Fixed Point Theory in Modular Function Spaces

Fixed Point Theory in Modular Function Spaces

Mohamed A. Khamsi • Wojciech M. Kozlowski

Fixed Point Theory in Modular Function Spaces

Mohamed A. Khamsi Department of Mathematical Sciences The University of Texas at El Paso El Paso, Texas USA

Wojciech M. Kozlowski School of Mathematics and Statistics University of New South Wales Sydney, New South Wales Australia

Department of Mathematics & Statistics King Fahd University of Petroleum & Minerals Dhahran Saudi Arabia

ISBN 978-3-319-14050-6 DOI 10.1007/978-3-319-14051-3

ISBN 978-3-319-14051-3 (eBook)

Mathematics Subject Classification (2010): 47H10, 47H09, 46E30, 46B20, 47H20, 47H30 Library of Congress Control Number: 2015931073 Springer Cham Heidelberg New York Dordrecht London c Springer International Publishing Switzerland 2015  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. 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. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)

Foreword

Because of its many diverse applications, fixed point theory has been a flourishing area of mathematical research for decades. S. Banach’s formulation of the contraction mapping principle in the early twentieth century signaled the advent of an intense interest in “metric” related aspects of the theory. The metric theory, often cast in a Banach space framework, has usually involved an intertwining of geometric and topological conditions. Fixed point theory in modular function spaces is closely related to the metric theory, in that it provides modular equivalents of norm and metric concepts. Modular spaces are extensions of the classical Lebesgue and Orlicz spaces, and in many instances conditions cast in this framework are more natural and more easily verified than their metric analogs. This book is devoted to a comprehensive treatment of what is currently known about fixed point theory in modular function spaces. A unified treatment of this subject was initiated in 1988 with the appearance of W. Kozlowski’s Marcel Dekker monograph Modular Function Spaces. Since the appearance of Kozlowski’s monograph, there have been numerous further developments, both in the theory of modular function spaces and in metric fixed point theory. This book takes full account of these developments. It covers the foundations of the theory, taking as a point of departure a review of the central themes of metric fixed point theory and the basic structures of modular function spaces. Anyone interested in these topics would want to have this book readily at hand. The book is essentially self-contained and should be easily accessible to students. Also it provides a valuable resource for those already involved in this avenue of research. Iowa City, May, 2014

W. A. Kirk

v

Preface

The single valued fixed point problem, which forms the basis of the fixed point theory, may be stated as : Let X be a set, A and B two nonempty subsets of X such that A∩B = 0, / and f : A → B be a map. When does a point x ∈ A such that f (x) = x, also called a fixed point of f , exist? And if yes, how many such points exist and how to find them? From the perspective of different settings, methods and applications, the fixed point theory is typically divided into the three major areas: • Topological fixed point theory • Metric fixed point theory • Discrete fixed point theory Historically, the boundary lines between the three areas were defined by the discovery of three major theorems: • Brouwer’s fixed point theorem [31] • Banach’s fixed point theorem [14] • Tarski’s fixed point theorem [198] In this book, we will focus mainly on the second area, although from time to time we may touch the other areas as well. Conceptually, the metric fixed point theory deals with situations where the set X is equipped with some kind of a method allowing to assign to every two elements of X a numeric value that measures the level of difference between them; in other words—a distance between them. We need to keep in mind that this idea of a “distance” can take many forms and does have not to be a distance in a popular meaning of the word, or a metric in the strict mathematical sense. Such a situation is very common when dealing with any quantifiable events in science and technology. Many problems like solving equations or finding stationary points of a time dependent system can be actually reduced to finding fixed points of a suitably defined mapping acting within a suitably selected set equipped with a suitably selected “distance.”

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Preface

The authors would like to invite the reader to join them in a journey taking them from a well-known base of classical fixed point theory in Banach and metric spaces (and yes, with Banach’s fixed point theorem as the starting point) to the world of the theory of fixed points of mappings defined in a class of spaces of measurable functions, known in the literature as modular function spaces. The results and methods of fixed point theory, applied to spaces of measurable functions, have been used extensively in the field of integral and differential equations. Since the 1930s many prominent mathematicians like Orlicz and Birnbaum recognized that using the methods of L p -norms alone created many complications and in some cases did not allow to solve some nonpower type integral equations. They considered spaces of functions with some growth properties different from the power type growth control provided by the L p -norms. The possibility of introducing the structure of a linear metric as well as the interesting properties of these spaces, later named Orlicz spaces, and many applications to differential and integral equations with kernels of nonpower types were among the reasons for the development of the theory of Orlicz spaces, their applications and generalizations. Using the apparatus of the modular function spaces we can go much further: the operator itself is used for the construction of a function modular and hence of a space in which this operator has required properties. These techniques together with relevant modular function space fixed point theorems can be efficiently applied to solving many mathematical problems. The aim of this book is to familiarize the readers with the main concepts and results of the fixed point theory in modular function spaces, as well as to encourage them to use these results in the course of their research activities. We would like to thank the University of Texas El Paso and the University of New South Wales for their continuing support during the preparation of this manuscript. We would like to acknowledge also the role of the King Fahd University of Petroleum and Minerals in extending a strong support from the beginning to the end, facilitating every means during the preparation of this book. We also owe a measure of gratitude to the Deanship of Scientific Research, for providing excellent opportunity of academic research activities. Dhahran, El Paso, Sydney November 2014

M. A. Khamsi W. M. Kozlowski

Contents

1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

2

Fixed Point Theory in Metric Spaces: An Introduction . . . . . . . . . . . . . 2.1 Banach Contraction Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Pointwise Lipschitzian Mappings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Caristi-Ekeland Extension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Some Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.1 ODE and Integral Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.2 Cantor and Fractal sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Metric Fixed Point Theory in Banach Spaces . . . . . . . . . . . . . . . . . . . 2.5.1 Classical Existence Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.2 The Normal Structure Property . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.3 The Demiclosedness Principle . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.4 Opial and Kadec-Klee Properties . . . . . . . . . . . . . . . . . . . . . . . 2.6 Ishikawa and Mann Iterations in Banach spaces . . . . . . . . . . . . . . . . . 2.7 Metric Convexity and Convexity Structures . . . . . . . . . . . . . . . . . . . . . 2.8 Uniformly Convex Metric Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.9 More on Convexity Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.10 Common Fixed Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5 5 7 12 13 13 14 15 16 19 23 25 31 34 37 43 44

3

Modular Function Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Foundations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Space Eρ , Convergence Theorems, and Vitali Property . . . . . . . . . . . 3.3 An Equivalent Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Compactness and Separability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6 Generalizations and Special Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6.1 General Definition of Function Modular . . . . . . . . . . . . . . . . . 3.6.2 Nonlinear Operator Valued Measures . . . . . . . . . . . . . . . . . . . 3.6.3 Nonlinear Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . . . .

47 47 58 65 67 69 71 71 72 75

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Contents

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Geometry of Modular Function Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Uniform Convexity in Modular Function Spaces . . . . . . . . . . . . . . . . . 4.2 Parallelogram Inequality and Minimizing Sequence Property . . . . . . 4.3 Uniform Noncompact Convexity in Modular Function Spaces . . . . . 4.4 Opial and Kadec–Klee Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

79 79 88 91 98

5

Fixed Point Existence Theorems in Modular Function Spaces . . . . . . . 111 5.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 5.2 Contractions in Modular Function Spaces . . . . . . . . . . . . . . . . . . . . . . 114 5.2.1 Banach Contraction Principle in Modular Function Spaces . . 114 5.2.2 Case of Uniformly Continuous Function Modulars . . . . . . . . 116 5.2.3 Case of Modular Function Spaces with Strong Opial Property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 5.2.4 Quasi-Contraction Mappings in Modular Function Spaces . . 118 5.3 Nonexpansive and Pointwise Asymptotic Nonexpansive Mappings . 121 5.3.1 Case of Uniformly Convex Function Modulars . . . . . . . . . . . 121 5.3.2 Normal Structure Property in Modular Function Spaces . . . . 127 5.3.3 Case of Uniformly Lipschitzian Mappings . . . . . . . . . . . . . . . 134 5.3.4 Common Fixed Point Theorems in Modular Function Spaces146 5.3.5 Asymptotic Nonexpansive Mappings in Modular Function Spaces Satisfying Δ2 -type Condition . . . . . . . . . . . . . . . . . . . . 153 5.3.6 KKM and Ky Fan Theorems in Modular Function Spaces . . 159 5.4 Applications to Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . 163

6

Fixed Point Construction Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 6.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 6.2 Demiclosedness Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172 6.3 Generalized Mann Iteration Process . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 6.4 Generalized Ishikawa Iteration Process . . . . . . . . . . . . . . . . . . . . . . . . 180 6.5 Strong Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184

7

Semigroups of Nonlinear Mappings in Modular Function Spaces . . . . 185 7.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 7.2 Fixed Point Existence for Semigroups of Nonexpansive Mappings . 186 7.3 Characterization of the Set of Common Fixed Points . . . . . . . . . . . . . 190 7.4 Convergence of Mann Iteration Processes . . . . . . . . . . . . . . . . . . . . . . 198 7.5 Convergence of Ishikawa Iteration Processes . . . . . . . . . . . . . . . . . . . . 202 7.6 Applications to Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . 205 7.7 Asymptotic Pointwise Nonexpansive Semigroups . . . . . . . . . . . . . . . . 211

8

Modular Metric Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219 8.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219 8.2 Banach Contraction Principle in Modular Metric Spaces . . . . . . . . . . 223 8.3 Nonexpansive Mappings in Modular Metric Spaces . . . . . . . . . . . . . . 230

