , μ ) is translationinvariant. Proposition 5.20 (Liang et al. [129]). Fix μ ∈ U∞Inv . If f ∈ PAP(X , μ ) then there exists a unique h ∈ AP(X ) and a unique ϕ ∈ PAP0 (X , μ ) such that f = h + ϕ . Proof. Suppose f ≡  0. Thus there exists a t ∈ R such that f (t) = 0. One can suppose that there exists δ > 0 such that f (0) > 2δ . Define Bδ := {τ ∈ R : f (t) − f (0) ≤ δ }. Using Lemma 4.8 it follows that there exist s1 , s2 , . . . , sm ∈ R such that m 

k=1

 sk + Bδ = R.

152

5 Pseudo-Almost Periodic Functions

Now f (t) ≥ f (0) − f (t) − f (0) ≥ δ for all t ∈ Bδ , and hence f (t − sk ) = f (t)|B ≥ δ δ

for all t + sk ∈ Bδ for k = 1, 2, . . . , m. Setting F(t) = f (t) + f (t − s1 ) + f (t − s2 ) + · · · + f (t − sm ) one can easily see that F(t) ≥ δ for all t ∈ R. Now for any r > 0, 1 μ (Qr )

 Qr

F(t)μ (t)dt ≥

δ μ (Qr )

 Qr

μ (t)dt = δ .

(5.4)

Using the fact that PAP0 (X , μ ) is translation-invariant and f ∈ PAP0 (X , μ ) it follows that f (t − sk ) ∈ PAP0 (X , μ ) for k = 1, 2, . . . , m. Namely, 1 μ (Qr )

 Qr

f (t) μ (t)dt = 0

and 1 μ (Qr )

 Qr

f (t − sk ) μ (t)dt = 0 for k = 1, 2, . . . , m.

And that is in contradiction with Eq. (5.4), and hence PAP(X , μ ) = AP(X ) ⊕ PAP0 (X , μ ). Proposition 5.21. Fix μ ∈ U∞Inv . Let f ∈ PAP(X , μ ). If g ∈ L1 (R), then f ∗ g, the convolution of f with g on R, belongs to PAP(X , μ ). Proof. Let f = h + ϕ where h ∈ AP(X ) and ϕ ∈ PAP0 (X, μ ). Now, f ∗ g = h ∗ g + ϕ ∗ g. It is enough to show that ϕ ∗ g ∈ PAP0 (X , μ ) as it has already been shown that h ∗ g ∈ AP(X ) (see Proposition 3.4). Now from ϕ ∈ PAP0 (X , μ ) and g ∈ L1 (R) it is clear that ϕ ∗ g ∈ BC(R, X ). By assumption, 1 lim r→∞ μ (Qr )

 Qr

ϕ (t) μ (t)dt = 0.

5.2 Weighted Pseudo-Almost Periodic Functions

153

Now setting L(r) :=

1 μ (Qr )



 +∞

Qr −∞

ϕ (t − s) |g(s)|μ (t)ds dt

it follows that 1 μ (Qr )

 Qr

(ϕ ∗ g)(t) μ (t)dt ≤ L(r) 1 = μ (Qr ) = =

 +∞ −∞

 +∞ −∞

 r  +∞ −r −∞

|g(s)|

ϕ (t − s) |g(s)|μ (t)ds dt

1 μ (Qr )



 Qr

ϕ (t − s) μ (t)dt

ds

|g(s)| Γ (s) ds,

where

Γr (s) =

1 μ (Qr )

 Qr

ϕ (t − s) μ (t)dt.

Since PAP(X , μ ) is translation-invariant it follows that Γr (s) → 0 as r → ∞. Using the fact that Γr is uniformly bounded, g ∈ L1 (R), and the Lebesgue Dominated Convergence Theorem, it follows that lim L(r) = 0, and hence ϕ ∗ g ∈ PAP0 (X , μ ). r→∞

Proposition 5.22 ([121]). Fix μ ∈ U∞Inv . Let f ∈ PAP(X , μ ) and write f = g + ϕ where g ∈ AP(X ) and ϕ ∈ PAP0 (X , μ ), then g(X ) ⊂ f (X ) and f ∞ ≥ g ∞ . Fix μ ∈ U∞Inv . If f , g ∈ PAP(X , μ ) and λ ∈ F, we can then show that f + g and λ f belong to PAP(X , μ ). If g is F-valued for instance, it can be easily shown that f g ∈ PAP(X , μ ). It is also clear that (PAP(X , μ ), · ∞ ) is a normed space. In the next theorem, we show that the normed vector space (PAP(X , μ ), · ∞ ) is in fact complete. Theorem 5.23. Fix μ ∈ U∞Inv . The space of weighted pseudo-almost periodic functions PAP(X , μ ) equipped with the sup-norm is a Banach space. Proof. The proof follows along the same lines as in the pseudo-almost periodic case and hence is omitted.

154

5 Pseudo-Almost Periodic Functions

5.2.2 Weighted Mean for Almost Periodic Functions Definition 5.24. Let μ , ν ∈ U∞ . If f : R → X is a bounded continuous function, we define its (doubly) weighted mean, if the limit exists, by 1 r→∞ μ (Qr )

M ( f , μ , ν ) := lim

 Qr

f (t)ν (t)dt.

In Liang et al. [129], the original question which is that of the existence of a weighted mean for almost periodic functions was raised. In particular, Liang et al. have shown that there exist weights μ ∈ U∞ for which a weighted mean for almost periodic functions may or may not exist. Our main objective here consists of investigating the question of the existence of a weighted mean for almost periodic functions. Namely, we give some sufficient conditions which do guarantee the existence of a weighted mean for almost periodic functions (Theorem 5.25). Under those conditions, it will be shown that both the weighted mean and the classical one are proportional. Theorem 5.25 (Diagana [68, 74]). Let μ , ν ∈ U∞ and suppose lim

r→∞

If f ∈ AP(X ) such that $ $ $ 1  $ $ $ eiλ t ν (t)dt $ = 0 lim $ r→∞ $ μ (Qr ) Qr $ for all 0 = λ ∈ σb ( f ), then the (doubly) weighted mean of f , 1 r→∞ μ (Qr )

M ( f , μ , ν ) = lim

 Qr

f (t)ν (t)dt

exists. Furthermore, M ( f , μ , ν ) = θμν M ( f ). Proof. If f is a trigonometric polynomial, say, f (t) =

n

∑ ak eiλkt ,

k=0

where ak ∈ X − {0} and λk ∈ R for k = 1, 2, . . . , n, then

σb ( f ) = {λk : k = 1, 2, . . . , n}.

ν (Qr ) = θμν . μ (Qr )

(5.5)

5.2 Weighted Pseudo-Almost Periodic Functions

155

Moreover, 1 μ (Qr )

 Qr

f (t)ν (t)dt = a0 = a0

1 ν (Qr ) + μ (Qr ) μ (Qr ) &



& Qr

' iλk t a e ν (t)dt ∑ k n

k=1

1 ν (Qr ) + ∑ ak μ (Qr ) k=1 μ (Qr ) n

 Qr

' eiλk t ν (t)dt .

Now    $ $ n  1  ν (Qr )  $ iλk t  ≤ ∑ ak $$ 1  f (t) ν (t)dt − a e ν (t)dt $ 0  μ (Qr ) Q μ (Qr )  k=1 μ (Qr ) Qr r which by Eq. (5.5) yields    1      μ (Qr ) Q f (t)ν (t)dt − a0 θμν  → 0 as r → ∞ r

and therefore M ( f , μ , ν ) = a0 θμν = θμν M( f ). If in the finite sequence of λk there exist λnk = 0 for k = 1, 2, . . . , l with am ∈ X − {0} for all m = nk (k = 1, 2, . . . , l), it can be easily shown that M ( f , μ , ν ) = θμν

l

∑ ank = θμν M( f ).

k=1

Now if f : R → X is an arbitrary almost periodic function, then for every ε > 0 there exists a trigonometric polynomial (Theorem 1.3) Pε defined by Pε (t) =

n

∑ ak eiλkt

k=1

where ak ∈ X and λk ∈ σb ( f ) such that f (t) − Pε (t) < ε for all t ∈ R. Proceeding as in Bohr [26] it follows that there exists r0 such that for all r1 , r2 > r 0 ,   

  1  Pε (t)ν (t)dt  μ (Q ) Qr1 Qr2 r2     = θμν M(Pε ) − M(Pε ) = 0 < ε .

1 μ (Qr1 )



Pε (t)ν (t)dt −

156

5 Pseudo-Almost Periodic Functions

In view of the above it follows that for all r1 , r2 > r0 ,   

1 μ (Qr1 )

 Qr1

f (t)ν (t)dt − ≤

  +

1 μ (Qr1 )



1 μ (Qr2 ) 1 μ (Qr1 )

1 Pε (t)ν (t)dt − μ (Qr2 ) Qr1 +

1 μ (Qr2 )

   

Qr2

Qr1

Qr2

Qr2

  f (t)ν (t)dt  f (t) − Pε (t) ν (t)dt   Pε (t)ν (t)dt  f (t) − Pε (t) ν (t)dt < 3ε .

Now for all r > r0 ,   

1 μ (Qr )

 Qr

f (t)ν (t)dt −

1 μ (Qr )



 ε  Pε (t)ν (t)dt  < 3 Qr

and hence M( f , μ , ν ) = M(Pε , μ , ν ) = M(Pε ) = M( f ). The proof is complete.

5.2.3 A Composition Theorem The next composition theorem is a generalization of Theorem 5.13. Theorem 5.26 (Diagana [66]). Let μ ∈ U∞ and let f ∈ PAP(R × X , μ ) satisfying the Lipschitz condition f (t, u) − f (t, v) ≤ L u − v for all u, v ∈ X , t ∈ R. If h ∈ PAP(X , μ ), then f (·, h(·)) ∈ PAP(X , μ ). Proof. Let f = g + ϕ where g ∈ AP(R × X ) and ϕ ∈ PAP0 (R × X , μ ). Similarly, let h = h1 + h2 , where h1 ∈ AP(X ) and h2 ∈ PAP0 (X , μ ). Clearly, f (·, h(·)) ∈ C(R, X ). Next, decompose f as follows: f (·, h(·)) = g(·, h1 (·)) + f (·, h(·)) − f (·, h1 (·)) + ϕ (·, h1 (·)). Using Theorem 3.30, one can easily see that g(·, h1 (·)) ∈ AP(X ). Now, set F(·) = f (·, h(·)) − f (·, h1 (·)). Clearly, F ∈ PAP0 (X , μ ). Indeed, for r > 0, 1 μ (Qr )

 Qr

F(s) μ (s)ds =

1 μ (Qr )

≤

L μ (Qr )

 Qr



Qr

f (s, h(s)) − f (s, h1 (s)) μ (s)ds h(s) − h1 (s) μ (s)ds

5.2 Weighted Pseudo-Almost Periodic Functions

≤

L μ (Qr )

157

 Qr

h2 (s) μ (s)ds,

and hence 1 r→∞ μ (Qr )



lim

Qr

F(s) μ (s)ds = 0.

To complete the proof we have to show that 1 r→∞ μ (Qr )



lim

Qr

ϕ (s, h1 (s)) μ (s)ds = 0.

As h1 ∈ AP(X ), h1 (R) is relatively compact. Thus for each ε > 0 there exists a finite number of open balls Bk = B(xk , ε /3L), centered at xk ∈ h1 (R) with radius for m

instance ε /3L with h1 (R) ⊂

Bk . Therefore, for 1 ≤ k ≤ m, the set Uk = {t ∈ R :

k=1

h1 (t) ⊂ Bk } is open and

R=

m

Uk .

k=1

Now, set Vk = Uk −

k−1

Ui

i=1

and V1 = U1 . Clearly, Vi



V j = 0/

for all i = j. Since ϕ ∈ PAP0 (X , μ ) there exists r0 > 0 such that 1 r→∞ μ (Qr )



lim

Qr

ϕ (s, xk ) μ (s)ds <

ε for r ≥ r0 3m

(5.6)

and k ∈ {1, 2, . . . , m}. Moreover, since g (g ∈ AP(R × X )) is uniformly continuous in R × h1 (R), one has g(t, xk ) − g(t, x) <

ε for x ∈ Bk , k = 1, 2, . . . , m. 3

(5.7)

158

5 Pseudo-Almost Periodic Functions

In view of the above and following the decompositions

ϕ (·, h1 (·)) = f (·, h1 (·)) − g(·, h1 (·)) and

ϕ (t, xk ) = f (t, xk ) − g(t, xk ) it follows that 1 μ (Qr )

 Qr

ϕ (s, h1 (s)) μ (s)ds =

m 1 ∑ μ (Qr ) k=1

≤

m 1 ∑ μ (Qr ) k=1

+ ≤



ϕ (s, h1 (s)) μ (s)ds

Qr

 (

Vk

m 1 ∑ μ (Qr ) k=1

m 1 ∑ μ (Qr ) k=1

(

Vk

ϕ (s, h1 (s)) − ϕ (s, xk ) μ (s)ds

Qr

 (

Vk

Qr

 (

Vk

+

m 1 ∑ μ (QT ) k=1

+

m 1 ∑ μ (Qr ) k=1

+

m 1 ∑ μ (Qr ) k=1

+

m 1 ∑ μ (Qr ) k=1 m

1 μ (Q r) k=1

+∑

f (s, h1 (s)) − f (s, xk ) μ (s)ds

Qr

 Vk

(

Qr

 Vk

(

Qr

 Vk

(

Qr

 Vk

(

Qr

 Vk

ϕ (s, xk ) μ (s)ds

(

Qr

ϕ (s, xk ) μ (s)ds g(s, h1 (s))−g(s, xk ) μ (s)ds L h1 (s) − xk μ (s)ds g(s, h1 (s))−g(s, xk ) μ (s)ds ϕ (s, xk ) μ (s)ds.

For each s ∈ Vk ∩ Qr , h1 (s) ∈ Bk in the sense that h1 (s) − xk <

ε 3L

for 1 ≤ k ≤ m. Clearly, from Eqs. (5.6)–(5.7) it easily follows that 1 μ (Qr )

 Qr

φ (s, h1 (s)) μ (s)ds ≤ ε

5.3 C(n) -Pseudo-Almost Periodic Functions

159

for r ≥ r0 , and hence 1 r→∞ μ (Qr )



lim

Qr

ϕ (s, h1 (s)) μ (s)ds = 0.

5.3 C(n) -Pseudo-Almost Periodic Functions This section is devoted to an important subclass of the class of pseudo-almost periodic functions called C(n) -pseudo-almost periodic functions, which was recently introduced in the literature by Diagana and Nelson [71]. Among other things, the concept of C(n) -pseudo-almost periodicity is a generalization of the C(n) -almost periodicity. In this section, we study some of the basic properties of these new functions.

5.3.1 Definitions and Properties Definition 5.27 (Diagana and Nelson [71]). A function f ∈ C(n) (R, X ) is called C(n) -pseudo-almost periodic (or n-differentiable pseudo-almost periodic) if it can be expressed as f = g+ϕ (n)

(n)

with g ∈ AP(n) (X ) and ϕ ∈ PAP0 (X ) where the space PAP0 (X ) is defined by (n)

PAP0 (X ) :=



f ∈ BC(n) (R, X ) : f (k) ∈ PAP0 (X ) for k = 0, 1, . . . , n .

The collection of all C(n) -pseudo-almost periodic functions will be denoted by PAP(n) (X ). Clearly, the following inclusions hold · · · → PAP(n+1) (X ) → PAP(n) (X ) → · · · → PAP(1) (X ) → PAP(X ). Example 5.28. Let α , β be real numbers such that αβ −1 is an irrational. Define the function f (t) = sin α t + sin β t +

1 . 2 + |t|

It is easily seen that f ∈ PAP(R) while f ∈ PAP(1) (R) as the function f is not differentiable at 0.

160

5 Pseudo-Almost Periodic Functions

Definition 5.29. A bounded jointly continuous function F : R × X → X , (t, x) → F(t, x) is said to be C(n) -pseudo-almost periodic in t ∈ R uniformly in x ∈ X whenever it can be expressed as F = G+Φ (n)

(n)

with G ∈ AP(n) (R × X ) and Φ ∈ PAP0 (R × X ) where the space PAP0 (R × X ) is defined by (n)

PAP0 (R × X ) :=



(k)

f ∈ BC(n) (R × X ) : Dt f ∈ PAP0 (R × X ) for k = 0, 1, . . . , n .

The collection of all such C(n) -pseudo-almost periodic functions will be denoted by PAP(n) (R × X ).   Theorem 5.30. The space PAP(n) (X ) equipped with the norm  · (n) is a Banach space. Proof. The proof follows along the same lines as in the cases of C(n) -almost periodicity and C(n) -almost automorphy and hence is omitted. Theorem 5.31. If f ∈ PAP(n) (X ) and if g ∈ L1 (R), then their convolution f ∗ g ∈ PAP(n) (X ). Proof. The proof follows along the same lines as in the cases of C(n) -almost periodicity and C(n) -almost automorphy and hence is omitted. Theorem 5.32. If f ∈ PAP(n) (X ) such that f (n+1) is uniformly continuous, then f ∈ PAP(n+1) (X ). Proof. Let f˜ := f (n) . Since f˜ is uniformly continuous, then for every ε > 0 there exists δ > 0 such that for all t, s ∈ R with |t − s| < δ , then f˜ (t) − f˜ (s) < ε . The rest of the proof follows slightly along the same lines as that of [142, Theorem 2.4.1]. For an arbitrary r ∈ R and δ > 1n , we have  1    n 1 ˜ ˜ ˜ f# (r + τ ) − f# (r) d τ . n f (r + ) − f (r) − f (r) = n n 0

Now it is easily seen that the sequence   1 gn (r) = n f˜(r + ) − f˜(r) ∈ PAP0 (X ) n converges uniformly to f˜ (r) on R. Therefore, using Lemma 5.8 it follows that f˜ ∈ PAP(X ), and hence f ∈ PAP(n+1) (X ). Theorem 5.33. Suppose that the Banach space X is uniformly convex. Let f = h + ϕ ∈ PAP(X ) with h ∈ AP(X ) and ϕ ∈ PAP0 (X ). Define the functions

5.3 C(n) -Pseudo-Almost Periodic Functions

 t

H(t) := 0

h(s)ds and Φ (t) :=

161

 t 0

ϕ (s)ds,

t ∈R

and suppose (a) H is bounded; (b) there exists ξ ∈ X such that Φ − ξ ∈ PAP0 (X ); and (c) f (k) is uniformly continuous for k = 0, 1, . . . , n. Then F belongs to PAP(n+1) (X ). Proof. From Theorem 5.12 it follows that F = H + Φ belongs to PAP(X ). Now since F = f is uniformly continuous, then from Theorem 1.16 it follows that F belongs to PAP(1) (X ). Now since F ∈ PAP(X ) and F = f is uniformly continuous it follows from Theorem 1.16 that F ∈ PAP(2) (X ). By induction it can be easily shown that F ∈ PAP(n+1) (X ).

5.3.2 Example Example 5.34 (Diagana and Nelson [71]). We give an example of a function f ∈ PAP(2) (R) such that f ∈ PAP(3) (R). Indeed, consider the function f defined by f (t) =

∞

sin(kt) 1 + . 4 k 1 + t2 k=1

∑

Setting

ϕ (t) =

∞

sin(kt) 1 and h(t) = , 4 k 1 + t2 k=1

∑

we claim that (i) ϕ ∈ AP(2) (R) while ϕ ∈ AP(3) (R); and (2) (ii) h ∈ PAP0 (R). Indeed, ϕ is a continuous 2π -periodic and hence is almost periodic. Similarly, the functions

ϕ (t) =

∞

∞ cos(kt) sin(kt) , ϕ (t) = − ∑ 3 k k2 k=1 k=1

∑

are continuous 2π -periodic and hence are almost periodic.

162

5 Pseudo-Almost Periodic Functions

Now ∞

cos(kt) k k=1

ϕ (t) = − ∑

is not continuous for all real numbers. Namely, ϕ is not defined at t = 0. Hence ϕ is not almost periodic. Therefore, ϕ ∈ AP(2) (R) while ϕ ∈ AP(3) (R). Similarly, it is readily seen that h belongs to PAP0 (R). Now the functions h (t) = −

2t −2 − 10t 2 , h (t) = 2 2 (1 + t ) (1 + t 2 )3

are bounded continuous functions with |h (t)| ≤ 2 and |h (t)| ≤ 12 for all t ∈ R. Further, it is easily seen that both h and h belong to PAP0 (R) and hence h ∈ (2) PAP0 (R). In view of the above it follows that f ∈ PAP(2) (R) while f ∈ PAP(3) (R).

5.4 S p -Pseudo-Almost Periodic Functions The concept of S p -pseudo-almost periodicity, which is a natural generalization of the notion of pseudo-almost periodicity, was introduced a few years ago by Diagana [54]. This section is devoted to the study of basic properties of these functions. Definition 5.35 (Diagana [54]). Let p ≥ 1. A function f ∈ BS p (X ) is called S p pseudo-almost periodic (or Stepanov-like pseudo-almost periodic) if it can be expressed as f = h + ϕ,     where hb ∈ AP L p ((0, 1), X ) and ϕ b ∈ PAP0 L p ((0, 1), X ) . In other words, a function f ∈ L p (R, X ) is said to be S p -pseudo-almost periodic if its Bochner transform f b : R → L p ((0, 1), X ) is pseudo-almost periodic in the sense that there exist two functions h, ϕ : R → X such that f = h + ϕ , where hb ∈ AP(L p ((0, 1), X )) and ϕ b ∈ PAP0 (L p ((0, 1), X )), that is,

5.4 S p -Pseudo-Almost Periodic Functions





1 r→∞ 2r lim

Qr

t

t+1

163

φ (σ ) p d σ

1/p dt = 0.

p (X ). The collection of such functions will be denoted by Spap

Proposition 5.36. If f ∈ PAP(X ), then f is S p -pseudo-almost periodic for any 1 ≤ p < ∞. Proof. Let f = h + ϕ where h ∈ AP(X ) and ϕ ∈ PAP0 (X ). First of all, note that f ∈ BS p (X ). It is enough to show that ϕ b ∈ PAP0 (L p ((0, 1), X )). For that, note that if q is chosen such that p−1 + q−1 = 1, then for r > 0,





1

0

Qr

ϕ (t + s) ds p

 1p

dt ≤ (2r)

1 q

≤ (2r)

1 q

)  

1

0

Qr

)  

p−1 p

1 q

= (2r) ϕ ∞

= (2r) ϕ ∞ p−1 p

= 2r ϕ ∞

*1

p

ϕ (s + t) . ϕ ∞p−1 ds

)   Qr

p−1 p

1 q

1

0

Qr

 * 1p ϕ (s + t) ds dt p

1

0

)  1

0

p

ϕ (s + t) ds dt p

ϕ (s + t) dt ds

Qr

1 2r

*1

*1

)   1 0

dt

*1



p

Qr

ϕ (s + t) dt ds

,

and hence 1 2r



 Qr

1 0

ϕ (t + s) p ds

 1p

p−1 p

dt ≤ ϕ ∞

)  1

0

1 2r

*1



p

Qr

ϕ (s + t) dt ds

Since PAP0 (X ) is translation-invariant it follows that 1 2r

 Qr

ϕ (t + s) dt → 0 as r → ∞

for all s ∈ [0, 1]. Using the Lebesgue Dominated Convergence Theorem it follows that )  1

0

1 2r

 Qr

*1

p

ϕ (s + t) dt ds

→ 0 as r → ∞,

.

164

5 Pseudo-Almost Periodic Functions

and hence





1 lim r→∞ 2r

Qr

1 0

ϕ (t + s) ds p

 1p

dt = 0.

Definition 5.37. A function F : R × X → X , (t, u) → F(t, u) with F(·, u) ∈ L p (R, X ) for each u ∈ X , is said to be S p -pseudo-almost periodic in t ∈ R uniformly in u ∈ X if t → F(t, u) is S p -pseudo-almost periodic for each u ∈ K where K ⊂ X is an arbitrary compact subset. This means, there exist two functions H, Φ : R × X → X such that F = H + Φ, where H b ∈ AP(R × L p ((0, 1), X )) and Φ b ∈ PAP0 (R × L p ((0, 1), X )), i.e., 1 r→∞ 2r





lim

Qr

t

t+1

Φ (σ , u) p d σ

1/p dt = 0

uniformly in u ∈ K where K ⊂ X is an arbitrary compact subset. The collection of those S p -pseudo-almost periodic functions F : R × X → X will p be denoted by Spap (R × X ). p Theorem 5.38 (Long and Ding [135]). Let f ∈ Spap (R × X ) for p > 1 and write f = g + h where gb ∈ AP(R × L p ((0, 1), X )) and hb ∈ PAP0 (R × L p ((0, 1), X )). p ) such that Suppose that g, h satisfy: there exists L ∈ BSr (R) with r ≥ max(p, p−1

g(t, x) − g(t, y) ≤ L(t) x − y and h(t, x) − h(t, y) ≤ L(t) x − y p for all x, y ∈ X and t ∈ R. If x = y + z ∈ Spap (X ) where yb ∈ AP(L p ((0, 1), X )) b p and z ∈ PAP0 (L ((0, 1), X )) and if there exists a set E ⊂ R of measurable zero such K = {y(t) : t ∈ R\E} ⊂ X is compact, then there exists q ∈ [1, p) such that q the function Γ : R → X defined by Γ (t) = f (t, x(t)) belongs to Spap (X ).

The proof of Theorem 5.38 requires the next lemma whose proof can be found in [135]. Lemma 5.39. Let K ⊂ X be compact and suppose that there exists L ∈ BSr (R) p with r ≥ max(p, p−1 ) such that f (t, x) − f (t, y) ≤ L(t) x − y

5.4 S p -Pseudo-Almost Periodic Functions

165

for all x, y ∈ X and t ∈ R, and f b ∈PAP0 (R × L p ((0, 1), X )). Then f# ∈ PAP0 (R)   where f#(t) :=  supu∈K f (t + ·, u)  , t ∈ R. p

p ) it follows that there Proof (Theorem 5.38). Using the fact that r ≥ max(p, p−1 −1 −1 exists q ∈ [1, p) such that r = pq(p − 1) . Now let q = pq and p = p(p − q )−1 . It is easy to see that p , q > 1 and p −1 + q −1 = 1. Now write

f (t, x(t)) = H(t) + I(t) + J(t) where H(t) = g(t, y(t)), I(t) = f (t, x(t)) − f (t, y(t)), and J(t) = h(t, y(t)). q Using, Theorem 3.72 it follows that H ∈ Sap (X ). To complete the proof, we have to show that I b , J b ∈ PAP0 (Lq ((0, 1), X )). Indeed, using the fact that zb ∈ PAP0 (L p ((0, 1), X )), we obtain 1 2r

 Qr

I b (t) q dt = ≤

1 2r 1 2r





Qr



1

0



Qr

≤ L Sr

1

0

1 2r



Qr

 I(t + s) q ds dt Lq (t + s) z(t + s) q ds

1 q

dt

zb (t) p dt

→ 0, r → ∞. Using the fact that h = f − g is Lipschitz with Lipschitz function L ∈ BSr (R) (and hence h is Lipschitz with Lipschitz function L ∈ BS p (R)) and Lemma 5.39, it can be shown that 1 r→∞ 2r



lim

Qr

     sup h(t + ·, u)  dt = 0. p

u∈K

In view of the above, 1 2r

 Qr

J b (t) q dt ≤

1 2r

=

1 2r

≤

1 2r

 Qr

 Qr

 Qr

J b (t) p dt 

1 0

h(t + s, y(t + s)) p ds

& 1  0

→ 0, r → ∞.

1

p

dt

 '1 p sup h(t + s, u) ds dt

u∈K

166

5 Pseudo-Almost Periodic Functions

Bibliographical Notes The results presented in this chapter are taken from various sources including Zhang [172–174], Diagana [63, 64, 66–69], Diagana et al. [53, 54, 57, 59, 62– 69, 71, 73, 74], Cuevas et al. [43–45], Ding et al. [79, 81, 82], Zhang [172–174], Ji and Zhang [121], Liang et al. [129], Fan et al. [95, 96], Agarwal et al. [6, 7], Ait et al. [9, 10], Pinto [150], Al-Islam et al. [11], Amir and Maniar [12], Cuevas et al. [8, 43, 44], Bugajewski et al. [33], Boukli-Hacenea and Ezzinbi [27, 28], Ezzinbi et al. [88, 89], Li et al. [128], Liang et al.[130], Long and Ding [135], N’Gu´er´ekata and Pankov [146], and Zhang and Li [175–177].

Chapter 6

Pseudo-Almost Automorphic Functions

The concept of pseudo-almost automorphy, which is a generalization of the notions of almost periodicity, almost automorphy, and that of the pseudo-almost periodicity, was introduced in the literature a few years ago by Xiao et al. [164]. Since then, such a powerful concept has generated several developments and extensions, see, e.g., [39, 77, 79, 88, 89, 131]. Among other things, the notion of pseudo-almost automorphy has been utilized to study the qualitative behavior to various differential and partial differential equations with pseudo-almost automorphic coefficients, see for instance [28, 39, 55, 71, 77, 79, 88, 89, 95, 117, 131, 164]. Our main task in this chapter consists of studying basic properties of pseudoalmost automorphic functions including some composition theorems as well as some of their recent extensions.

6.1 Pseudo-Almost Automorphic Functions 6.1.1 Basic Definitions Definition 6.1 (Xiao et al. [164]). A continuous function f : R → X is called pseudo-almost automorphic if it can be written as f = h + ϕ, where h ∈ AA(X ) and ϕ ∈ PAP0 (X ). The collection of pseudo-almost automorphic functions will be denoted by PAA(X ). The function h and ϕ are respectively called the almost automorphic and the ergodic components of f .

T. Diagana, Almost Automorphic Type and Almost Periodic Type Functions in Abstract Spaces, DOI 10.1007/978-3-319-00849-3 6, © Springer International Publishing Switzerland 2013

167

168

6 Pseudo-Almost Automorphic Functions

Example 6.2. Consider the function defined by f (t) = cos



 1 1 √ + . 1 + t2 sint + sin 2t

The function f is an example of a pseudo-almost automorphic function, which is not pseudo-almost periodic. Proposition 6.3. Let f ∈ PAA(X ). If g ∈ L1 (R), then f ∗ g, the convolution of f with g on R, belongs to PAA(X ). Proof. The proof is based on both Theorem 4.5 and Proposition 5.3.

6.1.2 Properties of Pseudo-Almost Automorphic Functions Proposition 6.4. The space PAA(X ) is translation-invariant. Proof. The proof is quite obvious as both AA(X ) and PAP0 (X ) are translationinvariant. If f , g ∈ PAA(X ) and λ ∈ F, then we can show that f + g and λ f belong to PAA(X ). If g is F-valued for instance, it can be easily shown that f g ∈ PAA(X ). It is also clear that (PAA(X ), · ∞ ) is a normed space. In the next theorem, we show that it is in fact complete. Theorem 6.5 (Xiao et al. [164]). The space (PAA(X ), · ∞ ) is a Banach space. Proof. Let f ∈ PAA(X ) and write f = g + ϕ,

(6.1)

where g ∈ AA(X ) and ϕ ∈ PAP0 (X ). Our first task consists of showing that the following identify holds: {g(t) : t ∈ R} ⊂ { f (t) : t ∈ R}.

(6.2)

For that, we will proceed by contradiction. So if Eq. (6.2) is not true, then there are t0 ∈ R and ε > 0 such that g(t0 ) − f (t) ≥ 2ε

(6.3)

for all t ∈ R. Let s1 , s2 , . . . , sm ∈ R as in Lemma 4.8 and write s˜k = sk − t0 for k = 1, 2, . . . , m,

η=

sup k=1,2,...,m

|s˜k |

6.1 Pseudo-Almost Automorphic Functions

169

and for T ∈ R such that |T | > η , we set   (k) Bε ,T = [−T + η − s˜k , T − η − s˜k ] ∩ t0 + Bε for k = 1, 2, . . . , m where Bε is as in Lemma 4.8. Using Lemma 4.8 it follows that m 

 (k) s˜k + BT,ε = [−T + η , T − η ]

k=1

and hence 2(T − η ) = mes[−T + η , T − η ]   m (k) ≤ ∑ mes s˜k + BT,ε k=1

≤

m

∑ mes

  (k) BT,ε

k=1

≤ m.

max

k=1,2,...,m

  (k) mesBT,ε

  ≤ m . mes [−T, T ] ∩ (t + 0 + Bε ) ,

(6.4)

(k)

by noticing the fact BT,ε ⊂ [−T, T ] ∩ (t + 0 + Bε ) for k = 1, 2, . . . , m. Using Eqs. (6.1) and (6.3) it follows that for t ∈ t0 + Bε , ϕ (t) = f (t) − g(t) ≥ g(t0 ) − f (t) − g(t) − g(t0 ) > ε. In view of the above and Eq. (6.4) it follows that 1 2r

 Qr

ϕ (t) dt ≥ ε (r − η )(mr)−1 → ε m−1 as r → ∞

which contradicts the fact that ϕ ∈ PAP0 (X ) and hence Eq. (6.2) holds. Let ( fn )n∈N ∈ PAA(X ) be a Cauchy sequence. Since ( fn )n∈N ∈ PAA(X ) write fn = gn + ϕn where (gn )n∈N ∈ AA(X ) and (ϕn )n∈N ∈ PAP0 (X ).