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243

Chapter 1

Introduction

Before we introduce the definition and the fundamental properties of modular function spaces, let us discuss a historical background, which will also provide a motivation for the introduction of the methods of modular function spaces into the fixed point theory. The results and methods of fixed point theory, applied to spaces of measurable functions, have been used extensively in the field of integral equations and integral inequalities. Since the 1930s, many prominent mathematicians like Orlicz and Birnbaum recognized that using the methods of L p -spaces alone created many complications and in some cases did not allow to solve some nonpower type integral equations, see [29]. They considered spaces of functions with some growth properties different from the power type growth control provided by the L p -norms. Orlicz and Birnbaum considered for instance, function spaces defined as follows: Lϕ = { f : R → R; ∃λ > 0 :

 R

ϕ (λ | f (x)|) dm(x) < ∞},

where ϕ : [0, ∞) → [0, ∞) was assumed to be a convex function increasing to infinity, that is, the function which to some extent behaves similarly to power functions ϕ (t) = t p . Let us mention two typical examples of such functions: ϕ1 (t) = et −t − 1 2 or ϕ2 (t) = et − 1. The possibility of introducing the structure of a linear metric in ϕ L as well as the interesting properties of these spaces, later named Orlicz spaces, and many applications to differential and integral equations with kernels of nonpower types were among the reasons for the development of the theory of Orlicz spaces, their applications, and generalizations. Consider, for example, the following Hammerstein nonlinear integral equation, which plays an important role in the elasticity theory:  1

f (x) = 0

k(x, y)ϕ ( f (y))dy,

where ϕ (u) is a function which increases more rapidly than an arbitrary power function. Krasnosel’skii and Rutickii, [147], showed that the Hammerstein operac Springer International Publishing Switzerland 2015  M. A. Khamsi, W. M. Kozlowski, Fixed Point Theory in Modular Function Spaces, DOI 10.1007/978-3-319-14051-3 1

1

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1 Introduction

tor defined by the right member of this integral equation does not operate in any of the L p spaces. And yet, they showed how to find an Orlicz space where the Hammerstein operator is well defined and posses properties allowing to use some fixed point theorems for solving the corresponding integral equation. Using the apparatus of the modular function spaces we can go much further: the operator itself is used for the construction of a function modular and hence of a space in which this operator has required properties. This technique together with a relevant modular function space fixed point theorem can be applied to solving integral equations like Urysohn integral equation. An additional importance for applications of modular function spaces consists in the richness of structure of modular function spaces that, besides being Banach spaces (or F-spaces in a more general settings), are equipped with modular equivalents of norm or metric notions, and also are equipped with almost convergence everywhere and convergence in submeasure. In many situations in integral equations, approximation and fixed point theory, modular type conditions are much more natural, and modular type assumptions can be more easily verified than their metric or norm counterparts. There are also important results that can be proved only using the apparatus of modular function spaces. From this perspective, the fixed point theory in modular function spaces should be considered as complementary to the fixed point theory in normed and metric spaces. The theory of contractions and nonexpansive mappings defined on convex subsets of Banach spaces has been well-developed since the 1960s (see, for example, [32, 58, 79, 82, 85, 120, 122]), and generalized to metric spaces (see, for example, [13, 84, 111]), and modular function spaces (see, for example, [109, 116, 117]). The corresponding fixed point results were then extended to larger classes of mappings like asymptotic mappings [112, 125], pointwise contractions [121] and asymptotic pointwise contractions, and nonexpansive mappings [96, 114, 115, 127, 128]. While Banach contraction principle and its generalizations usually provide a fixed point construction method as limits of orbits, the proofs of the fixed point existence results for nonexpansive mappings and their generalizations are of the existential nature, and do not describe any algorithms for constructing fixed points of an asymptotic pointwise nonexpansive mapping. It is well known that the fixed point construction iteration processes for generalized nonexpansive mappings have been successfully used to develop efficient and powerful numerical methods for solving various nonlinear equations and variational problems, often of great importance for applications in various areas of pure and applied science. In [133, 138], the author proved convergence to fixed points of some iterative algorithms applied to asymptotic pointwise nonexpansive mappings in Banach spaces and the existence of common fixed points of semigroups of pointwise Lipschitzian mappings in Banach spaces [134]. Recently, the weak and strong convergence of such processes to common fixed points of semigroups of mappings in Banach spaces was demonstrated in

1 Introduction

3

[137, 144]. The pioneering work on the convergence of Mann and Ishikawa algorithms in modular function spaces can be found in [54]. Our approach is based on this paper. We would like to emphasize that all convergence theorems presented in this book, define constructive algorithms that can be actually implemented. When dealing with specific applications of these theorems, one should take into consideration how additional properties of the mappings, sets, and modulars involved, can influence the actual implementation of the algorithms defined in this book. The existence of common fixed points for families of contractions and nonexpansive mappings in Banach spaces have been investigated since the early 1960s, see, for example, [19, 20, 32, 35, 57, 153]. The asymptotic approach for finding common fixed points of semigroups of Lipschitzian (but not pointwise Lipschitzian) mappings has been also investigated for some time, see, for example, [199]. It is worthwhile mentioning the recent studies on the special case, when the parameter set for the semigroup is equal to {0, 1, 2, 3, ...} and Tn = T n , the n-th iterate of an asymptotic pointwise nonexpansive mapping, that is, such a T : C → C that there exists a sequence of functions αn : C → [0, ∞) with T n (x) − T n (y) ≤ αn (x)x − y. In [128], the authors proved the existence of fixed points for asymptotic pointwise contractions and asymptotic pointwise nonexpansive mappings in Banach spaces, while the paper [96] extended this result to metric spaces, and the papers [114, 115] to the case of modular function spaces. The existence of common fixed points for general semigroups of mappings acting in modular function spaces was proved in [135]. The book is structured as follows: Chapter 2 provides an introduction to the general metric fixed point theory. The aim of this chapter is to use the classical fixed point results for contractions and nonexpansive mappings acting in metric or normed spaces, to set the scene for developing the fixed point theory in modular function spaces. The reader will be able to see many similarities but at the same time will be able to appreciate significant differences. Readers familiar with the classical theory may skip reading Chap. 2 and jump straight to Chap. 3, while the beginners can actually use Chap. 2 as a “cheat sheet” summarizing key results of the fixed point theory in metric spaces. Chapter 3 introduces the foundation of the theory of modular function spaces defined by convex regular function modulars. The results discussed in this chapter are used throughout the rest of the book. This chapter also presents the reader with an exhaustive list of examples which frequently reoccur in later parts of the book. Section 3.4 “Generalizations and Special Cases” summarizes a more general approach which allows nonconvex function modulars and extends the setting to spaces of function with values in Banach spaces. While these results are not necessary for understanding the contents of the remaining chapters, the general framework introduced in this section may help to appreciate the full potential of the theory of modular function spaces. The readers who are interested in an active research in

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1 Introduction

this area, can use this chapter for possible generalizations of the results presented in the book and for setting new research directions. In Chap. 4 we discuss some fundamental notions of the geometry of modular functions spaces. Hence, we introduce several types of modular uniform convexity, as well as modular versions of Opial and Kadec—Klee properties. We will direct readers’ attention to the fact that even though these properties formally resemble their Banach space counterparts, they are different and frequently they are satisfied, while their norm equivalents do not. Similarly, as in the fixed point theory in Banach spaces, these properties will be extensively used in proofs of our fixed point theorems. Chapter 5 is probably the most important part of this book as it presents a long list of fixed point existence theorems for nonlinear mappings acting in modular functions spaces. We cover a range of different types of mappings including ρ -contractions and their pointwise asymptotic versions and ρ -nonexpansive mappings and pointwise asymptotic ρ -nonexpansive mappings, under various assumptions on the function modular ρ and on the sets these mappings are defined in. In addition some examples and applications to the theory of differential equations are provided. In this chapter, like in the whole book, we deal only with the fixed point theory for single-valued mappings acting in modular function spaces. At the time of this publication, very little was known about the set-valued case, which definitely creates an interesting direction for further research. Chapter 6 is devoted to the fixed point construction processes and their convergence. These results form an important sequel to the existential results of the previous chapter since proving existence without any means to approximate fixed points would be of limited use for applications. This area of research is in its early stages and there are many untouched topics like implicit iterative methods or possible generalizations of the Halpern type method to the case of modular function spaces. Chapter 7 discusses at length, the question of existence and construction of common fixed points of semigroups of nonlinear mappings in modular functions spaces. This chapter also demonstrates how solutions to some initial value problems, defined by nonlinear differential equations, form semigroups of ρ -nonexpansive mappings in modular functions spaces and interpret the meaning of common fixed points in terms of stationary points of corresponding processes. The structure of the set of common fixed points is also investigated in this chapter. The final chapter, Chap. 8, provides a sneak preview into the fixed point theory for nonlinear mappings in modular metric spaces, which in essence is an attempt to generalize the modular function space theory to the nonlinear case, that is to remove an assumption that the base space is a vector space. These results are quite recent and they open a wide field for further research.

Chapter 2

Fixed Point Theory in Metric Spaces: An Introduction

This chapter is primarily intended to serve as an introduction to metric fixed point theory. It will set the foundation for the coming chapters. In terms of content this chapter overlaps in places with the following popular books on fixed point theory by A. Aksoy and M. A. Khamsi [5], by K. Goebel and W. A. Kirk [81], by J. Dugundji and A. Granas, [67], by M. A. Khamsi and W. A. Kirk [111], by I.A. Rus, A. Petrusel, and G. Petrusel [186], and by E. Zeidler [203]. Material on the general theory of Banach space geometry is drawn from many sources but the following books by B. Beauzamy [16], and by J. Diestel [59] are worth a special mention.