170

6 Pseudo-Almost Automorphic Functions

Using Eq. (6.2) it follows that (gn )n∈N is a Cauchy sequence. Now since the space (AA(X ), · ∞ ) is a Banach space it follows that there exists g ∈ AA(X ) such that gn − g ∞ → 0 as n → ∞. Setting ϕ = f − g one can easily see that ϕn − ϕ ∞ → 0 as n → ∞, which yields ϕ ∈ BC(R, X ). Moreover, we have the following: 1 2r

 Qr

ϕ (t) dt =

1 2r

1 ≤ 2r

 

Qr

ϕ (t) − ϕn (t) + ϕ (t) dt

1 ϕ (t) − ϕn (t) dt + 2r Qr

≤ ϕ − ϕn ∞ +

1 2r



Qr

 Qr

ϕn (t) dt

ϕn (t) dt.

Letting r → ∞ it follows that 1 lim r→∞ 2r

 Qr

ϕ (t) dt ≤ ϕ − ϕn ∞ .

And letting n → ∞ in the previous inequality it follows that 1 r→∞ 2r



lim

Qr

ϕ (t) dt = 0.

The proof is complete. Theorem 6.6. The decomposition of a pseudo-almost automorphic function is unique. Proof. Suppose f = g1 + ϕ1 and f = g2 + ϕ2 where g1 , g2 ∈ AA(X ) and ϕ1 , ϕ2 ∈ PAP0 (X ). Consequently, 0 = (g1 − g2 ) + (ϕ1 − ϕ2 ). Using Eq. (6.2) it follows that (g1 − g2 ) ⊂ {0} and hence g1 = g2 which yields ϕ1 = ϕ2 . Therefore, the decomposition of a pseudo-almost automorphic function is unique. Definition 6.7. A jointly bounded continuous function f : R × X → X is called pseudo-almost automorphic if F = G + Φ , where G ∈ AA(X × R) and Φ ∈ PAP0 (R × X ). The collection of such functions will be denoted by PAA(R × X ). The next composition result makes use of the uniform continuity rather than the Lipschitz condition. Theorem 6.8 (Liang et al. [132]). Let f = g+ ϕ ∈ PAA(R×X ) with g ∈ AA(R× X ), ϕ ∈ PAP0 (R × X ) and suppose that the following assumptions hold, (i) (t, x) → g(t, x) is uniformly continuous in any bounded subset K ⊂ X uniformly for t ∈ R. (ii) (t, x) → ϕ (t, x) is uniformly continuous on every bounded subset K ⊂ X uniformly for t ∈ R.

6.1 Pseudo-Almost Automorphic Functions

171

If x(t) ∈ PAA(X ), then f (·, x(·)) ∈ PAA(X ). The proof of Theorem 6.8 requires the following technical lemma. Lemma 6.9 ([128, 132]). Let f ∈ BC(R, X ). The function f ∈ PAP0 (X ) if and only if for any ε > 0, 1 mes(Mr,ε ( f )) = 0, r→∞ 2r lim

where mes (·) stands for the Lebesgue measure and Mr,ε ( f ) := {t ∈ Qr : f (t) ≥ ε }. Proof. Necessity—Suppose that there exists ε > 0 such that 1 mes(Mr,ε ( f )) = 0. 2r

lim

r→∞

Thus there exists δ > 0 such that lim

rn →∞

1 mes(Mrn ,ε ( f )) ≥ δ 2rn

for some rn > n. Now 1 2rn



1 f (t) dt = 2rn Qrn

 

Mrn ,ε

f (t) dt +

≥

1 2rn

≥

1 mes(Mrn ,ε ( f ))ε 2rn

Mrn ,ε

 Qrn \Mrn ,ε

f (t) dt

f (t) dt

≥ ε δ , which contradicts the fact that f ∈ PAP0 (X ). Sufficiency—Obviously, f ∞ ≤ M and for any ε > 0, there exists r0 > 0 such that for r > r0 ,

ε 1 mes(Mr,ε ( f )) < . 2r M+1

172

6 Pseudo-Almost Automorphic Functions

Then 1 2r



1 f (t) dt = 2r Qr

 Mr,ε

f (t) dt +

 Qr \Mr,ε

 f (t) dt

ε M 1 mes(Mr,ε ( f )) + (2r − mes(Mr,ε ( f ))) · 2r 2r M+1 ε Mε + ≤ M+1 M+1 = ε, ≤

and hence 1 r→∞ 2r



lim

Qr

f (t) dt = 0.

Proof (Theorem 6.8). Using the facts that f ∈ PAA(R × X ) and x ∈ PAA(X ), it follows that f = g + ϕ and x = α + β , where g ∈ AA(R × X ), ϕ ∈ PAP0 (R × X ), α ∈ AA(X ), β ∈ PAP0 (X ). Now f (t, x(t)) = g(t, α (t)) + f (t, x(t)) − g(t, α (t)) = g(t, α (t)) + f (t, x(t)) − f (t, α (t)) + φ (t, α (t)). Let G(t) = g(t, α (t)),

Φ (t) = f (t, x(t)) − f (t, α (t)) + ϕ (t, α (t)). Using assumption (i) it follows that G ∈ AA(X ) by Lemma 6.9. In order to show that f (·, x(·)) ∈ PAA(R × X ) it is enough to prove Φ ∈ PAP0 (X ). Let us first show that f (t, x(t)) − f (t, α (t)) ∈ PAP0 (X ). Indeed, the function t → f (t, x(t)) − f (t, α (t)) is bounded and continuous. Further, using Lemma 2.1, it is enough to show that lim

r→∞

1 mes(Mr,ε ( f (t, x(t)) − f (t, α (t)))) = 0. 2r

Now using the facts that x(t), α (t) are bounded, there exists a bounded subset K ⊂ X such that x(R), α (R) ⊂ K. From assumptions (i) and (ii), f is uniformly continuous on the bounded subset K ⊂ X uniformly for t ∈ R. Thus for each ε > 0, there exists δ > 0 such that x, y ∈ K and x − y ≤ δ yield f (t, x) − f (t, y) < ε , for all t ∈ R.

6.1 Pseudo-Almost Automorphic Functions

173

Now 1 1 mes(Mr,ε ( f (t, x(t)) − f (t, α (t)))) ≤ mes(Mr,δ (x(t) − α (t))) 2r 2r 1 = mes(Mr,δ (β (t))), 2r where β (t) = x(t) − α (t). Since β (t) ∈ PAP0 (X ), Lemma 6.9 yields, for δ , lim

r→∞

1 mes(Mr,δ (β (t))) = 0. 2t

Thus lim

r→∞

1 mes(Mr,ε ( f (t, x(t)) − f (t, α (t)))) = 0, 2r

and hence f (t, x(t)) − f (t, α (t)) ∈ PAP0 (X ). We next show that ϕ (t, α (t)) ∈ PAP0 (X ). Since t → ϕ (t, x(t)) is continuous in Qr , then it is uniformly continuous on Qr . Set I = α (Qr ). Clearly, I is compact in R, and so there exists finite open balls Ok , (k = 1, 2, . . . , m) with center xk ∈ I and radius δ small enough such that I⊂

m

Ok

k=1

and ϕ (t, α (t)) − ϕ (t, xk ) <

ε , α (t) ∈ Ok , t ∈ Qr . 2

The set Bk = {t ∈ Qr : α (t) ∈ Ok } is open in Qr and Qr = E1 = B1 ,

Ek = Bk \

k−1

j=1

Then Ei ∩ E j = 0/ when i = j, 1 ≤ i, j ≤ m. Now {t ∈ Qr : ϕ (t, α (t)) ≥ ε }

m

k=1 Bk .

B j (2 ≤ k ≤ m).

(6.5) Let

174

⊂

6 Pseudo-Almost Automorphic Functions m

{t ∈ Ek : ϕ (t, α (t)) − ϕ (t, xk ) + ϕ (t, xk ) ≥ ε }

k=1

⊂

m 

t ∈ Ek : ϕ (t, α (t)) − ϕ (t, xk ) ≥

k=1

ε  ε  t ∈ Ek : ϕ (t, xk ) ≥ . 2 2

Using Eq. (6.5) it follows that {t ∈ Ek : ϕ (t, α (t)) − ϕ (t, xk ) ≥ ε2 } are empty for all 1 ≤ k ≤ m. Therefore,   m 1 1 mes(Mr,ε (ϕ (t, x(t)))) ≤ ∑ mes Mr, ε (ϕ (t, xk )) . 2 2r k=1 2r Since ϕ ∈ PAP0 (R × X ), and   1 mes Mr, ε (ϕ (t, xk )) → 0 2 2r

as r → ∞,

we have lim

r→∞

1 mes (Mr,ε (ϕ (t, α (t))) = 0. 2r

That is ϕ (t, α (t)) ∈ PAP0 (X ). Hence G(t) ∈ AA(X ) and Φ (t) ∈ PAP0 (X ). That is f (·, x(·)) ∈ PAA(X ), which completes the proof. Theorem 6.10. Let f = G + Φ ∈ PAA(R × X ) where G ∈ AA(R × X ) and Φ ∈ PAP0 (R × X ). Suppose that there exists a constant L > 0 such that f (t, x) − f (t, y) ≤ L x − y for all x, y ∈ X and t ∈ R. If ϕ ∈ PAA(X ), then the function F(t) = f (t, φ (t)) ∈ PAA(X ). Proof. Let f ∈ PAA(R × X ) and write f = G + H where G ∈ AA(R × X ) and H ∈ PAP0 (R × X ). Moreover, write ϕ = ϕ1 + ϕ2 with ϕ1 ∈ AA(X ) and ϕ2 ∈ PAP0 (X ). Now f (t, ϕ (t)) ≤ L ϕ ∞ + f (t, 0) ≤ L ϕ ∞ + G(t, 0) + H(t, 0) ≤ L ϕ ∞ + G(·, 0) + H(·, 0) , and hence f (·, ϕ (·)) ∈ BC(R × X ), and f (·, ϕ (·)) = G(·, ϕ1 (·)) + f (·, ϕ (·)) − G(·, ϕ1 (·))

6.1 Pseudo-Almost Automorphic Functions

175

= G(·, ϕ1 (·)) + f (·, ϕ (·)) − f (·, ϕ1 (·)) + H(·, ϕ1 (·)). In view of Theorem 4.17 it easily follows that G(·, ϕ1 (·)) ∈ AA(X ). Now, using the fact that f is Lipschitzian and ϕ2 ∈ PAP0 (X ) it follows that f (·, ϕ (·)) − f (·, ϕ1 (·)) ∈ PAP0 (X ). To complete the proof we must show that 1 r→∞ 2r



lim

Qr

H(t, ϕ1 (t)) dt = 0

but this has already been done in the proof of Theorem 5.13. Theorem 6.11 (Liang et al. [131]). Let f = g + ϕ ∈ PAA(R × X ) where g ∈ AA(R × X ), ϕ ∈ PAP0 (R × X ). Suppose that the following conditions hold: (i) The function x → g(t, x) is uniformly continuous on all bounded subsets of X uniformly in t ∈ R. (ii) There exists a nonnegative function μ ∈ L p (R) (1 ≤ p ≤ ∞) such that f (t, x) − f (t, y) ≤ μ (t) x − y for all x, y ∈ X and t ∈ R. If x(t) ∈ PAA(X ), then f (·, x(·)) ∈ PAA(X ). Proof. Let f ∈ PAA(R × X ) and x ∈ PAA(X ) and write f = g + ϕ and x = α + β , where g ∈ AA(R × X ), ϕ ∈ PAP0 (R × X ), α ∈ AA(X ), β ∈ PAP0 (X ). Now f (t, x(t)) = g(t, α (t)) + f (t, x(t)) − f (t, α (t)) + φ (t, α (t)). Set G(t) := g(t, α (t)) and Φ (t) = f (t, x(t)) − f (t, α (t)) + φ (t, α (t)). Using assumption (i) it follows that G(t) ∈ AA(X ). In order to show that f (·, x(·)) ∈ PAA(X ), it is sufficient to prove that Φ (t) ∈ PAP0 (R, X ). Let us first show that f (t, x(t)) − f (t, α (t)) ∈ PAP0 (X ). Obviously, t → f (t, x(t)) − f (t, α (t)) is a bounded continuous function. Clearly, it can be assumed that f (t, x(t)) − f (t, α (t)) ≤ M, ∀t ∈ R. Using the facts that the functions x(t), α (t) are bounded, there exists a bounded subset B ⊂ R such that x(R), α (R) ⊂ B. Using assumption (ii), for all ε > 0, x − y ≤ ε , yields f (t, x) − f (t, y) < ε μ (t),

for all t ∈ R.

176

6 Pseudo-Almost Automorphic Functions

Using the fact that β ∈ PAP0 (R, X ) and Lemma 6.9, we obtain that 1 mes(Mr,ε (β (t))) = 0, r→∞ 2r lim

and hence 1 2r

 Qr

f (t, x(t)) − f (t, α (t)) dt



=

1 2r

≤

ε M mes(Mr,ε (β (t)))+ 2r 2r

Mr,ε (β (t))

f (t, x(t))− f (t, α (t)) dt+  Qr

1 2r

 Qr \Mr,ε (β (t))

f (t, x(t))−f (t, α (t)) dt

μ (t)dt.

There are three cases to consider: Case 1—If p = 1, we can see

ε 2r

 Qr

μ (t)dt ≤

ε 2r

 +∞ −∞

μ (t)dt ≤

ε μ L1 (R) 2r

.

Case 2—If p = ∞, we can see

ε 2r

 Qr

μ (t)dt ≤ ε μ L∞ (R) .

Case 3—If 1 < p < ∞, then

ε 2r

 Qr

μ (t)dt ≤

ε 2r

 Qr

μ p (t)dt

1 

1

p

dt Qr

q

=

ε μ L p (R) 1

(2r)1− q

.

where q = p(p − 1)−1 . In view of the above, 1 r→∞ 2r



lim

Qr

f (t, x(t)) − f (t, α (t)) dt = 0.

Let us now show that t → ϕ (t, α (t)) ∈ PAP0 (X ) . Let ε > 0. Using the fact that x → g(t, x) is uniformly continuous on all bounded subsets uniformly for t ∈ R, there is a δ > 0 such that g(t, x) − g(t, y) ≤ ε for all x, y ∈ B with x − y ≤ δ . Setting δ0 = min{ε , δ }, we obtain ϕ (t, x) − ϕ (r, x) ≤ f (t, x) − f (t, y) + g(t, x) − g(t, y) ≤ ε (μ (t) + 1), for all x, y ∈ B with x − y ≤ δ0 .

6.1 Pseudo-Almost Automorphic Functions

177

Set I = α (Qr ). Clearly I ⊂ R is compact as the image of a compact set under a continuous function is compact. Thus, there exists a finite number of open balls Ok , (k = 1, 2, . . . , m) with center xk ∈ I and radius δ0 small enough such that m

I⊂

Ok

k=1

and ϕ (t, α (t)) − ϕ (t, xk ) < ε (μ (t) + 1), α (t) ∈ Ok , t ∈ [−r, r]. Suppose ϕ (t, x p ) = max { ϕ (t, xk ) }, 1≤k≤m

where p is an index number among {1, 2, . . . , m}. The set Bk = {t ∈ Qr : α (t) ∈ Ok } is open in Qr and Qr =

m

Bk .

k=1

Let E1 = B1 ,

Ek = Bk \

k−1

B j (2 ≤ k ≤ m).

j=1

Now Ei ∩ E j = φ when i = j, 1 ≤ i, j ≤ m. Observe 1 2r

 Qr

ϕ (t, α (t)) dt =

1 2r

 m

k=1 Bk



≤

1 m ∑ r k=1

≤

1 m ∑ 2r k=1

≤ε+

ε 2r

Bk

ϕ (t, α (t)) dt

ϕ (t, α (t)) − ϕ (t, xk ) + ϕ (t, xk ) dt

 Bk

ε (μ (t) + 1)dt +

 Qr

μ (t)dt +

1 2r

 Qr

1 m ∑ 2r k=1

 Bk

ϕ (t, xk ) dt

ϕ (t, x p ) dt

178

6 Pseudo-Almost Automorphic Functions

Using similar arguments as above, it follows that 1 r→∞ 2r



lim

Qr

ϕ (t, α (t)) dt = 0,

and hence t → ϕ (t, α (t)) ∈ PAP0 (X ). Consequently, G ∈ AA(X ) and Φ ∈ PAP0 (X ). Therefore, t → f (t, x(t)) ∈ PAA(X ).

6.2 Weighted Pseudo-Almost Automorphic Functions The concept of weighted-pseudo-almost automorphy, which is a natural generalization of the notion of weighted pseudo-almost periodicity, was introduced in the literature by Blot et al. [19]. Since then, such a notion has generated various developments and extensions, see, e.g. [19, 65, 68, 69, 129]. This section is devoted to the study of basic properties of these new functions including their composition.

6.2.1 Basic Definitions Definition 6.12 (Blot et al. [19]). Fix μ ∈ U∞ . A continuous function f : R → X is called weighted pseudo-almost automorphic if it can be written as f = h + ϕ , where h ∈ AP(X ) and ϕ ∈ PAP0 (X , μ ). The function h and ϕ are respectively called the almost automorphic and the weighted ergodic components of f . The collection of weighted pseudo-almost periodic functions will be denoted by PAA(X , μ ). Definition 6.13. Fix μ ∈ U∞ . A jointly continuous function f : R × X → X is called weighted pseudo-almost automorphic if F = G + Φ , where G ∈ AA(X × R) and Φ ∈ PAP0 (R × X , μ ). The collection of such functions will be denoted by PAA(R × X , μ ). We have previously seen that the decomposition for pseudo-almost automorphic functions is unique (Theorem 6.1). Unfortunately, this is not always the case for weighted pseudo-almost automorphic functions. As in the case of weighted pseudo-almost periodic functions, here also the uniqueness of the decomposition of weighted pseudo-almost automorphic functions depends upon the translationinvariance of the weighted ergodic space PAP0 (X , μ ). Proposition 6.14. Fix μ ∈ U∞Inv . Let f ∈ PAA(X , μ ). If g ∈ L1 (R), then f ∗ g, the convolution of f with g on R, belongs to PAA(X , μ ). Proof. The proof is based on both Theorem 4.5 and Proposition 5.21 and hence is omitted.

6.2 Weighted Pseudo-Almost Automorphic Functions

179

Using similar arguments as in the proof of Proposition 5.19 we can establish the following: Proposition 6.15. Fix μ ∈ U∞Inv . Let f ∈ PAA(X , μ ) and write f = h + ϕ where h ∈ AA(X ) and ϕ ∈ PAP0 (X , μ ), then g(X ) ⊂ f (X ). Proposition 6.16. Fix μ ∈ U∞Inv . If f ∈ PAA(X , μ ) then there exists a unique h ∈ AA(X ) and a unique ϕ ∈ PAP0 (X , μ ) such that f = h + ϕ . Proof. The proof makes use of similar ideas as that of Proposition 5.20 and hence is omitted. Fix μ ∈ U∞Inv . If f , g ∈ PAA(X , μ ) and λ ∈ F, we can then show that f + g and λ f belong to PAA(X , μ ). If g is F-valued for instance, it can be easily shown that f g ∈ PAA(X , μ ). It is also clear that (PAA(X , μ ), · ∞ ) is a normed space. In the next theorem, we show that the normed vector space (PAA(X , μ ), · ∞ ) is in fact complete. Theorem 6.17. Fix μ ∈ U∞Inv . The space of pseudo-almost automorphic functions PAA(X , μ ) equipped with the sup-norm is a Banach space. Proof. The proof follows along the same lines as in the pseudo-almost automorphic case and hence is omitted.

6.2.2 A Composition Theorem Theorem 6.18. Let μ ∈ U∞ and let f ∈ PAA(R × X , μ ) satisfying the Lipschitz condition f (t, u) − f (t, v) ≤ L u − v for all u, v ∈ X , t ∈ R. If h ∈ PAA(X , μ ), then f (·, h(·)) ∈ PAA(X , μ ). Proof. Let f = g + ϕ where g ∈ AA(R × X ) and ϕ ∈ PAP0 (R × X , μ ). Similarly, let h = h1 + h2 , where h1 ∈ AA(X ) and h2 ∈ PAP0 (X , μ ). Clearly, f (·, h(·)) ∈ C(R, X ). Next, decompose f as follows: f (·, h(·)) = g(·, h1 (·)) + f (·, h(·)) − f (·, h1 (·)) + ϕ (·, h1 (·)). Using Theorem 4.16, one can easily see that g(·, h1 (·)) ∈ AA(X ). Now, set F(·) = f (·, h(·)) − f (·, h1 (·)). Clearly, F ∈ PAP0 (X , μ ). Indeed, for r > 0, 1 μ (Qr )

 Qr

F(s) μ (s)ds =

1 μ (Qr )

≤

L μ (Qr )

 Qr



Qr

f (s, h(s)) − f (s, h1 (s)) μ (s)ds h(s) − h1 (s) μ (s)ds

180

6 Pseudo-Almost Automorphic Functions

≤

L μ (Qr )

 Qr

h2 (s) μ (s)ds,

and hence 1 r→∞ μ (Qr )



lim

Qr

F(s) μ (s)ds = 0.

To complete the proof we have to show that 1 r→∞ μ (Qr )



lim

Qr

ϕ (s, h1 (s)) μ (s)ds = 0.

but this has already been done in the proof of Theorem 5.26.

6.3 C(n) -Pseudo-Almost Automorphic Functions An important subclass of the class of pseudo-almost periodic functions is that of C(n) -pseudo-almost automorphic functions, which was introduced in the literature by Diagana and Nelson [71]. In this section, we study basic properties of these functions.

6.3.1 Basic Definitions Definition 6.19 (Diagana and Nelson [71]). A function f ∈ BC(n) (R, X ) is called C(n) -pseudo-almost automorphic (respectively, C(n) -compact pseudo-almost automorphic) if it can be expressed as f = g+φ, (n)

where g ∈ AA(n) (X ) and φ ∈ PAP0 (X ) (respectively, f = g + φ , where g ∈ (n)

KAA(n) (X ) and φ ∈ PAP0 (X )). The collection of such functions is denoted by PAA(n) (X) (respectively, KPAA(n) (X)).

The function g appearing in Definition 6.19 will be called the C(n) -almost automorphic (respectively, the C(n) -compact almost automorphic) component of f . The following inclusions hold: · · · → PAA(n+1) (X ) → PAA(n) (X ) → · · · → PAA(1) (X ) → PAA(X )

6.3 C(n) -Pseudo-Almost Automorphic Functions

181

and · · · → KPAA(n+1) (X ) → KPAA(n) (X ) → · · · → KPAA(1) (X ) → PAA(X ). Definition 6.20. A bounded continuous function F : R × X → Y is said to belong to PAA(n) (R × X ) if it can be expressed as F = G + Φ, (n)

where G ∈ AA(n) (R × X ) and Φ ∈ PAP0 (R × X ). The collection of such functions is denoted by PAA(n) (R × X ).

6.3.2 Properties of C(n) -Pseudo-Almost Automorphic Functions   Theorem 6.21. The space PAA(n) (X ) equipped with the norm  · (n) is a Banach space. Theorem 6.22. If f ∈ PAA(n) (R) and if g ∈ L1 (R), then their convolution f ∗ g ∈ PAA(n) (R). Theorem 6.23. If f ∈ PAA(n) (X ) such that f (n+1) is uniformly continuous, then f ∈ PAA(n+1) (X ). Theorem 6.24. Suppose that the Banach space X is uniformly convex. Let f = h + ϕ ∈ PAA(X ) with h ∈ AA(X ) and ϕ ∈ PAP0 (X ). Define the functions  t

H(t) := 0

h(s)ds and Φ (t) :=

 t 0

ϕ (s)ds,

t ∈R

and suppose (a) H is bounded; (b) there exists ξ ∈ X such that Φ − ξ ∈ PAP0 (X ); and (c) f (k) is uniformly continuous for k = 0, 1, . . . , n. Then F belongs to PAA(n+1) (X ). Proof. From Theorem 5.12 it follows that F = H + Φ belongs to PAA(X ). Now since F = f is uniformly continuous, then according to Theorem 4.22 it follows that F belongs to PAA(1) (X ). Now since F ∈ PAA(X ) and F = f is uniformly continuous it follows from Theorem 4.22 that F ∈ PAA(2) (X ). By induction it can be easily shown that F ∈ PAA(n+1) (X ).

182

6 Pseudo-Almost Automorphic Functions

6.4 S p -Pseudo-Almost Automorphic Functions The concept of S p -pseudo-almost automorphy was introduced a few years ago by Diagana [55]. Such a concept is a natural generalization of the notion of S p -pseudoalmost periodicity. This section is devoted to properties of these functions.

6.4.1 Basic Definitions Definition 6.25 (Diagana [55]). Let p ≥ 1. A function f ∈ BS p (X ) is called S p pseudo-almost automorphic (or Stepanov-like pseudo-almost automorphic) if it can be expressed as f = h + ϕ,     where hb ∈ AA L p ((0, 1), X ) and ϕ b ∈ PAP0 L p ((0, 1), X ) . p The collection of such functions will be denoted by Spaa (X ). In other words, a function f ∈ L p (R, X ) is said to be S p -pseudo-almost automorphic if its Bochner transform f b : R → L p ((0, 1), X ) is pseudo-almost automorphic in the sense that there exist two functions h, ϕ : R → X such that f = h + ϕ , where hb ∈ AA(L p ((0, 1), X )) and ϕ b ∈ PAP0 (L p ((0, 1), X )), that is, 1 lim r→∞ 2r





t+1

t

Qr

φ (σ ) d σ

1/p dt = 0.

p

Proposition 6.26. If f ∈ PAA(X ), then f is S p -pseudo-almost automorphic for any 1 ≤ p < ∞. Proof. The proof follows along the same lines as that of Proposition 5.36 and hence omitted. Definition 6.27. A function F : R × X → X , (t, u) → F(t, u) with F(·, u) ∈ L p (R, X ) for each u ∈ X , is said to be S p -pseudo-almost automorphic in t ∈ R uniformly in u ∈ X if t → F(t, u) is S p -pseudo-almost automorphic for each u ∈ K where K ⊂ X is a bounded subset. This means, there exist two functions H, Φ : R × X → X such that F = H + Φ , where H b ∈ AA(R × L p ((0, 1), X )) and Φ b ∈ PAP0 (R × L p ((0, 1), X )), i.e., 1 r→∞ 2r





lim

Qr

t

t+1

Φ (σ , u) p d σ

1/p

uniformly in u ∈ K where K ⊂ X is a bounded subset.

dt = 0

6.4 S p -Pseudo-Almost Automorphic Functions

183

The collection of those S p -pseudo-almost automorphic functions F : R × X → X p will be denoted by Spaa (R × X ).

6.4.2 A Composition Theorem The next composition theorem is due to Fan et al. [95]. Although its proof is highly technical, we will reproduce it for the sake of clarity. p Theorem 6.28 (Fan et al. [95]). Let f = g + h ∈ Spaa (R × X ) for p > 1 such b p b that h ∈ AA(R × L ((0, 1), X )) and h ∈ PAP0 (R × L p ((0, 1), X )). Moreover, we suppose that p (a) There exists a nonnegative function L ∈ BSr (R) with r ≥ max{p, p−1 } such that for all u, v ∈ X and t ∈ R,

f (t, u) − f (t, v) ≤ L(t) u − v . p (R, X ) and t ∈ R, (b) There exists a constant L > 0 such that for all u, v ∈ Lloc



1 0

1/p g(t + s, u(s)) − g(t + s, v(s)) ds p

≤L

 0

1

1/p u(s) − v(s) ds p

.

p If ϕ=α +β ∈ Spaa (X )where α b∈AA(L p ((0, 1), X )) and β b ∈ PAP0 (L p ((0, 1), X )) and such that K = {α (t) : t ∈ R} is compact, then there exists q ∈ [1, p) such that q t → f (t, ϕ (t)) belongs to Spaa (X ).

The proof of Theorem 6.28 requires the next technical lemma, which is also due to Fan et al. [95]. p Lemma 6.29 (Fan et al. [95]). If f = g + h ∈ Spaa (R × X ) with gb ∈ AA(R × p b p L ((0, 1), X )), h ∈ PAP0 (R × L ((0, 1), X )) and if there exists a nonnegative p function L ∈ BSr (R) with r ≥ max{p, p−1 } such that for all u, v ∈ X and t ∈ R,

f (t, u) − f (t, v) ≤ L(t) u − v ,

(6.6)

then the function g satisfies g(t + ·, u) − g(t + ·, v) p ≤ L S p u − v . p (R × X ) it follows that f b (·, u) ∈ Proof. Using the fact that f = g + h ∈ Spaa p b p PAA(L ((0, 1), X )), g (·, u) ∈ AA(L ((0, 1), X )), hb (·, u) ∈ PAP0 (L p ((0, 1), X )) for each u ∈ X . Using Eq. (6.2), it follows that

{ f (t + ·, u);t ∈ R} ⊃ {g(t + ·, u);t ∈ R}

in L p ((0, 1), X ),

184

6 Pseudo-Almost Automorphic Functions

for each u ∈ X , which by Eq. (6.6) yields for all u, v ∈ X and t ∈ R g(t + ·, u) − g(t + ·, v) p ≤ sup f (t + ·, u) − f (t + ·, v) p ≤ L S p u − v . t∈R

Proof (Theorem 6.28). The proof follows that given by Fan et al. [95]. Indeed, pq p using the fact that r ≥ p−1 , it follows that there exists q ∈ [1, p) such that r = p−q . Setting p , p−q

p =

p , q

q =

one can easily see that p , q > 1 and p1 + q1 = 1. Write f = g + h, where gb ∈ AA(R × L p ((0, 1), X )) and hb ∈ PAP0 (R × p L ((0, 1), X )). Similarly, write ϕ = α + β , where α b ∈ AA(L p ((0, 1), X )) and β b ∈ PAP0 (L p ((0, 1), X )). Now f (t, ϕ (t)) = g(t, α (t)) + f (t, ϕ (t)) − g(t, α (t)) = g(t, α (t)) + f (t, ϕ (t)) − f (t, α (t)) + h(t, α (t)). Set Γ (·) = f (·, ϕ (·)) = g(·, α (·)) + f (·, ϕ (·)) − f (·, α (·)) + h(·, α (·)). To complete the proof, we need to show that Γ b ∈ PAA(Lq ((0, 1), X )). p p Note that g satisfies condition (b). Further, g ∈ Saa (R × X ), α ∈ Saa (X ) and the set K = {α (t) : t ∈ R} is compact. Thus using Theorem 4.53 it follows that p g(·, α (·)) ∈ Saa (X ). Setting Ω (·) := f (·, ϕ (·)) − f (·, α (·)), we claim that Ω b ∈ PAP0 (Lq ((0, 1), X )). Indeed, using (a) it follows that Ω (t + ·) q = ≤ ≤



t+1

t



t+1

t

  t+1 t

≤ L Sr

f (s, ϕ (s)) − f (s, α (s)) q ds Lq (s) β (s) q ds Lr (s) ds



t+1

t

1/q

1/q

1/(p q)  (

t+1

t

β (s) p ds

1/p



β (s) p ds)1/(q q)

,

which, for any r > 0, yields 1 2T

 QT

Ω (t + ·) q dt ≤

L Sr 2T



 QT

t

t+1

β (s) p ds

1/p

dt.