2.1 Banach Contraction Principle In 1922, Banach [14] published his fixed point theorem, also known as the Banach Contraction Principle, which uses the concept of Lipschitz mappings . Definition 2.1. Let (M, d) be a metric space. The map T : M → M is said to be Lipschitzian if there exists a constant k > 0 (called Lipschitz constant) such that   d T (x), T (y) ≤ k d(x, y) for all x, y ∈ M. A Lipschitzian mapping with a Lipschitz constant k < 1, is called contraction . Moreover if the Lipschitz constant k = 1, the Lipschitzian mapping is called nonexpansive . Observe that all contractions and nonexpansive mappings are uniformly continuous. Theorem 2.1 (Banach Contraction Principle). Let (M, d) be a complete metric space and let T : M → M be a contraction mapping, with Lipschitz constant k < 1. Then T has a unique fixed point ω in M, and for each x ∈ M, we have lim T n (x) = ω .

n→∞

c Springer International Publishing Switzerland 2015  M. A. Khamsi, W. M. Kozlowski, Fixed Point Theory in Modular Function Spaces, DOI 10.1007/978-3-319-14051-3 2

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2 Fixed Point Theory in Metric Spaces: An Introduction

Moreover, for each x ∈ M, we have d (T n (x), ω ) ≤

kn d (T (x), x) . 1−k

Proof. Since T is a contraction mapping we know that for each x ∈ M and n ≥ 1, d(T n (x), T n+1 (x)) ≤ k d(T n−1 (x), T n (x)) ≤ kn d(x, T (x)). Let n, m ∈ N, with n ≥ 1, we get d(T n (x), T n+m (x)) ≤

n+m

∑

d(T i (x), T i+1 (x)) ≤

n+m

i=n

∑ ki d(x, T (x)),

i=n

for any x ∈ M. Since k < 1, we get n+m

∑ ki = kn

i=n

kn 1 − km+1 ≤ . 1−k 1−k

Hence d(T n (x), T n+m (x)) ≤

kn d(x, T (x)). 1−k

(2.1)

This forces {T n (x)} to be a Cauchy sequence. Since M is complete, then lim T n (x) = ωx exists in M. Note that a priori the limit point depends on the starting n→∞ point x. Since T is continuous

ωx = lim T n (x) = lim T n+1 (x) = T (ωx ). n→∞

n→∞

Thus ωx is a fixed point of T , for any x ∈ M. Next we show that ωx is independent of the starting point x. Let y ∈ M and ωy be the fixed point of T associated to y. Then d(ωx , ωy ) = d(T (ωx ), T (ωy )) ≤ k d(ωx , ωy ). Since k < 1, we get d(ωx , ωy ) = 0, i.e., ωx = ωy . This shows that T has one fixed point ω and for any x ∈ M, we have lim T n (x) = ω . If we let m → ∞ in the inequality n→∞ (2.1), we get kn d(x, T (x)), d(T n (x), ω ) ≤ 1−k for any x ∈ M and n ∈ N. This completes the proof of Theorem 2.1. 

An easy implication of the Banach Contraction Principle is the following theorem. Theorem 2.2. Let (M, d) be a complete metric space. Suppose T : M → M is a mapping for which T N is a contraction mapping, for some positive integer N ≥ 1. Then T has a unique fixed point.

2.2 Pointwise Lipschitzian Mappings

7

Proof. Since T N is a contraction, Theorem 2.1 implies the existence of a unique fixed point x0 of T N . Since T N (T (x0 )) = T N+1 (x0 ) = T (T N (x0 )) = T (x0 ), then T (x0 ) is also a fixed point of T N . Since T N has only one fixed point, we conclude that T (x0 ) = x0 . 

It is not clear in general whether T has a fixed point whenever T N has a fixed point. Note that fixed points of T N are also known as periodic points of T . Remark 2.1. We all have learned that the origins of the metric contraction principle and, ergo, metric fixed point theory itself, rest in the method of successive approximations for proving existence and uniqueness of solutions of differential equations. This method is associated with the names of such celebrated nineteenth century mathematicians as Cauchy, Liouville, Lipschitz, Peano, and specially Picard. In fact the iterative process used in the proof of the Banach Contraction Theorem bears the name of Picard iterates. It is quite interesting to know that in 1429, Al-Kashani [23] published a book entitled: “The Calculator’s Key”, where he used Picard iterates. In fact, Al-Kashani set the stage for the so-called numerical techniques some 600 years ago. He was keen to develop ideas with practical matters, like the approximate values of sin(1o ) which enabled scientists after him to come up with very good approximations to the circumference of the Earth.

2.2 Pointwise Lipschitzian Mappings The notion of pointwise Lipschitzian mappings was introduced in [125, 127, 128]. The main motivation behind this generalization is to find a larger class of mappings for which the conclusion of Banach’s celebrated contraction mapping theorem is still valid. Pointwise Lipschitzian mappings are defined as follows. Definition 2.2. Let (M, d) be a metric space. A mapping T : M → M is called a pointwise Lipschitzian mapping if there exists a mapping α : M → [0, +∞) such that d(T (x), T (y)) ≤ α (x) d(x, y), for any x, y ∈ M. The map T is called pointwise contraction if α (x) < 1, for any x ∈ M. Note that if T is a pointwise contraction, then it is continuous although not necessarily uniformly continuous. Moreover, if α (x) = 0 for some x ∈ M, then T is a constant map. It is easy to prove that a pointwise contraction T : M → M has at most one fixed point, and if x0 is its fixed point, then the orbit {T n (x)} converges to x0 , for each x ∈ M. Indeed, we have d(x0 , T n (x)) ≤ α (x0 )n d(x0 , x)

8

2 Fixed Point Theory in Metric Spaces: An Introduction

for any x ∈ M. The above conclusion follows because α (x0 ) < 1. It is not clear how to prove the existence of the fixed point from the convergence of the orbits which is the case in the classical proof given to the Banach Contraction Principle, Theorem 2.1. In order to extend the main conclusion of [128] to metric spaces, the authors in [96] needed the following definition. Definition 2.3. Let (M, d) be a metric space. A subset C of M is admissible if it is a nonempty intersection of closed balls. The family of all admissible subsets of M is denoted by A (M). We will say that A (M) is compact if any family (Aα )α ∈Γ of Aα = 0/ for any finite elements of A (M), has a nonempty intersection provided α ∈F

subset F ⊂ Γ . Moreover for any nonempty subset A of M, the cover of A is defined by  cov(A) = {B : B is a closed ball and A ⊂ B}. The admissible subsets were introduced in metric fixed point theory because of their similarities with convex subsets in linear vector spaces. Banach Contraction Principle for pointwise contraction mappings may be stated as: Theorem 2.3. Let M be a bounded metric space. Assume that the convexity structure A (M) is compact. Let T : M → M be a pointwise contraction. Then T has a unique fixed point x0 . Moreover the orbit {T n (x)} converges to x0 , for each x ∈ M. Proof. Since A (M) is compact, there exists a minimal nonempty K ∈ A (M) such that T (K) ⊂ K. It is easy to check that cov(T (K)) = K. Let a ∈ K, then we have K ⊂ B(a, ra (K)), where ra (K) = sup{d(a, x) : x ∈ K}. Since T is a pointwise contraction, there exists a mapping α : M → [0, 1) such that d(T (x), T (y)) ≤ α (x)d(x, y) for any x, y ∈ M. In particular, we have then T (K) ⊂ B(T (a), α (a)ra (K)), which implies K = cov(T (K)) ⊂ B(T (a), α (a)ra (K)). So rT (a) (K) ≤ α (a)ra (K). This will force diam(K) = 0. Indeed let a ∈ K and define Ka = {x ∈ K : rx (K) ≤ ra (K)}. Clearly Ka is not empty. Moreover, we have Ka =

 x∈K

B(x, ra (K)) ∩ K ∈ A (M).

2.2 Pointwise Lipschitzian Mappings

9

And since rT (x) (K) ≤ α (x)rx (K), for any x ∈ K, we get T (Ka ) ⊂ Ka . The minimality behavior of K implies Ka = K. In particular we have rx (K) = ra (K) for any x ∈ K. Hence diam(K) = ra (K), for any a ∈ K, i.e., a is a diametral point of K. Hence diam(K) ≤ α (a)diam(K). And since α (a) < 1, we get diam(K) = 0, i.e., K is reduced to one point which is fixed by T . The remaining conclusion of the theorem follows from the general properties of pointwise contractions. 

As we did in the case of Banach’s Contraction Principle, we extend Theorem 2.3 to the case when iterates are pointwise contractions. First we need the following definition. Definition 2.4. Let (M, d) be a metric space. A mapping T : M → M is called an asymptotic pointwise mapping if there exists a sequence of mappings αn : M → [0, ∞) such that d(T n (x), T n (y)) ≤ αn (x)d(x, y), for any y ∈ M. (i) If {αn } converges pointwise to α : M → [0, 1), then T is called an asymptotic pointwise contraction . (ii) If lim sup αn (x) ≤ 1, then T is called asymptotic pointwise nonexpansive . n→∞

(ii) If lim sup αn (x) ≤ k, with 0 < k < 1, then T is called a strongly asymptotic n→∞

pointwise contraction . Let (M, d) be a metric space. We will say that a function Φ : M → [0, ∞) is convex if {x : Φ (x) ≤ r} ∈ A (M), for any r ≥ 0. Also we define a type to be a function Φ : M → [0, ∞) defined as

Φ (u) = lim sup d(xn , u) n→∞

where {xn } is a bounded sequence in M. Types are very useful in the study of the geometry of Banach spaces and the existence of fixed point of mappings. We will say that A (M) is T-stable if types are convex. We have the following lemma. Lemma 2.1. Let M be a metric space. Assume that A (M) is compact and T-stable. Then for any type Φ , there exists x0 ∈ M such that

Φ (x0 ) = inf{Φ (x) : x ∈ M}. Proof. Let Φ be a type. Then there exists a bounded sequence {xn } in M such that Φ (x) = lim sup d(xn , x), for any x ∈ M. Set Φ0 = inf Φ (x). Then the set x∈M

n→∞

 1 Mn = x ∈ M : Φ (x) ≤ Φ0 + , n 

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2 Fixed Point Theory in Metric Spaces: An Introduction

is a nonempty admissible subset of M because A (M) is convex. Since {Mn } is a Mn is not empty. Note decreasing sequence and A (M) is compact, then M∞ = n≥1

that for any x ∈ M∞ we have Φ (x) = Φ0 . 