6.4 S p -Pseudo-Almost Automorphic Functions

185

Using the fact that β b ∈ AA0 (L p ((0, 1), X )) it follows that 1 T →∞ 2T



lim

QT

Ω (t + ·) q dt = 0,

which yields Ω b ∈ PAP0 (Lq ((0, 1), X )). Letting H(·) := h(·, α (·)), our last task consists of showing that H b ∈ PAP0 (Lq ((0, 1), X )). Indeed, using assumptions (a)–(b) it follows that h(t + ·, u(·)) − h(t + ·, v(·)) q ≤ f (t + ·, u(·)) − f (t + ·, v(·)) q + g(t + ·, u(·)) − g(t + ·, v(·)) q  1 q 1/q  1 1/q ≤ L (t + s) u(s) − v(s) q ds +L u(s) − v(s) q ds 0

0

≤( L Sr + L) u(·) − v(·) p

(6.7)

p (R, X ). for all t ∈ R, u, v ∈ Lloc Using the fact that the set K = {α (t) : t ∈ R} is compact, for each ε > 0, there exists a finite number of open balls Ok (k = 1, 2, . . . , m) centered at xk ∈ K and radius ε such that

{α (t) : t ∈ R} ⊂

m

Ok .

k=1

Now choose the set Bk such that Bk = {s ∈ R : α (s) ∈ Ok } and R=

m

Bk .

k=1



Let E1 = B1 , Ek = Bk \ k−1 / when i = j, j=1 B j , (2 ≤ k ≤ m). Then Ei ∩ E j = 0 1 ≤ i, j ≤ m. Define the step function x : R → X by x(s) = xk , s ∈ Ek , i = 1, 2, . . . , m. It is no hard to see that α (s) − x(s) ≤ ε for all s ∈ R. Using Eq. (6.7), it follows that 1 2T ≤

1 2T

 

QT

QT

h(t + ·, α (t + ·)) q dt  

t+1

t

≤( L Sr + L)ε +

h(s, α (s)) − h(s, x(s)) q ds 1 2T m



m

(

QT k=1 [t,t+1] Ek

1 2T k=1

≤( L Sr + L)ε + ∑

(∑



 QT

 t

t+1

1/q

+(

 t+1 t

 h(s, x(s)) q ds)1/q dt

h(s, xk ) q ds)1/q dt

h(s, xk ) p ds

1/p

dt.

186

6 Pseudo-Almost Automorphic Functions

Now using the fact that ε is arbitrary and that hb ∈ PAP0 (R × L p ((0, 1), X )), it follows that 1 T →∞ 2T



lim

QT

h(t + ·, α (t + ·)) q dt = 0.

(6.8)

Consequently, using Eq. (6.8), we get H b ∈ PAP0 (Lq ((0, 1), X )). The proof is complete.

(n)

6.5 S p -Pseudo-Almost Automorphic Functions (n)

The class of S p -pseudo-almost automorphic functions is due to Diagana et al. (n) [73]. Those functions generalize S p -pseudo-almost periodic functions in a natural fashion. This section is devoted to a brief introduction to these functions. (n),p

(n)

Definition 6.30. The space Spaa (X ) of S p -pseudo-almost automorphic functions (or Stepanov-like C(n) -pseudo-almost automorphic) consists of all   (n) f ∈ BS p (X ) such that ( f (k) )b ∈ PAA L p ((0, 1), X ) for k = 0, 1, . . . , n. (n)

(n)

In other words, a function f ∈ BS p (X ) is said to be S p -pseudo-almost automorphic if f = h+ϕ such that for every sequence of real numbers (s m )m∈N , there exists a subsequence (n) (sm )m∈N and a function g ∈ BS p (X ) such that 

t+1

t



(k)

(k)

1/p

h (sm + s) − g (s) ds

t+1

t

(k)

p

(k)

→ 0, and 1/p

g (s − sm ) − h (s) ds p

→0

for k = 0, 1, . . . , n, as m → ∞ pointwise on R, and 1 lim r→∞ 2r for k = 0, 1, . . . , n.

 Qr

 t

1/p  ϕ (k) (σ ) p d σ dt = 0

t+1 

(n)

6.5 S p -Pseudo-Almost Automorphic Functions

187

Similarly, Definition 6.31. A function F : R × X →  X , (t, u) → F(t, u) with F(·, u) ∈ (n),p (n) Spaa (X ) for each u ∈ K where K ⊂ X is a bounded subset, is said to be S p pseudo-almost automorphic. (n)

The collection of those S p -pseudo-almost automorphic functions F : R × X → Y (n),p will be denoted by Spaa (R × X ). The following inclusions hold: (n+2),p

· · · → Spaa

(n+1),p

(X ) → Spaa

(n),p

(1),p

p (X ) → Spaa (X ) → · · · → Spaa (X ) → Spaa (X ).

(n),p

Theorem 6.32. If f ∈ Spaa (R) and if g ∈ L1 (R), then their convolution f ∗ g ∈ (n),p Spaa (R). (n),p

Proof. Let f ∈ Spaa (R) and let g ∈ L1 (R). Let f = h + ϕ such that (h(k) )b =     (hb )(k) ∈ AA L p (0, 1; R) and (ϕ (k) )b = (ϕ b )(k) ∈ PAP0 L p ((0, 1), R) for k = 0, 1, . . . , n. To complete the proof it suffices to show that   [(h ∗ g)(k) ]b ∈ AA L p ((0, 1), R) and   [(ϕ ∗ g)(k) ]b ∈ PAP0 L p ((0, 1), R) for all k = 0, 1, 2, . . . , n. (k) b (k) b (k) b Indeed, using the fact  p [(h ∗ g) ] = [h ∗ g] = (h ) ∗ g it easily follows (k) b that [(h ∗ g) ] ∈ AA L ((0, 1), R) for all k = 0, 1, 2, . . . , n. Similarly, from [(ϕ ∗ g)(k) ]b = [ϕ (k) ∗ g]b = (ϕ (k) )b ∗ g it easily follows that   [(ϕ ∗ g)(k) ]b ∈ PAP0 L p ((0, 1), R) for all k = 0, 1, 2, . . . , n. (n),p

Proposition 6.33. The space Spaa (X ) equipped with the norm · p,(n) is a Banach space. (n),p

Proof. The proof is based on the fact that Spaa (X ) is a closed subspace of (n) BS p (X ). (n),p

Proposition 6.34. If f ∈ PAA(n) (X ), then f ∈ Spaa (X ). That is, PAA(n) (X ) ⊂ (n),p Spaa (X ). (n)

Proof. Let f = h + ϕ where h ∈ AA(n) (X ) and ϕ ∈ PAP0 (X ). Clearly, f (k) ∈ (n)

BS p (X ) for k = 0, 1, . . . , n and hence f ∈ BS p (X ). To complete the proof it suffices to show that (h(k) )b ∈ AA(L p ((0, 1; X )) and (ϕ (k) )b ∈ PAP0 (L p (((0, 1), X ))

188

6 Pseudo-Almost Automorphic Functions

for k = 0, 1, 2, . . . , n. Using the fact that AA(X ) ⊂ AS p (X ) for p ∈ [1, ∞) it follows that (h(k) )b ∈ AA(L p (((0, 1), X )) for k = 0, 1, 2, . . . , n. Similarly, if k ∈ PAP0 (X ), then kb ∈ PAP0 (L p (((0, 1), X )) (see Proposition 6.26). Using that it readily follows that (ϕ (k) )b ∈ PAP0 (L p (((0, 1), X )) for k = 0, 1, 2, . . . , n.

Bibliographical Notes The results presented in this chapter and some of their proofs were based on several sources. Among them are the following: Diagana et al. [55, 72, 76, 77], Diagana and Nelson [71, 75], Diagana et al. [73], Blot et al. [19], Boukli-Hacenea and Ezzinbi [27, 28], Cieutat and Ezzinbi [39], Fan et al. [95, 96], Ezzinbi et al. [88, 89, 91, 93], Liang et al. [129–132], N’Gu´er´ekata [146], and Xiao et al. [165]. There are certainly lots of unanswered questions on these new classes of functions. Here is a list of questions, which may be of interest to people working in connection with these function spaces and related topics. 1. The concepts of weighted pseudo-almost automorphy, S p -pseudo-almost auto(n),p morphy, C(n) -pseudo-almost automorphy, and Spaa -pseudo-almost automorphy are yet to be understood. More investigations on these function spaces are needed. Once that is done, one may look into a weighted theory for these spaces. 2. Study S p -pseudo-almost automorphic, weighted S p -pseudo-almost automorphic, (n),p and Spaa -pseudo-almost automorphic functions in the case when the exponent p = p(x) is a (positive) function of x. This question is relevant especially that Lebesgue spaces with variable exponents, L p(x) , are now well understood. 3. If μ , ν ∈ U∞ , what are the connections between PAA(X , μ ), PAA(X , ν ) and the space PAA(X , μν ) where μν is the product of both weights defined by (μν )(t) = μ (t)ν (t) for all t ∈ (0, ∞). Similarly, what are the connections between PAA(X , μ ), PAA(X , ν ), and the space PAA(X , μ ∗ ν ) where μ ∗ ν is the convolution of μ with ν defined by (μ ∗ ν )(t) = for all t ∈ R?

 ∞ 0

μ (t − s)ν (s)ds

Chapter 7

Existence Results for Some Second-Order Differential Equations

7.1 Introduction Let α ∈ (0, 1). Fix once and for all a separable infinite dimensional complex Hilbert space (H , ·, ·, · ). In this chapter we study the existence of almost periodic (respectively, almost automorphic) solutions to the classes of nonautonomous damped second-order differential equations d2u du + b(t)B + a(t)Au = h(t, u), 2 dt dt

(7.1)

where A : D(A) ⊂ H → H is a self-adjoint linear operator on H whose spectrum consists of isolated eigenvalues 0 < λ1 < λ2 < · · · < λl → ∞ as l → ∞ with each eigenvalue having a finite multiplicity γ j equals to the multiplicity of the corresponding eigenspace, B : D(B) ⊂ H → H is a closed linear operator such that the following holds Hα := (H , D(A))α ,∞ ⊂ D(B),

(7.2)

the functions a, b : R → C are almost periodic (respectively, almost automorphic) and h : R × Hα → H , (t, u) → h(t, u) is almost periodic (respectively, almost automorphic) in t ∈ R uniformly in u ∈ Hα . To investigate the existence of almost periodic (respectively, almost automorphic) solutions to Eq. (7.1), we study the (nonautonomous) Sobolev-type second-order differential equations with operator coefficients given by

T. Diagana, Almost Automorphic Type and Almost Periodic Type Functions in Abstract Spaces, DOI 10.1007/978-3-319-00849-3 7, © Springer International Publishing Switzerland 2013

189

190

7 Existence Results for Some Second-Order Differential Equations

) * d du + g(t, Bu(t)) = w(t)Au(t) + f (t, Bu(t)), t ∈ R dt dt

(7.3)

where A, B are the same as in Eq. (7.1), the function w : R → C given by w(t) = ρ (t)eiθ (t) for all t ∈ R is assumed to be almost periodic (respectively, almost π automorphic) and further there exist three constants ρ0 , ρ1 > 0 and θ0 ∈ (0, ) 4 satisfying the following conditions

ρ0 ≤ ρ (t) ≤ ρ1 for all t ∈ R,

(7.4)

π + 2θ0 < θ (t) < 2π − 2θ0 for all t ∈ R,

(7.5)

and

and the functions f , g : R × D(A) → H are almost periodic (respectively, almost automorphic) in the first variable uniformly in the second one. To deal with Eq. (7.3), our strategy consists of rewriting it as a nonautonomous first-order differential equation on D(A) × H and then study the obtained first-order differential equation with the help of dichotomy tools as well as the well-known Schauder fixed-point theorem. Indeed, assuming that u ∈ D(A) is twice differentiable and setting

z :=

u



u + g(t, Bu)

,

then Eq. (7.3) can be rewritten in the form dz = A (t)z + F(t, z(t)), t ∈ R, dt

(7.6)

where A (t) is the family of 2×2-operator matrices defined by  A (t) =

0 I w(t)A 0 ,

t ∈ R.

(7.7)

Clearly, D(A (t)) = D(A) × H is constant in t ∈ R. Notice that the term F appearing in Eq. (7.6) is defined on R × Xα for some α ∈ (0, 1) by  F(t, z(t)) =

−g(t, Bu) f (t, Bu)

,

7.2 Existence of Almost Periodic Solutions

191

where Xα is the real interpolation space of order (α , ∞) between X and D(A (t)) is explicitly given by Xα = (X , D(A (t)))α ,∞ = (H × H , D(A) × H )α ,∞ = Hα × H . If we suppose that b : R → C is differentiable, one can then easily see that Eq. (7.1) is a special case of Eq. (7.3). Indeed, Eq. (7.1) can be rewritten as ) * d du + g(t, ˜ Bu(t)) = w(t)Au(t) ˜ + f˜(t, Bu(t)), t ∈ R dt dt

(7.8)

where w(t) ˜ = −a(t), g(t, ˜ Bu) = b(t)Bu, and f˜(t, Bu) = h(t, u)+ b (t)Bu for all t ∈ R. To deal with the above-mentioned issues, we will make extensive use of ideas developed in [15,50,103,125], dichotomy tools, and the well-known Schauder fixedpoint theorem. For more on abstract second-order differential and related issues, we refer the reader to [35, 36, 118, 139, 166–168].

7.2 Existence of Almost Periodic Solutions 7.2.1 First-Order Differential Equations 7.2.1.1

Preliminaries

Recall that BC(R, Xα ) for α ∈ (0, 1) stands for the Banach space of all bounded continuous functions f : R → Xα when equipped with the α -sup norm f α ,∞ := sup f (t) α t∈R

for f ∈ BC(R, Xα ). Our main task here consists of studying the existence of almost periodic mild solutions to the nonautonomous first-order differential equations du = A(t)u + F(t, u), t ∈ R, dt

(7.9)

where A(t) : D(A(t)) ⊂ X → X is a family of closed linear operators and F : R × Xα → X is almost periodic in the first variable uniformly in the second one. To deal with Eq. (7.9) we need the following assumptions: (C.1) The linear operators {A(t)}t∈R satisfy the Acquistapace–Terreni Conditions. (C.2) There exists 0 ≤ α < β < 1 such that

192

7 Existence Results for Some Second-Order Differential Equations

Xαt = Xα and Xβt = Xβ for all t ∈ R, with uniform equivalent norms. Definition 7.1. Under assumption (C.1), a function v : R → Xα is said to be a mild solution to Eq. (7.9) provided that v(t) = U(t, s)v(s) +

 t s

U(t, τ )F(τ , v(τ ))d τ

(7.10)

for all t ≥ s with t, s ∈ R. It is not difficult to see that if the function F : R × Xα → X is bounded, then u(t) =

 t −∞

U(t, s)P(s)F(s, u(s))ds −

 ∞ t

UQ (t, s)Q(s)F(s, u(s))ds.

(7.11)

for all t ∈ R is a mild solution to Eq. (7.9). Fix α , β , real numbers, satisfying 0 < α < β < 1. To study the existence of almost periodic solutions to Eq. (7.9), in addition to the previous assumptions, we suppose that the injection Xα → X is compact and that the following additional assumptions hold: (C.3) R(ω , A(·)) ∈ AP(B(X , Xα )). (C.4) The function F : R × Xα → X is almost periodic in the first variable uniformly in the second one. The function u → F(t, u) is uniformly continuous on any bounded subset K of Xα for each t ∈ R. Finally,     F(t, u) ≤ M (u ), ∞ α ,∞ where M : R+ → R+ is a continuous, monotone increasing function satisfying M (r) = 0. r→∞ r lim

We denote by S the nonlinear integral operator defined by (Su)(t) = (S1 u)(t) − (S2 u)(t), where (S1 u)(t) = and

 t −∞

U(t, s)P(s)F(s, u(s))ds,

(7.12)

7.2 Existence of Almost Periodic Solutions

(S2 u)(t) =

 ∞ t

193

UQ (t, s)Q(s)F(s, u(s))ds.

Lemma 7.2. If (C.1) holds and if α , β be real numbers such that 0 < α < β < 1, then there exists two constants R , D > 0 such that for every x ∈ Xβ ,     A(t)U(t, s)P(s)x ≤ R e− δ4 (t−s) (t − s)−β −1 x, α and

    A(t)U˜ Q (t, s)Q(s)x ≤ D e−δ (s−t) x, α

t >s

t ≤ s.

(7.13)

(7.14)

  Proof. Let x ∈ X . First of all, note that A(t)U(t, s)xα ≤ K(t − s)−(1+α ) x for all s < t such that 0 < t − s ≤ 1 and α ∈ [0, 1] (see Lunardi [137, Corollary 6.1.8, pages 219–222] for details). Letting t − s ≥ 1, we obtain,     A(t)U(t, s)x = A(t)U(t,t − 1)U(t − 1, s)x α α     ≤ A(t)U(t,t − 1)B(X ,Xα ) U(t − 1, s)x   ≤ MKeδ e−δ (t−s) x   = K1 e−δ (t−s) x = K1 e−

3δ 4

(t−s)

  δ (t − s)β +1 (t − s)−β −1 e− 4 (t−s) x.

3δ

Now since e− 4 (t−s) (t − s)β +1 → 0 as t → ∞, this quantity is necessarily bounded and so there exists c4 (β ) > 0 such that     A(t)U(t, s)x ≤ c4 (β )(t − s)−β −1 e− δ4 (t−s) x. α Choose θ ∈ (0, 1) such that 0 < θ < α < β . Then according to [137, Corollary 6.1.8, pages 219–222], for any x ∈ Xβ ,   A(t)U(t, s)x ≤ K (t − s)−(1+α −θ ) x θ α for all s < t such that 0 < t − s ≤ 1. Now, if 0 < t − s ≤ 1, we obtain     A(t)U(t, s)x = A(t)U(t, t + s )U( t + s , s)x α α 2 2    t +s t +s  )B(X ,Xα ) U( , s)xθ ≤ A(t)U(t, θ 2 2  t − s − θ δ  t − s −α −1+θ   ≤ K2 c(θ ) e− 4 (t−s) x 2 2   δ = c5 (θ , α )(t − s)−α −1 e− 4 (t−s) x

194

7 Existence Results for Some Second-Order Differential Equations

  δ ≤ c5 (θ , α )(t − s)−β −1 e− 4 (t−s) x. In summary, there exists a constant R > 0 such that     A(t)U(t, s)x ≤ R (t − s)−1−β e− δ4 (t−s) x α for all t > s. Let x ∈ X . Since the restriction of A(s) to R(Q(s)) is a bounded linear operator it follows that     A(s)U˜ Q (t, s)Q(s)x ≤ c U˜ Q (t, s)Q(s)x α α   δ (s−t)   x ≤ cm(α )e   = D eδ (s−t) x for t ≤ s by using Eq. (2.34).

7.2.1.2

Existence of Almost Periodic Mild Solutions

Lemma 7.3 (Diagana[50]). Under assumptions (C.1)–(C.2)–(C.4), then the integral operator S : BC(R, Xα ) → BC(R, Xα ) is well defined and continuous. Proof. We first show that S1 (BC(R, Xα )) ⊂ BC(R, Xα ). Indeed, using Eq. (2.33) it follows that for all v ∈ BC(R, Xα ),    S1 v(t) ≤ α

≤

t −∞

 t

−∞

  δ c(α )(t − s)−α e− 2 (t−s) F(s, v(s))ds   δ c(α )(t − s)−α e− 2 (t−s) M (vα ,∞ )ds

  = M (vα ,∞ )c(α )(2δ −1 )1−α Γ (1 − α ), and hence       S1 v = sup S1 v(t)α ≤ s(α )M (vα ,∞ ), α ,∞ t∈R

where s(α ) = c(α )(2δ −1 )1−α Γ (1 − α ). It remains to prove that S1 is continuous. For that consider an arbitrary sequence of functions un∈ BC(R, Xα ) which converges uniformly to some u ∈ BC(R, Xα ),  that is, un − uα ,∞ → 0 as n → ∞. Now

7.2 Existence of Almost Periodic Solutions

 

t −∞

195

 U(t, s)P(s)[F(s, un (s)) − F(s, u(s))] dsα ≤ c(α )

 t −∞

  δ (t − s)−α e− 2 (t−s) F(s, un (s)) − F(s, u(s)) ds.

Now, using the continuity of F and the Lebesgue Dominated Convergence Theorem we conclude that  

t

−∞

 U(t, s)P(s)[F(s, un (s)) − F(s, u(s))] dsα → 0 as n → ∞ ,

  and hence S1 un − S1 u∞,α → 0 as n → ∞. The proof for S2 is similar to that of S1 and hence omitted. For S2 , one makes use of Eq. (2.34) rather than Eq. (2.33). The proof is complete.   Let γ ∈ (0, 1] and let BCγ (R, Xα ) = {u ∈ BC(R, Xα ) : uα ,γ < ∞}, where   u

α ,γ

  = sup u(t)α + γ

  u(t) − u(s) $γ α . $ sup $ $ t − s t,s∈R, t=s

  The space BCγ (R, Xα ) equipped with the norm  · α ,γ is a Banach space, which in fact is the Banach space of all bounded continuous H¨older functions from R to Xα whose H¨older exponent is γ . Lemma 7.4 (Diagana [50]). Under assumptions (C.1)–(C.2)–(C.4), then the integral operator S = S1 − S2 maps bounded sets of BC(R, Xα ) into bounded sets of BCγ (R, Xα ) for some 0 < γ < 1, where S1 , S2 are the integral operators introduced previously. Proof. Let u ∈ BC(R, Xα ) and let g(t) = F(t, u(t)) for each t ∈ R. Then we have     S1 u(t) ≤ k(α )Su(t) α β ≤ k(α )

 t  −∞

 U(t, s)P(s)g(s) ds β

≤ k(α )c(β )

 t −∞

  δ e− 2 (t−s) (t − s)−β g(s)ds

 &   ≤ M (uα ,∞ ) k(α )c(β )

− β

2d σ ' δ 0 & '   ≤ M (uα ,∞ ) k(α )c(β )(2−1 δ )1−β Γ (1 − β ) , and hence

+∞

−σ

e

2σ δ

196

7 Existence Results for Some Second-Order Differential Equations

)

*

−1

S1 u α ,∞ ≤ k(α )c(β )(2 δ )

1−β

  Γ (1 − β ) M (uα ,∞ ).

Similarly,     S2 u(t) ≤ k(α )Ru(t) α β ≤ k(α )

 ∞

 UQ (t, s)Q(s)g(s) ds β

t

≤ k(α )m(β )

 ∞

  e−δ (s−t) g(s)ds

t

  ≤ M (uα ,∞ )k(α )m(β )2−1 δ −1 , and hence   Su α ,∞ ≤ p(α , β , δ )M (uα ,∞ ). Let t1 < t2 . Clearly,   Su(t2 ) − Su(t1 )  ≤

α

t2

t1

  t2

=

t1

 ≤

t2

t1

U(t2 , s)P(s)g(s)ds + U(t2 , s)P(s)g(s) ds +

 t1 & −∞

'  U(t2 , s) −U(t1 , s) P(s)g(s) dsα

 t1   t2 ∂ U(τ , s) −∞

  U(t2 , s)P(s)g(s) dsα + 

t1 t1

−∞



∂τ

t2

t1

  d τ P(s)g(s) dsα

  A(τ )U(τ , s)P(s)g(s)d τ dsα

= N1 + N2 . Clearly, N1 ≤

 t2  t1

 U(t2 , s)P(s)g(s) ds α

≤ c(α )

 t2 t1

  δ (t2 − s)−α e− 2 (t2 −s) g(s) ds

 t2   δ ≤ c(α )M (uα ,∞ ) (t2 − s)−α e− 2 (t2 −s) ds t1

 t2     ≤ c(α )M ( u α ,∞ ) (t2 − s)−α ds t1

  ≤ (1 − α ) c(α )M (uα ,∞ )(t2 − t1 )1−α . −1

7.2 Existence of Almost Periodic Solutions

197

Similarly, N2 ≤

 t1   t2  −∞

t1

  A(τ )U(τ , s)P(s)g(s) d τ ds α

 t1   t2

≤R

−∞

t1

   δ (τ − s)−β −1 e− 4 (τ −s) g(s)d τ ds

   ≤ R M (uα ,∞ )

t2   t1 −∞

t1

  ≤ R M (u

) α ,∞

 t2 t1

 δ (τ − s)−β −1 e− 4 (τ −s) ds d τ

(τ − t1 )−β −1



  ≤ 4δ −1 R M (uα ,∞ )(t2 − t1 )1−β .

∞

τ −t1

 δ e− 4 r dr d τ

Now    S2 u(t2 ) − S2 u(t1 ) ≤ D α

t2

  e− δ (s−t1 ) g(s)ds

t1

+ D

 ∞   t2 t2

t1

   e− δ (s−τ ) g(s) d τ ds

  ≤ N(α , δ ) (t2 − t1 )M (uα ,∞ ), where N(α , δ ) is a positive constant. Consequently, letting γ = 1 − β it follows that     $ $ Su(t2 ) − Su(t1 ) ≤ s(α , β , δ )M (u )$t2 − t1 $γ α α ,∞ where s(α , β , δ ) is a positive constant. Therefore, for each u ∈ BC(R, Xα ) such that u(t) α ≤ R for all t ∈ R, then Su belongs to BCγ (R, Xα ) with Su(t) α ≤ R4 for all t ∈ R, where R4 depends on R. Lemma 7.5. The integral operator S = S1 − S2 maps bounded sets of AP(Xα ) into bounded sets of BC1−β (R, Xα ) ∩ AP(Xα ). Similarly, the next lemma is a consequence of [103, Proposition 3.3]. Lemma 7.6. The set BC1−β (R, Xα ) is compactly contained in BC(R, X ), that is, the canonical injection id : BC1−β (R, Xα ) → BC(R, X ) is compact, which yields id : BC1−β (R, Xα ) ∩ AP(Xα ) → AP(Xα ) is compact, too.

198

7 Existence Results for Some Second-Order Differential Equations

Theorem 7.7. Suppose assumptions (C.1)–(C.4) hold, then Eq. (7.9) has an almost periodic mild solution. Proof. Let us recall that in view of Lemma 7.4, we have Su α ,∞ ≤ p(α , β , δ )M ( u α ,∞ ) and Su(t2 ) − Su(t1 ) α ≤ s(α , β , δ )M ( u α ,∞ )|t2 − t1 | for all u ∈ BC(R, Xα ), t1 ,t2 ∈ R with t1 = t2 , where p(α , β , δ ), s(α , β , δ ) are positive constants. Consequently, u ∈ BC(R, Xα ) and u α ,∞ < R yield Su ∈ BC1−β (R, Xα ) and Su α < R1 where R1 = c(α , β , δ )M (R). Therefore, there exists r > 0 such that for all R ≥ r, the following hold:   S BAP(Xα ) (0, R) ⊂ BBC1−β (R,Xα ) (0, R) ∩ BAP(Xα ) (0, R).

(7.15)

In view of the above, it follows that S : D → D is continuous and compact, where D is the ball in AP(Xα ) of radius R with R ≥ r. Using the Schauder fixed-point (Theorem 1.98) it follows that S has a fixed-point, which obviously is an almost periodic mild solution to Eq. (7.9).

7.2.2 Sobolev-Type Second-Order Differential Equations 7.2.2.1

Introduction

Let X = H 2 = H × H be equipped with its natural topology. To study the existence of almost periodic solutions to Eq. (7.3), we suppose that similar assumptions as (C.1)–(C.3) hold for the operator A (t). Further, we suppose that the injection Hα → H is compact and that the following additional assumptions hold: (C.5) The function f , g : R × Hα → H are almost periodic in the first variable uniformly in the second one. The functions u → f (t, u) and u → g(t, u) are uniformly continuous on any bounded subset K of Hα for each t ∈ R. Finally, 

f (t, Bu) 2∞ + g(t, Bu) 2∞

1/2

≤ M ( u α ,∞ ),

for all u ∈ Hα , where M : R+ → R+ is a continuous, monotone increasing function satisfying

7.2 Existence of Almost Periodic Solutions

199

lim

r→∞

M (r) = 0. r

(C.6) The linear operator B : Hα → H is bounded in the sense there exists K > 0 such that Bu ≤ K u α for all u ∈ Hα .

7.2.2.2

Existence of Almost Periodic Mild Solutions

Theorem 7.8. Under previous assumptions and if (C.5)–(C.6) hold, then Eq. (7.3) has at least one almost periodic mild solution u ∈ Hα .

u Proof. For all z := ∈ D = D(A (t)) = D(A) × H , we obtain the following v

(for the spectral decomposition of the self-adjoint operator A : D(A) ⊂ H → H we refer to Appendix A.1): A (t)z =

∞

∑ An (t)Pn z,

n=1

where Pn := and

 An (t) :=

 En 0 0 En

0

1

w(t)λn 0

, n ≥ 1,

, n ≥ 1, t ∈ R.

(7.16)

Now, the characteristic equation for An (t) is given by

λ 2 − λn w(t) = λ 2 − λn ρ (t)eiθ (t) = 0,

(7.17)

from which we obtain its eigenvalues given by

λ1n (t) =

  θ (t) θ (t) λn ρ (t)ei 2 and λ2n (t) = λn ρ (t)e−i 2 ,

 and hence σ (An (t)) = λ1n (t), λ2n (t) . Noticing thatA (t) is invertible for all t ∈ R and using Eq. (7.5) it follows that π , π such that there exists θ ∈ 2 Sθ ∪ {0} ⊂ ρ (A (t)) . Now since λ1n and λ2n are distinct and each of them is of multiplicity one, then An (t) is diagonalizable. Further, it is not difficult to see that

200

7 Existence Results for Some Second-Order Differential Equations

An (t) = Kn−1 (t)Jn (t)Kn (t), where Jn (t), Kn (t) and Kn−1 (t) are respectively given by

Jn (t) =

 

1 1 λ1n (t) 0 , K , (t) = n λ1n (t) λ2n (t) 0 λ2n (t)

and Kn−1 (t)



n 1 −λ2 (t) 1 . = n λ1 (t) − λ2n (t) λ1n (t) −1

It can be shown that there exists K > 0 such R(λ , A (t)) ≤

K 1 + |λ |

for all λ ∈ Sθ and t ∈ R. Now D = D(A (t)) is constant in t. Also, A (t) is invertible with A (t)

−1



0 w(t)−1 A−1 , = I 0

t ∈ R.

Therefore, for t, s, r ∈ R, one has 

 A (t) − A (s) A (r)−1

 0 0 = , 0 w(r)−1 (w(t) − w(s))I

and hence assuming that there exist M0 ≥ 0 and μ ∈ (0, 1] such that $μ $ $ $ $ $ $ $ $w(t) − w(s)$ ≤ M0 $t − s$

(7.18)

it follows that there exists M > 0 such that $μ    $  $   $  (A (t) − A (s))A (r)−1 z ≤ M $t − s$ z. In view of the above, the family of linear operators {A (t)}t∈R satisfy Acquistapace–Terreni conditions. Now, for every t ∈ R, the family of linear operators A (t) generate an analytic semigroup (eτ A (t) )τ ≥0 on X given by

7.2 Existence of Almost Periodic Solutions

eτ A (t) z =

201

∞

∑ Kn (t)−1 Pn eτ Jn Pn Kn (t)Pn z, z ∈ X .

n=0

On the other hand, we have eτ A (t) z =

∞

∑ Kn (t)−1 Pn eτ Jn Pn Kn (t)Pn Pn z ,

n=0

with for each z =

z1  z2

⎛ n ⎞ ⎛ ⎞2  eλ1 (t)τ E z1  0 n   τ Jn 2  ⎝ ⎠ ⎝ ⎠ e Pn z =   n (t)τ λ  z2  0 e 2 En ≤ eλ1 (t)τ En z1 2 + eλ2 (t)τ En z2 2 n

n

≤ e2ℜe(λ1 (t))τ z 2 . n

Clearly, using Eq. (7.5) it follows that 

 θ (t)  2   π + θ0 ≤ λn ρ0 cos 2  = − λn ρ0 sin(θ0 )  ≤ − λ1 ρ0 sin(θ0 ).

ℜe(λ1n (t)) =

Setting, δ =



λn ρ (t) cos

λ1 ρ0 sin(θ0 ) > 0 it follows that there exists C0 > 0 such that eτ A (t) ≤ C0 e−δ τ ,

τ ≥ 0.

(7.19)

Arguing as in [15] it follows that the evolution family (U(t, s))t≥s is exponentially stable. Clearly, a similar assumption as (C.1) for A (t) holds, too. It remains to check that a similar assumption as (C.3) for A (t) holds. Now since t → w(t) and t → w(t)−1 are almost periodic it follows that t → A (t)−1 is almost periodic with respect to operator topology. Using Theorem 7.13 it follows that Eq. (7.3) has at least one almost periodic mild solution.