We have the following fixed point theorem for strongly asymptotic pointwise contractions in metric spaces. Theorem 2.4. Let M be a fixed bounded metric space. Assume that A (M) is compact. Let T : M → M be a strongly asymptotic pointwise contraction. Then T has a unique fixed point x0 . Moreover any orbit {T n (x)} converges to x0 . Proof. First note that T has, at most, one fixed point. Indeed, let a, b ∈ M be two fixed points of T . Then we have d(a, b) = d(T n (a), T n (b)) ≤ αn (a) d(a, b). If we let n go to infinity, we get d(a, b) ≤ k d(a, b) for some k ∈ (0, 1). This will force d(a, b) = 0. Let us fix x ∈ M and define the type

Φ (u) = lim sup d(T n (x), u), for u ∈ M. n→∞

Since A (M) is compact, then

Ω (x) =



  cov {T k (x) : k ≥ n} = 0. /

n≥1

Let ω ∈ Ω (x). Then for every n, m ∈ N we have d(T m+n+h (x), T m+h (x)) ≤ αh (T m (x))d(T n (x), T m (x)). If we let n go to infinity, we get

Φ (T m+h (x)) ≤ αh (T m (x))Φ (T m (x)). Next we let h go to infinity to get lim sup Φ (T n (x)) ≤ k Φ (T n (x)) n→∞

for some k ∈ (0, 1), which easily implies that lim sup Φ (T n (x)) = 0. Notice that n→∞

Φ (ω ) ≤ lim sup Φ (T n (x)) = 0. n→∞

Indeed let u ∈ M, then for any ε > 0, then there exists n0 ≥ 1 such that for any n ≥ n0 d(T n (x), u) ≤ Φ (u) + ε .

2.2 Pointwise Lipschitzian Mappings

11

In particular we have T n (x) ∈ B(u, Φ (u) + ε ), for any n ≥ n0 . So   Ω (x) ⊂ cov {T n (x) : n ≥ n0 } ⊂ B(u, Φ (u) + ε ), which implies ω ∈ B(u, Φ (u) + ε ). This is true for any ε > 0. Hence for any u ∈ M we have d(ω , u) ≤ Φ (u), which implies that

Φ (ω ) = lim sup d(T n (x), ω ) ≤ lim sup Φ (T n (x)). n→∞

n→∞

Therefore we have Φ (ω ) = 0 which implies that {T n (x)} converges to ω . Since T is continuous, then ω is a fixed point of T . Since T has at most one fixed point, then the limit of the orbit {T n (x)} is independent of x ∈ M. Therefore T has a unique fixed point x0 and any orbit converges to x0 . 

Let us relax now the strong behavior of T but assume that types are convex to obtain the following result. Theorem 2.5. Let M be a bounded metric space. Assume that A (M) is compact and T-stable. Let T : M → M be an asymptotic pointwise contraction. Then T has a unique fixed point x0 . Moreover the orbit {T n (x)} converges to x0 , for each x ∈ M. Proof. As we did in the beginning of the proof of Theorem 2.4, we can easily show that T has at most one fixed point. Let x ∈ M and define the type

Φ (u) = lim sup d(T n (x), u), for each u ∈ M. n→∞

Since A (M) is compact and T-stable, then by Lemma 2.1 there exists x0 ∈ M such that Φ (x0 ) = inf {Φ (u) : u ∈ M}. Let us show that Φ (x0 ) = 0. Indeed we have d(T n+m (x), T m (x0 )) ≤ αm (x0 ) d(T n (x), x0 ), for any n, m ≥ 1. If we let n go to infinity, we get

Φ (T m (x0 )) ≤ αm (x0 ) Φ (x0 ), which implies

Φ (x0 ) = inf {Φ (u) : u ∈ M} ≤ Φ (T m (x0 )) ≤ αm (x0 ) Φ (x0 ). If we let m go to infinity, we get Φ (x0 ) ≤ α (x0 )Φ (x0 ). Since α (x0 ) < 1, we get Φ (x0 ) = 0, which implies that {T n (x)} converges to x0 . This will force x0 to be a fixed point of T . Since we already noticed that T has at most one fixed point, then T has a fixed point x0 and any orbit converges to x0 . 

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2 Fixed Point Theory in Metric Spaces: An Introduction

2.3 Caristi-Ekeland Extension This is one of the most interesting extensions of the Banach Contraction Principle. In order to understand its context, let us first go over the proof of Theorem 2.1 given by Caristi [40, 41] . Let T : M → M be a contraction with Lipschitz constant k < 1. Then we have d(T (x), T 2 (x)) ≤ k d(x, T (x)), for any x ∈ M. Adding d(x, T (x)) to both sides of the above inequality yields d(x, T (x)) + d(T (x), T 2 (x)) ≤ d(x, T (x)) + k d(x, T (x)) which can be rewritten d(x, T (x)) − k d(x, T (x)) ≤ d(x, T (x)) − d(T (x), T 2 (x)). This in turn is equivalent to d(x, T (x)) ≤

1  d(x, T (x)) − d(T (x), T 2 (x)) , 1−k

for any x ∈ M. Now define the function ϕ : M → R+ by setting

ϕ (x) =

1 d(x, T (x)). 1−k

This gives us the basic inequality d(x, T (x)) ≤ ϕ (x) − ϕ (T (x)), for any x ∈ M. As a generalization to contraction mappings, Caristi [41] and Ekeland [73] considered mappings T : M → M which satisfy the following property d (x, T (x)) ≤ ϕ (x) − ϕ (T (x)),

x ∈ M,

where ϕ : M → R+ = [0, +∞). Both Caristi and Ekeland investigated this new class of mappings to find out when a fixed point exists. Recall that the function ϕ is said to be lower semicontinuous(l.s.c.), if for any sequence {xn } ⊂ M, if lim xn = x and lim ϕ (xn ) = r, then ϕ (x) ≤ r.

n→∞

n→∞

Theorem 2.6 (Ekeland Variational Principle, 1974). Let (M, d) be a complete metric space and ϕ : M → R+ l.s.c. Define the Brønsted partial order : x y ⇔ d(x, y) ≤ ϕ (x) − ϕ (y), x, y ∈ M. Then (M, ) has a maximal element.

2.4 Some Applications

13

Theorem 2.7 (Caristi Fixed Point Theorem, 1975). Let (M, d) be a complete metric space and ϕ : M → R+ l.s.c. Suppose T : M → M satisfies: d(x, T (x)) ≤ ϕ (x) − ϕ (T (x)), x ∈ M. Then T has a fixed point. Although both theorems have different settings, in fact they are equivalent, [171]. The proof of Caristi-Ekeland’s theorems is based on the discrete technique: Zorn’s lemma and Axiom of Choice. There are some attempts to find a pure metric proof of Caristi’s fixed point theorem, without success so far.

2.4 Some Applications There are many known examples of the Banach Contraction Principle. Here, we will discuss only two of them.

2.4.1 ODE and Integral Equations Consider the integral equation f (x) = λ

 x a

K(x,t) f (t)dt + φ (x)

for a fixed real number λ , where K(x,t) is continuous on [a, b] × [a, b]. Consider the metric space C [a, b] of continuous real-valued functions defined on [a, b]. Consider the map T : C [a, b] → C [a, b] defined by T ( f )(x) = λ

 x a

K(x,t) f (t)dt + φ (x).

For f1 , f2 ∈ C [a, b], we have d (T n ( f1 ), T n ( f2 )) ≤ |λ |n M n

(b − a)n d( f1 , f2 ) n!

where d( f1 , f2 ) = max{| f1 (x) − f2 (x)| : x ∈ [a, b]}, and M = max {|K(x,t)| : (x,t) ∈ [a, b] × [a, b]} . Clearly there exists n ≥ 1 such that T n is a contraction, which implies that the above integral equation has a unique solution f (x). In general, the map T may not be a contraction on [a, b]. Bielecki [26] discovered another way to find a remedy to this

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2 Fixed Point Theory in Metric Spaces: An Introduction

problem. Indeed, for λ > 0, set  f λ = max

a≤x≤b



 e−λ (x−a) | f (x)| ,

it is now possible to prove that for any f1 , f2 ∈ C [a, b], we have dλ (T ( f1 ), T ( f2 )) = T ( f1 ) − T ( f2 )λ ≤

M M  f1 − f2 λ = d ( f1 , f2 ), λ λ λ

where M = max |K(x, y)| is as before. It is then clear that for λ sufficiently large a≤x,y≤b

T is a contraction for the new distance dλ .

2.4.2 Cantor and Fractal sets Let (M, d) be a complete metric space and let M denote the family of all nonempty bounded closed subsets of M, and let C denote the subfamily of M consisting of all compact sets. For A ∈ M and ε > 0 define the ε -neighborhood of A to be the set Nε (A) = {x ∈ M : dist(x, A) < ε } where dist(x, A) = inf d(x, y). Now for A, B ∈ M , set y∈A

H(A, B) = inf{ε > 0 : A ⊆ Nε (B) and B ⊆ Nε (A)}. Then (M , H) is a metric space, and H is called the Hausdorff metric on M . Notice that if (M, d) is complete, then (M , H) is also complete. Let Ti : M → M, i = 1, . . . , n, be a family of contractions. Define the map T ∗ : C → C by T ∗ (X) =

n

Ti (X).

i=1

Then T ∗ is a contraction and its Lipschitz constant is smaller than the maximum of all Lipschitz constants of the mappings Ti , i = 1, . . . , n. Then Banach Contraction Principle implies the existence of a unique nonempty compact subset X of M such that X=

n

Ti (X).

i=1

As an application of this, consider the real interval [0, 1] and the two contractions 2 1 1 T1 (x) = x and T2 (x) = x + . 3 3 3

2.5 Metric Fixed Point Theory in Banach Spaces

15

Then the compact X which satisfies X = T1 (X) ∪ T2 (X) is the well-known Cantor set. Moreover this fixed point set is the limit of the iterate of [0, 1].

2.5 Metric Fixed Point Theory in Banach Spaces The formal definition of Banach spaces is due to Banach himself. But examples like the finite dimensional vector space Rn was studied prior to Banach’s formal definition. In 1912, Brouwer [31] proved the following: Theorem 2.8 (Brouwer Fixed Point Theorem). Let B be a closed ball in Rn . Then any continuous mapping T : B → B has at least one fixed point. This theorem has a long history. The ideas used in its proof were known to Poincare as early as 1886. In 1909, Brouwer proved the theorem for n = 3, and in 1910, Hadamard gave the first proof for arbitrary n. Brouwer gave another proof in 1912. All of these are older results than the Banach Contraction Principle. Though the two theorems are different in nature, they bear some similarities. A combination of the two led to the so-called metric fixed point theorem in Banach spaces. Indeed, in Brouwer’s theorem, the convexity, compactness, and the continuity of T are crucial, while the Lipschitz behavior of the contraction and completeness are crucial in Banach’s fixed point theorem. In infinite normed linear vector spaces, we lose the compactness of the bounded closed convex sets (like closed balls). So if we assume completeness, we get Banach spaces. On these, we have another natural topology (other than the norm topology), that is the weak topology. A weakly compact convex set does not need to be compact for the norm. The best example here is the Hilbert space, where any bounded closed convex set is weakly compact. The Lipschitz condition considered is when the Lipschitz constant is equal to 1, i.e., nonexpansive mappings. The metric fixed point problem in Banach spaces becomes: Let X be a Banach space, and C a nonempty bounded closed convex subset of X. When does any nonexpansive mapping T : C → C have a fixed point? Other interesting problems closely related to this one are: • The structure of the fixed points sets [34] • The approximation of fixed points. Recognition of fixed point theory for nonexpansive mappings as a noteworthy avenue of research almost surely dates from the 1965 publication of 1. Kirk’s Theorem [120]. Let K be a weakly-compact convex subset of a Banach space X. Assume that K has the normal structure property (see Definition 2.5 in Section 2.5.2), then any nonexpansive mapping T : K → K has a fixed point.

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2 Fixed Point Theory in Metric Spaces: An Introduction

2. Browder-G¨ohde’s Theorem [32, 85]. If K is a bounded closed convex subset of a uniformly convex Banach space X and if T : K → K is nonexpansive, then T has a fixed point. Moreover the fixed point set of T is a closed convex subset of K. While studying a paper by Brodskii and Milman [30], Kirk [120] was able to discover the above stated result. Let us mention that the normal structure property is an old concept, not directly related to nonexpansive mappings (for instance it is known that Hilbert space and uniformly convex Banach spaces have the normal structure property). In fact, the proofs given independently by Browder [32] and G¨ohde [85] of their results do not use this property at all, but refer directly to the concept of the uniform convexity.

2.5.1 Classical Existence Results Let K be a bounded closed convex subset of a Banach space (X, .) and T : K → K be a nonexpansive mapping. Let ε ∈ (0, 1) and x0 ∈ K. Define Tε by Tε (x) = ε x0 + (1 − ε )T (x),

for any x ∈ K.

It is easy to check that Tε (K) ⊂ K (since K is convex) and it is a contraction. The Banach Contraction Principal implies the existence of a unique point xε ∈ K such that xε = ε x0 + (1 − ε )T (xε ). which implies xε − T (xε ) = ε x0 − T (xε ) ≤ ε diam(K) = ε sup{x − y : x, y ∈ K}. Therefore we have inf x − T (x) = 0.

x∈K

This property is known as the approximate fixed point property . Therefore, any nonexpansive mapping defined on a bounded closed convex subset of Banach space has this property. By taking a sequence {εn } which goes to 0, we generate a sequence of points {xn } from K such that lim xn − T (xn ) = 0.

n→∞

Such a sequence is called an approximate fixed point sequence (a.f.p.s.). In order to grasp the ideas behind the proofs of the existence of fixed point for nonexpansive mappings in the Hilbert and uniformly convex spaces, we need the concept of asymptotic center of a sequence discovered by Edelstein [71]. This concept is very useful whenever one deals with sequential approximations in Banach spaces. Let {xn } be a bounded sequence in a Banach space X, and let C be a closed convex

2.5 Metric Fixed Point Theory in Banach Spaces

17

subset of X. Consider the functional τ : C → [0, ∞) defined by

τ (x) = lim sup xn − x. n→∞

Usually we use the notation τ (x) = r(x, {xn }). The infimum of τ (x) over C is called the asymptotic radius of {xn } and denoted by r(C, {xn }), i.e.,

 r(C, {xn }) = inf r(x, {xn }) = inf lim sup xn − x . x∈C

The set

x∈C

n→∞

 A(C, {xn }) = x ∈ C : r(x, {xn }) = r(C, {xn })

is the set of all asymptotic centers of {xn }. If {xn } converges to x ∈ C, then A(C, {xn }) = {x}. In general, the set A(C, {xn }) is not reduced to one point. Theorem 2.9. Every bounded sequence in a uniformly convex Banach space X has a unique asymptotic center with respect to any closed convex subset of X. Proof. Let X be a uniformly convex Banach space. Let {xn } be a bounded sequence in X, and let C be a closed convex subset of X. Set R = r(C, {xn }). Without loss of generality, we may assume R > 0. For any ε > 0, set Cε = {x ∈ C : r(x, {xn }) ≤ R + ε }. Since the function x → r(x, {xn }) is continuous and convex, we conclude that Cε is a nonempty closed convex subset of C. For any x, y ∈ Cε , we have x − y ≤ 2(R + ε ). Hence Cε is bounded. Since X is uniformly convex, it is reflexive. Hence Cε is weakly compact. Set Cn = C1/n , for n ≥ 1. The sequence {Cn } is decreasing, which  Cn is a nonempty closed bounded convex implies by weak compactness that n≥1

subset of C. It is easy to check that 

A(C, {xn }) =

Cn .

n≥1

Let us finish the proof of Theorem 2.9 by showing that A(C, {xn }) has at most one point. Let x, y ∈ A(C, {xn }). Then we have lim sup x − xn  = lim sup y − xn  = R. n→+∞

n→+∞

Let η > 0. Then there exists n0 ≥ 1 such that sup x − xn  ≤ R + η , and sup y − yn  ≤ R + η .

n≥n0

n≥n0

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2 Fixed Point Theory in Metric Spaces: An Introduction

Assume that x = y. Set ε = x − y > 0. Since X is uniformly convex, then we have  



x+y  ε    2 − xn  ≤ (R + η ) 1 − δX R + η for any n ≥ n0 , where δX is the modulus of uniform convexity of X. Since A(C, {xn }) is convex, we get  

 

x+y  x+y ε   , {xn } = lim sup  − xn  ≤ (R + η ) 1 − δX , R=r 2 2 R+η n→+∞ which implies

δX

ε R+η

 ≤

η , R+η

for any η > 0. Fix η0 > 0. Since δX is nondecreasing, we get



 ε ε η 0 < δX ≤ δX ≤ , R + η0 R+η R+η for any η < η0 . This is clearly a contradiction which forces x = y.



In the Hilbert case, we can prove even more. Theorem 2.10. In a Hilbert space H, the weak limit of a weakly convergent sequence coincides with its asymptotic center with respect to H. Proof. Let H be a Hilbert space. Denote by <, > the scalar product of H. Let {xn } be a sequence which converges weakly to x in H. For any y ∈ H, we have y − xn 2 = y − x2 + 2 < y − x, x − xn > +x − xn 2 , for any n ≥ 1. Since {xn } converges weakly to x, we get lim < y − x, x − xn >= 0. n→+∞

Hence lim sup y − xn 2 = y − x2 + lim sup x − xn 2 . n→+∞

(HE)

n→+∞

Therefore we have

 lim sup x − xn  = inf lim sup y − xn  : y ∈ H , n→+∞

n→+∞

i.e., x ∈ A(H, {xn }). Since H is uniformly convex, then we must have A(H, {xn }) = {x}. 

Remark 2.2. In any Hilbert space H, the asymptotic center of a bounded sequence {xn } with respect to H belongs to the closed convex hull of {xn }. Indeed, let {xn } be

2.5 Metric Fixed Point Theory in Banach Spaces

19

a bounded sequence in a Hilbert space H. Let x be the asymptotic center of {xn } with respect to H. Let C = conv({xn }) be the closed convex hull of {xn }. Let P : H → C be the nearest point projection. Using the nonexpansive behavior of P in any Hilbert space, we get xn − P(x) ≤ xn − x, for any n ≥ 1. Hence lim sup xn − P(x) ≤ lim sup xn − x = r(H, {xn }). n→+∞

n→+∞

Since x is the asymptotic center of {xn }, we must have x = P(x). Note that this property does not necessarily hold in (even uniformly convex) Banach spaces.

Theorem 2.11. Let C be a bounded closed convex subset of a uniformly convex Banach space X. Let T : C → C be a nonexpansive mapping. (a) Let {xn } be an a.f.p.s. of T in C. Let z ∈ C be its asymptotic center with respect to C, then it is quite easy to check that T (z) is also an asymptotic center of {xn } with respect to C. By the uniqueness of the asymptotic center, we get T (z) = z. (b) Let x ∈ X and consider the orbit of x under T , i.e., {T n (x)}. Let z ∈ C be its asymptotic center with respect to C, then T (z) is also an asymptotic center of {T n (x)} with respect to C. By the uniqueness of the asymptotic center, we get T (z) = z.