202

7 Existence Results for Some Second-Order Differential Equations

7.2.3 Damped Second-Order Evolution Equations 7.2.3.1

Preliminaries

In order to study Eq. (7.1) we still suppose that the injection Hα → H is compact and that the following additional assumptions hold: (h1 ) The function a : R → C is given such that a(t) = −ρ˜ (t)eiθ (t) = ρ˜ (t)ei(π +θ (t)) π for all t ∈ R is almost periodic and there exist ρ˜ 0 , ρ˜ 1 > 0 and θ˜0 ∈ (0, ) such 4 that ˜

˜

ρ˜ 0 ≤ ρ˜ (t) ≤ ρ˜ 1 and

π + 2θ˜0 < θ˜ (t) < 2π − 2θ˜0 for all t ∈ R. (h2 ) There exist L0 > 0 and μ ∈ (0, 1] such that |a(t) − a(s)| ≤ L0 |t − s|μ for all s,t ∈ R. (h3 ) The function b : R → C is uniformly continuous, almost periodic, and differentiable. (h4 ) h : R×Hα → H is almost periodic in the first variable uniformly in the second one. The function u → h(t, u) is uniformly continuous on any bounded subset K of Hα for each t ∈ R. Finally,  1/2    u ), h(t, u) + b (t)Bu2 + b(t)Bu 2 ≤ Q( ∞ α ,∞ ∞ where Q : R+ → R+ is a continuous, monotone increasing function satisfying lim

r→∞

7.2.3.2

Q(r) = 0. r

Existence of Almost Periodic Solutions

Theorem 7.9. Under previous assumptions and if (h1 )–(h2 )–(h3 )–(h4 )–(C.6) hold, then Eq. (7.1) has at least one almost periodic solution u ∈ Hα . Proof. It suffices to let w(t) = −a(t), g(t, Bu) = b(t)Bu, and f (t, Bu) = h(t, u) + b (t)Bu for all t ∈ R in Eq. (7.3) and make use of Theorem 7.8.

7.2 Existence of Almost Periodic Solutions

203

7.2.4 The Linear Beam Equation Let Ω ⊂ RN (N ≥ 1) be an open bounded subset with C2 boundary ∂ Ω and let H = L2 (Ω ) equipped with its natural topology. To illustrate the previous abstract results, we study the existence of almost periodic solutions to the following boundary value problem known as the linear beam equation,

∂ 2u ∂u + b(t, x) + a(t, x)Δ 2 u = h(t, x, u), ∂ t2 ∂t Δ u(t, x) = u(t, x) = 0,

t ∈ R, x ∈ Ω

t ∈ R, x ∈ ∂ Ω

(7.20) (7.21)

where b : R × Ω → C satisfies (h3 ), a : R × Ω → C satisfies (h1 )–(h2 ), and h : R × Ω × L2 (Ω ) → L2 (Ω ) satisfies (h4 ). Define the linear operator A and B as follows: Au = Δ 2 u for all u ∈ D(A) = H02 (Ω ) ∩ H 4 (Ω ), and Bu = u for all u ∈ D(B) = L2 (Ω ). Proposition 7.10. If α ∈ (0, 1) and q ∈ (0, ∞], then the injection 

L2 (Ω ), H02 (Ω ) ∩ H 4 (Ω )

 α ,q

→ L2 (Ω )

is compact. Proof. Since the injection H02 (Ω ) ∩ H 4 (Ω ) → L2 (Ω ) is compact, one obtains the result by taking Y = L2 (Ω ) and X = H02 (Ω ) ∩ H 4 (Ω ) and using Proposition A.1 (see Appendix A.1). The proof is complete. In view of the above it follows that the injection   Hα = L2 (Ω ), H02 (Ω ) ∩ H 4 (Ω )

α ,∞

→ L2 (Ω )

(7.22)

is compact. Consequently, taking into account the previous assumptions it follows that the boundary value problem Eqs. (7.20) and (7.21) has at least one almost periodic mild solution.

204

7 Existence Results for Some Second-Order Differential Equations

7.3 Existence of Almost Automorphic Solutions 7.3.1 First-Order Differential Equations 7.3.1.1

Introduction

In this section we make extensive use of the techniques utilized in the previous section to study and establish the existence of almost automorphic solutions to the nonautonomous differential equations (Eq. 7.9). For that we will make extensive use of the compact almost automorphy as it is more appropriate in this context. Let α , β ∈ (0, 1) such that 0 < α < β < 1. Further, we still suppose that the injection Xα → X is compact and that the following additional assumptions hold: (D.4) R(ω , A(·)) ∈ KAA(B(X , Xα )). (D.5) The function F : R × Xα → X is such that t → F(t, p) belongs to KAA(X ) for all p in a bounded subset of Xα . The function p → F(t, p) is uniformly continuous on any bounded subset K of Xα for each t ∈ R. Finally,     F(t, p) ≤ M ( p ), ∞ α ,∞ where M : R+ → R+ is a continuous, monotone increasing function satisfying lim

r→∞

M (r) = 0. r

Let γ ∈ (0, 1] and let BCγ (R, X ) be the collection of all bounded continuous functions R into X equipped with the distance:

δ (p, q) =

∞

ρn (p, q)

∑ 2−n 1 + ρn (p, q)

n=1

where, for r = p − q, 

δn (p, q) = δn (r, 0) = r C[−n,n] + γ . sup

 r(t) − r(s) : t, s ∈ [−n, n], t = s . |t − s|γ

In this subsection, BCγ (R, X ) will be viewed as a locally convex Fr´echet space equipped with the metric δ (see [104]). Also, the symbol S stands for the integral operator given in Eq. (7.12).

7.3 Existence of Almost Automorphic Solutions

7.3.1.2

205

Existence of Almost Automorphic Mild Solutions

The proof of the next lemma follows along the same lines as that of Lemma 7.4 and hence is omitted. Lemma 7.11. The integral operator S maps bounded sets of KAA(Xα ) into bounded sets of BC1−β (R, Xα ) ∩ KAA(Xα ). Similarly, the next lemma is a consequence of [103, Proposition 3.3] (see also [104]). Lemma 7.12. The set BC1−β (R, Xα ) is compactly contained in BC(R, X ), that is, the canonical injection id : BC1−β (R, Xα ) → BC(R, X ) is compact, which yields id : BC1−β (R, Xα ) ∩ KAA(Xα ) → KAA(Xα ) is compact, too. Theorem 7.13. Suppose assumptions (C.1)–(D.4)–(D.5) hold, then Eq. (7.9) has a mild solution, which belongs to KAA(Xα ). Proof. Using Lemma 7.4, we obtain Su α ,∞ ≤ p(α , β , δ )M ( u α ,∞ ) and Su(t2 ) − Su(t1 ) α ≤ s(α , β , δ )M ( u α ,∞ )|t2 − t1 | for all u ∈ BC(R, Xα ), t1 ,t2 ∈ R with t1 = t2 , where p(α , β , δ ), s(α , β , δ ) are positive constants. Thus if u ∈ BC(R, Xα ) and u α < R, then Su ∈ BC1−β (R, Xα ) with δ (Su, 0) < R1 where R1 = c(α , β , δ )M (R). Consequently, there exists r > 0 such that for all R ≥ r, the following hold:   S BKAA(Xα ) (0, R) ⊂ BBC1−β (R,Xα ) (0, R) ∩ BKAA(Xα ) (0, R).

(7.23)

In view of the above, it follows that S : D → D is continuous and compact, where D is the ball in KAA(Xα ) of radius R with R ≥ r for the topology defined by the distance δ . Using the Schauder fixed-point (in locally convex Fr´echet spaces) it follows that Eq. (7.9) has a mild solution, which belongs to KAA(Xα ).

206

7 Existence Results for Some Second-Order Differential Equations

7.3.2 Sobolev-Type Second-Order Evolution Equations In this subsection we make use of the techniques used in the almost periodic case to study and obtain through the Schauder’s fixed-point theorem, the existence of almost automorphic solutions to the second differential equation (Eq. 7.3). To study the existence of almost automorphic solutions to Eq. (7.3), in addition to the previous assumptions, we suppose that the injection Hα → H is compact and that the following additional assumptions hold: (D.6) The functions f , g : R × Hα → H belong to KAA(Hα , H ). The functions u → f (t, u) and u → g(t, u) are uniformly continuous on any bounded subset K of Hα for each t ∈ R. Finally, 1/2

 f (t, Bu) 2∞ + g(t, Bu) 2∞

≤ M ( u α ,∞ ),

for all u ∈ Hα , where M : R+ → R+ is a continuous, monotone increasing function satisfying lim

r→∞

M (r) = 0. r

Theorem 7.14. Under previous assumptions and if (D.6) holds, then Eq. (7.3) has a mild solution, which belongs to KAA(Hα ). Proof. The proof follows along the same lines as in the proof of Theorem 7.8 and hence is omitted.

7.3.3 Damped Second-Order Evolution Equations In order to study Eq. (7.1) we still suppose that the injection Hα → H is compact and that the following additional assumptions hold: ˜ ˜ (h 1 ) The function a : R → C is given such that a(t) = −ρ˜ (t)eiθ (t) = ρ˜ (t)ei(π +θ (t)) π belongs to KAA(C) and there exist ρ˜ 0 , ρ˜ 1 > 0 and θ˜0 ∈ (0, ) such that 4

ρ˜ 0 ≤ ρ˜ (t) ≤ ρ˜ 1 and

π + 2θ˜0 < θ˜ (t) < 2π − 2θ˜0 for all t ∈ R.

7.3 Existence of Almost Automorphic Solutions

207

(h 2 ) The function b : R → C which belongs to KAA(C), is uniformly continuous, and differentiable. (h 3 ) h : R × Hα → H belongs to KAA(Hα , H ). The function u → h(t, u) is uniformly continuous on any bounded subset K of Hα for each t ∈ R. Finally, 

  h(t, u) + b (t)Bu2 + b(t)Bu 2 ∞ ∞

1/2

  ≤ Q(uα ,∞ ),

where Q : R+ → R+ is a continuous, monotone increasing function satisfying lim

r→∞

Q(r) = 0. r

Theorem 7.15. Under previous assumptions and if (h 1 )–(h2 )–(h 2 )–(h 3 ) hold, then Eq. (7.1) has at least one mild solution, which belongs to KAA(Hα ).

7.3.4 The Linear Beam Equation Let Ω ⊂ RN (N ≥ 1) be a open bounded subset with C2 boundary ∂ Ω and let H = L2 (Ω ) equipped with its natural topology. We study the existence of almost automorphic solutions to the beam equation given by

∂ 2u ∂u + a(t, x)Δ 2 u = h(t, x, u), + b(t, x)Δ 2 ∂t ∂t Δ u(t, x) = u(t, x) = 0,

t ∈ R, x ∈ Ω

t ∈ R, x ∈ ∂ Ω

(7.24) (7.25)

where b : R × Ω → C satisfies (h 2 ), a : R × Ω → C satisfies (h 1 )-(h2 ), and h : R × Ω × L2 (Ω ) → L2 (Ω ) satisfies (h 3 ). Proceeding as in the almost periodic setting one can show that the boundary value problem Eqs. (7.24) and (7.25) has at least one mild solution, which belongs to KAA(Hα ).

Bibliographical Notes The material presented in this chapter is slightly based on Diagana [51,52]. Relevant references to this chapter include the following: Goldstein and N’Gu´er´ekata [103, 104], Lunardi [137], Chicone and Latushkin [38], and Engel and Nagel [90].

Chapter 8

Existence Results to Some Integrodifferential Equations

8.1 Introduction In this chapter we study the existence of asymptotically almost automorphic mild solutions to the abstract partial neutral integrodifferential equation d D(t, ut ) = AD(t, ut ) + dt

 t 0

B(t − s)D(s, us )ds + g(t, ut ), t ∈ [σ , σ + a), (8.1)

uσ = ϕ ∈ B,

(8.2)

where A, B(t) : D(A) ⊂ X → X are densely defined closed linear operators with a common domain D(A), which is independent of t; the history ut : (−∞, 0] → X , defined by ut (θ ) := u(t + θ ) belongs to an abstract phase space B defined axiomatically, f , g are functions subject to some additional conditions, and D(t, ϕ ) = ϕ (0) + f (t, ϕ ). For that, we will make extensive use of the concept of compact asymptotically almost automorphy and the so-called resolvent of operators. In this chapter, we assume that the abstract Cauchy problem x (t) = Ax(t) +

 t 0

B(t − s)x(s)ds,

t ≥ 0,

x(0) = x0 ∈ X ,

(8.3) (8.4)

has an associated resolvent operator (R(t))t≥0 .

T. Diagana, Almost Automorphic Type and Almost Periodic Type Functions in Abstract Spaces, DOI 10.1007/978-3-319-00849-3 8, © Springer International Publishing Switzerland 2013

209

210

8 Existence Results to Some Integrodifferential Equations

Definition 8.1. A one-parameter family (R(t))t≥0 of bounded linear operators from X into itself is called a strongly continuous resolvent operator for Eqs. (8.3) and (8.4) whenever the following conditions hold: (a) R(0) = I and the function t → R(t)x is continuous on [0, ∞) for every x ∈ X . (b) R(t)D(A) ⊂ D(A) for all t ≥ 0 and for x ∈ D(A), the function t → AR(t)x is continuous on [0, ∞) and the function t → R(t)x is continuously differentiable on [0, ∞). (c) For x ∈ D(A), the next resolvent equations hold, d R(t)x = AR(t)x + dt d R(t)x = R(t)Ax + dt

 t 0

 t 0

B(t − s)R(s)xds,

t ≥ 0,

(8.5)

R(t − s)B(s)xds.

t ≥ 0.

(8.6)

In addition to the above-mentioned assumptions, we assume that the resolvent operator (R(t))t≥0 is exponentially stable, that is, there exist positive constants M˜ and δ such that ˜ −δ t R(t) ≤ Me for every t ≥ 0. For more on partial integrodifferential equations, resolvent of operators and related issues, we refer the interested reader to [47, 105, 106, 113, 114]. Here we define the phase space B axiomatically, using ideas and notations developed in [116]. More precisely, B will denote the vector space of functions defined from (−∞, 0] into X endowed with a seminorm denoted by · B and such that the following axioms hold: A. If x : (−∞, σ + b) → X with b > 0, is continuous on [σ , σ + b) and xσ ∈ B, then for each t ∈ [σ , σ + b), we have (i) xt is in B, (ii) x(t) ≤ H xt B , (iii) xt B ≤ K(t − σ ) sup{ x(s) : σ ≤ s ≤ t} + M(t − σ ) xσ B , where H > 0 is a constant, and K, M : [0, ∞) → [1, ∞) are functions such that K(·) and M(·) are respectively continuous and locally bounded, and H, K, M are independent of x(·). A1. If x(·) is a function as in (A), then xt is a B-valued continuous function on [σ , σ + b). B. The space B is complete. C2 . If (ϕ n )n∈N is a sequence of continuous functions with compact support defined from (−∞, 0] into X , which converges to ϕ uniformly on compact subsets of (−∞, 0], then ϕ ∈ B and ϕ n − ϕ B → 0 as n → ∞.

8.2 Existence of Asymptotically Almost Automorphic Solutions

211

In what follows L > 0 stands for a constant such that ϕ B ≤ L . sup ϕ (θ ) θ ≤0

for every ϕ ∈ BC((−∞, 0], X ) (see [116, Proposition 7.1.1] for details). Definition 8.2. Let S(t) : B → B be the c0 -semigroup defined by S(t)ϕ (θ ) = ϕ (0) on [−t, 0] and S(t)ϕ (θ ) = ϕ (t + θ ) on (−∞, −t]. The phase space B is called a fading memory if S(t)ϕ B → 0 as t → ∞ for each ϕ ∈ B with ϕ (0) = 0. We suppose that there is a constant K > 0 such that max{K(t), M(t)} ≤ K

(8.7)

for each t ≥ 0. Note that Eq. (8.7) is satisfied, for example, if the phase space B is a fading memory (see [116, Proposition 7.1.5] for details).

8.2 Existence of Asymptotically Almost Automorphic Solutions In this section we establish the existence of asymptotically almost automorphic solutions to the partial neutral integrodifferential system Eq. (8.1). For that, we first need to establish some preliminary results. Lemma 8.3. Let (R(t))t≥0 be a one-parameter family of bounded linear operators ˜ −δ t for all t ≥ 0 and let f ∈ KAAA(X ). If ϑ is the on X satisfying R(t) ≤ Me function defined by

ϑ (t) :=

 t 0

R(t − s) f (s)ds, t ≥ 0,

then ϑ ∈ KAAA(X ). Proof. Suppose f = k + h where k ∈ KAA(X ) and h ∈ C0 (R+ , X ). Then

ϑ (t) =

 t −∞

R(t − s)k(s)ds +

 0 −∞

R(t − s)k(s)ds +

 t 0

R(t − s)h(s)ds

= w(t) + q(t) where w(t) = q(t) =

 t −∞

 t 0

R(t − s)k(s)ds +

R(t − s)h(s)ds.

 0 −∞

R(t − s)k(s)ds,

212

8 Existence Results to Some Integrodifferential Equations

For a given sequence (σn )n∈N of real numbers, fix a subsequence (sn )n∈N , and a function v ∈ BC(R, X ) such that k(t + sn ) converges to v(t) in X , and v(t − sn ) converges to k(t) in X , uniformly on compact sets of R. From Bochner’s criterion related to integrable functions and the estimate ˜ −δ (t−s) k(s) R(t − s)k(s) = R(t − s) k(s) ≤ Me

(8.8)

it follows that the function s → R(t − s)k(s) is integrable over (−∞,t) for each t ∈ R. Furthermore, since w(t + sn ) =

 t −∞

R(t − s)k(s + sn )ds, t ∈ R, n ∈ N,

using the estimate Eq. (8.8) and the Lebesgue Dominated Convergence Theorem, it follows that w(t + sn ) converges to z(t) =

 t −∞

R(t − s)v(s)ds

as n → ∞, for each t ∈ R. It remains to show that the convergence is uniform on all compact subsets of R and that q(·) ∈ C0 (R+ , X ). Let K ⊂ R be an arbitrary compact and let ε > 0. Since h ∈ C0 (R+ , X ) and k(·) ∈ KAA(X ), there exist a constant L and Nε such L that K ⊂ [ −L 2 , 2 ] with  ∞ L 2

e−δ s ds < ε ,

k(s + sn ) − v(s) ≤ ε ,

n ≥ Nε , s ∈ [−L, L],

h(s) ≤ ε ,

s ≥ L.

Now, for each t ∈ K, w(t + sn ) − z(t) ≤ ≤

 t −∞

R(t − s) k(s + sn ) − v(s) ds

 −L −∞

+

˜ −δ (t−s) k(s + sn ) − v(s) ds Me

 t −L

≤ 2 k ∞

˜ −δ (t−s) k(s + sn ) − v(s) ds Me  ∞ t+L

˜ −δ s ds + ε Me

 ∞ 0

˜ −δ s ds Me

8.2 Existence of Asymptotically Almost Automorphic Solutions

≤ 2 k ∞

 ∞

≤ε

L 2

˜ −δ s ds + ε Me

˜ 2M k ∞+

 ∞ 0

213

 ∞

˜ −δ s ds Me

0

 ˜ −δ s ds , Me

which proves that the convergence is uniform on K, by the fact that the last estimate is independent of t ∈ K. Proceeding as previously, one can similarly prove that z(t − sn ) converges to w uniformly on all compact subsets of R. Next, let us show that q(·) ∈ C0 (R+ , X ). For all t ≥ 2L we obtain q(t) ≤ ≤

 0 −∞

 0

˜ −δ (t−s) k(s) ds + Me

−∞

+ ≤ M˜

 t

R(t − s) k(s) ds +

 t/2 0

 ∞ L 2

0

R(t − s) h(s) ds

 t t/2

˜ −δ (t−s) h(s) ds Me

˜ −δ (t−s) h(s) ds Me

e−δ s ds k ∞ + ε

 ˜ ≤ ε M k + ∞

∞

0

 t t/2

˜ −δ s ds + M˜ Me

 ˜ −δ s ds + M h ˜ Me ∞ .

 ∞ L 2

e−δ s ds h ∞

This completes the proof. Lemma 8.4. If u ∈ KAA(X ), then the function s → us belongs to KAA(B). Moreover, if B is a fading memory space and u ∈ C(R, X ) is such that u0 ∈ B and u|[0,∞) ∈ KAAA(X ), then t → ut ∈ KAAA(B). Proof. For a given sequence (s n )n∈N of real numbers, fix a subsequence (sn )n∈N of (s n )n∈N and a function v ∈ BC(R, X ) such that u(s + sn ) → v(s) uniformly on compact subsets of R. Since B satisfies axiom C2 , from [116, Proposition 7.1.1], we infer that us+sn → vs in B for each s ∈ R. Similarly, one can prove that vt−sn converges to ut in B. Thus, the function s → us belongs to AA(B). Since B satisfies axiom C2 , from [116, Proposition 7.1.1], we infer that us+sn → vs in B for each s ∈ R. Let K ⊂ R be an arbitrary compact and let L > 0 such that K ⊂ [−L, L]. For ε > 0, fix Nε ,L ∈ N such that u(s + sn ) − v(s) ≤ ε , u−L+sn − v−L ≤ ε ,

s ∈ [−L, L],

214

8 Existence Results to Some Integrodifferential Equations

whenever n ≥ Nε ,L . In view of the above, for t ∈ K and n ≥ Nε ,L we get ut+sn − vt B ≤ M(L + t) u−L+sn − v−L B + K(L + t) .

sup u(θ + sn ) − v(θ )

θ ∈[−L,L]

≤ 2Kε . In view of the above, ut+sn converges to vt uniformly on K. Let u ∈ C(R, X ) is such that u0 ∈ B and u|[0,∞) ∈ KAAA(X ), therefore, u admits a decomposition u = z + k where z ∈ KAA(X ), k ∈ C0 (R+ , X ) and k0 = u0 . As z belongs to KAA(X ) from the previous fact we conclude that zt ∈ KAA(B). As k(t) → 0 when t → ∞ and the phase space has fading memory, from [116, Proposition 7.1.3] we conclude that kt B → 0 when t → ∞, we then get that s → us belongs to KAAA(B). The rest of this section is devoted to the existence of asymptotically almost automorphic solutions to the partial neutral system Eqs. (8.1) and (8.2). For that, in addition to the previous assumptions, we require that, (L.1) The functions f , g are X -valued and belongs to KAAA(R × B) and there exist continuous and nondecreasing functions Lg , L f : [0, ∞) → [0, ∞) such that for r ≥ 0 and for all x B ≤ r, y B ≤ r, f (t, x) − f (t, y) ≤ L f (r) x − y B , g(t, x) − g(t, y) ≤ Lg (r) x − y B , for all t ∈ R. We adopt the notion of mild solutions for Eq. (8.1) from the one given in [112]. Definition 8.5. A function u : (−∞, σ + a) → X for a > 0 is a mild solution of the neutral integrodifferential system Eq. (8.1) on [σ , σ + a), if uσ ∈ B, the restriction of u to the interval [σ , σ + a) is continuous and u(t) = R(t)(ϕ (0) + f (σ , ϕ )) − f (t, ut ) +

 t σ

R(t − s)g(s, us )ds,

for every t ∈ [σ , σ + a). Theorem 8.6. Assume that the phase space B is a fading memory space and that f (·) and g(·) satisfy H1 . If L f (0) = Lg (0) = 0 and f (t, 0) = g(t, 0) = 0 for every t ∈ R, then there exists γ0 > 0 such that for each ϕ satisfying ϕ B ≤ γ0 there exists a mild solution u(·, ϕ ) to (8.1) on [0, ∞) such that u(·, ϕ ) ∈ KAAA(X ) and u0 (·, ϕ ) = ϕ .

8.2 Existence of Asymptotically Almost Automorphic Solutions

215

Proof. Let r > 0 and λ ∈ (0, 1) be such that ˜ ˜ λ + L f (λ r)λ + L f ((λ + 1)K)(λ + 1) + MLg ((λ + 1)K) (λ + 1)K < 1. Θ = MH δ We claim that the assertion holds for γ0 = λ r. Let ϕ such that ϕ B ≤ γ0 . Define the space  D = x ∈ KAAA(X ) : x(0) = ϕ (0), x(t) ≤ r for t ≥ 0 and equip it with the metric defined by d(u, v) = u − v for all x, y ∈ D. Define the operator Γ : D → C(R+ , X ) by

Γ u(t) = R(t)(ϕ (0) + f (0, ϕ )) − f (t, u˜t ) +

 t 0

R(t − s)g(s, u˜s )ds, t ≥ 0,

where u˜ : R → X is such that u˜0 = ϕ on (−∞, 0) and u˜ = u on [0, ∞). From the properties of (R(t))t≥0 , f (·) and g(·), we deduce that Γ u(·) is well defined and that Γ u ∈ C([0, ∞), X ). Moreover, from Lemma 8.3, Lemma 8.4, and the fact that R(·) is exponentially stable, it follows that Γ u ∈ KAAA(X ). Next, we prove that Γ (·) is a contraction from D into D. First of all, note that Γ u(0) = ϕ (0). Moreover, if u ∈ D and t ≥ 0, we then get ˜ λ r + ML ˜ f (λ r)λ r + L f ((λ + 1)K)(λ + 1)Kr Γ u(t) ≤ MH +

 t 0

˜ −δ (t−s) Lg ((λ + 1)K)(λ + 1)Krds Me

˜ λ r + ML ˜ f (λ r)λ r + L f ((λ + 1)K)(λ + 1)Kr ≤ MH M˜ Lg ((λ + 1)K)(λ + 1)Kr δ ≤ Θr +

≤ r, where the inequality u˜t ≤ (λ + 1)Kr has been utilized. Therefore, Γ (D) ⊂ D.

216

8 Existence Results to Some Integrodifferential Equations

On the other hand, for u, v ∈ D, we see that Γ u(t) − Γ v(t) ≤ f (t, u˜t ) − f (t, v˜t ) +

 t 0

˜ g ((λ + 1)Kr)e−δ (t−s) u˜s − v˜s B ds ML

  ˜ g ((λ + 1)Kr) ML ≤ L f ((λ + 1)Kr) + K u − v , δ which shows that Γ (·) is a contraction from D into D. We now conclude by using the Banach fixed-point principle (Theorem 1.96).

8.3 Existence Result for Some Neutral Integrodifferential Equations 8.3.1 Introduction In this section we make use of our previous abstract existence results to study the existence and uniqueness of an asymptotically almost automorphic mild solution to a concrete partial neutral integrodifferential equation with unbounded delay.

8.3.2 Asymptotically Almost Automorphic Solutions Let Ω ⊂ R2 be an open subset whose boundary ∂ Ω is sufficiently regular and let X = H01 (Ω ) × L2 (Ω ). Consider the linear operator A whose domain is given by D(A) = (H 2 (Ω ) ∩ H01 (Ω )) × H01 (Ω ) and A

  x y = y α (0)x − β (0)y

where α (·), β (·) are real valued functions of class C2 on [0, ∞) such that α (0) > 0, and β (0) > 0. We know from Chen [37] that the linear operator A given above is the infinitesimal generator of a uniformly exponentially stable c0 -semigroup (T (t))t≥0

8.3 Existence Result for Some Neutral Integrodifferential Equations

217

˜ γ > 0 such on H01 (Ω ) × L2 (Ω ). In the sequel, we will assume that there exists M, that ˜ −γ t T (t) ≤ Me for all t > 0. Let B(t) = AF(t) where F : H01 (Ω ) × L2 (Ω ) → H01 (Ω ) × L2 (Ω ) is the operator family defined by ⎛ ⎜ F = (Fi j ) = ⎜ ⎝

0

0

α (t) −β (t) + β (0) α (0)

⎞

⎟ ⎟, α (t) ⎠ α (0)

and suppose max{ F22 (t) , F21 (t) } ≤

γ −γ t e , 2M

t ≥ 0,

max{ F22 (t) , F21 (t) } ≤

γ 2 −γ t e , 4M 2

t ≥ 0.

From the results in Grimmer [105], we know that the abstract integrodifferential system x (t) = Ax(t) +

 t 0

AF(t − s)x(s)ds,

has an associated uniformly exponentially stable resolvent of operators (R(t))t≥0 on H01 (Ω ) × L2 (Ω ) with ˜ R(t) ≤ Me

−γ 2t

for t ≥ 0. Let h : (−∞, −r) → R be a positive (Lebesgue) integrable function and assume that there exists a nonnegative and locally bounded function γ on (−∞, 0] such that h(ξ + θ ) ≤ γ (ξ )h(θ ), for all ξ ≤ 0 and θ ∈ (−∞, −r) \ Nξ , where Nξ ⊆ (−∞, −r) is a set with Lebesgue measure zero. Let r ≥ 0 and 1 ≤ p < ∞. Let the phase space B be defined by, B := Cr × L p (h, H01 (Ω ) × L2 (Ω )), where the space Cr × L p (h, H01 (Ω ) × L2 (Ω )) consists of the

218

8 Existence Results to Some Integrodifferential Equations

collection of all functions ϕ : (−∞, 0] → H01 (Ω ) × L2 (Ω ) such that ϕ is continuous on [−r, 0], Lebesgue-measurable, and h ϕ p is Lebesgue integrable on (−∞, −r). The seminorm of · B is defined by ϕ B := sup{ ϕ (θ ) : −r ≤ θ ≤ 0} +



−r −∞

h(θ ) ϕ (θ ) p d θ

1/p .

Under the previous assumptions, the phase space B satisfies the axioms: (A), (A-1), (B), and (C2), see [116, Theorem 1.3.8]. Moreover, when r = 0 we have that H = 1, M(t) = γ (−t)1/2 and K(t) = 1 +



0

−t

L=



h(θ ) d θ

1/2 ,

h(s)ds





sup | γ (s)1/2 | + 1 +

K=

for t ≥ 0,

0 −∞



1/2

0

−∞

s≤0

h(θ )d θ

1/2 .

Consider the neutral system

∂ ∂t

 x(t, ξ ) +

−∞

= A x(t, ξ ) + + +

 t 0



t

a2 (t − s)x(s, ξ )ds 

 t −∞

a2 (t − s)x(s, ξ )ds

 AF(t − s) x(s, ξ ) +

 t −∞

s

−∞

 a2 (s − u)x(u, ξ )du ds

a1 (t − s)u(s, ξ )ds,

(8.9)

where a1 , a2 : R → R are continuous functions with  12

 0 2 a1 (s) ds L1 = <∞ −∞ h(s) and L2 =



a22 (s) ds −∞ h(s) 0

Setting f , g : R × B → H01 (Ω ) × L2 (Ω ) by

 12

< ∞.

8.3 Existence Result for Some Neutral Integrodifferential Equations

f (t, ϕ )(ξ ) =

 0 −∞

219

a2 (s)ϕ (s)(ξ )ds,

D(t, ϕ )(ξ ) = ϕ (0)(ξ ) + f (t, ϕ )(ξ ), g(t, ϕ )(ξ ) =

 0

−∞

a1 (s)ϕ (s)(ξ )ds,

one can rewrite Eq. (8.9) as a neutral integrodifferential system of the form Eq. (8.1). Moreover, f , g are bounded linear operators satisfying g ≤ L1 and f ≤ L2 . The next result is a straightforward consequence of Theorem 8.6 and for this reason we will omit its proof. Theorem 8.7. Under previous assumptions, then Eq. (8.9) has a unique mild solution u ∈ KAAA(X ) whenever K is small enough.

Bibliographical Notes The material presented in this chapter is taken from Diagana et al. [48]. Other relevant references to this chapter include Diagana and Hern`andez [59], Ding et al. [80], Hern´andez and Henr´ıquez [110], Hern´andez [111], Hern´andez and Santos [112, 114], Hern´andez and Pelicer [115], Wu and Xia [161], Wu and Xia [162], and Wu [163]. An interesting question consists of studying a nonautonomous version of the previous work.

Chapter 9

Existence of C(m) -Pseudo-Almost Automorphic Solutions

In this chapter we study and obtain the existence of C(m) -pseudo-almost automorphic solutions to some classes of first-order, second-order, and higher-order differential equations with operator coefficients whose forcing term is C(m) -pseudoalmost automorphic. For that, we will make extensive use of various tools including analytic semigroup and exponential dichotomy techniques.