2.5.2 The Normal Structure Property The reason behind separating the Normal Structure Property from the other results is the important role it played during the first 20 years (since 1965) of the theory. In order to appreciate this property, let us give more information on nonexpansive mappings in Banach spaces. Indeed, let C be a weakly compact convex subset of a Banach space X. Let T : C → C be nonexpansive. The first pioneers of the metric fixed point theory were mostly concerned about the existence of fixed points. So assume that T fails to have a fixed point in C. Since C is weakly compact, there exists (by the Zorn Lemma) a minimal nonempty closed convex subset K of C invariant under T , i.e., T (K) ⊂ K. The central research of the metric fixed point theory in Banach spaces always orbited around the discovery of new properties of such minimal convex sets. The first properties are due to Kirk. Theorem 2.12. [120] Under the above assumptions and notations, we have (a) conv(T (K)) = K. (b) sup z − x = diam(K), for any x ∈ K. z∈K

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2 Fixed Point Theory in Metric Spaces: An Introduction

Proof. Let C be a weakly compact convex subset of a Banach space X. Let T : C → C be nonexpansive. Let K be a minimal T -invariant closed convex subset of C. Set K0 = conv(T (K)). Since K is T -invariant and convex, then we have K0 ⊂ K ⊂ C. Hence T (K0 ) ⊂ T (K) ⊂ K0 , i.e., K0 is T -invariant. By minimality, we get K = K0 = conv(T (K)). Next we prove (b). Let x ∈ K. Set r = sup z − x. Set z∈K

K0 = {y ∈ K : sup z − y ≤ r}. z∈K

Then we have K0 = K ∩



 B(z, r) ,

z∈K

where B(z, r) is the closed ball in X centered at z with radius r. Since x ∈ K0 , we deduce that K0 is a nonempty closed convex subset of K. Let us show that K0 is T -invariant. Let y ∈ K0 . Then K ⊂ B(y, r). Since T is nonexpansive, we get T (K) ⊂ B(T (y), r). Since balls are convex, we get conv(T (K)) ⊂ B(T (y), r). The conclusion of (a) implies that K ⊂ B(T (y), r). Therefore, we must have T (y) ∈ K0 , i.e., K0 is T -invariant. By minimality of K, we conclude that K = K0 . Hence sup y − z ≤ sup z − x, z∈K

z∈K

for any y ∈ K. Since x was arbitrarily taken in K, we get sup y − z = sup z − x, z∈K

z∈K

for any y ∈ K. Hence   diam(K) = sup y − z = sup sup y − z = sup z − x, y,z ∈K

y∈K

z∈K

z∈K

for any x ∈ K. This completes the proof of Theorem 2.12.



Definition 2.5. A closed convex subset C of a Banach space X is said to have the normal structure property if any bounded convex subset K of C which contains more than one point, contains a nondiametral point , i.e., there exists a point x0 ∈ K such that sup x0 − x < diam(K) . x∈K

We will also say that X has the normal structure property if any bounded closed convex subset has the normal structure property. The normal structure property forbids the conclusion (b) in Theorem 2.12 to hold (see [21]).

2.5 Metric Fixed Point Theory in Banach Spaces

21

Throughout this section we will use the following notations: • diam(C) = sup x − y x,y∈C

• rz (C) = sup z − x x∈C

• R(C) = inf rx (C) x∈C

• C (C) = {z ∈ C : rz (C) = r(C)} The number R(C) and the set C (C) are called the Chebyshev radius and Chebyshev center of C, respectively. In light of the discussion above and Theorem 2.12, we get the following fundamental theorem [120]. Theorem 2.13 (Kirk’s Fixed Point Theorem). Let X be a Banach space and suppose that C is a nonempty weakly compact convex subset of X which has the normal structure property. Then any nonexpansive mapping T : C → C has a fixed point. Remark 2.3. When X is strictly convex, and under the assumptions of Theorem 2.13, the fixed point set of T , F(T ), is convex. Indeed, let x, y ∈ F(T ). Then we have      

      x − T x + y  = T (x) − T x + y  ≤ x − x + y  = x − y .      2 2 2  2 Similarly we have

Hence

 

  y − T x + y  ≤ x − y .   2 2



  

    x − T x + y  = y − T x + y  = x − y .     2 2 2

Since X is strictly convex, we must have 

x+y x+y = , T 2 2 x+y i.e., ∈ F(T ). In fact, the convexity of the fixed point set of nonexpansive 2 mappings characterizes Banach spaces which are strictly convex (see [104]). Since the publication of Kirk’s fixed point theorem in 1965, people intensified their investigation of the normal structure property. For example, it was proved that uniformly convex Banach spaces have this property. Let X be a Banach space. Define the characteristic (or coefficient) of uniform convexity of the Banach space X by

ε0 (X) = sup{ε ∈ [0, 2] : δX (ε ) = 0},

22

where

2 Fixed Point Theory in Metric Spaces: An Introduction

    x+y  : x ≤ 1, y ≤ 1, x − y ≥ ε , x, y ∈ X , δX (ε ) = inf 1 −   2 

is the modulus of uniform convexity of X. Recall that X is uniformly convex if and only if ε0 (X) = 0, i.e., δX (ε ) > 0, for any ε ∈ (0, 2]. And X is said to be uniformly non-square if and only if ε0 (X) < 2. James [99] studied these spaces extensively (see also [76]). Assume that ε0 (X) < 1, and let ε ∈ (ε0 (X), 1). Then for any nonempty bounded closed convex subset C of X not reduced to one point, there exists a point x ∈ C such that r(x,C) ≤ (1 − δ (ε )) diam(C). 

So we have N(X) = sup

r(C) diam(C)

 ≤ 1 − δ (ε0 (X))

where the supremum is taken over all nonempty bounded closed convex subset of X with more than one point. The number N(X), introduced by Bynum [39], is known as the coefficient of uniform normal structure property. So we have the normal structure property and even more, uniform normal structure property, i.e., N(X) < 1. Therefore, uniformly convex Banach spaces have a stronger geometric property. Since its early connection with the fixed point property, the normal structure property had occupied a central role in the study of metric fixed point theory. At one point, it was asked whether the two properties are equivalent. In order to show that it is not the case, one has to dig deeper into the properties of the minimal convex sets associated to nonexpansive mappings. Independently, Goebel [78] and Karlovitz [102] proved the following: Theorem 2.14. [78, 102] Let K be a subset of a Banach space X which is minimal with respect to being nonempty, weakly compact, convex, and T -invariant for some nonexpansive mapping T, and suppose {xn } ⊆ K is an a.f.p.s., that is, lim xn − T (xn ) = 0. Then for each x ∈ K, we have n→∞

lim x − xn  = diam(K).

n→∞

Proof. Let K be a subset of a Banach space X which is minimal with respect to being nonempty, weakly compact, convex, and T -invariant for some nonexpansive mapping T . Let {xn } be an a.f.p.s of T in K. Let U be a nontrivial ultrafilter over N. Since K is bounded, then lim x − xn  = l(x) exists for any x ∈ K. It is obvious U

that the function l : K → [0, +∞) is continuous convex function. Moreover we have lim x − xn  = lim x − T (xn ), U

U

since {xn } is an a.f.p.s. of T . Since T is nonexpansive, we get l(T (x)) ≤ l(x), for any x ∈ K. Fix x ∈ K. Set Kx = {y ∈ K : l(y) ≤ l(x)}. Obviously we have x ∈ Kx . Using

2.5 Metric Fixed Point Theory in Banach Spaces

23

the properties of l and T , we conclude that Kx is a closed nonempty convex subset of K which is T -invariant. Using the minimality of K, we conclude that Kx = K, for any x ∈ K. Clearly this will force the function l to be constant, i.e., there exists r ≥ 0 such that lim x − xn  = r, U

for any x ∈ K. Clearly we have r ≤ diam(K). Assume that r = diam(K). Let ε ∈ (0, diam(K)−r). Since U is not a trivial ultrafilter, then for any y1 , y2 , · · ·, y p ∈ K, there exists n ≥ 1 such that yi − xn  ≤ r + ε , i = 1, 2, · · · , p, for any p ≥ 1. Hence the family {B(y, r + ε )} of closed balls inK has the finite    intersection property. Since K is weakly compact, then K∞ = K B(x, r + ε ) x∈K

is not empty. For any x ∈ K∞ , we have sup x − z ≤ r + ε < diam(K). This is the z∈K

sought contradiction with the property (b) of Theorem 2.12. Hence r = diam(K). Since U was an arbitrary nontrivial ultrafilter over N, we conclude that lim x − xn  n→∞ exists and lim x − xn  = diam(K). n→∞



Recall again that it was this property that allowed Karlovitz to prove that the normal structure property and the fixed point property are not equivalent.

2.5.3 The Demiclosedness Principle A fundamental result in the theory of nonexpansive mappings is Browder-G¨ohde’s demiclosedness principle [32, 85]. Definition 2.6. Let X be a Banach space and K ⊆ X. A mapping f : K → X is demiclosed (at y) if the conditions {x j } converges weakly to x and { f (x j )} converges strongly to y then x ∈ K and f (x) = y. The demiclosedness principle asserts that if K is a closed convex subset of a uniformly convex Banach space X and if T : K → X is nonexpansive, then I − T is demiclosed. This fact is proved in [32] using sharp properties of the modulus of convexity and a clever subsequential thinning process. In this section, we give the proof of this result in the case of Hilbert spaces. Theorem 2.15. Let K be a bounded closed convex subset of a Hilbert space H. Suppose T : K → K is nonexpansive. Then I − T is demiclosed at 0.