9.1 Existence Results for Some Autonomous Higher-Order Differential Equations 9.1.1 Introduction In the book by Xiao and Liang [168], a basic theory for the well posedness of the Cauchy problem for higher abstract differential equations with time-independent operator coefficients given by n−1

u(n) (t) + ∑ Ak u(k) (t) = 0, t ≥ 0,

u(k) (0) = uk , 0 ≤ k ≤ n − 1

(9.1)

k=0

where A0 , A1 , . . . , An−1 are linear operators on some phase space, was constructed. The techniques utilized by Xiao and Liang are based on a direct treatment of Eq. (9.1), which will differ from those that will be used in this book. Fix α ∈ (0, 1), m ∈ N, and p ∈ [0, ∞]. In this section it goes back to studying the existence of C(m) -pseudo-almost automorphic solutions to the autonomous higherorder differential equations given by n−1

u(n) (t) + ∑ Ak u(k) (t) + Au(t) = f (t), t ∈ R,

(9.2)

k=1

T. Diagana, Almost Automorphic Type and Almost Periodic Type Functions in Abstract Spaces, DOI 10.1007/978-3-319-00849-3 9, © Springer International Publishing Switzerland 2013

221

222

9 Existence of C(m) -Pseudo-Almost Automorphic Solutions

where A : D(A) ⊂ X → X and Ak : D(Ak ) ⊂ X → X (k = 1, . . . , n−1) are densely defined closed linear operators on a Banach space X , and the forcing term f : R → DA (α + m, p) with DA (α + m, p) being a real interpolation space, is C(m) -pseudoalmost automorphic. In this setting, we are exclusively interested in the special case Ak = Aαk (if it exists) with the exponents 0 ≤ αk < 1 for k = 1, 2, . . . , n − 1, which has been investigated in the book by Xiao and Liang [168] in the case when f = 0 and

α1 > α2 − α1 > · · · > 1 − αn−1 > 0. It should be mentioned that some other special cases of Eq. (9.2), which have been treated in the literature include, but are not limited to, the work by Chen and Triggiani [35, 36] and that of Huang [118, 119]. In order to deal with Eq. (9.2), our strategy consists of rewriting it as a first-order autonomous differential equation in Z = X n = D(A) × X × · · · × X involving the family of n × n-operator matrix A . Indeed, if u is differential n times, setting z := (u, u , . . . , u(n−1) )T , then Eq. (9.2) can be rewritten in the space Z in the form z (t) = A z(t) + F(t), t ∈ R,

(9.3)

where A is the n × n-operator matrix defined by ⎛

0 I ⎜ 0 0 ⎜ ⎜ A =⎜ . . ⎜ ⎝ . . −A −A1

0 I . . .

. 0 . 0 . . . I . −An−1

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

(9.4)

whose domain D(A ) is given by D := D(A) × D(A1 ) × D(A2 ) × · · · × D(An−1 ), and the function F appearing in Eq. (9.3) is defined by F(t) = (0, 0, . . . , f (t))T . Next, we make extensive use of Lemma 2.72 to study a generic first-order differential equation in the form Eq. (9.3) under some appropriate assumptions on A . Finally, we go back to Eq. (9.2) and study the existence of C(m+n−1) -pseudoalmost automorphic solutions to it.

9.1.2 First-Order Differential Equations Fix m ∈ N. Recall that BC(m) (R, X ) stands for the collection of all functions f ∈ C(m) (R, X ) such that m

f (m) := sup ∑ f (k) (t) < ∞. t∈R k=0

9.1 Existence Results for Some Autonomous Higher-Order Differential Equations

223

Similarly, if the Banach space (X , · ) is replaced with the real interpolation space (DA (α + m, p), · α +m,p ), then the norm of BC(m) (R, DA (α + m, p)) will be denoted by f (m),(α +m,p) . Fix α ∈ (0, 1), m ∈ N, and p ∈ [1, ∞). In this subsection we study the existence of C(m) -pseudo-almost automorphic solutions to the first-order equation, u (t) = Au(t) + F(t), t ∈ R,

(9.5)

where A is a sectorial linear operator with associated analytic semigroup (T (t))t≥0 , and F : R → X is a continuous function satisfying some additional conditions. In addition to the above, we suppose that the following assumptions hold: (A.1) The linear operator A : D(A) ⊂ X → X is sectorial. Moreover, the analytic semigroup (T (t))t≥0 associated with A is exponentially stable, say, Eq. (2.13) holds. (A.2) The function t → F(t) belongs to PAA(m) (DA (α + m, p)). Definition 9.1. Under assumption (A.1), a continuous function u : R → X is said to be a mild solution to Eq. (9.5) provided that u(t) = T (t − s)u(s) +

 t s

T (t − τ )F(τ )d τ ,

∀t ≥ s, t, s ∈ R.

(9.6)

Lemma 9.2. Under assumption (A.1), if F : R → X is a bounded continuous function, then u given by u(t) =

 t −∞

T (t − s)F(s)ds

(9.7)

for all t ∈ R, is the unique bounded mild solution to Eq. (9.5). Proof. Let the fact that u given in Eq. (9.7) is a bounded mild solution to Eq. (9.5) is clear. Now let u, v be two bounded mild solutions to Eq. (9.5). Setting w = u − v, one can easily see that w is bounded and that w(t) = T (t − s)w(s) for all t ≥ s. Now using (A.1) it follows that w(t) = T (t − s)P(s)w(s) ≤ Me−δ (t−s) w(s) ≤ Me−δ (t−s) w ∞ for all t ≥ s. Now let (sn )n∈N be a sequence of real numbers such that sn → −∞ as n → ∞. Clearly, for any fixed t ∈ R, there exists a subsequence (snk )k∈N of (sn )n∈N such that snk < t for all k ∈ N. In view of the above, letting k → ∞ yields w(t) = u(t)−v(t) = 0. Therefore, u = v.

9 Existence of C(m) -Pseudo-Almost Automorphic Solutions

224

Theorem 9.3. Under assumptions (A.1)–(A.2), then Eq. (9.5) has a unique C(m) pseudo-almost automorphic solution u ∈ DA (α + m, p). Proof. According to Lemma 9.2, the only bounded mild solution to Eq. (9.5) is given by Eq. (9.7). Furthermore, the k-derive of u exists and is given by u(k) (t) =

 t

k−1

−∞

Ak T (t − s)F(s)ds + ∑ Ak− j−1 F ( j) (t)

(9.8)

j=0

for all t ∈ R. Let F = g1 + g2 ∈ PAA(m) (DA (α + m, p)) where g1 ∈ AA(m) (DA (α + m, p)) and (m) g2 ∈ PAP0 (DA (α + m, p)). Now set Sg j (t) :=

 t

T (t − s)g j (s)ds for j = 1, 2.

−∞

  We first show that S AA(m) (DA (α + m, p)) ⊂ AA(m) (DA (α + m, p)). Indeed, since g1 ∈ AA(m) (DA (α + m, p)), for every sequence of real numbers (τn )n∈N there exist a subsequence (τn )n∈N and an m-differentiable function h1 such that (k)

(k)

h1 (t) := lim g1 (t + τn ) n→∞

is well defined for each t ∈ R, and (k) lim h (t − τn ) n→∞ 1

(k)

= g1 (t)

for each t ∈ R for a fixed k = 0, 1, . . . , m. Clearly (Sg1 )(k) (t+τn )−(Sh1 )(k) (t) =

= =

)

t+τn

−∞

)

t

−∞

 t −∞

T (t+τn −s)g1 (s)ds−

 t −∞

*(k) T (t−s)h1 (s)ds

  T (t − s) g1 (s + τn ) − h1 (s) ds

*(k)

  Ak T (t − s) g1 (s + τn ) − h1 (s) ds

  k−1 ( j) ( j) + ∑ Ak− j−1 g1 (t + τn ) − h1 (t) j=0

=

 t −∞

  Ak T (t − s) g1 (s + τn ) − h1 (s) ds

9.1 Existence Results for Some Autonomous Higher-Order Differential Equations

225

  +Ak−1 g1 (t + τn ) − h1 (t)   k−1 ( j) ( j) + ∑ Ak− j−1 g1 (t + τn ) − h1 (t) . j=1

Using Lemma 2.72 it follows that  t −∞

   Ak T (t−s) g1 (s+τn )−h1 (s) ds ≤

t

−∞

H(t−s) g1 (s+τn )−h1 (s) α +k−1,p ds

≤ C θ0 g1 (t + τn ) − h1 (t) (m),(α +m,p) . Setting dm =

m−1

∑ A− j−1 B(X ) < ∞ it follows that

j=1 k−1

∑ Ak− j−1



 ( j) ( j) g1 (t + τn ) − h1 (t) ≤

j=1

k−1

∑ A− j−1 B(X ) . g1

( j)

( j)

(t + τn ) − h1 (t) α +k−1,p

j=1

≤ C

m−1

∑ A− j−1 B(X ) g1 (t + τn ) − h1 (t) (m),(α +m,p)

k=1

≤ C g1 (t + τn ) − h1 (t) (m),(α +m,p) .

m−1

∑ A− j−1 B(X )

j=1

= C dm g1 (t + τn ) − h1 (t) (m),(α +m,p) .

Finally,   Ak−1 g1 (t + τn ) − h1 (t) = g1 (t + τn ) − h1 (t) α +k−1,p ≤ C g1 (t + τn ) − h1 (t) (m),(α +m,p) . Consequently,   Sg1 (t+τn )−Sh1 (t) (m) ≤ (m + 1)C 1+θ0 +dm g1 (t + τn )−h1 (t) (m),(α +m,p) → 0 as n → ∞. Hence (Sh1 )(k) (t) = lim (Sg1 )(k) (t + τn ) n→∞

for all t ∈ R for a fixed k = 0, 1, . . . , m. Thus, (Sh1 )(k) belongs to AA(DA (α + m, p)) for k = 0, 1, . . . , m.

9 Existence of C(m) -Pseudo-Almost Automorphic Solutions

226

Using similar arguments as above, one can easily see that (Sg1 )(k) (t) = lim (Sh1 )(k) (t − τn ) n→∞

for all t ∈ R for a fixed k = 0, 1, .. . , m. Hence, (Sg1 )(k) belongs to AA(DA (α +m, p))  for k = 0, 1, . . . , m. Therefore, S AA(m) (DA (α + m, p)) ⊂ AA(m) (DA (α + m, p)).   (m) (m) We next show that S PAP0 (DA (α + m, p) ⊂ PAP0 (DA (α + m, p)). Clearly, (m) Sg2 ∈ BC (R, DA (α + m, p)). Now, for each k = 0, 1, . . . , m, we have 1 lim r→∞ 2r

 Qr

(Sg2 )(k) (t) dt ≤ I + J + K

where 

1 r→∞ 2r

I = lim

Qr

g2 (t − s) α +k−1,p ds

=0 (m)

by using the fact g2 ∈ PAA0 (DA (α + m, p)). Similarly J=

k−1

1 r→∞ 2r

∑ A− j−1 . lim

j=1

≤

k−1

∑ A

− j−1

j=1

. lim

r→∞



 ∞ 0

 ∞

Qr 0



( j)

H(s) g2 (t − s) k dsdt

1 H(s) 2r

 Qr

( j) g2 (t − s) α +k−1,p dt

ds

=0 by using the fact that 1 r→∞ 2r



( j)

lim

Qr

g2 (t − s) α +k,p dt = 0

for all s ∈ R and applying the Lebesgue Dominated Convergence theorem. Finally 1 K = lim r→∞ 2r

 Qr

g2 (t) α +k−1,p dt = 0.

(A.3) The function t → F(t) belongs to PAA(m) (DA (α + m, p)). Corollary 9.4. Under assumptions (A.1)–(A.3), then Eq. (9.5) has a unique C(m+1) pseudo-almost automorphic solution u ∈ DA (α + m, p).

9.1 Existence Results for Some Autonomous Higher-Order Differential Equations

227

9.1.3 Higher-Order Differential Equations Theorem 9.5. Under similar assumptions as (A.1)–(A.2) for the matrix operator A and the vector-function F, then Eq. (9.2) has a unique C(m+n−1) -pseudo-almost automorphic mild solution u ∈ DA (α + m, p). Proof. Following the same lines as in the proof of Theorem 9.3 it follows that Eq. (9.3) has a unique C(m) -pseudo-almost automorphic solution given by t → z(t) := (w(t), w (t), w (t), . . . , w(n−1) (t))T . Thus, t → w(t),t → w (t), . . . ,t → w(n−1) (t) ∈ PAA(m) (DA (α + m, p)) and hence w ∈ PAA(m+n−1) (DA (α + m, p)) is the unique solution to Eq. (9.2). (A.4) The function t → f (t) belongs to PAP(m) (DA (α + m, p)). Corollary 9.6. Under assumptions (A.1)–(A.4), then Eq. (9.2) has a unique C(m+1) pseudo-almost periodic solution u ∈ DA (α + m, p).

9.1.4 Second-Order Differential Equations Fix α ∈ (0, 1), m ∈ N, and 1 ≤ p ≤ ∞. Let 0 ≤ γ < 1 and consider the nonautonomous second-order abstract differential equation du d2u + aAu = f (t), t ∈ R + bAγ dt 2 dt

(9.9)

where A : D(A) ⊂ H → H is a self-adjoint linear operator whose spectrum consists of isolated eigenvalues 0 < λ1 < λ2 < · · · < λn → ∞ with each eigenvalue having a finite multiplicity γ j equal to the multiplicity of the corresponding eigenspace, a, b > 0, and the function f : R → Hα +m,p is C(m) pseudo-almost automorphic, where Hα +m,p = DA (α + m, p). Our main objective here is to set some reasonable assumptions so that Eq. (9.9) can fit into our previous setting. Indeed, consider the polynomial associated with Eq. (9.9) given by γ

Rt (λ ) = λ 2 + bλnγ λ + aλn ,

t ∈ R,

where (λn )n≥1 is the sequence of eigenvalues for the self-adjoint operator A. Denote its roots by

9 Existence of C(m) -Pseudo-Almost Automorphic Solutions

228

ξkn = ηkn + iνkn for k = 1, 2 and n ≥ 1. We suppose that the roots ξ1n and ξ1n are simple and that the following crucial assumptions hold:

γ0 := sup [max(|ν1n |, |ν2n |)] < ∞,

(9.10)

n≥1,t∈R

and there exists δ0 > 0 such that sup[max(η1n , η2n )] ≤ −δ0 < 0.

(9.11)

n≥1

The assumptions Eqs. (9.10) and (9.11) will ensure that the operator A appearing in Eq. (9.13) is sectorial. We now rewrite Eq. (9.9) as a first-order differential equation on the appropriate space and then study that first-order equation and next go back to our second-order equation. Indeed, assuming that u is twice differentiable and setting z := (u, u )T , then Eq. (9.9) can be rewritten on X := D(A) × H in the following form: dz = A z + F(t), t ∈ R, dt

(9.12)

where A is the 2×2-operator matrix defined by

A =

 0 I . −aA −bAγ

(9.13)

Here n = 2 and A1 = bAγ . Now, the domain D(A ) = D(A) × D(Aγ ). The term F appearing in Eq. (9.3) is defined on R by F(t) = (0, f (t))T . Theorem 9.7. Under previous assumptions, then Eq. (9.9) has a unique C(m+1) pseudo-almost automorphic solution u ∈ DA (α + m, p). Proof. For all z := (u, v) ∈ D = D(A (t)) = D(A) × D(Aγ ), then (see Appendix for the spectral decomposition of the self-adjoint operator A): Az=

∞

∑ An Pn z,

n=1

9.1 Existence Results for Some Autonomous Higher-Order Differential Equations

229

where,

Pn :=

  En 0 0 1 , and An := γ , n ≥ 1. −aλn −bλn 0 En

The characteristic equation for An is given by

λ 2 + bλ λnγ + λn a = 0.

(9.14)

Noticing that A is invertible (by assumption a = 0) and using Eqs. (9.10) and π  , π such that Sθ ∪ {0} ⊂ ρ (A ) . On the (9.11) it follows that there exists θ ∈ 2 other hand, since ξ1n and ξ2n are distinct, then An is diagonalizable and hence An = Kn−1 Jn Kn , where Jn , Kn and Kn−1 are respectively given by ⎛ n ⎞ − ξ2 1 1 ⎠ , Kn = ⎝ ⎠ , Kn−1 = ⎝ ⎠. Jn = ⎝ ξ1n − ξ2n n n n n 0 ξ2 ξ1 ξ2 ξ1 −1 ⎛

ξ1n 0

⎞

⎛

1 1

⎞

It can be shown that: (a) there exists K > 0 such that: R(λ , A ) ≤

K for all λ ∈ Sθ . 1 + |λ |

(b) the operator A is sectorial. (c) the analytic semigroup (T (t))t≥s associated with A is exponentially stable. Using Theorem 9.5 it follows that Eq. (9.9) has a unique C(m+1) -pseudo-almost automorphic mild solution u ∈ DA (α + m, p). Corollary 9.8. Under assumptions (A.1)–(A.4), then Eq. (9.9) has a unique C(m+1) pseudo-almost periodic solution u ∈ DA (α + m, p).

9.1.5 An Autonomous Beam Equation Fix α ∈ (0, 1), m ∈ N and 1 ≤ p ≤ ∞. Let Ω ⊂ RN be an open bounded subset with sufficiently smooth boundary ∂ Ω . In this subsection, we make use of the results of the previous subsection to study and obtain the existence and uniqueness of a C(m+1) -pseudo-almost automorphic mild solution to the so-called linear beam equation given by

∂ 2u ∂u + aΔ 2 u = f (t), − bΔ 2 ∂t ∂t

t ∈ R, x ∈ Ω

(9.15)

9 Existence of C(m) -Pseudo-Almost Automorphic Solutions

230

Δ u(t, x) = u(t, x) = 0,

t ∈ R, x ∈ ∂ Ω

(9.16)

where a, b > 0, and f : R → DΔ 2 (α + m, p) is C(m) -pseudo-almost automorphic. Here we let H = L2 (Ω ) and equip it with its natural topology and let A be the linear operator given by Au = Δ 2 u for all u ∈ D(A) = H02 (Ω ) ∩ H 4 (Ω ). For each α ∈ (0, 1), the corresponding real interpolation space is DΔ 2 (α + m, p) equipped with its corresponding natural norm. In addition to the above, we also make the following assumptions: let α1 = 12 and suppose √ (A.5) b > 2 a. Now the polynomial associated with the beam equation (see Eqs. (9.15) and (9.16)) is given by 1

λ 2 + bλ p2 λ + aλ p = 0,

(9.17)

where (λ p ) p≥1 is the sequence of eigenvalues for the self-adjoint operator A = Δ 2 . Now (A.5) yields     λ p b2 − 4a ≥ λ1 b2 − 4a > 0. Consequently, the roots of Eq. (9.17) are real numbers and are given by

λ1p := 2−1

1

1   √  √  λ p − b + d and λ2p := 2−1 λ p − b − d

where d := b2 − 4a. Clearly λ2p < λ1p < 0. Moreover,

λ1p ≤ 2−1

  √  λ1 − b + d .

Now √ −b + d =

b2 − d √ −b − d 4a √ = −b − d 4a ≤− b

9.2 Existence Results for Some Nonautonomous Higher-Order Differential Equations

231

and hence  λ1p ≤ −2 λ1 ab−1 . Setting  δ0 = 2 λ1 ab−1 and in view of the above it follows that Eqs. (9.10) and (9.11) hold. Therefore, the beam equation has a unique C(m+1) -pseudo-almost automorphic mild solution u ∈ DΔ 2 (α + m, p). Corollary 9.9. Under assumptions (A.1)–(A.4)–(A.5), then the linear beam system Eqs. (9.15) and (9.16) has a unique C(m+1) -pseudo-almost automorphic mild solution u ∈ DA (α + m, p).

9.2 Existence Results for Some Nonautonomous Higher-Order Differential Equations 9.2.1 Introduction This section is a generalization of the previous one, that is, we study the case when the operators involved are time-dependent. Namely, we study the existence of C(m) -pseudo-almost automorphic (respectively, C(m) -pseudo-almost periodic) mild solutions to the higher-order nonautonomous differential equations n−1

u(n) (t) + ∑ Ak (t)u(k) (t) + A0 (t)u(t) = f (t), t ∈ R,

(9.18)

k=1

where Ak (t) : D(Ak (t)) ⊂ X → X (k = 0, 1, . . . , n−1) is a family of densely defined closed linear operators on a Banach space X , with domains Dk := D(Ak (t)) for k = 0, 1, . . . , n − 1, that are independent of t ∈ R, and the forcing term f : R → X is pseudo-almost automorphic. In order to study the above-mentioned problem, our strategy, as in the previous section, consists of rewriting Eq. (9.18) as a first-order nonautonomous differential equation in Z = X n = D(A) × X × · · · × X . Indeed, if u is differentiable n times, setting z := (u, u , . . . , u(n−1) )T , then Eq. (9.18) can be rewritten in the space Z in the form z (t) = A (t)z(t) + F(t), t ∈ R,

(9.19)

9 Existence of C(m) -Pseudo-Almost Automorphic Solutions

232

where A (t) is the family of n × n-time-dependent operator matrices defined by ⎛

0 I ⎜ 0 0 ⎜ ⎜ A (t) := ⎜ . . ⎜ ⎝ . . −A0 (t) −A1 (t)

⎞ 0. 0 ⎟ I . 0 ⎟ ⎟ . . . ⎟ ⎟ ⎠ . . I . . −An−1 (t)

(9.20)

whose domains D(A (t)) given by D := D(A0 (t))×D(A1 (t))×D(A2 (t))× · · · ×D(An−1 (t)) = D0 ×D1 ×D2 × · · · ×Dn−1 are independent of t ∈ R as each Dk := D(Ak (t)) for k = 0, 1, . . . , n − 1 is assumed to be independent of t ∈ R. Moreover, the function F appearing in Eq. (9.19) is defined by F(t) = (0, 0, . . . , f (t))T . Next we study and obtain the existence of a C(1) -pseudoalmost automorphic solution to a generic first-order differential equation of the form Eq. (9.19) under some appropriate assumptions and then go back to the study of Eq. (9.18). It should be mentioned that even in the autonomous setting, as pointed out in the excellent book by Xiao–Liang [168], further assumptions on the coefficient operators should be given if one would like to have well posedness. Moreover, in the nonautonomous, it is in general not interesting to go beyond the study of the first derivative of the solutions z to Eq. (9.19) as the k-derivative z(k) of z with k ≥ 2 involves expressions, which may or may not exist. Indeed, under some appropriate conditions it can be shown that z(t) =

 t −∞

U (t, s)F(s)ds

for all t ∈ R, is the unique bounded mild solution to Eq. (9.19), where U (t, s) is the evolution family associated with A (t). Formally, the first and second derivatives of z are given by z (t) =

 t −∞

A (t)U (t, s)F(s)ds + F(t)

and z (t) = +

 t −∞

 t

−∞

A 2 (t)U (t, s)F(s)ds A (t)U (t, s)F(s)ds + A (t)F(t) + F (t).

Looking at the expression of z and by induction one can easily see that it is in general not interesting to go beyond the study of z as z(k) with k ≥ 2 involves

9.2 Existence Results for Some Nonautonomous Higher-Order Differential Equations

233

A (k−1) (t) the (k − 1)-derivative with respect to t of the operators A (t), which may or may not exist. That is why instead of studying the existence of C(m) pseudo-almost automorphic mild solutions to Eq. (9.19) for m ≥ 2, we will only study its C(1) -pseudo-almost automorphic mild solutions.

9.2.2 First-Order Differential Equations Fix α ∈ (0, 1). In this subsection we study the existence of C(n) -pseudo-almost automorphic solutions to Eq. (9.18). For that, as we have mentioned in the introduction, it is enough to study Eq. (9.19). In order to investigate Eq. (9.19), the first step consists of studying a generic nonautonomous first-order equation of the form: u (t) = A(t)u(t) + F(t), t ∈ R,

(9.21)

where A(t) is a family of densely defined closed linear operators with constant domains D = D(A(t)) in t ∈ R, and F : R → X is a continuous function. For that, we will need the following assumptions: (B.1) The linear operators {A(t)}t∈R whose domains are constant in t satisfy the ! " Acquistapace–Terreni conditions. We then let U = U(t, s) : (t, s) ∈ T denote the evolution family associated with the family of linear operators A(t). (B.2) The evolution family U = {U(t, s) : (t, s) ∈ T} is exponentially stable, that is, there exists constants N, δ > 0 such that U(t, s) ≤ Ne−δ (t−s) for all t, s ∈ R, t > s. (B.3) t → F(t) is not only Xα -valued but also is pseudo-almost automorphic, where Xα = (X , D)α ,∞ is the real interpolation space of order (α , ∞) between X and D; and (B.4) The functions R × R → (t, s) → U(t, s)u and R × R → X , (t, s) → A(t)U(t, s)v belong to bAA(T, X ) for all u, v ∈ Xα . For details on the set T as well as the space bAA(T, X ), see Appendix A. Remark 9.10. If an evolution family U = {U(t, s) : (t, s) ∈ T} has an exponential dichotomy, we then define

Γ (t, s) :=

⎧ ⎨ U(t, s)P(s), ⎩

if t ≥ s, t, s ∈ R,

−UQ (t, s)Q(s), if s > t, t, s ∈ R.

234

9 Existence of C(m) -Pseudo-Almost Automorphic Solutions

Lemma 9.11. Suppose (B.1) holds and that U = {U(t, s) : (t, s) ∈ T} has exponential dichotomy with constants N and δ . If F : R → X is a bounded continuous function, then u given by u(t) =

 ∞ −∞

Γ (t, s)F(s)ds

(9.22)

for all t ∈ R is the unique bounded mild solution to Eq. (9.21). Proof. The fact that u given in Eq. (9.22) is a bounded mild solution to Eq. (9.21) is clear, see, e.g., [38, Chap. 4]. Now let u, v be two bounded mild solutions to Eq. (9.21). Setting w = u − v, one can easily see that w is bounded and that w(t) = U(t, s)w(s) for all (t, s) ∈ T. Now using property (a) from exponential dichotomy (Definition 2.80) it follows that P(t)w(t) = P(t)U(t, s)w(s) = U(t, s)P(s)w(s), and hence P(t)w(t) = U(t, s)P(s)w(s) ≤ Ne−δ (t−s) w(s) ≤ Ne−δ (t−s) w ∞ for all (t, s) ∈ T. Now, given t ∈ R with t ≥ s, if we let s → −∞, we then obtain that P(t)w(t) = 0, that is, P(t)u(t) = P(t)v(t). Since t is arbitrary it follows that P(t)w(t) = 0 for all t ≥ s. Similarly, from w(t) = U(t, s)w(s) for all t ≥ s and property (a) from exponential dichotomy (Definition 2.80) it follows that Q(t)w(t) = Q(t)U(t, s)w(s) = U(t, s)Q(s)w(s), and hence UQ (s,t)Q(t)w(t) = Q(s)w(s) for all t ≥ s. Moreover, Q(s)w(s) = UQ (s,t)Q(t)w(t) ≤ Ne−δ (t−s) w ∞ for all t ≥ s. Now, given s ∈ R with t ≥ s, if we let t → +∞, we then obtain that Q(s)w(s) = 0, that is, Q(s)u(s) = Q(s)v(s). Since s is arbitrary it follows that Q(s)w(s) = 0 for all t ≥ s. The proof is complete. Theorem 9.12. Under assumptions (B.1)–(B.4), then Eq. (9.21) has a unique C(1) pseudo-almost automorphic mild solution.

9.2 Existence Results for Some Nonautonomous Higher-Order Differential Equations

235

Proof. From Lemma 9.11 it is clear that the only bounded mild solution to (9.21) is given by u(t) =

 t −∞

U(t, s)F(s)ds for all t ∈ R.

(9.23)

Furthermore, the derivative of the bounded mild solution u appearing in (9.23) exists and is given by u (t) =

 t −∞

A(t)U(t, s)F(s)ds + F(t) for all t ∈ R.

(9.24)

Let F = g1 + g2 ∈ PAA(Xα ) where g1 ∈ AA(Xα ) and g2 ∈ PAP0 (Xα ). Set Sg j (t) :=

 t −∞

U(t, s)g j (s)ds for j = 1, 2.

  Our first objective consists of showing that S AA(Xα ) ⊂ AA(1) (X ). Using the fact that g1 ∈ AA(Xα ), for every sequence of real numbers (τn )n∈N there exist a subsequence (τn )n∈N and a function h1 such that h1 (t) := lim g1 (t + τn ) n→∞

is well defined for each t ∈ R, and lim h1 (t − τn ) = g1 (t)

n→∞

for each t ∈ R. Now (Sg1 )(t + τn ) − (Sh1 )(t) = = =

 t+τn −∞

 t −∞

 t −∞

+

U(t + τn , s)g1 (s)ds −

 t −∞

U(t, s)h1 (s)ds

U(t + τn , s + τn )g1 (s + τn )ds −

 t −∞

U(t, s)h1 (s)ds.

U(t + τn , s + τn )(g1 (s + τn ) − h1 (s))ds

 t −∞

U(t + τn , s + τn ) −U(t, s))h1 (s)ds.

From (B.2) and the Lebesgue Dominated Convergence Theorem, one can easily show that

9 Existence of C(m) -Pseudo-Almost Automorphic Solutions

236

  

t

−∞

   U(t + τn , s + τn )(g1 (s + τn ) − h1 (s))ds ≤

t −∞

NM e−δ (t−s)

    g1 (s + τn ) − h1 (s) ds → 0 as n → ∞. α

From (B.4) and the Lebesgue Dominated Convergence Theorem, it follows   

t −∞

  (U(t + τn , s + τn ) −U(t, s))h1 (s)ds → 0 as n → ∞

and hence (Sh1 )(t) = lim (Sg1 )(t + τn ) n→∞

for all t ∈ R. Using similar arguments as above one obtains that (Sg1 )(t) = lim (Sh1 )(t − τn ) n→∞

for all t ∈ R, which yields, t → (Sg1 )(t) belongs to AA(X ). Now (1)

(1)

(Sg1 ) (t+τn )−(Sh1 ) (t) = =

)

t+τn

−∞

 t −∞

−

U(t+τn , s)g1 (s)ds−

 t −∞

*(1) U(t, s)h1 (s)ds

A(t+τn )U(t+τn , s+τn )g1 (s + τn )ds + g1 (t + τn )

 t −∞

A(t)U(t, s)h1 (s)ds − h1 (t).

Write  t −∞

A(t + τn )U(t + τn , s + τn )g1 (s + τn )ds − = +

 t −∞

 t

−∞

 t −∞

A(t)U(t, s)h1 (s)ds

A(t + τn )U(t + τn , s + τn )(g1 (s + τn ) − h1 (s))ds

(A(t + τn )U(t + τn , s + τn ) − A(t)U(t, s))h1 (s)ds.

Using Lemma 2.83 and the Lebesgue Dominated Convergence Theorem, one can easily show that

9.2 Existence Results for Some Nonautonomous Higher-Order Differential Equations

  

t

−∞

237

  A(t + τn )U(t + τn , s + τn )(g1 (s + τn ) − h1 (s))ds ≤

 t −∞

    Hα ,δ (t − s)g1 (s + τn ) − h1 (s) ds. α

→ 0 as n → ∞. Using (B.4) and the Lebesgue Dominated Convergence Theorem it follows   

t

−∞

  (A(t + τn )U(t + τn , s + τn ) − A(t)U(t, s))h1 (s)ds → 0 as n → ∞

and hence (Sh1 )(1) (t) = lim (Sg1 )(1) (t + τn ) n→∞

for all t ∈ R. Using similar arguments as above, one can easily see that (Sg1 )(1) (t) = lim (Sh1 )(1) (t − τn ) n→∞

for all t ∈ R. In view of the above, Sg1 belongs to AA(1) (X ). The next step consists of   (k) showing that S PAP0 (Xα ) ⊂ PAP0 (X ) for k = 0, 1. Clearly, Sg2 ∈ BC(k) (R, X ) for k = 0, 1. Using the fact that g2 ∈ PAP0 (Xα ) and assumption (B.2) it can be easily shown that Sg2 ∈ PAP0 (X ). Now from Lemma 2.83 it follows that 1 2r

 Qr

  

  ∞    1    A(t)U(t, s)g2 (s)dsdt ≤ Hα ,δ (s)g2 (t − s) dsdt 2r α Qr 0 −∞

   +∞   1   Hα ,δ (s) ≤ g2 (t − s) dt ds. 2r Qr α 0 t

Now 1 r→∞ 2r



lim

Qr

    g2 (t − s) dt = 0, α

as t → g2 (t − s) ∈ PAP0 (Xα ) for every s ∈ R. One completes the proof by using the Lebesgue Dominated Convergence Theorem. In summary, (Sg2 )(k) ∈ PAP0 (X ) for k = 0, 1, which completes the proof.

238

9 Existence of C(m) -Pseudo-Almost Automorphic Solutions

9.2.3 Higher-Order Differential Equations To study Eq. (9.18), we need the following assumptions: (B.5) The domains Dk := D(Ak (t)) for k = 0, 1, . . . , n − 1 are assumed to be constant in t. The linear operators {A ! (t)}t∈R satisfy the Acquistapace– " Terreni conditions. We then let U = U (t, s) : (t, s) ∈ T denote the evolution family associated with the family of linear operators A (t). (B.6) The evolution family U = {U (t, s) : (t, s) ∈ T} is exponentially stable, that is, there exists constants N, δ > 0 such that U (t, s) ≤ Ne−δ (t−s) for all t, s ∈ R, t > s. (B.7) t → f (t) is not only Xαn -valued but also is pseudo-almost automorphic, where Xαn = (X , Dn−1 )α ,∞ is the real interpolation space of order (α , ∞) between X and Dn−1 ; and (B.8) The functions R×R → Z, (t, s) → U (t, s)u and (t, s) → A (t)U (t, s)v belong to bAA(T, Z) for all u, v ∈ Zα . Theorem 9.13. Let α ∈ (0, 1). Under assumptions (B.5)–(B.8), then Eq. (9.18) has a unique C(n) -pseudo-almost automorphic mild solution. Proof. Following the same lines as in the proof of Theorem 9.12 it follows that Eq. (9.19) has a unique C(1) -pseudo-almost automorphic solution given by the mapping t → z(t) := (w(t), w (t), w (t), . . . , w(n−1) (t))T . Thus, t → w(t),t → w (t), . . . ,t → w(n−1) (t) ∈ PAA(1) (X ) and hence w ∈ PAA(n) (X ) is the unique solution to Eq. (9.18).