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2 Fixed Point Theory in Metric Spaces: An Introduction

Proof. Let {xn } be a sequence in K weakly convergent to x which is an a.f.p.s. for T , i.e., lim xn − T (xn ) = 0. Since K is weakly closed, x ∈ K. Let us prove that n→+∞

x ∈ F(T ), i.e., T (x) = x. Using the equation (HE) in the proof of Theorem 2.10, we get (HE) lim sup y − xn 2 = y − x2 + lim sup x − xn 2 , n→+∞

n→+∞

for any y ∈ H. In particular, we have lim sup T (x) − xn 2 = T (x) − x2 + lim sup x − xn 2 . n→+∞

n→+∞

Since {xn } is an a.f.p.s. for T , then we have lim sup T (x) − xn 2 = lim sup T (x) − T (xn )2 ≤ lim sup x − xn 2 , n→+∞

n→+∞

n→+∞

because T is nonexpansive. Hence T (x) − x2 + lim sup x − xn 2 ≤ lim sup x − xn 2 , n→+∞

n→+∞

which implies T (x) − x2 ≤ 0, i.e., T (x) = x. 

Remark 2.4. The equation (HE) is fundamental and is at the heart of the proof of Theorem 2.15. So it is of no surprise that whenever a similar formula holds, we have a similar conclusion. In fact, this theorem may be adapted to other topologies than the weak topology. Indeed, in [27] it is proved that if { fn } is a sequence of L p uniformly bounded functions on a measure space, and if fn → f almost everywhere, then (AEQ) lim inf || fn − g|| p = lim inf || fn − f || p + || f − g|| p , n→∞

n→∞

for all p ∈ (0, ∞) and g ∈ Let K be a nonempty subset of L p which is closed almost everywhere. Let T : K → K be nonexpansive. Let { fn } be a sequence of L p uniformly bounded functions on a measure space such that fn → f almost everywhere. Assume that { fn } is an a.f.p.s. for T . Then we have f ∈ K. Moreover from the equation (AEQ), we get Lp.

lim inf || fn − T ( f )|| p = lim inf || fn − f || p + || f − T ( f )|| p . n→∞

n→∞

Since { fn } is is an a.f.p.s. for T and T is nonexpansive, we get lim inf || fn − T ( f )|| p = lim inf ||T ( fn ) − T ( f )|| p ≤ lim inf || fn − f || p . n→∞

n→∞

n→∞

As in the proof of Theorem 2.15, we will get T ( f ) = f , i.e., T is demiclosed at 0. The fundamental application of Browder-G¨ohde’s demiclosedness principle in fixed point theory may now be stated.

2.5 Metric Fixed Point Theory in Banach Spaces

25

Theorem 2.16. [32, 85] If K is a bounded closed convex subset of a uniformly convex Banach space X and if T : K → K is nonexpansive, then T has a fixed point. Moreover, the fixed point set of T is a closed convex subset of K. Proof. Since X is strictly convex, then F(T ) is convex. Moreover, since T is continuous, then F(T ) is closed. We are only left to prove that F(T ) is not empty. Since K is bounded and convex, then T has an a.f.p.s. {xn } in K. Since X is reflexive, then K is weakly compact. Hence, a subsequence {xφ (n) } of {xn } exists which is weakly convergent to some x ∈ K. The demiclosedness principle will then imply T (x) = x. 

The equations (HE), (AEQ) and similar equations [110] play a major role in the geometry of Banach spaces in connection with the fixed point theory. This was initially noted by Radon [180] and Riesz [184, 185] who introduced a property known as Radon-Riesz property, also known as Kadec-Klee property, which will be a subject of our discussion in the next section.

2.5.4 Opial and Kadec-Klee Properties The Opial property finds its roots in [173], where Opial strengthened a previous result of Browder and Petryshyn [33] by proving the weak convergence of the iterate of a nonexpansive mapping defined within Hilbert spaces. The Kadec-Klee property, also known as Radon-Riesz [180, 184, 185] property or property (H) [94], deals with the problem of when the weak convergence implies the strong convergence in Banach spaces. In particular, this property is satisfied on the unit sphere of Hilbert spaces. Both properties continue to play a major role in the study of the metric fixed point property. The original definitions of both properties are closely linked to the weak topology. Let X be a Banach space. Let τ be a topological vector space topology on X that is weaker than the norm topology. Throughout this section, we will assume that the topology τ is lower semi continuous with respect to the norm, that is, if {xn } τ -converges to x ∈ X, then ||x|| ≤ lim inf ||xn ||. n→∞

It is easy to see that this will happen if and only if the closed balls are sequentially τ -closed. Definition 2.7. We will say that X satisfies τ -Opial condition if for every bounded {xn } ∈ X that τ -converges to x ∈ X, then lim inf ||xn − x|| < lim inf ||xn − y||, n→∞

n→∞

for every y = x. We will say that X satisfies the uniform τ -Opial condition if for every R > 0, for every ε > 0 there exists η > 0 such that for every {xn } ∈ X which

26

2 Fixed Point Theory in Metric Spaces: An Introduction

τ -converges to x ∈ X and for every y ∈ X, we have lim inf ||xn − x|| + η ≤ lim inf ||xn − y||, n→∞

n→∞

provided lim inf ||xn − x|| ≤ R and ||x − y|| ≥ ε . n→∞

Opial’s property plays an important role in the study of τ -convergence of iterates nonexpansive mappings and of the asymptotic behavior of nonlinear semigroups [148, 173, 177]. Clearly uniform τ -Opial condition implies τ -Opial condition. In his original paper, Opial [173] showed the demiclosedness principle for nonexpansive mappings in Banach spaces which satisfy the Opial condition. Theorem 2.17. Let X be a Banach space which satisfies the Opial condition for the weak topology. Let C be a nonempty closed convex subset of X, and T : C → X be a nonexpansive mapping. Then T is demiclosed, i.e., for any {xn } ⊂ C such that {xn } is weakly convergent to x0 ∈ C, and {xn − T (xn )} is strongly convergent to y0 ∈ X, we have x0 − T (x0 ) = y0 . Proof. Let {xn } ⊂ C be such that {xn } is weakly convergent to x0 ∈ C, and {xn − T (xn )} is strongly convergent to y0 ∈ X. Since T is nonexpansive, we get lim inf xn − x0  ≥ lim inf T (xn ) − T (x0 ) = lim inf xn − y0 − T (x0 ). n→∞

n→∞

n→∞

Since X satisfies the Opial condition, we get x0 = y0 + T (x0 ), or x0 − T (x0 ) = y0 .

 Note that this proof can be easily adapted to any topology τ as described above. As a direct application of this result, we get the Opial property version of Theorem 2.16, which we mentioned in the previous sections. Theorem 2.18. [32, 85] Let X be a Banach space which satisfies the Opial condition for the weak topology. Let C be a nonempty weakly compact convex subset of X, and T : C → C be a nonexpansive mapping. Then T has a fixed point. Another important application of Theorem 2.17 is the answer to a long-standing question whether there exists a larger class of mappings for which the orbits converge, for some topology, to a fixed point. Recall that a map T is called asymptotically regular whenever we have lim T n+1 (x) − T n (x) = 0. Opial [173] proved the n→∞ following result. Theorem 2.19. Let X be a uniformly convex Banach space which satisfies the Opial condition for the weak topology. Let C be a nonempty closed convex subset of X, and T : C → C be a nonexpansive asymptotically regular mapping with a nonempty fixed point set. Then for any x ∈ C, the sequence {T n (x)} is weakly convergent to a fixed point of T .

2.5 Metric Fixed Point Theory in Banach Spaces

27

Proof. Let x ∈ C. Set

τ (z) = lim sup T n (x) − z, z ∈ C. n→+∞

Since T has a nonempty fixed points set, then {T n (x)} is bounded. Indeed, let us fix y ∈ F(T ), then by the nonexpansiveness of T , T n (x) − y ≤ x − y, which means that {T n (x)} ⊂ B(y, x − y). Since X is uniformly convex, then it is reflexive which implies that C is weakly compact. Hence {T n (x)} has a subsequence which converges weakly. Let z0 ∈ C be a weak cluster point of {T n (x)}, i.e., there exists {T ψ (n) (x)} a subsequence which converges weakly to z0 . We claim that z0 ∈ F(T ). Indeed, since T is asymptotically regular, we have lim T ψ (n) (x)−T ψ (n)+1 (x)=0. n→+∞

Using Theorem 2.17, we get T (z0 ) = z0 . We will show that z0 is an asymptotic center of {T n (x)}. Indeed since z0 ∈ F(T ), we have

τ (z0 ) = lim sup T n (x) − z0  = lim T n (x) − z0  = lim inf T ψ (n) (x) − z0 . n→+∞

n→+∞

n→+∞

Using the Opial property, we get

τ (z0 ) ≤ lim inf T ψ (n) (x) − z n→+∞

≤ lim sup T ψ (n) (x) − z n→+∞

≤ lim sup T n (x) − z = τ (z), n→+∞

for any z ∈ C. Hence τ (z0 ) = min τ (z), i.e., z0 is an asymptotic center of {T n (x)}. z∈C

Using Theorem 2.9, z0 is the unique asymptotic center of {T n (x)}. Therefore, {T n (x)} has one weak-cluster point, i.e., {T n (x)} is weakly convergent. Note that its weak limit is its asymptotic center which is a fixed point of T . 

The first nontrivial example of asymptotically regular mappings was given by Ishikawa [98]. Indeed, Ishikawa noted that if D is bounded and convex, and if T : D → D is nonexpansive, then the mapping f = (I + T )/2 has the same fixed point set as T and is asymptotically regular. Before we give proof of such a statement, we need the following technical lemma. Lemma 2.2. [111] Suppose {xn } and {yn } are sequences in a Banach space X which satisfy for all n ∈ N: 1 (i) xn+1 = (xn + yn ), 2 (ii) yn+1 − yn  ≤ xn+1 − xn .