9.2.4 Second-Order Damped Differential Equations Fix α ∈ (0, 1), γ ∈ [0, 1). Consider the nonautonomous damped second-order abstract differential equation u + b(t)Aγ u + a(t)Au = f (t), t ∈ R

(9.25)

where A : D(A) ⊂ H → H is a self-adjoint (possibly unbounded) linear operator on H whose spectrum consists of isolated eigenvalues 0 < λ1 < λ2 < · · · < λl → ∞ as l → ∞

9.2 Existence Results for Some Nonautonomous Higher-Order Differential Equations

239

with each eigenvalue having a finite multiplicity γ j equal to the multiplicity of the corresponding eigenspace, f : R → Hα is pseudo-almost automorphic, with Hα = (H , D(A))α ,∞ being the real interpolation space of order (α , ∞) between H and D(A), and the functions a, b : R → R are continuous and ω -periodic (ω > 0) in the sense that a(t + ω ) = a(t) and b(t + ω ) = b(t) for all t ∈ R and satisfy: • there exist K1 , K0 > 0 and 0 < κ ≤ 1 such that |b(t) − b(s)| ≤ K1 |s − t|κ and |a(t) − a(s)| ≤ K0 |s − t|κ for all t, s ∈ R; and • there exist a0 , b0 > 0 with inf a(t) = a0 and inf b(t) = b0 .

t∈R

t∈R

The main objective here is to set some reasonable assumptions so that Eq. (9.25) can fit into our abstract setting. For that, consider the polynomial associated with Eq. (9.25) given by γ

Rt (λ ) = λ 2 + b(t)λnγ λ + a(t)λn ,

t ∈ R,

where (λn )n≥1 is the sequence of eigenvalues of A. Denote the roots of the above-mentioned polynomial by

ξkn (t) = ηkn (t) + iνkn (t) for k = 1, 2, n ≥ 1, and t ∈ R. We next suppose that the roots ξ1n (t) and ξ2n (t) are simple and that the following crucial assumptions hold:

γ0 := sup [max(|ν1n (t)|, |ν2n (t)|)] < ∞,

(9.26)

n≥1,t∈R

and there exists δ0 > 0 such that sup [max(η1n (t), η2n (t))] ≤ −δ0 < 0.

n≥1,t∈R

(9.27)

9 Existence of C(m) -Pseudo-Almost Automorphic Solutions

240

The assumptions (9.26) and (9.27) will ensure that the operators A (t) appearing in Eq. (9.29) satisfy (2.26) from the Acquistapace–Terreni conditions. Rewrite Eq. (9.25) as a first-order differential equation on the appropriate space. Indeed, assuming that u is twice differentiable and setting z := (u, u )T , then Eq. (9.25) can be rewritten in X := D(A) × H in the following form: dz = A (t)z + F(t), t ∈ R, dt

(9.28)

where A (t) is the family of 2×2-operator matrices defined by ⎛

0

I

⎞

⎠, A (t) = ⎝ γ −a(t)A −b(t)A

t ∈ R.

(9.29)

The domains D(A (t)) = D(A) × D(Aγ ) are constant in t ∈ R and term F in Eq. (9.19) is given by F(t) = (0, f (t))T . Here we take A1 (t) = b(t)Aγ and A0 (t) = a(t)A. Theorem 9.14. Under previous assumptions, then Eq. (9.25) has a unique C(2) pseudo-almost automorphic mild solution. Proof. We make use of the spectral decomposition of A given in Appendix A. For all z := (u, v) ∈ D = D(A (t)) = D(A) × D(Aγ ), we have A (t)z =

∞

∑ An (t)Pn z,

n=1

where

Pn :=

  En 0 0 1 , and An (t) := γ , n ≥ 1, t ∈ R. −a(t)λn −b(t)λn 0 En

The characteristic equation for An (t) is given by

λ 2 + b(t)λ λnγ + λn a(t) = 0.

(9.30)

Noticing that A (t) is invertible for all t ∈ R (by assumptiona(t) = 0 for all π  , π such that t ∈ R) and using (9.26) and (9.27) it follows that there exists θ ∈ 2 {0} ⊂ ρ (A (t)) . Moreover, it can be shown [11] that Sθ ∪ (a) there exists K > 0 such that: R(λ , A (t)) ≤ for all λ ∈ Sθ and t ∈ R;

K 1 + |λ |

9.2 Existence Results for Some Nonautonomous Higher-Order Differential Equations

241

(b) the family of operators {A (t)}t∈R satisfy Acquistapace–Terreni conditions; and (c) the evolution family (U (t, s))t≥s associated with A (t) is exponentially stable. From the ω -periodicity of both a and b it easily follows that A (t + ω ) = A (t) for all t ∈ R, which yields U (t + ω , s + ω ) = U (t, s) and A (t + ω )U (t + ω , s + ω ) = A (t)U(t, s) for all (t, s) ∈ T. That is, assumption (B.8) holds. Using Theorem 9.13 it follows that Eq. (9.25) has a C(2) -pseudo-almost automorphic mild solution.

9.2.5 A Nonautonomous Beam Equation Fix α ∈ (0, 1) and γ = 12 , and let Ω ⊂ RN be an open bounded subset with sufficiently smooth boundary ∂ Ω . Let H = L2 (Ω ) and equip it with its natural topology. We study and obtain the existence and uniqueness of a C(2) -pseudo-almost automorphic mild solution to the nonautonomous linear beam equation given by

∂ 2u ∂u + a(t, x)Δ 2 u = f (t), − b(t, x)Δ 2 ∂t ∂t Δ u(t, x) = u(t, x) = 0,

t ∈ R, x ∈ Ω

t ∈ R, x ∈ ∂ Ω

(9.31) (9.32)

where a, b : R × Ω → R are ω -periodic (ω > 0) in the sense that a(t + ω , x) = a(t, x) and b(t + ω , x) = b(t, x) for all t ∈ R and x ∈ Ω , (t, x) → a(t, x), b(t, x) are jointly continuous, x → a(t, x), b(t, x) are differentiable for all t ∈ R, and f : R → Hα is pseudo-almost automorphic, where Hα = (L2 (Ω ), H02 (Ω ) ∩ H 4 (Ω ))α ,∞ is the real interpolation space of order (α , ∞) between L2 (Ω ) and H02 (Ω ) ∩ H 4 (Ω ). Moreover, we suppose that the following assumptions hold: (B.9) if we set

inf

t∈R,x∈Ω

b(t, x) = b0 > 0,

inf

t∈R,x∈Ω

a(t, x) = a0 > 0, then

 b0 > 2 a ∞ .

9 Existence of C(m) -Pseudo-Almost Automorphic Solutions

242

Let A be the linear operator given by Au = Δ 2 u for all u ∈ D(A) = H02 (Ω ) ∩ H 4 (Ω ). The polynomial associated with the nonautonomous beam equation (see Eqs. (9.31) and (9.32)) is given by 1

λ 2 + b(t, x)λn2 λ + a(t, x)λn = 0,

t ∈ R, x ∈ Ω

(9.33)

where (λn )n≥1 is the sequence of eigenvalues for the self-adjoint operator A = Δ 2 . Now (B.9) yields     λn b2 (t, x) − 4a(t, x) ≥ λ1 b20 − 4 a ∞ > 0 for all t ∈ R and x ∈ Ω . Consequently, the roots of Eq. (9.33) are real numbers and are given by     λ1n (t, x) := 2−1 λn − b(t, x) + d(t, x) and

λ2n (t, x) := 2−1

    λn − b(t, x) − d(t, x)

for all x ∈ Ω and t ∈ R, where d(t) := b2 (t, x) − 4a(t, x). Clearly λ2n (t, x) < λ1n (t, x) < 0 for all x ∈ Ω and t ∈ R. Moreover,     λ1n (t, x) ≤ 2−1 λ1 − b(t, x) + d(t, x) for all x ∈ Ω and t ∈ R. Now −b(t, x) +

 d(t, x) =

b2 (t, x) − d(t, x)  −b(t, x) − d(t, x)

=

4a(t, x)  −b(t, x) − d(t, x)

≤−

4a0 b ∞

for all x ∈ Ω , and t ∈ R and hence  4a0 λ1n (t, x) ≤ − λ1 2 b ∞ for all x ∈ Ω and t ∈ R.

9.2 Existence Results for Some Nonautonomous Higher-Order Differential Equations

243

Setting

δ0 =



λ1

4a0 2 b ∞

and in view of the above it follows that Eqs. (9.26) and (9.27) hold. Therefore, the nonautonomous beam equation (9.31) and (9.32) has a C(2) -pseudo-almost automorphic mild solution.

Bibliographical Notes The results presented in this chapter are mainly based on the following sources: Diagana and Nelson [75] (for the autonomous case) and Diagana [76] (for the nonautonomous case). Additional relevant references to this chapter include Lunardi [137], Chicone and Latushkin [38], and Engel and Nagel [90].

Chapter 10

Pseudo-Almost Periodic Solutions to Some Third-Order Differential Equations

10.1 Introduction Motivated by the recent work by Diagana [49, 52, 61], in this chapter using the Schauder fixed-point theorem (Theorem 1.98), the Banach fixed-point principle (Theorem 1.96), and the dichotomy techniques, we study the problem which consists of the existence of pseudo-almost periodic (respectively, weighted pseudoalmost periodic) solutions to the nonautonomous third-order differential equations ' d & u + g(t, Bu(t)) = w(t)Au(t) + f (t,Cu(t)), t ∈ R dt

(10.1)

where the following preliminary assumptions will be made: 1. A : D(A) ⊂ H → H is a self-adjoint linear operator on the Hilbert space H whose spectrum consists of isolated eigenvalues 0 < λ1 < λ2 < · · · < λl → ∞ as l → ∞ with each eigenvalue having a finite multiplicity γ j equals to the multiplicity of the corresponding eigenspace; 2. the algebraic sum of the (possibly unbounded) linear operators B and C defined by B +C : D(B) ∩ D(C) ⊂ H → H is assumed to be a nontrivial linear operator and the following holds:   Hα := H , D(A) α ,∞ ⊂ D(B) ∩ D(C);

(10.2)

3. the function w : R → R given by w(t) = −ρ (t) for all t ∈ R is assumed to be almost periodic and further there exist two constants ρ0 , ρ1 > 0 satisfying:

ρ0 ≤ ρ (t) ≤ ρ1 for all t ∈ R; and T. Diagana, Almost Automorphic Type and Almost Periodic Type Functions in Abstract Spaces, DOI 10.1007/978-3-319-00849-3 10, © Springer International Publishing Switzerland 2013

(10.3)

245

246

10 Pseudo-Almost Periodic Solutions to Some Third-Order Differential Equations

4. the functions f , g : R × D(A) → H are pseudo-almost periodic (respectively, weighted pseudo-almost periodic) in the first variable uniformly in the second variable. As an immediate application, we study and obtain the existence of pseudo-almost periodic solutions to the nonautonomous third-order differential equations u + B(t)u + A(t)u = h(t, u),

(10.4)

where A(t) = q(t)A, and B(t) = p(t)B for each t ∈ R, the functions p, q : R → R are almost periodic, A, B are the same as in Eq. (10.1), and the function h : R × Hα → H , (t, u) → h(t, u) is pseudo-almost periodic in t ∈ R uniformly in u ∈ Hα . To deal with Eq. (10.1), we rewrite it as a nonautonomous first-order differential equation in D(A) × H × H and next study the obtained first-order differential equation with the help of the well-known Schauder fixed-point principle. Indeed, assuming that u ∈ D(A) is three times differentiable and setting ⎛

u

⎞

⎜ ⎜ z := ⎜ ⎜ ⎝

u

⎟ ⎟ ⎟, ⎟ ⎠

u + g(t, Bu) then Eq. (10.1) can be rewritten in X := D(A) × H × H in the following form dz = A (t)z + F(t, z(t)), t ∈ R, dt

(10.5)

where A (t) is the family of 3×3-operator matrices defined by ⎛

0

I 0

⎞

⎟ ⎜ ⎟ ⎜ ⎟ ⎜ A (t) = ⎜ 0 0 I ⎟ , ⎟ ⎜ ⎠ ⎝ w(t)A 0 0

t ∈ R.

(10.6)

Clearly, domain D(A (t)) = D(A) × H × H is constant in t ∈ R. The vector-valued function F appearing in Eq. (10.5) is defined on R × Xα for some α ∈ (0, 1) by ⎛

0

⎞

⎟ ⎜ ⎜ −g(t, Bu) ⎟ ⎟, ⎜ F(t, z(t)) = ⎜ ⎟ ⎠ ⎝ f (t,Cu)

10.2 Existence of Pseudo-Almost Periodic Solutions

247

where Xα is the real interpolation space of order (α , ∞) between X and D(A (t)) which is explicitly given by     Xα = X , D(A (t)) α ,∞ = H ×H ×H , D(A)×H ×H α ,∞ = Hα ×H ×H . Clearly, if p : R → R is differentiable, one can easily check that Eq. (10.4) is a special case of Eq. (10.1). Indeed, Eq. (10.4) can be rewritten as ) * d du + g(t, ˜ Bu(t)) = w(t)Au(t) ˜ + f˜(t, Bu(t)), t ∈ R dt dt

(10.7)

where C = B, w(t) ˜ = −q(t), g(t, ˜ Bu) = p(t)Bu, and f˜(t, Bu) = h(t, u) + p (t)Bu for all t ∈ R. Once we rewrite Eq. (10.7) in the form Eq. (10.5), its corresponding vectorvalued function F which we denote by F˜ is defined on R × Xα for some α ∈ (0, 1) by ⎛ ⎜ ˜ z(t)) = ⎜ F(t, ⎝

0 −p(t)Bu

⎞ ⎟ ⎟. ⎠

h(t, u) + p (t)Bu

10.2 Existence of Pseudo-Almost Periodic Solutions 10.2.1 First-Order Differential Equations Let α , β are real numbers such that 0 < α < β < 1. The bound of the injection Xβ → Xα will be denoted by c, that is, u(t) α ≤ c u(t) β for all u ∈ Xβ . Consider the nonautonomous evolution equation u (t) = A(t)u(t) + F(t, u(t)), t ∈ R,

(10.8)

where F : R × Xα → X is jointly continuous. The study of Eq. (10.8) requires the following additional assumptions: (E.1) The linear operators {A(t)}t∈R whose domains are constant in t satisfy the ! " Acquistapace–Terreni conditions. Let U = U(t, s) : t, s ∈ R, t ≥ s denote the evolution family associated with the family of linear operators A(t). (E.2) The evolution family U(t, s) is compact for t > s and is exponentially stable, that is, there exists constants N, δ > 0 such that U(t, s) ≤ Ne−δ (t−s) for t ≥ s.

248

10 Pseudo-Almost Periodic Solutions to Some Third-Order Differential Equations

(E.3) R(ω , A(·)) ∈ AP(B(Xα , X )). (E.4) The function F : R × Xα → X is pseudo-almost periodic in the first variable uniformly in the second one. For each bounded subset K ⊂ Xα , F(R, K) is bounded. Moreover, the function u → F(t, u) is uniformly continuous on any bounded subset K of Xα for each t ∈ R. Finally, we suppose that there exists L > 0 such that sup t∈R, u α ≤L

F(t, u) ≤

L , e(β )

where e(β ) := cc(β )δ β Γ (1 − β ). (E.5) Let (un )n∈N ⊂ PAP(Xα ) be uniformly bounded and uniformly convergent in every compact subset of R. Then F(·, un (·)) is relatively compact in BC(R, Xα ). Remark 10.1. Under assumption (E.2), it can be shown that for each given t ∈ R and τ > 0, the family {U(·, s) : s ∈ (−∞,t − τ )} is equi-continuous in t for the uniform operator topology. Recall the notion of a mild solution for Eq. (10.8). Definition 10.2. Under assumption (E.1), a continuous function u : R → Xα is said to be a mild solution to Eq. (10.8) provided that u(t) = U(t, s)u(s) +

 t s

U(t, τ )F(τ , u(τ ))d τ

(10.9)

for each ∀t ≥ s, t, s ∈ R. Let us indicate that if F : R × Xα → X is a jointly continuous bounded function, then it can be easily shown that u satisfying u(t) =

 t −∞

U(t, s)F(s, u(s))ds.

(10.10)

for all t ∈ R, is a mild solution to Eq. (10.8). Set (Su)(t) =

 t −∞

U(t, s)P(s)F(s, u(s))ds.

We need the following Lemma. Lemma 10.3 (Diagana [49]). Under assumptions (E.1)–(E.3), the mapping S : BC(R, Xα ) → BC(R, Xα ) is well defined and continuous. Proof. First of all, S(BC(R, Xα )) ⊂ BC(R, Xα ). Indeed, setting g(t) := F(t, u(t)) and using Proposition 2.81, we obtain

10.2 Existence of Pseudo-Almost Periodic Solutions

249

    Su(t) ≤ cSu(t) α β ≤c

 t  −∞

 U(t, s)g(s) ds β  t

≤ cc(β )

−∞

  δ e− 2 (t−s) (t − s)−β g(s)ds

   ≤ cc(β )g∞

+∞

e−σ

0

2σ δ

−β

  ≤ cc(β )δ β Γ (1 − β )g∞ ,

2d σ δ

and hence     Su(t) ≤ e(β )g α ∞ for all t ∈ R. To complete the proof, we need to show that S is continuous. For that consider an arbitrary sequence of functions un ∈ BC(R, Xα ) which converges uniformly to some u ∈ BC(R, Xα ), that is,   un − u

α ,∞

→0

as n → ∞.

Now  

t −∞

 U(t, s)P(s)[F(s, un (s)) − F(s, u(s))] dsα ≤ c(α )

 t −∞

  δ (t − s)−α e− 2 (t−s) F(s, un (s)) − F(s, u(s)) ds.

Now, using the continuity of F and the Lebesgue Dominated Convergence Theorem we conclude that  

t

−∞

 U(t, s)P(s)[F(s, un (s)) − F(s, u(s))] dsα → 0 as n → ∞ ,

and hence   Sun − Su

α ,∞

→0

as n → ∞. The proof of the next lemma follows along the same lines as in [82, 128] and hence is omitted.

250

10 Pseudo-Almost Periodic Solutions to Some Third-Order Differential Equations

Lemma 10.4. Let Bα = {u ∈ PAP(Xα ) : u α ≤ L}. Under assumptions (E.1)– (E.2)–(E.4), then the functions in S(Bα ) are equicontinuous on R. Theorem 10.5. Suppose assumptions (E.1)–(E.5) hold, then Eq. (10.8) has at least one pseudo-almost periodic mild solution Proof. First of all, note that using the proof of Lemma 10.3 one can easily show that S(Bα ) ⊂ Bα . In view of Lemmas 10.3 and 10.4, it remains to show that V = {Su(t) : u ∈ Bα } is a relatively compact subset of Xα for each t ∈ R. For that, fix t ∈ R and consider an arbitrary ε > 0. Clearly, (Sε u)(t) :=

 t−ε −∞

U(t, s)F(s, u(s))ds, u ∈ Bα

= U(t,t − ε )

 t−ε −∞

U(t − ε , s)F(s, u(s))ds, u ∈ Bα

= U(t,t − ε )(Su)(t − ε ), u ∈ Bα and hence Vε := {Sε u(t) : u ∈ Bα } is relatively compact in Xα as U(t,t − ε ) is compact by assumption. Now Su(t) −U(t,t − ε )

 t−ε −∞

U(t − ε , s)F(s, u(s))ds α

≤ c Su(t) −U(t,t − ε ) ≤c

 t t−ε

≤ cc(β )

=

−∞

U(t − ε , s)F(s, u(s))ds β

U(t, s)F(s, u(s)) β ds  t

δ

t−ε

cc(β )L ≤ e(β ) ≤

 t−ε

e− 2 (t−s) (t − s)−β F(s, u(s)) ds

 ε 0

 cc(β )L ε

e(β )

0

δ

e− 2 σ σ −β d σ

σ −β d σ

cc(β )L ε 1−β . (1 − β )e(β )

10.2 Existence of Pseudo-Almost Periodic Solutions

251

Using the facts that Bα is a closed convex subset of PAP(Xα ) and that S(Bα ) ⊂ Bα , one can see that co S(Bα ) ⊂ Bα . Hence the following inclusions hold: S(co S(Bα )) ⊂ S(Bα ) ⊂ co S(Bα ). Moreover, one can easily check that {u(t) : u ∈ co S(Bα )} is relatively compact in Xα for each fixed t ∈ R and that functions in co S(Bα ) are equicontinuous on R. By the well-known Arzel`a-Ascoli theorem, the restriction of co S(Bα ) to any compact subset I of R is relatively compact in C(I, Xα ). In view of the above, it follows that S : co S(Bα ) → co S(Bα ) is continuous and compact. Using the Schauder fixed-point theorem (Theorem 1.98) it follows that S has a fixed-point, which obviously is an almost periodic mild solution to Eq. (10.8).

10.2.2 Third-Order Differential Equations To study the existence of almost periodic solutions to Eq. (10.1), in addition to the previous assumptions, we suppose that the following additional assumption holds: (E.6) The linear operators B,C : Hα → H are bounded. Let K > 0 be their bound, that is,     Bu ≤ K u

α

    and Cu ≤ K uα

for all u ∈ Hα . Theorem 10.6. Under previous assumptions and if (E.4)–(E.6) hold, then Eq. (10.1) has at least one pseudo-almost periodic solution u ∈ Hα . Proof. We make use of the spectral decomposition of A given in Appendix A. For all ⎛ ⎞ u ⎝ z := v ⎠ ∈ D = D(A (t)) = D(A) × H × H , w

252

10 Pseudo-Almost Periodic Solutions to Some Third-Order Differential Equations

we obtain the following: A (t)z =

∞

∑ An (t)Pn z,

n=1

where ⎛

En 0 0

⎞

⎟ ⎜ ⎟ ⎜ ⎟ ⎜ Pn := ⎜ 0 En 0 ⎟ , n ≥ 1, ⎟ ⎜ ⎠ ⎝ 0 0 En and ⎛

⎞ 10 ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ An (t) := ⎜ 0 0 1⎟ , n ≥ 1, t ∈ R. ⎜ ⎟ ⎝ ⎠ w(t)λn 0 0 0

Clearly, the characteristic equation for An (t) is given by − λ 3 + λn w(t) = −λ 3 − λn ρ (t) = 0,

(10.11)

from which we obtain its eigenvalues given by    2π 2π λ1n (t) = − 3 λn ρ (t), λ2n (t) = 3 λn ρ (t) ei 3 , and λ3n (t) = 3 λn ρ (t)e−i 3  and therefore σ (An (t)) = λ1n (t), λ2n (t), λ3n (t) . In view of the above it follows that there exists θ ∈

π 2

 , π such that

Sθ ∪ {0} ⊂ ρ (A (t)) .

π π + ε with ε ∈ (0, ) would be fine. 2 6 It is also clear that λ1n , λ2n , λ3n are distinct and each of them is of multiplicity one, then An (t) is diagonalizable. Further, it is not difficult to see that An (t) = Kn−1 (t)Jn (t)Kn (t), where Jn (t), Kn (t) and Kn−1 (t) are respectively given by More precisely, any θ of the form θ =

10.2 Existence of Pseudo-Almost Periodic Solutions

⎛

λ1n (t)

⎜ ⎜ ⎜ Jn (t) = ⎜ 0 ⎜ ⎝ 0

0

λ2n (t) 0

⎛

⎞

0

253

1

1

1

⎞

⎟ ⎜ ⎟ ⎟ ⎜ ⎟ ⎟ ⎜ n ⎟ n n 0 ⎟ , Kn (t) = ⎜ λ1 (t) λ2 (t) λ3 (t) ⎟ . ⎟ ⎜ ⎟ ⎠ ⎝ ⎠ n n 2 n 2 n 2 [λ1 (t)] [λ2 (t)] [λ3 (t)] λ3 (t)

For λ ∈ Sθ and z ∈ X , one has ∞

∑ (λ − An (t))−1 Pn z

R(λ , A (t))z =

n=1 ∞

∑ Kn (t)(λ − Jn (t))−1 Kn−1 (t)Pn z.

=

n=1

It is not hard to see that there exists K > 0 such that   R(λ , A (t)) ≤

K $ $ 1 + $λ $

for all λ ∈ Sθ and t ∈ R. Clearly, D = D(A (t)) is constant in t. Moreover, A (t) is invertible with ⎛ A (t)−1

0 0 w(t)−1 A−1

⎜ ⎜ ⎜ = ⎜I 0 ⎜ ⎝ 0I

0

⎞ ⎟ ⎟ ⎟ ⎟, ⎟ ⎠

t ∈ R.

0

Therefore, for t, s, r ∈ R, one has 

 A (t) − A (s) A (r)−1 ⎞ ⎛ 00 0 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ = ⎜0 0 0 ⎟, ⎟ ⎜ ⎠ ⎝ −1 0 0 w(r) (w(t) − w(s))IH

and hence assuming that there exist M0 ≥ 0 and μ ∈ (0, 1] such that $μ $ $ $ $ $ $ $ $w(t) − w(s)$ ≤ M0 $t − s$ it follows that there exists M > 0 such that

(10.12)

254

10 Pseudo-Almost Periodic Solutions to Some Third-Order Differential Equations

$μ    $  $   $  (A (t) − A (s))A (r)−1 z ≤ M $t − s$ z.  Therefore, the family of linear operators A (t)

t∈R

satisfy Acquistapace–Terreni

conditions. For every t ∈ R, the family of linear operators A (t) generate an analytic semigroup (eτ A (t) )τ ≥0 on X given by eτ A (t) z =

∞

∑ Kn (t)−1 Pn eτ Jn Pn Kn (t)Pn z, z ∈ X .

n=0

On the other hand, we have  τ A (t)  e z =

∞











∑ Kn (t)−1 Pn eτ Jn Pn Kn (t)Pn Pn z,

n=0

 z1 with for each z =

z2 z2

⎛ n  eλ1 (t)τ E 0 0 n  ⎜ ⎜   τJ n ⎜ e n Pn z2 =  0 eλ2 (t)τ En 0 ⎜ ⎜ ⎝  n  0 0 eλ3 (t)τ En

⎞ ⎛ ⎞2 z1   ⎟ ⎜ ⎟ ⎟ ⎜ ⎟ ⎟ ⎜ ⎟ ⎟ ⎜z2 ⎟ ⎟ ⎜ ⎟ ⎠ ⎝ ⎠  0 z2 

 n 2  n 2  n 2 ≤ eλ1 (t)τ En z1  + eλ2 (t)τ En z2  + eλ3 (t)τ En z2   2 n ≤ e2ℜe(λ2 (t))τ z . Clearly, using Eq. (10.3) it follows that  3

 2π  λn ρ (t) cos 3  3 λn ρ (t) =− 2  3 λ1 ρ0 ≤− 2

ℜe(λ2n (t)) =

Setting δ =

 3

λ1 ρ0 > 0, it follows that there exists C0 > 0 such that  τ A (t)  e  ≤ C0 e−δ τ ,

τ ≥ 0.

(10.13)

10.3 Existence of Weighted Pseudo-Almost Periodic Solutions

255

Arguing as in [15] it follows that the evolution family (U(t, s))t≥s is exponentially stable and hence (E.2) holds. Using the fact that t → w(t) and t → w(t)−1 are almost periodic it follows that t → A (t)−1 is almost periodic with respect to operator topology. Using Theorem 10.5 it follows that Eq. (10.1) has at least one almost periodic mild solution.

10.2.3 Some General Third-Order Differential Equations Suppose F˜ satisfies (E.4)–(E.5) and that the following assumptions hold: (hh1 ) The function q : R → C is given such that q(t) = ρ˜ (t) for all t ∈ R is almost periodic and there exist ρ˜ 0 , ρ˜ 1 > 0 such that

ρ˜ 0 ≤ ρ˜ (t) ≤ ρ˜ 1 for all t ∈ R. (hh2 ) There exist L0 > 0 and μ ∈ (0, 1] such that $ $ $ $ $q(t) − q(s)$ ≤ L0 $t − s$μ for all s,t ∈ R. (hh3 ) The function p : R → C is uniformly continuous, almost periodic, and differentiable. The proof of the next theorem is now clear. Theorem 10.7. Under previous assumptions and if (hh1 )–(hh2 )–(hh3 )–(E.6) hold, then Eq. (10.4) has at least one pseudo-almost periodic solution u ∈ Hα .

10.3 Existence of Weighted Pseudo-Almost Periodic Solutions 10.3.1 First-Order Differential Equations In this section, we fix a weight μ ∈ U∞Inv (hence PAP(X , μ ) is translation-invariant) and study the existence of weighted pseudo-almost periodic solutions to Eq. (10.8). Here we utilize dichotomy techniques and the well-known Banach fixed-point principle (Theorem 1.96). To study the existence of weighted pseudo-almost periodic solutions to Eq. (10.8) we will assume that the following assumptions hold:

256

10 Pseudo-Almost Periodic Solutions to Some Third-Order Differential Equations

(E.7) The evolution family U = {U(t, s)}t≥s generated by A(·) has an exponential dichotomy with constants N, δ > 0 and dichotomy projections P(t) for t ∈ R. (E.8) There exists 0 ≤ α < 1 such that Xαt = Xα for all t ∈ R, with uniform equivalent norms. (H.9) The function F : R × X → X belongs to PAP(R × X , μ ). Moreover, the functions F are uniformly Lipschitz with respect to the second argument in the following sense: there exists K > 0 such that F(t, u) − F(t, v) ≤ K u − v for all u, v ∈ X and t ∈ R. If 0 < α < 1, then the nonnegative constant k will denote the bounds of the embedding Xα → X , that is, x ≤ k x α for all x ∈ Xα . Note that under assumptions (E.1)–(E.9), it can be easily shown that Eq. (10.8) has a unique mild solution given by u(t) =

 t −∞

U(t, s)P(s)F(s, u(s))ds −

 ∞ t

UQ (t, s)Q(s)F(s, u(s))ds

for each t ∈ R. In what follows, we denote by Γ1 and Γ2 , the nonlinear integral operators defined by (Γ1 u)(t) :=

 t −∞

U(t, s)P(s)F(s, u(s))ds, and

and (Γ2 u)(t) :=

 ∞ t

UQ (t, s)Q(s)F(s, u(s))ds.

Lemma 10.8. Under assumptions (E.1)–(E.3)–(E.7)–(E.8)–(E.9), the integral operators Γ1 and Γ2 defined above map PAP(Xα , μ ) into itself. Proof. Let u ∈ PAP(Xα , μ ). Setting h(t) = F(t, u(t)) and using the theorem of composition of weighted pseudo-almost periodic functions (Theorem 5.26) it follows that h ∈ PAP(X , μ ). Now write h = φ + ζ where φ ∈ AP(X ) and ζ ∈ PAP0 (X , μ ). The nonlinear integral operator Γ1 u can be rewritten as

10.3 Existence of Weighted Pseudo-Almost Periodic Solutions

(Γ1 u)(t) =

 t −∞

U(t, s)P(s)ϕ (s)ds +

 t −∞

257

U(t, s)P(s)ζ (s)ds.

Set

Φ (t) =

 t −∞

U(t, s)P(s)ϕ (s)ds

and

Ψ (t) =

 t −∞

U(t, s)P(s)ζ (s)ds

for each t ∈ R. The next step consists of showing that Φ ∈ AP(Xα ) and Ψ ∈ PAP0 (Xα , μ ). Obviously, Φ ∈ AP(Xα ). Indeed, since ϕ ∈ AP(X ), for every ε > 0 there exists l(ε ) > 0 such that for every interval of length l(ε ) contains a τ with the property ϕ (t + τ ) − ϕ (t) < ε C for each t ∈ R, where C = Now

δ 1−α with Γ being the classical Γ function. c(α )21−α Γ (1 − α )

Φ (t + τ ) − Φ (t) = = =

 t+τ −∞

 t

−∞

 t

−∞

− + =

−∞

U(t, s)P(s)ϕ (s)ds  t −∞

U(t, s)P(s)ϕ (s)ds

U(t + τ , s + τ )P(s + τ )ϕ (s + τ )ds

−∞

 t

−∞

−∞

 t

U(t + τ , s + τ )P(s + τ )ϕ (s + τ )ds −

 t

 t

+

U(t + τ , s)P(s)ϕ (s)ds −

U(t + τ , s + τ )P(s + τ )ϕ (s)ds U(t + τ , s + τ )P(s + τ )ϕ (s)ds −

 t −∞

U(t, s)P(s)ϕ (s)ds

  U(t + τ , s + τ )P(s + τ ) ϕ (s + τ ) − ϕ (s) ds

 t  −∞

 U(t + τ , s + τ )P(s + τ ) −U(t, s)P(s) ϕ (s)ds.

Using [16, 138] it follows that   t & '   2 ϕ ∞    −∞ U(t + τ , s + τ )P(s + τ ) −U(t, s)P(s) ϕ (s)ds ≤ δ ε . α

258

10 Pseudo-Almost Periodic Solutions to Some Third-Order Differential Equations

Similarly, using Eq. (2.33), it follows that   t    ≤ ε.  U(t + τ , s + τ )P(s + τ )( ϕ (s + τ ) − ϕ (s))ds   −∞ α

Therefore,  2 ϕ ∞  ε for each t ∈ R, Φ (t + τ ) − Φ (t) α < 1 + δ and hence, Φ ∈ AP(Xα ). To complete the proof for Γ1 , we have to show that Ψ ∈ PAP0 (Xα , μ ). First, note that s → Ψ (s) is a bounded continuous function. It remains to show that 1 r→∞ μ (Qr )



lim

Qr

Ψ (t) α μ (t)dt = 0.

Again using Eq. (2.33) it follows that 1 r→∞ μ (Qr )



lim

Qr

c(α ) r→∞ μ (Qr )

Ψ (t) α μ (t)dt ≤ lim



 +∞

δ

s−α e− 2 s

Qr 0

× ζ (t − s) μ (t)dsdt ≤ lim c(α ) r→∞

 +∞

δ

s−α e− 2 s

0

1 μ (Qr )

 Qr

× ζ (t − s) μ (t)dtds. Set

Γs (T ) =

1 μ (Qr )

 Qr

ζ (t − s) μ (t)dt.

Since PAP0 (X , μ ) is translation-invariant, it follows that t → ζ (t − s) belongs to PAP0 (X , μ ) for each s ∈ R, and hence 1 lim r→∞ μ (Qr )

 Qr

ζ (t − s) μ (t)dt = 0

for each s ∈ R. One completes the proof by using the well-known Lebesgue Dominated Convergence Theorem and the fact Γs (r) → 0 as r → ∞ for each s ∈ R. The proof for Γ2 u(·) is similar to that of Γ1 u(·). However one makes use of Eq. (2.34) rather than Eq. (2.33). Theorem 10.9. Under assumptions (E.1)–(E.3)–(E.7)–(E.8)–(E.9), then Eq. (10.8) has a unique weighted pseudo-almost periodic mild solution whenever K is small enough.

10.3 Existence of Weighted Pseudo-Almost Periodic Solutions

259

Proof. Consider the nonlinear operator M defined on PAP(Xα , μ ) by Mu(t) =

 t −∞

U(t, s)P(s)F(s, u(s))ds −

 ∞ t

UQ (t, s)Q(s)F(s, u(s))ds

for each t ∈ R. In view of Lemma 10.8, it follows that M maps PAP(Xα , μ ) into itself. To complete the proof one has to show that M has a unique fixed-point. If v, w ∈ PAP(Xα , μ ), then Γ1 (v)(t) − Γ1 (w)(t) α ≤ ≤

 t −∞

 t

U(t, s)P(s) [F(s, v(s)) − F(s, w(s))] α ds δ

−∞

c(α )(t − s)−α e− 2 (t−s) F(s, v(s)) − F(s, w(s)) ds

≤ Kc(α )

 t −∞

≤ kKc(α )

δ

(t − s)−α e− 2 (t−s) v(s) − w(s) ds

 t

−∞

δ

(t − s)−α e− 2 (t−s) v(s) − w(s) α ds

≤ kKc(α )21−α Γ (1 − α )δ α −1 v − w α ,∞ , and Γ2 (v)(t) − Γ2 (w)(t) α ≤ ≤ ≤

 ∞ t

 ∞ t

 ∞ t

UQ (t, s)Q(s) [F(s, v(s)) − F(s, w(s))] α ds m(α )eδ (t−s) F(s, v(s)) − F(s, w(s)) ds m(α )Keδ (t−s) v(s) − w(s) ds

≤ km(α )K

 ∞ t

eδ (t−s) v(s) − w(s) α ds

≤ Kkm(α ) v − w α ,∞

 +∞

eδ (t−s) ds

t

= Kkm(α )δ −1 v − w α ,∞ , where u α ,∞ := sup u(t) α . t∈R

Combining the previous estimates it follows that Mv − Mw ∞,α ≤ KC(α , δ ). v − w α ,∞ ,

260

10 Pseudo-Almost Periodic Solutions to Some Third-Order Differential Equations

where C(α , δ ) = km(α )δ −1 + kc(α )21−α Γ (1 − α )δ α −1 > 0 is a constant, and hence if the Lipschitz K is small enough, then Eq. (10.8) has a unique solution, which obviously is its only weighted pseudo-almost periodic mild solution.

10.3.2 Third-Order Differential Equations To study the existence of almost periodic solutions to Eq. (10.1), in addition to the previous assumptions, we suppose that the following additional assumption holds: (E.6) The linear operators B,C : Hα → H are bounded. Let K > 0 be their bound, that is,     Bu ≤ K u

α

    and Cu ≤ K uα

for all u ∈ Hα . Following along the same lines as in the proof of Theorem 10.10 one can easily establish the next result. Theorem 10.10. Under previous assumptions and if (E.6)–(E.9) hold, then Eq. (10.1) has at least one weighted pseudo-almost periodic solution u ∈ Hα .

Bibliographical Notes The results presented in this chapter are based on the following sources: Diagana [49, 52, 61, 69], Ding et al. [82], and Li et al. [128]. Additional relevant references to this chapter include Goldstein and N’Gu´er´ekata [103, 104], Al-Islam et al. [11], Lunardi [137], Chicone and Latushkin [38], and Engel and Nagel [90].

Chapter 11

Pseudo-Almost Automorphic Solutions to Some Sobolev-Type Equations

11.1 Introduction Let p, q > 1 such that q = 1 − p−1 . This chapter is devoted to the search of a pseudo-almost automorphic mild solution to the class of Sobolev-type differential equations given by ) * d u(t) + f (t, u(t)) = A(t)u(t) + g (t, u(t)) , t ∈ R, dt

(11.1)

where A(t) : D ⊂ X → X for t ∈ R is a family of densely defined closed linear p operator on a domain D, independent of t, and f , g : R×X → X belong to Spaa (R× X ) ∩C(R × X ). The main result of this chapter, to some extent, generalizes most of the known results on the existence of pseudo-almost automorphic (respectively, pseudo-almost periodic) solutions to differential equations of the Eq. (11.1), in particular those studied in [54]. Sobolev-type differential equations have various applications in particular in wave propagations or in dynamic of fluids [86]. Various formulations of these equations can be found in literature (see [30, 133]). This work will be heavily based on the recent progress made by Xiao et al. [95, 96], notably on the composition of S p -pseudo-almost automorphic functions as well as the existence of pseudo-almost automorphic solutions to differential equations with S p -pseudo-almost automorphic coefficients. To illustrate the abstract results of this chapter, the existence of pseudo-almost automorphic solutions to the heat equation with a negative time-dependent diffusion coefficient will be investigated.

T. Diagana, Almost Automorphic Type and Almost Periodic Type Functions in Abstract Spaces, DOI 10.1007/978-3-319-00849-3 11, © Springer International Publishing Switzerland 2013

261

262

11 Pseudo-Almost Automorphic Solutions to Some Sobolev-Type Equations

11.2 Existence of Pseudo-Almost Automorphic Solutions To study Eq. (11.1) we will among other things suppose that there exists a Banach space (Y , · Y ) such that the embedding (Y , · Y ) → (X , · ) is continuous. The bounded of such an injection will be denoted by C. In addition to the above, we assume that the following additional assumptions hold: (G.1) The system u (t) = A(t)u(t), t ≥ s,

u(s) = ϕ ∈ X

(11.2)

has an associated evolution family of operators {U(t, s) : t ≥ s with t, s ∈ R}. Further, we assume that the domains of the operators A(t) are constant in t, that is, D(A(t)) = D = Y for all t ∈ R and that the evolution family U(t, s) is asymptotically stable in the sense that there exist some constants M, δ > 0 such that U(t, s) ≤ Me−δ (t−s) for all t, s ∈ R with t ≥ s. (G.2) The function s → A(s)U(t, s) defined from (−∞,t) into B(R × Y ) is strongly measurable and there exist a measurable function H : (0, ∞) → (0, ∞) with H ∈ L1 (0, ∞) and a constant ω > 0 such that A(s)U(t, s)y B(Y ,X ) ≤ e−ω (t−s) H(t − s) y , t, s ∈ R, t > s, y ∈ Y . (G.3) The function R × R → X , (t, s) → U(t, s)y ∈ bAA(T, Y ) uniformly for y∈X. (G.4) The function R × R → X , (t, s) → A(s)U(t, s)y ∈ bAA(T, X ) uniformly for y∈Y . p (G.5) The function f is Y -valued and belongs to Spaa (R × X ) and g is X -valued p and belongs to Spaa (R × X ) ∩C(R × X ). Moreover, there exists L > 0 such that f (t, u) − f (t, v) Y ≤ L u − v for all u, v ∈ X and t ∈ R, and g(t, u) − g(t, v) ≤ L u − v for all u, v ∈ X and t ∈ R.

11.2 Existence of Pseudo-Almost Automorphic Solutions

263

11.2.1 Existence Results To study the existence and uniqueness of pseudo-almost automorphic solutions to Eq. (11.1) we first introduce the notion of mild solution, which has been adapted from the one given in Diagana et al. [53, Definition 3.1]. Definition 11.1. A continuous function u : R → X is said to be a mild solution to Eq. (11.1) provided that the function s → A(s)U(t, s) f (s, u(s)) is integrable on (s,t), and   u(t) = − f (t, u(t)) +U(t, s) u(s) + f (s, u(s)) −

 t s

A(s)U(t, s) f (s, u(s))ds +

 t

U(t, s)g(s, u(s))ds s

for t ≥ s and for all t, s ∈ R. Under assumptions (G.1)–(G.2), it can be easily shown that the function u given by u(t) = − f (t, u(t)) +

 t −∞

U(t, s)g(s, u(s))ds −

 t −∞

A(s)U(t, s) f (s, u(s))ds

for each t ∈ R is a mild solution to Eq. (11.1). Lemma 11.2. Under assumptions (G.1)–(G.3)–(G.5), then the nonlinear integral operator Γ defined by (Γ u)(t) :=

 t −∞

U(t, s)g(s, u(s))ds

maps PAA(X ) into PAA(X ). Proof. Let u ∈ PAA(X ). Using Theorem 6.28 it follows that  G(t) :=  g(t, u(t)) p b ∈ AA L p (0, 1)) and ϕ b ∈ belongs to S (X ). Now let G = h + ϕ , where h paa  PAP0 L p (0, 1)) . Consider for each k = 1, 2, . . ., the integral Vk (t) = =

 k k−1

 k

k−1

U(t,t − ξ )g(t − ξ )d ξ U(t,t − ξ )h(t − ξ )d ξ +

 k k−1

U(t,t − ξ )ϕ (t − ξ )d ξ

and set Yk (t) =

 k k−1

U(t,t − ξ )h(t − ξ )d ξ

264

11 Pseudo-Almost Automorphic Solutions to Some Sobolev-Type Equations

and  k

Xk (t) =

k−1

U(t,t − ξ )ϕ (t − ξ )d ξ .

Let us show that Yk ∈ AA(X ). For that, letting r = t − ξ one obtains Yk (t) =

 t−k+1 t−k

U(t, r)h(r)dr for each t ∈ R.

From (G.1) it follows that the function s → U(t, r)h(r) is integrable over (−∞,t) for each t ∈ R. Now using the H¨older’s inequality, it follows that Yk (t) ≤

 t−k+1 t−k

≤M ≤M ≤M ⎡

U(t, r)h(r) dr

 t−k+1 t−k



e−δ (t−r) h(r) dr

t−k+1

−qδ (t−r)

e t−k



1/q  dr .

t−k+1

t−k

k

e−qδ s ds

k−1

4

≤ ⎣e−δ k M

q

1/p h(r) dr p

1/q h S p

⎤ 1 + eqδ ⎦ h S p . qδ

Using the fact that 4 M

q

1 + eqδ qδ

∞

∑ e−δ k < ∞

k=1

∞

we deduce from the well-known Weierstrass theorem that the series

k=1

uniformly convergent on R. Furthermore, Y (t) :=

 t −∞

U(t, s)h(s)ds =

∞

∑ Yk (t),

k=1

Y ∈ C(R, X ), and Y (t) ≤

∞

∑ Yk (t) ≤ K1 h S p ,

k=1

where K1 > 0 is a constant.

∑ Yk (t) is

11.2 Existence of Pseudo-Almost Automorphic Solutions

265

Fix k ∈ N. Let (sm )m∈N be a sequence of real numbers. Since U(t, s)x ∈ bAA(R × p R, Y ) and h ∈ Saa (X ), for every sequence (sm )m∈N there exists a subsequence p (smn )k∈N of (sm )m∈N and functions U1 and v ∈ Saa (X ) such that lim U(t + smn , s + smn )x = U1 (t, s)x, t, s ∈ R, x ∈ X ,

(11.3)

lim U1 (t − smn , s − smn )x = U(t, s)x, t, s ∈ R, x ∈ X ,

(11.4)

lim h(t + smn + ·) − v(t + ·) S p = 0, for each t ∈ R,

(11.5)

lim v(t − smn + ·) − h(t + ·) S p = 0, for each t ∈ R.

(11.6)

n→∞

n→∞

and n→∞

n→∞

Define Tk (t) =

 k k−1

U1 (t,t − ξ )h(t − ξ )d ξ

and Zk (t) =

 k k−1

U(t,t − ξ )v(t − ξ )d ξ .

Now  k       Yk (t+smn ) − Zk (t) ≤  U(t+smn ,t+smn −ξ ) h(t+smn −ξ )−v(t−ξ ) d ξ   k−1  k       + U(t + smn ,t + smn − ξ ) −U(t,t − ξ ) v(t − ξ )d ξ   k−1

= Ink (t) + Jnk (t) where   Ink (t) :=  

k

k−1

    U(t + smn ,t + smn − ξ ) h(t + smn − ξ ) − v(t − ξ ) d ξ  

and Jnk (t) :=

  k        k−1 U(t + smn ,t + smn − ξ ) −U(t,t − ξ ) v(t − ξ )d ξ  .

266

11 Pseudo-Almost Automorphic Solutions to Some Sobolev-Type Equations

Then using the H¨older’s inequality we get Ink (t) ≤ M

 k

≤ K2

k−1



e−δ ξ h(t + smn − ξ ) − v(t − ξ ) d ξ 1/p

k

k−1

h(t + smn − ξ ) − v(t − ξ ) p d ξ

where K2 > 0 is a constant. Now using Eq. (11.5) it follows that Ink (t) → 0 as n → ∞ for each t ∈ R. Similarly, using the Lebesgue Dominated Convergence Theorem and Eq. (11.3) it follows that Jnk (t) → 0 as n → ∞ for each t ∈ R. Now, Yk (t + smn ) − Zn (t) → 0 as n → ∞. Similarly, using Eqs. (11.4) and (11.6) it can be shown that Zk (t − smn ) −Yk (t) → 0 as n → ∞. Therefore each Yk ∈ AA(X ) for each k and hence their uniform limit Y (t) ∈ AA(X ), by using [142, Theorem 2.1.10]. Let us show that each Xn ∈ PAP0 (X ). For that, note that Xk (t) ≤ M

 t−k+1 t−k

⎡

4

≤ ⎣e−δ k M

≤ K3



e−δ (t−r) ϕ (r) dr

q

1 + eqδ qδ

t−k+1

t−k

⎤ ⎦



t−k+1

t−k

1/p ϕ (r) p dr

1/p ϕ (r) p dr

where K3 > 0 is a constant. Now 1 2r

 Qr

Xk (t) dt ≤

K3 2r

 Qr



t−k+1

t−k

1/p ϕ (s) p ds

dt.

Letting r → ∞ in the previous inequality it follows that Xk ∈ PAP0 (X ), as ϕ b ∈ PAP0 (L p (0, 1)). Furthermore,  t

X(t) :=

−∞

U(t, s)ϕ (s)ds =

∞

∑ Xk (t),

k=1

11.2 Existence of Pseudo-Almost Automorphic Solutions

267

X ∈ C(R, X ), and ∞

X(t) ≤

∑ Xk (t) ≤ K4 ϕ S p ,

k=1

where K4 > 0 is a constant. Consequently the uniform limit X(t) =

∞

∑ Xk (t)

k=1

belongs to PAP0 (X ). Therefore, Γ u(t) = X(t) +Y (t) ∈ PAA(X ). Lemma 11.3. Under assumptions (G.1)–(G.2)–(G.4)–(G.5), then the nonlinear integral operator Λ defined by (Λ u)(t) :=

 t −∞

A(s)U(t, s) f (s, u(s))ds

maps PAA(X ) into itself whenever the series

∞

∑

n=1

converges.



n

−ω s

e

1/q q

H(s) ds

n−1

Proof. Let u ∈ PAA(X ). Using the composition of pseudo-almost automorphic p functions it follows that F(t) := f (t, u(t)) belongs to PAA(Y ) ⊂ Spaa (Y ) ⊂ p Spaa (X ). The proof is, up to some slight modifications, similar to that of   b ∈ AA L p ((0, 1) and ϕ b ∈ Lemma 11.2. Indeed, write F = h + ϕ , where h   PAP0 L p ((0, 1) . Consider for each k = 1, 2, . . ., the integral vk (t) = =

 n k−1

 k

k−1

A(t − ξ )U(t,t − ξ )g(t − ξ )d ξ A(t − ξ )U(t,t − ξ )h(t − ξ )d ξ +

 k k−1

A(t − ξ )U(t,t − ξ )ϕ (t − ξ )d ξ

and set Wk (t) =

 k k−1

A(t − ξ )U(t,t − ξ )h(t − ξ )d ξ

and Zk (t) =

 k k−1

A(t − ξ )U(t,t − ξ )ϕ (t − ξ )d ξ .

268

11 Pseudo-Almost Automorphic Solutions to Some Sobolev-Type Equations

Let us show that Wk ∈ AA(X ). For that, letting r = t − ξ one obtains Wk (t) =

 t−k+1 t−k

A(r)U(t, r)h(r)dr for each t ∈ R.

From (G.2) it follows that the function s → A(r)U(t, r)h(r) is integrable over (−∞,t) for each t ∈ R. Now using the H¨older’s inequality, it follows that Wk (t) ≤ ≤ ≤

 t−k+1 t−k



t−k+1

t−k



e−ω (t−r) H(t − r) h(r) dr

k

e−ω (t−r) H q (t − r)dr

e−qω s H(s)q ds

1/q  .

t−k+1

t−k

1/p h(r) p dr

1/q h S p .

k−1

Using the fact that the series given by 

k

e−qω s H(s)q ds

1/q

k−1

converges, we then deduce from the well-known Weierstrass theorem that the series ∞

∑ Wk (t)

k=1

is uniformly convergent on R. Furthermore, W (t) :=

 t −∞

A(s)U(t, s)h(s)ds =

∞

∑ Wk (t),

k=1

W ∈ C(R, X ), and W (t) ≤

∞

∑ Yk (t) ≤ K5 h S p ,

k=1

where K5 > 0 is a constant. Fix k ∈ N. Let (sm )m∈N be a sequence of real numbers. Since A(s)U(t, s)x ∈ p p bAA(R×R, X ) and h ∈ Saa (Y ) ⊂ Saa (X ), for every sequence (sm )m∈N there exists p p a subsequence (smn )k∈N of (sm )m∈N and functions Θ1 and v ∈ Saa (Y ) ⊂ Saa (X ) such that lim A(s + smn )U(t + smn , s + smn )x = Θ (t, s)x, t, s ∈ R, x ∈ X ,

n→∞

(11.7)

11.2 Existence of Pseudo-Almost Automorphic Solutions

269

lim Θ (t − smn , s − smn )x = A(s)U(t, s)x, t, s ∈ R, x ∈ X ,

(11.8)

lim h(t + smn + ·) − v(t + ·) S p = 0, for each t ∈ R,

(11.9)

lim v(t − smn + ·) − h(t + ·) S p = 0, for each t ∈ R.

(11.10)

n→∞

and n→∞ n→∞

Define Tk (t) =

 k k−1

Θ (t,t − ξ )h(t − ξ )d ξ

and Zk (t) =

 k k−1

A(t − ξ ))U(t,t − ξ )v(t − ξ )d ξ .

Now Wk (t + smn ) − Zk (t)  k      A(t + s − ξ )U(t + s ,t + s − ξ ) h(t + s − ξ ) − v(t − ξ ) dξ  |≤ mn mn mn mn   k−1

 k       + A(t+s − ξ )U(t+s ,t+s − ξ ) − A(t − ξ )U(t,t − ξ ) v(t − ξ )d ξ mn mn mn  k−1  = Lnk (t) + Mnk (t), where

 

Lnk (t):=  

k k−1

    A(t+smn −ξ )U(t+smn ,t + smn − ξ ) h(t + smn − ξ ) − v(t − ξ ) d ξ  

and Mnk (t)

 k   :=   k−1 A(t + smn − ξ )U(t + smn ,t + smn − ξ )    −A(t − ξ )U(t,t − ξ ) v(t − ξ )d ξ  .

Then using the H¨older’s inequality we get Lnk

≤ ≤

 k k−1



e−ωξ H(ξ ) h(t + smn − ξ ) − v(t − ξ ) Y d ξ

k

−ωξ

e k−1

H (ξ )d ξ q

1/q 

k

k−1

1/p h(t + smn − ξ ) − v(t − ξ ) Yp

dξ

.

270

11 Pseudo-Almost Automorphic Solutions to Some Sobolev-Type Equations

Now using Eq. (11.9) it follows that Lnk (t) → 0 as n → ∞ for each t ∈ R. Similarly, using the Lebesgue Dominated Convergence theorem and Eq. (11.7) it follows that Mnk (t) → 0 as n → ∞ for each t ∈ R. Now, Wk (t + smn ) − Zk (t) → 0 as n → ∞. Similarly, using Eqs. (11.8) and (11.10) it can be shown that Zk (t − smn ) −Wk (t) → 0 as n → ∞. Therefore each Wk ∈ AA(X ) for each k and hence it uniform limit W (t) ∈ AA(X ). Let us show that each Zk ∈ PAP0 (X ). For that, note that Zk (t) ≤ ≤

 t−k+1 t−k



k

e−ω (t−r) H(t − r) ϕ (r) Y dr

−ω s

e

1/q  q

t−k+1

H(s) ds

k−1

t−k

1/p ϕ (r) Yp

dr

and hence Zk ∈ PAP0 (X ), as ϕ b ∈ PAP0 (L p ((0, 1)). Furthermore,  t

Z(t) :=

−∞

A(s)U(t, s)ϕ (s)ds =

∞

∑ Zk (t),

k=1

Z ∈ C(R, X ), and Z(t) ≤

∞

∑ Zk (t) ≤ K7 ϕ S p ,

k=1

where K7 > 0 is a constant. Consequently the uniform limit Z(t) =

∞

∑ Zk (t) ∈ PAP(X ).

k=1

Therefore, Λ u(t) = W (t) + Z(t) ∈ PAA(X ). In addition to the previous assumptions, we suppose that the series ∞

∑

n=1



n

e−ω s H(s)q ds

1/q

n−1

converges. Theorem 11.4. Under assumptions (G.1)–(G.5), then Eq. (11.1) has a unique mild solution u ∈ PAA(X ) whenever L is small enough.

11.2 Existence of Pseudo-Almost Automorphic Solutions

271

Proof. Consider the nonlinear operator Γ defined by (Π u)(t) = − f (t, u(t)) +

 t −∞

U(t, s)g(s, u(s))ds −

 t −∞

A(s)U(t, s) f (s, u(s))ds

for each t ∈ R. Using the proofs of Lemmas 11.2 and 11.3 as well as the composition of pseudoalmost automorphic function for Lipschitzian function [131, Theorem 2.4], one can easily see that Λ maps PAA(X ) into PAA(X ). To complete the proof, it suffices to apply the Banach fixed-point theorem to the nonlinear operator Π . For that, note that for all u, v ∈ PAA(X ), Π u − Π v ∞ ≤ d u − v ∞ ) where d := L

M δ −1 +C

 1+

0

∞

−ω s

e

* H(s)ds .

Therefore, Eq. (11.1) has a unique fixed-point u ∈ PAA(X ) whenever L is small enough, i.e., d < 1, or )

 −1 L < M δ +C 1 +

∞

e−ω s H(s)ds

 *−1

0

.

11.2.2 Existence Results for the Heat Equation Fix p > 1. Let Ω ⊂ RN (N ≥ 1) be an open bounded subset with C2 boundary Γ = ∂ Ω and let X = L2 (Ω ) equipped with its natural topology · 2 . We study the existence and uniqueness of a pseudo-almost automorphic solution to the heat equation with a negative time-dependent diffusion coefficient given by

∂ [u(t, x) + F(t, u(t, x))] = −a(t, x)Δ u(t, x) + G(t, u(t, x)), in R × Ω ∂t u = 0, on R × Γ

(11.11) (11.12)

where F, G : R × L2 (Ω → L2 (Ω ) are S p -pseudo-almost automorphic and jointly continuous, the function (t, x) → a(t, x) is jointly continuous, x → a(t, x) is differentiable for all t ∈ R, t → a(t, x) is ω -periodic (ω > 0) in the sense that a(t + ω , x) = a(t, x) for all t ∈ R and x ∈ Ω , and the following assumptions hold:

272

(G.6)

11 Pseudo-Almost Automorphic Solutions to Some Sobolev-Type Equations

inf

t∈R,x∈Ω

a(t, x) = m0 > 0, and

(G.7) there exists d > 0 and 0 < μ ≤ 1 such that |a(t, x) − a(s, x)| ≤ d|s − t|μ for all t, s ∈ R uniformly in x ∈ Ω . The problem is quite interesting as the system given by Eqs. (11.11) and (11.12) models among other things the heat conduction in the domain R × Ω ⊂ R × RN . Namely, solutions u(t, x) to this system represent the temperature at position x ∈ Ω at time t ∈ R. Define the linear operators A(t) appearing in Eqs. (11.11) and (11.12) as follows: A(t)u = −a(t, x)Δ u for all u ∈ D(A(t)) = D = H01 (Ω ) ∩ H 2 (Ω ). Under previous assumptions, it is clear that the operators A(t) defined above are invertible and satisfy Acquistapace–Terreni conditions. Clearly, the system 

u (t) = A(t)u(t), t ≥ s, u(s) = ϕ ∈ L2 (Ω ),

(11.13)

has an associated evolution family (U(t, s))t≥s on L2 (Ω ), which satisfies: there exist ω0 > 0 and M ≥ 1 such that U(t, s) B(L2 (Ω )) ≤ Me−ω0 (t−s) for every t ≥ s. Moreover, since A(t + ω ) = A(t) for all t ∈ R, it follows that U(t + ω , s + ω ) = U(t, s) and A(s + ω )U(t + ω , s + ω ) = A(s)U(t, s) for all t, s ∈ R with t ≥ s. Therefore (t, s) → U(t, s)w belongs to bAA(T, L2 (Ω )) uniformly in w ∈ L2 (Ω ) and (t, s) → A(s)U(t, s)w belongs to bAA(T, D) uniformly in w ∈ D. It is also clear that (G.2) holds. In this section, we take Y = (D, · G (Δ )) ) where · G (Δ ) is the graph norm of the N-dimensional Laplace operator Δ with domain D defined by u G (Δ ) = u 2 + Δ u 2 for all u ∈ D. Clearly, the bound of the embedding H01 (Ω ) ∩ H 2 (Ω ) → L2 (Ω ) is C = 1. We need the following additional assumption: (G.8) The function F is H01 (Ω ) ∩ H 2 (Ω )-valued and belongs to PAA(R × L2 (Ω )) p and G ∈ Spaa (R × L2 (Ω )) ∩C(R × L2 (Ω )). Moreover, there exits L > 0 such that F(t, u) − F(t, v) G (Δ ) ≤ L u − v 2

11.2 Existence of Pseudo-Almost Automorphic Solutions

273

for all u, v ∈ L2 (Ω ) and t ∈ R, and G(t, u) − G(t, v) 2 ≤ L u − v 2 for all u, v ∈ L2 (Ω ) and t ∈ R. Theorem 11.5. Under assumptions (G.6)–(G.7)–(G.8), then the heat equation with time-dependent diffusion coefficient given by Eqs. (11.11) and (11.12) has a unique solution u ∈ PAA(L2 (Ω )) whenever L is small enough.

Bibliographical Notes The results presented in this chapter are inspired by the following sources: Diagana [54, 72, 77, 78]. Additional relevant references to this chapter include Diagana et al. [55, 62, 65] and Xiao et al. [95, 96].

Chapter 12

Stability Results for Some Higher-Order Difference Equations

12.1 Introduction The main motivation of this chapter comes from Diagana et al. [58], in which not only a basic theory for almost periodic sequences on Z+ was introduced and studied but also discrete dichotomy techniques were utilized to find various sufficient conditions for the existence of globally attracting almost periodic solutions to some first-order nonautonomous system of difference equations. Furthermore, Diagana et al. [58] subsequently applied their abstract results to study discretely reproducing populations with and without overlapping generations. In this chapter we study the existence of almost periodic (respectively, almost automorphic) solutions to the higher-order difference equations n−1

x(t + n) + ∑ Ar (t)x(t + r) + A0 (t)x(t) = f (t, x), t ∈ Z+ ,

(12.1)

r=1

where Ar (t) : X → X for r = 0, 1, . . . , n − 1 are sequences of bounded linear operators on the Banach space X , and the forcing term f : Z+ × X → X is almost periodic (respectively, almost automorphic) in the first variable uniformly in the second one, and satisfies some additional conditions. For that, our strategy is simple and consists of rewriting Eq. (12.1) as a nonautonomous first-order system of difference equation on X n = X × X × · · · × X . Next, we study the resulting first-order difference equation and then go back to the study of Eq. (12.1). Indeed, setting y := (x(t), x(t + 1), . . . , x(t + n − 1))T , then Eq. (12.1) can be rewritten in X n in the form y(t + 1) = A (t)y(t) + F(t, y(t)), t ∈ Z+ ,

T. Diagana, Almost Automorphic Type and Almost Periodic Type Functions in Abstract Spaces, DOI 10.1007/978-3-319-00849-3 12, © Springer International Publishing Switzerland 2013

(12.2)

275

276

12 Stability Results for Some Higher-Order Difference Equations

where A (t) is the family of time-dependent sequence matrices defined by ⎛

0 I ⎜ 0 0 ⎜ ⎜ A (t) := ⎜ . . ⎜ ⎝ . . −A0 (t) −A1 (t)

0. 0 I . 0 . . . . . I . . −An−1 (t)

⎞ ⎟ ⎟ ⎟ ⎟, ⎟ ⎠

t ∈ Z+ ,

(12.3)

and the function F appearing in Eq. (12.2) is defined by F(t, y) = (0, 0, . . . , f (t, x))T . The study of almost periodic (respectively, almost automorphic) difference equations is an important topic as almost periodicity (respectively, almost automorphy) is more likely to accurately describe many phenomena occurring in our daily life than periodicity, see, e.g., Henson et al. [108].

12.2 Almost Periodic Higher-Order Difference Equations 12.2.1 First-Order Difference Equations In order to study the existence of almost periodic solutions to x(t + 1) = A(t)x(t) + h(t, x(t)), t ∈ Z+ ,

(12.4)

we make use of the fundamental solutions of its associated homogeneous equation x(t + 1) = A(t)x(t), t ∈ Z+ ,

(12.5)

to examine almost periodic solutions to the difference equations x(t + 1) = A(t)x(t) + g(t), t ∈ Z+

(12.6)

where g : Z+ → X and h : Z+ × X → X are almost periodic. In Eq. (12.5), the classical definition of dichotomy does not apply, whether the state transition matrix t−1

X(t, s) = ∏ A(r) r=s

is not invertible. To address such an issue, we will make extensive use of the concept of regular (discrete) exponential dichotomy proposed by Henry [109].

12.2 Almost Periodic Higher-Order Difference Equations

277

Definition 12.1. Equation (12.5) is said to have a regular exponential dichotomy if there exist projection operators P(t) with t ∈ Z+ and positive constants M and β ∈ (0, 1) such that the following four conditions are satisfied, (a) A(t)P(t) = P(t + 1)A(t); (b) The operator A(t) |R(I−P(t)) is an isomorphism from R(I − P(t)) onto R(I − P (t + 1)); (c) X(t, r)P(r)x ≤ M β t−r x for 0 ≤ r ≤ t, x ∈ X ; and (d) X(r,t)(I − P(t))x ≤ M β t−r x , for 0 ≤ r ≤ t, x ∈ X . By repeated application of [(a), Definition 12.1], we obtain P(t)X(t, s) = X(t, s)P(s),

t ≥ s.

(12.7)

Definition 12.2. The hull H(x) of a sequence x is defined by  H(x) = x˜ : there exists a sequence α in Z+ with Tα x = x˜ .

Similarly, for an operator function A, we define  H(A) = A˜ | there exists a sequence α in Z+ with Tα A = A˜ , where Tα A = A˜ means that ˜ lim A(t + α (t)) = A(t).

t→∞

We need the following assumptions: (K.1) (K.2) (K.3) (K.4)

Equation (12.5) has a regular exponential dichotomy. ˜ ∈ H(A(t)). A(t) g : Z+ → X is almost periodic. h : Z+ × X → X , (t, x) → h(t, x) is almost periodic in t ∈ Z+ uniformly on x ∈ B where B ⊂ X is an arbitrary bounded subset. (K.5) h : Z+ × X → X is Lipschitz in the sense that there exists L > 0 such that h(t, x) − h(t, y) ≤ L x − y for all x, y ∈ X and t ∈ Z+ . Theorem 12.3. Under assumptions (K.1)–(K.2), the homogeneous difference equation ˜ x(t + 1) = A(t)x(t) has a regular exponential dichotomy with the same projections and constants.

278

12 Stability Results for Some Higher-Order Difference Equations

˜ Then Xi (t) = X(t + αi ) is a fundamental matrix for the equation Proof. Let Tα A = A. x(t + 1) = A(t + αi )x(t) and satisfies regular exponential dichotomy with the same projection P(i) and same constants M and β . One may take subsequences so that Xi (0) converges to Y0 . For a suitable subsequence, Xi (t) converges to a solution Y (t) of ˜ x(t + 1) = A(t)x(t). Then Y (0) = Y0 and Y (t) satisfy the conditions of regular exponential dichotomy. Theorem 12.4. Under assumptions (K.1)–(K.2)–(K.3), then Eq. (12.6) has an almost periodic solution given by x(t) =

t−1

∑

X(t, r + 1)P(r + 1)g(r) −

r=−∞

∞

∑ X(t, r + 1)(I − P(r + 1))g(r),

(12.8)

r=t

where X(t, r)P(r) = 0 for r > t and g(r) = 0 for r < 0. Proof. It can be easily checked that x(t) defined by Eq. (12.8) is indeed a solution to Eq. (12.6). Moreover, 

t−1

∑

x(t) ≤

X(t, r + 1)P(r + 1) +

r=−∞

 ≤



∑ X(t, r + 1)(I − P(r + 1))

g

r=t

t−1

∞

r=0

r=t

≤M

∞



∑ Mβ t−r−1 + ∑ Mβ r+1−t

g

1+β g . 1−β

Let {α˜ } and {γ˜} be arbitrary sequences of nonnegative integers, and let {α } ⊂ ˜ {α } and {γ } ⊂ {γ˜} be their common subsequences. Then Tα +γ A = Tγ Tα A and Tα +γ g = Tγ Tα g.

12.2 Almost Periodic Higher-Order Difference Equations

Now x(t + αi ) =

t+αi −1

∑

X(t + αi , r + 1)P(r + 1)g(r)

r=−∞

−

=

∞

∑

X(t + αi , r + 1)(I − P(r + 1))g(r)

r=t+αi

t−1

∑

X(t + αi , s + αi + 1)P(s + αi + 1)g(s + αi )

s=−∞

−

∞

∑ X(t + αi , s + αi + 1)(I − P(s + αi + 1))g(s + αi )

s=t

=

t−1

∑

A(t + αi − 1) · · · A(s + αi + 1)P(s + αi + 1)g(s + αi )

s=−∞ ∞

− ∑ A(t + αi − 1) · · · A(s + αi + 1)(I − P(s + αi + 1))g(s + αi ). s=t

lim x(t + αi ) = (Tα x)t =

i→∞

t−1

∑

˜ − 1) · · · A(s ˜ + 1)P(s ˜ + 1)g(s) A(t ˜

s=−∞

−

∞

˜ − 1) · · · A(s ˜ + 1)(I − P(s ˜ + 1))g(s) ˜ ∑ A(t

s=t t−1

∑

=

(Tα A)t−1 · · · (Tα A)s+1 (Tα P)s+1 (Tα g)s

s=−∞

−

∞

∑ (Tα A)t−1 · · · (Tα A)s+1 (I − Tα P)s+1 (Tα g)s .

s=t

Moreover,   Tγ Tα x t =

t−1

∑

s=−∞



       Tγ Tα A t−1 · · · Tγ Tα A s+1 Tγ Tα P s+1 Tγ Tα g s

t−1         − ∑ Tγ Tα A t−1 · · · Tγ Tα A s+1 I − Tγ Tα P s+1 Tγ Tα g s



s=t

 = Tγ +α x t , and hence {x(t)}t∈Z+ ∈ AP(Z+ ).

279

280

12 Stability Results for Some Higher-Order Difference Equations

Corollary 12.5. Under assumptions (K.1)–(K.2)–(K.3), if the zero solution of Eq. (12.5) is uniformly asymptotically stable, then Eq. (12.6) has a unique globally asymptotically stable almost periodic solution, given by x(t) =

t−1





t−1

∑ ∏ A(s)

g(r).

s=r

r=0

Moreover, Mβ g . 1−β

x(t) ≤

Proof. Let y(t) be a solution of Eq. (12.6) with y(0) = y0 . Then, t−1





y(t) = X(t)y0 + ∑

r=0

t−1

∏ A(s)

g(r).

s=r

Therefore, y(t) = γ (t) + x(t), where γ (t) is a null sequence. Thus, y(t) is an asymptotically almost periodic solution of Eq. (12.6). By Lemma 3.60, y(t) ∈ AP(Z+ ) implies that y = x. Hence, x(t) =

t−1



t−1

∑ ∏ A(s)

r=0

g(r)

s=r

is the only almost periodic solution of Eq. (12.6). Moreover, it is easy to see that x(t) ≤

Mβ g . 1−β

Theorem 12.6. Under assumptions (K.1)–(K.2)–(K.4)–(K.5), then Eq. (12.4) has a unique globally asymptotically stable almost periodic solution if Mβ L < 1. 1−β

(12.9)

Proof. Consider the Banach space AP(Z+ ) equipped with the sup-norm. By Theorem 4.42, if ϕ ∈ AP(Z+ ) then h(t, ϕ (t)) ∈ AP(Z+ ). Let

Γ : AP(Z+ ) → AP(Z+ ) be the nonlinear operator defined by

12.2 Almost Periodic Higher-Order Difference Equations

(Γ ϕ ) (t) :=

t−1



t−1

∑ ∏ A(s)

r=0

281

h(r, ϕ (r)).

s=r

By Theorem 12.4, Γ is well defined. Moreover, for ϕ , ψ ∈ AP(Z+ ), (Γ ϕ ) (t) − (Γ ψ ) (t) ≤

Mβ h(t, ϕ (t)) − h(t, ψ (t)) , 1−β

and Γ ϕ − Γ ψ ∞ ≤

Mβ L ϕ − ψ ∞ . 1−β

Γ is a contraction whenever L is small enough, that is, Mβ L < 1. 1−β Using the Banach fixed-point theorem, we obtain that Γ has a unique fixed point, x. Moreover, x is the globally asymptotically stable almost periodic solution of Eq. (12.4).

12.2.2 Existence Results for the Beverton–Holt Model Lots of interesting models arising in population dynamics can, to some extent, be modeled through the following abstract difference equation: x(t + 1) = at x(t) + f (t, x(t)),

(12.10)

where x(t) is the total size of the population in generation t, at ∈ (0, 1) for all t ∈ Z+ is the survival function, and f is the recruitment function. In this setting we consider the case when the recruitment function f is a Beverton–Holt recruitment function given by f (t, x(t)) =

(1 − at )μ Kt x(t) , (1 − at )Kt + (μ − 1 + at )x(t)

where t → Kt is the carrying capacity and μ > 1 is the growth rate. (N.1) Both t → at and t → Kt are almost periodic. (N.2) at ∈ (0, 1), Kt > 0 and μ > 1. We have

(12.11)

282

12 Stability Results for Some Higher-Order Difference Equations

Theorem 12.7. [58] Under assumptions (N.1)–(N.2), Eq. (12.10) has a unique globally asymptotically stable almost periodic solution whenever " ! sup at |t∈Z+ <

1 . μ +1

Proof. Clearly, | f (t, x) − f (t, y)| ≤

(1 − at )2 μ Kt 2 |x − y| (1 − at )2 Kt2 + (μ − 1 + at )(1 − at )Kt (x + y) + (μ − 1 + at )2 xy

≤ μ |x − y| , and hence f is Lipschitz with " constant L = μ . ! Lipschitz Let M = 1 and β = sup γt |t∈Z+ . Then, ! " sup γt |t∈Z+ <

1 μ +1

which yields " ! μ . sup γt |t∈Z+ Mβ L " <1 ! = 1−β 1 − sup γt |t∈Z+ and by Theorem 12.6, Eq. (12.10) has a unique globally asymptotically stable almost periodic solution.

12.2.3 Higher-Order Difference Equations Consider the associated homogeneous equation associated with Eq. (12.2) given by y(t + 1) = A (t)y(t), t ∈ Z+ .

(12.12)

We need the following assumptions: (K.6) Equation (12.12) has a regular exponential dichotomy. (K.7) A˜(t) ∈ H(A (t)). (K.8) f : Z+ × X → X , (t, x) → f (t, x) is almost periodic in t ∈ Z+ uniformly on x ∈ B where B ⊂ X is an arbitrary bounded subset. (K.9) f : Z+ × X → X is Lipschitz in the sense that there exists L > 0 such that f (t, x) − f (t, y) ≤ L x − y for all x, y ∈ X and t ∈ Z+ .

12.3 Almost Automorphic Higher-Order Difference Equations

283

Remark 12.8. Note that (K.8) yields F : Z+ × X n → X n , (t, y) → F(t, y) is almost periodic in t ∈ Z+ uniformly on y ∈ Bn := B × B × · · · × B where B ⊂ X is an arbitrary bounded subset. Similarly, (K.9) yields F : Z+ × X n → X n is Lipschitz, that is, F(t, x) − F(t, y) ≤ L x − y for all x, y ∈ X n and t ∈ Z+ . Corollary 12.9. Under assumptions (K.6)–(K.7)–(K.8)–(K.9), if the zero solution of Eq. (12.2) is uniformly asymptotically stable, Eq. (12.1) has a unique globally asymptotically stable almost periodic solution whenever L is small enough. Proof. Following the same lines as in the proof of Theorem 12.6 it follows that Eq. (12.2) has a unique globally asymptotically stable almost periodic solution given by the mapping t → z(t) := (x(t), x(t + 1), x(t + 2), . . . , x(t + n − 1))T whenever L is small enough. Therefore, Eq. (12.1) has a unique globally asymptotically stable almost periodic solution t → x(t) whenever L is small enough.

12.3 Almost Automorphic Higher-Order Difference Equations 12.3.1 Introduction The main objective in this section consists of generalizing the results of the previous section to the almost automorphic setting. Namely, we present conditions upon which Eq. (12.1) has a unique globally asymptotically stable almost automorphic solution.

12.3.2 First-Order Difference Equations If Eq. (12.5) has a discrete dichotomy, then we define the so-called Green function G associated with it by

G(t, r) =

⎧ ⎨ X(t, r)P(r) ⎩

if r ≤ t,

X(r,t)(I − P(t)) if t ≤ r.

284

12 Stability Results for Some Higher-Order Difference Equations

In view of the above, we have G(t, s) ≤

⎧ ⎨ M β t−r if r ≤ t, ⎩

M β t−r if t ≤ r.

Our setting requires the following assumptions: (L.1) (L.2) (L.3) (L.4)

Equation (12.5) has a regular exponential dichotomy. t → A(t) is almost automorphic. g : Z → X is almost automorphic. h : Z × X → X , (t, x) → h(t, x) is almost automorphic in t ∈ Z uniformly on x ∈ B where B ⊂ X is an arbitrary bounded subset.

Theorem 12.10. Under assumptions (L.1)–(L.2), (t, s) → G(t, s)Y ∈ bAAd (Td , X ) uniformly in Y ∈ B where B in an arbitrary bounded subset of X . Proof. Since t → A(t) is almost automorphic, for every sequence {h (n)}n∈N ⊂ Z there exists a subsequence {h(n)}n∈N ⊂ Z such that ˜ lim A(t + h(n)) = A(t)

n→∞

is well defined for each t ∈ Z, and ˜ − h(n)) = A(t) lim A(t

n→∞

for each t ∈ Z. Using similar techniques as in the proof of Theorem 12.3, it can be easily shown that the homogeneous difference equation ˜ x(t + 1) = A(t)x(t) has a regular exponential dichotomy with the same projections and constants as for Eq. (12.5). The rest of the proof is clear. We have Theorem 12.11. Under assumptions (L.1)–(L.2)–(L.3), Eq. (12.6) has an almost automorphic solution given by x(t) =

t−1

∑

r=−∞

X(t, r + 1)P(r + 1)g(r) −

∞

∑ X(t, r + 1)(I − P(r + 1))g(r),

(12.13)

r=t

where X(t, r)P(r) = 0 for r > t and g(r) = 0 for r < 0. Proof. Clearly, z given in Eq. (12.13) a solution to Eq. (12.6). We now set z = M(g) − N(g) where

12.3 Almost Automorphic Higher-Order Difference Equations t−1

∑

M(g)(t) :=

285

X(t, r + 1)P(r + 1)g(r)

r=−∞

and ∞

N(g)(t) = ∑ X(t, r + 1)(I − P(r + 1))g(r). r=t

Let us show that t → M(g)(t) is almost automorphic. Indeed, since g is almost automorphic, for every sequence (τn )n∈N ⊂ Z there exists a subsequence (σn )n∈N ⊂ Z such that

ϕ (t) := lim g(t + σn ) n→∞

is well defined for each t ∈ Z, and lim ϕ (t − σn ) = g(t)

n→∞

for each t ∈ Z. Similarly, since G(t, s)Y ∈ bAAd (Td , X ) uniformly for all Y in any bounded subset of X (Theorem 12.10), there exists a subsequence (τn )n∈N ⊂ Z of (σn )n∈N ⊂ Z such that ˜ s)Y := lim G(t + τn , s + τn )Y G(t, n→∞

(12.14)

is well defined for each t ∈ Z, and ˜ − τn , s − τn )Y = G(t, s)Y lim G(t

n→∞

for each (t, s) ∈ Td . We have M(g)(t + τn ) − M(ϕ )(t) =

t+τn −1

∑

X(t + τn , r + 1)P(r + 1)g(r)

r=−∞

−

t−1

∑

X(t, r + 1)P(r + 1)ϕ (r)

r=−∞

=

t−1

∑

X(t + τn , r + 1 + τn )P(r + τn + 1)g(r + τn )

r=−∞

−

t−1

∑

r=−∞

X(t, r + 1)P(r + 1)ϕ (r)

(12.15)

286

12 Stability Results for Some Higher-Order Difference Equations t−1

∑

=

  X(t + τn , r + 1 + τn )P(r + 1 + τn ) g(r + τn ) − ϕ (r)

r=−∞

+



t−1

∑

 X(t + τn , r + 1 + τn )P(r + 1 + τn ) − X(t, r + 1)P(r + 1) ϕ (r).

r=−∞

Clearly,  

t−1

∑

& ' X(t + τn , r + 1 + τn )P(r + 1 + τn ) g(r + τn ) − ϕ (r)  → 0 as n → ∞.

r=−∞

From Eq. (12.14) or Theorem 12.10 it follows that  

t−1

∑



  X(t+τn , r+1+τn )P(r+1+τn ) − X(t, r + 1)P(r + 1) ϕ (r) → 0 as n → ∞,

r=−∞

and hence lim M(g)(t + τn ) = M(ϕ )(t)

n→∞

for each t ∈ Z. Using similar ideas as the previous ones, one can easily see that lim M(ϕ )(t − τn ) = M(g)(t)

n→∞

for each t ∈ Z. Corollary 12.12. Under assumptions (L.1)–(L.2)–(L.3), if the zero solution of Eq. (12.5) is uniformly asymptotically stable, then Eq. (12.6) has a unique globally asymptotically stable almost automorphic solution, given by x(t) =

t−1



t−1



∑ ∏ A(s)

g(r).

s=r

r=0

Moreover, x(t) ≤

Mβ g . 1−β

Proof. Let w be a solution of Eq. (12.6) with initial condition w(0) = w0 . Clearly, t−1

w(t) = X(t, 0)w0 + ∑

r=0



t−1

∏ A (s) s=r

g(r),

t ∈ Z+

12.3 Almost Automorphic Higher-Order Difference Equations

287

and hence w(t) = b(t) + z(t), where b = (b(t))t∈Z+ ∈ N(Z+ ). Thus, w(t) is an asymptotically almost automorphic solution to Eq. (12.6). In view of Lemma 4.45 it follows that w(t) ∈ AA(Z) yields w = z. Hence,  z(t) =

t−1

t−1

r=0

s=r

∑ ∏ A (s)

g(r)

is the only almost automorphic solution of Eq. (12.6). Clearly, z(t) ≤

Mβ g . 1−β

Theorem 12.13. Under assumptions (L.1)–(L.2)–(L.4)–(K.5), Eq. (12.4) has a unique globally asymptotically stable almost automorphic solution if Mβ L < 1. 1−β

(12.16)

Proof. Consider the Banach space AA(Z+ ) equipped with the sup-norm. By Theorem 4.42, if ϕ ∈ AA(Z+ ) then h(t, ϕ (t)) ∈ AA(Z+ ). Let

Γ : AA(Z+ ) → AA(Z+ ) be the nonlinear operator defined by (Γ ϕ ) (t) :=

t−1



t−1

∑ ∏ A(s)

r=0

h(r, ϕ (r)).

s=r

Clearly, Γ is well defined. Moreover, for ϕ , ψ ∈ AA(Z+ ), (Γ ϕ ) (t) − (Γ ψ ) (t) ≤

Mβ h(t, ϕ (t)) − h(t, ψ (t)) , 1−β

and Γ ϕ − Γ ψ ∞ ≤

Mβ L ϕ − ψ ∞ . 1−β

Γ is a contraction whenever L is small enough, that is, Mβ L < 1. 1−β

288

12 Stability Results for Some Higher-Order Difference Equations

Using the Banach fixed-point theorem, we obtain that Γ has a unique fixed point, x. Moreover, x is the globally asymptotically stable almost automorphic solution of Eq. (12.4).

12.3.3 Higher-Order Difference Equations This setting requires the following additional assumptions: (L.5) t → A (t) is almost automorphic. (L.6) f : Z × X → X , (t, x) → f (t, x) is almost automorphic in t ∈ Z uniformly on x ∈ B where B ⊂ X is an arbitrary bounded subset. Corollary 12.14. Under assumptions (K.6)–(K.9)–(L.5)–(K.6), if the zero solution of Eq. (12.2) is uniformly asymptotically stable, then Eq. (12.1) has a unique globally asymptotically stable almost automorphic solution whenever L is small enough. Proof. Following the same lines as in the proof of Theorem 12.13 it follows that Eq. (12.2) has a unique globally asymptotically stable almost automorphic solution given by the mapping t → z(t) := (x(t), x(t + 1), x(t + 2), . . . , x(t + n − 1))T whenever L is small enough. Therefore, Eq. (12.1) has a unique globally asymptotically stable almost automorphic solution t → x(t) whenever L is small enough.

Bibliographical Notes The results presented in this chapter are either taken from or inspired by the results by Diagana et al. [58] and Diagana [60]. Additional relevant references to this chapter include Ky Fan [94], Halanay and Rasvan [107], Pennequin [149], Muchnik et al. [140], and Jajte [120].

Appendix A

A.1 Spectral Decomposition of the Self-Adjoint Operator A Let H stand for an infinite dimensional separable complex Hilbert space equipped with the norm · and the inner product ·, ·. And if A : D(A) ⊂ H → H is a self-adjoint (possibly unbounded) linear operator on H whose spectrum consists of isolated eigenvalues 0 < λ1 < λ2 < · · · < λl → ∞ as l → ∞ with each eigenvalue having a finite multiplicity γ j equals to the multiplicity of the corresponding eigenspace, then we will often be using its spectral decomposition given below. Let {ekj } be a (complete) orthonormal sequence of eigenvectors associated with the eigenvalues {λ j } j≥1 . Clearly, for each u ∈ D(A) where  D(A) := u ∈ H :

∞

2



∑ λ j2 E j u

<∞ ,

j=1

we have the spectral decomposition, Au =

∞

∑ λj

j=1

γj

∑ u, ekj ekj =

k=1

∞

∑ λ jE ju

j=1

γj

where E j u =

∑ u, ekj ekj .

k=1

Note that {E j } j≥1 is a sequence of orthogonal projections on H . Moreover, each u ∈ H can be written as: u=

∞

∑ E j u.

j=1

T. Diagana, Almost Automorphic Type and Almost Periodic Type Functions in Abstract Spaces, DOI 10.1007/978-3-319-00849-3, © Springer International Publishing Switzerland 2013

289

290

Appendix A

It should also be mentioned that the operator −A is the infinitesimal generator of an analytic semigroup {T (t)}t≥0 , which is explicitly expressed in terms of those orthogonal projections E j by, for all u ∈ H , T (t)u =

∞

∑ e−λ j t E j u.

j=1

In addition, the fractional powers Ar (r ≥ 0) of A exist and are given by  D(Ar ) =

   2 u ∈ H : ∑ λ j2r E j u < ∞ ∞

j=1

and Ar u =

∞

∑ λ j2r E j u,

∀u ∈ D(Ar ).

j=1

A.2 Proof of an Interpolation Theorem Proposition A.1 (Sawano). Let (X , · ) and (Y , · 0 ) be Banach spaces such that X is compactly embedded into Y : X ⊂⊂ Y . Then (Y , X )θ ,q0 ⊂⊂ Y whenever 0 < q0 ≤ ∞ and 0 < θ < 1. Proof. Suppose that we are given a bounded subset {xl }∞ l=1 ⊂ (Y , X )θ ,1 . We shall show that {xl }∞ is convergent in Y if we pass to a subsequence. l=1 We choose xl, j,1 ∈ X and xl, j,2 ∈ Y so that xl = xl, j,1 + xl, j,2 , xl, j,1 + 2 j xl, j,2 0 ≤ 2K(xl , 2 j ). A passage to a subsequence allows us to assume that {xl, j,1 }∞ l=1 is convergent in Y for each j ∈ Z. We claim that {xl }∞ l=1 is convergent in Y . To prove this, we let l1 , l2 ∈ N. Then we have xl1 − xl2 0 ≤ xl1 , j,1 − xl2 , j,1 0 + xl1 , j,2 − xl2 , j,2 0 ≤ xl1 , j,1 − xl2 , j,1 0 + xl1 , j,2 0 + xl2 , j,2 0   ≤ C xl1 , j,1 − xl2 , j,1 0 + 2− jθ x (Y ,X )θ ,1 .

A.3 Bi-Almost Automorphy and Positive Bi-Almost Automorphy

291

If we let l1 , l2 → ∞, then we have lim sup xl1 − xl2 0 ≤ C2− jθ x (Y ,X )θ ,1 l1 ,l2 →∞

for all j ∈ Z. If we let j → ∞, then we have lim sup xl1 − xl2 0 ≤ 0. l1 ,l2 →∞

Consequently the compactness of (Y , X )θ ,q0 ⊂⊂ Y was established.

A.3 Bi-Almost Automorphy and Positive Bi-Almost Automorphy Continuous Case Let T be the set defined by: T := {(t, s) ∈ R × R : t ≥ s}. Definition A.2. [165] A jointly continuous function F(t, s) : R × R → X is called bi-almost automorphic if for every sequence of real numbers (s n ) we can extract a subsequence (sn ) such that G(t, s) := lim F(t + sn , s + sn ) n→∞

is well defined for all t, s ∈ R, and lim G(t − sn , s − sn ) = F(t, s)

n→∞

for all t, s ∈ R. The collection of such functions is denoted by bAA(R × R, X ). Definition A.3. A jointly continuous function F : T → X is called positively bialmost automorphic if for every sequence of real numbers (s n )n∈N , we can extract a subsequence (sn )n∈N such that G(t, s) := lim F(t + sn , s + sn ) n→∞

is well defined for (t, s) ∈ T, and lim G(t − sn , s − sn ) = F(t, s)

n→∞

for each (t, s) ∈ T. The collection of such functions will be denoted by bAA(T, X ).

292

Appendix A

Discrete Case: We now introduce the concept of bi-almost automorphy (respectively, positively bi-almost automorphy) for sequences defined on Z × Z. Such notions is utilized in Chap. 12. Definition A.4. A sequence L(t, s) : Z × Z → X is called bi-almost automorphic if for every sequence {h (n)}n∈N ⊂ Z there exists a subsequence {h(n)}n∈N ⊂ Z such that L (t, s) := lim L(t + h(n), s + h(n)) n→∞

is well defined for all t, s ∈ Z, and lim L (t − h(n), s − h(n)) = L(t, s)

n→∞

for all t, s ∈ Z. The collection of such sequences is denoted by bAAd (Z × Z, X ). We now introduce the concept of positively bi-almost automorphic sequences. For that, let Td be the set defined by:  Td := (t, s) ∈ Z × Z : t ≥ s . Definition A.5. A sequence L : Td → X is called positively bi-almost automorphic if for every numbers (h (n))n∈N ⊂ Z, we can extract a subsequence (h(n)) such that L (t, s) := lim L(t + h(n), s + h(n)) n→∞

is well defined for (t, s) ∈ Td , and lim L (t − h(n), s − h(n)) = L(t, s)

n→∞

for each (t, s) ∈ Td . The collection of such sequences will be denoted by bAAd (Td , X ). Obviously, every bi-almost automorphic sequence is positively bi-almost automorphic with the converse being false. Example A.6. Classical examples of bi-almost automorphic sequences L include those which are of the form L(t, s) = h(t − s) for all (t, s) ∈ Td , where h = (h(t))t∈Z is periodic, that is, there exists 0 = ω ∈ Z such that h(t + ω ) = h(t) for all t ∈ Z.

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Index

AAP(X ), 97 AP(X ), 80 BC(R, X ), 31 BC(m) (R, X ), 32 C(Ω ), 45 C(m) -pseudo-almost automorphic, 180, 221, 224 C(m) (Ω ), 45 C(n) -almost automorphic, 120, 180 C(n) -almost periodic, 94 C(n) -compact pseudo-almost automorphic, 180 C(n) -pseudo-almost periodic, 159 DA (α ), 67 DA (α + n, p), 67 DA (α , p), 67 H k (Ω ), 29, 34 L p (Ω ), 26 (m) PAA0 (DA (α + m, p)), 226 PAP(X ), 167 S p -pseudo-almost automorphic, 182 S p -pseudo-almost periodic, 162, 163 (n) S p -pseudo almost automorphic, 187 (n) S p -almost automorphic, 138 (n) S p -almost periodicity, 109 W k,p (Ω ), 29 ρ (A) 57 σ (A), 57 ε -net, 6 ε -period, 80 ε -translation, 80 bAA(T, X ), 233 c0 -group, 64 c0 -semigroup, 61, 216 c0 -semigroup of contractions, 64 cl(X ), 57 AAA(Z), 131

AAA(X ), 122 AA(X ), 112 AA(n) (X ), 120 AP(n) (X ), 94 KAAA(X ), 122, 127 KAA(B), 213 KAA(X ), 112, 122, 125 KAA(n) (X ), 120 PAA(X ), 168 PAP(X ), 141, 142 PAP0 (X ), 141, 167 WAA(X ), 115 WKAA(X ), 116 accumulation point, 8 Acquistapace–Terreni conditions, 73, 76, 191, 200, 233, 238, 247, 254, 272 adherent point, 8 adjoint operator, 48 almost automorphic, 111, 133, 178, 190, 283 almost automorphic component, 167, 178 almost automorphic sequence, 128 almost periodic, 80, 82, 84, 100, 142, 281 analytic semigroup, 65, 67, 69, 223, 229, 254 Arzel`a–Ascoli theorem, 22, 251 asymptotic almost automorphic, 131, 209 asymptotically almost automorphic, 121 asymptotically almost periodic, 96

Banach fixed-point theorem, 17, 216, 255, 281 Banach space, 1, 23, 94, 97, 115, 120, 153, 275 beam equation, 229 Bessel’s Inequality, 37 Beverton–Holt model, 281 Beverton–Holt recruitment function, 281

T. Diagana, Almost Automorphic Type and Almost Periodic Type Functions in Abstract Spaces, DOI 10.1007/978-3-319-00849-3, © Springer International Publishing Switzerland 2013

301

302 bidual space, 25 bijection, 16 Bochner transform, 106, 163 Bohr spectrum, 90 boundary, 8 bounded operator, 43, 51, 53 Brouwer fixed point theorem, 32

canonical injection, 25 cardinality, 15 carrying capacity , 281 Cauchy sequence, 7, 13 Cauchy-Bunyakowsky-Schwarz inequality, 34 closed ball, 6 closed operator, 55 closed set, 8 compact, 53, 126 compact almost automorphic, 112 compact asymptotically almost automorphic, 121 compact metric space, 21 compact operator, 52 compact resolvent, 59 compactness, 20 complete metric space, 13 completeness, 7 composition theorem, 146, 170 continuous function, 11 convergence, 7 convolution, 81 corrective term, 96 countable, 15

dense, 13 derivative operator, 54 diameter, 6 dichotomy projections, 256 distance, 2 dual space, 24

eigenspace, 227, 239, 245 equi-continuous, 22 ergodic component, 142, 167 evolution equation, 247 evolution family, 72 exponentially stable, 72, 210

fading memory, 211 finite rank operator, 53 Fourier coefficients, 90

Index Fourier exponents, 90 Fourier series, 87, 91 fractional powers, 65 globally asymptotically stable, 280, 282, 283, 288 graph, 55 graph norm, 60 Green function, 283 growth rate, 281 H¨older’s inequality, 26 heat equation, 271 higher-order, 231 Hilbert space, 1, 33, 34, 48 Hille–Yosida Theorem, 64 history function, 209 homeomorphism, 16 hull, 277 hyperbolic semigroup, 70 infinitesimal generator, 62 injective, 15 inner product, 33 integrodifferential equation, 209 interior point, 8 intermediate space, 65 inverse operator, 50 kernel, 45 Laplace operator, 54 Lebesgue dominated convergence theorem, 212, 226, 235, 249, 258, 266 Lipschitz condition, 146, 170 Lipschitz constant, 17 mean, 143 mean value, 87 metric, 2 metric space, 1, 2 mild solution, 192 Minkowski’s Inequality, 27 multiplication operator, 44 neighborhood, 8 neutral integrodifferential, 214 norm, 5 normed vector space, 5

Index one-paremeter family, 211 one-to-one, 15 onto, 15 open ball, 6 open set, 8 operator matrix, 222 orthogonal decomposition, 41 orthogonal system, 38 orthogonality, 35 orthonormal base, 37, 53

parallelogram law, 33 phase space, 211 pointwise convergence, 12 polarization identity, 34 pre-image, 11 principal term, 96 projection, 39 pseudo-almost automorphic, 167, 261 pseudo-almost periodic, 141, 245, 255, 261 Pythagorean theorem, 36

range, 45 real interpolation space, 191 recruitment function, 281 reflexive, 25 regular exponential dichotomy, 276 relatively compact, 83 resolvent, 51, 56 resolvent operators, 209

Schauder fixed point theorem, 33, 190, 246, 251 sectorial operator, 60, 65 self-adjoint operator, 49, 58, 227 separable, 39 separable space, 15 sequential compactness, 20 sequentially compact, 20 Sobolev space, 29, 30, 34, 35

303 spectrum, 51, 56 sphere, 6 Stepanov bounded function, 106 Stepanov-like C(n) -almost periodic, 109 Stepanov-like C(n) -pseudo almost automorphic, 186 Stepanov-like almost periodic, 106 strongly continuous semigroup, 61 subcovering, 20 sup-norm, 31 surjective, 15 survival function, 281 symmetric operator, 49

totally bounded, 6 translate, 79 translation-invariant, 80, 151 triangle inequality, 2, 8 trigonometric polynomial, 80

unbounded operator, 53 uncountable, 15 uniform continuity, 12, 83 uniform convergence, 12 uniformly asymptotically stable, 280, 283 uniformly convex, 25

Volterra operator, 44

weak convergence, 24, 25 weakly compact, 25 Weierstrass theorem, 264 weight, 149 weighted ergodic component, 150, 178 weighted mean, 154 weighted pseudo-almost automorphic, 178 weighted pseudo-almost periodic, 149–151, 245, 259 well-posedness, 221, 232