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2 Fixed Point Theory in Metric Spaces: An Introduction

Then for all i, n ∈ N,   n yi − xi  ≤ 2n yi − xi  − yi+n − xi+n  + yi+n − xi  . 1+ 2 Proof. The proof is by induction on n. Observe that the lemma is trivially true for all i if n = 0, and make the inductive assumption that the theorem holds for a given n ∈ N and all i. This gives (upon replacing i with i + 1 in the inequality)   n yi+1 − xi+1  ≤ 2n yi+1 − xi+1 −yi+n+1 − xi+n+1  +yi+n+1 − xi+1  . 1+ 2 Next observe that by (i), the triangle inequality, and (ii) respectively:  yi+n+1 − xi+1  ≤ (1/2) yi+n+1 − xi  + yi+n+1 − yi   n ≤ (1/2) yi+n+1 − xi  + ∑ yi+k+1 − yi+k  k=0 n

≤ (1/2) yi+n+1 − xi  + ∑ xi+k+1 − xi+k  . 

k=0

Combined with the previous inequality, this gives   n yi+1 − xi+1  ≤ 2n yi+1 − xi+1  − yi+n+1 − xi+n+1  1+ 2  n +(1/2) yi+n+1 − xi  + ∑ xi+k+1 − xi+k  . k=0

Thus

  n yi+1 − xi+1  ≤ 2n+1 yi+1 − xi+1  − yi+n+1 − xi+n+1  2 1+ 2  n + yi+n+1 − xi  + ∑ xi+k+1 − xi+k  . k=0

Next we observe that (i) and (ii) imply yn+1 − xn+1  ≤ yn+1 − yn  + yn − xn+1  ≤ xn+1 − xn  + yn − xn+1  = yn − xn  . This means that the sequence {yn − xn } is monotone nonincreasing. Thus n

1

n

∑ xi+k+1 − xi+k  = 2 ∑ yi+k − xi+k  ≤

k=0

k=0

(n + 1) yi − xi  , 2

2.5 Metric Fixed Point Theory in Banach Spaces

29

and we have   n yi+1 − xi+1  ≤ 2n+1 yi+1 − xi+1  − yi+n+1 − xi+n+1  2 1+ 2  + yi+n+1 − xi  + ((n + 1)/2) yi − xi   = 2n+1 yi − xi  − yi+n+1 − xi+n+1  +2n+1 yi+1 − xi+1  +[((n + 1)/2) − 2n+1 ] yi − xi  + yi+n+1 − xi   n − 2n+1 ≤ 0, Since 2 1 + 2    n − 2n+1 yi+1 − xi+1  + 2n+1 − ((n + 1)/2) yi − xi  2 1+ 2   n − 2n+1 yi − xi  + 2n+1 − ((n + 1)/2) yi − xi  ≥ 2 1+ 2  = 1 + (n + 1)/2 yi − xi  , and we have   1 + (n + 1)/2 yi − xi  ≤ 2n+1 yi+1 − xi+1  − yi+n+1 − xi+n+1  + yi+n+1 − xi  . This completes the proof. 

Using Lemma 2.2, we get Ishikawa’s result on asymptotically regular mappings. Theorem 2.20. Let K be a bounded convex subset of a Banach space and suppose T : K → K is nonexpansive. Then the mapping f : K → K defined by setting f (x) = (x + T (x))/2 is asymptotically regular. Proof. Select x0 ∈ K, let y0 = f (x0 ), and having defined xn take yn = f (xn ) and set xn+1 = (xn + yn )/2, n = 1, 2, · · ·. It is easy to see that the resulting sequences {xn }, {yn } satisfy the assumptions of Lemma 2.2. Thus for each i, n ∈ N,   n yi − xi  ≤ 2n yi − xi  − yi+n − xi+n  1+ 2 + yi+n − xi  . Since {xn − yn } = {xn − f (xn )} is monotone nonincreasing, there exists a number r ≥ 0 such that lim xn+1 − f (xn+1 ) = r. Now let i → ∞ in the above inequality. n→∞ Then  n r ≤ diam(K). 1+ 2 Clearly this implies r = 0. 

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2 Fixed Point Theory in Metric Spaces: An Introduction

Remark 2.5. A slight modification of the preceding argument yields the fact that each of the mappings fα = (1 − α )I + α T are asymptotically regular for each α ∈ (0, 1). Moreover, it can be shown that the convergence in Theorem 2.20 is uniform in the sense that the rate of convergence does not depend on the initial point x0 , nor indeed on the mapping T. Details may be found in [81]. Let us introduce another important property which played a major role in the study of metric fixed point theory [68, 69, 94]. Definition 2.8. We will say that X satisfies τ -Kadec-Klee property if for some ε > 0 there exists η > 0 such that for every {xn } in the unit ball of X which is τ -convergent to x, we have ||x|| ≤ 1 − η , provided that sep{xn } = inf{||xn − xm || : n = m} > ε . We will say that X satisfies τ - uniform Kadec-Klee property if the above holds for every ε . The most popular topology that one thinks of in Banach spaces, which satisfies most of the above desired properties, is the weak topology. In fact, the ideas described in this section should be extended to a larger class of topologies. Example 2.1. We have seen in Remark 2.4 that if { fn } is a sequence of L p -uniformly bounded functions on a measure space, and if fn → f almost everywhere, then lim inf || fn || p = lim inf || fn − f || p + || f || p , n→∞

n→∞

for all p ∈ (0, ∞). It is easy to check that L p satisfy the uniform Opial condition and the uniform Kadec-Klee property [110] for the topology of convergence almost everywhere. It is not hard to see that this conclusion still holds for convergence in measure. But L p ([0, 1], dx), where dx is the Lebesgue measure, for p ∈ [1, ∞), p = 2, fails to satisfy the Opial condition for the weak topology [173]. The connection between the metric fixed point property, Opial, and Kadec-Klee properties is given in the following theorem. Theorem 2.21. Let X be a Banach space and τ be a topology as described above. (i) Assume that X satisfies the τ -Opial condition. Let D be a τ -sequentially compact bounded convex subset of X. Then D has the fixed point property for nonexpansive mappings, i.e., any T : D → D nonexpansive has a fixed point. (ii) Assume that X satisfies the τ -uniform Kadec-Klee property. Then X has τ normal structure property.

2.6 Ishikawa and Mann Iterations in Banach spaces

31

2.6 Ishikawa and Mann Iterations in Banach spaces Another aspect in the study of metric fixed point theory is the theory of approximation of fixed point, when it exists. This theory is very important for the applications. Very often, the solution to a problem may be translated to be a fixed point. Its existence may be insured by, for example, physical inspections. Therefore, for such problems, the question of existence of the fixed point is not an issue. Since the explicit solution may not be available, one has to consider a method of approximating such a solution. As Banach Contraction Principle suggests, one may think of the iterates of the given map. Unfortunately, such iterates do not converge in general. Because of this, mathematicians started to think of other type of iterations. Here we focus on two of them that attracted most of the attention. Let X be a Banach space. Let C be a nonempty closed convex subset of X. Let T : C → C be a map. One of the first iteration processes was introduced by Halpern [91] (HI) xn+1 = αn+1 x0 + (1 − αn+1 )T (xn ), n ≥ 0, where the initial guess x0 ∈ C is taken arbitrarily and the sequence {αn } is in the interval [0, 1]. Then another iteration was introduced by Mann [161] which is defined by (MI) xn+1 = αn xn + (1 − αn )T (xn ), n ≥ 0, where the initial guess x0 ∈ C is taken arbitrarily and the sequence {αn } is in the interval [0, 1]. Yet another iteration process is referred to as Ishikawa’s [97] process which is defined recursively by   (II) xn+1 = αn xn + (1 − αn )T βn xn + (1 − βn )T (xn ) , n ≥ 0, where the initial guess x0 ∈ C is taken arbitrarily and the sequences {αn } and {βn } are in the interval [0, 1]. In general, not much is known regarding the convergence of the iterations (MI) and (II) unless the underlying space X has some convenient geometric properties. Theorem 2.22. [91, 200] Let {bn }n≥0 be a sequence in [0, +∞) such that i=n

(i) Bn = ∑ bi → +∞ as n → +∞; i=0

1 i=n (ii) ∑ |bi − bi−1 | → 0 as n → +∞. Bn i=1 Let K be a closed convex subset of a Hilbert space H and T : K → K be nonexpansive such that F(T ) is not empty. Then for any x ∈ K, the sequence {xn } generated by

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2 Fixed Point Theory in Metric Spaces: An Introduction

the iteration (HI), with αn = nearest point P(x) ∈ F(T ).

bn and x0 = x, is convergent in the norm to the unique Bn

Proof. Without loss of generality, we may assume that 0 ∈ F(T ). We know that in a strictly convex Banach space, the set of fixed points of a nonexpansive mapping is convex. Hence F(T ) is a closed convex subset of H. In this case, the nearest point projection map P : H → F(T ) exists. In particular, we have x − P(x) ≤ x − y, y ∈ F(T ). Recall that P(x) is characterized by the inequality y − P(x), x − P(x) ≤ 0, y ∈ F(T ).

(NPP)

where ·, · is the scalar product of H. Fix x ∈ H. Since 0 ∈ F(T ), then we have T (z) ≤ z, for any z ∈ H. Using this inequality, we see that xn  ≤ x, for any n ≥ 0, where {xn } is generated by the iteration (HI), with x0 = x. By induction, we prove that   i=n x i=n bi bi−1 (HW) xn − xn−1  ≤ 2 ∑ |bi − bi−1 | + ∑ Bi−1 , n ≥ 0 Bn i=0 i=1 where b−1 = 0 and x−1 = 0. Indeed, it is clear that the inequality (HW) holds for n = 0. Assume it holds for n ≥ 0. Then we have xn+1 − xn  ≤ |αn+1 − αn | x + |αn+1 − αn | T (xn−1 ) + (1 − αn+1 ) T (xn ) − T (xn−1 ) ≤ |αn+1 − αn | x + |αn+1 − αn | xn−1  + (1 − αn+1 ) xn − xn−1  ≤ 2|αn+1 − αn | x + (1 − αn+1 ) xn − xn−1  

|bn+1 − bn | bn bn+1 x + ≤2 Bn+1 Bn Bn+1   i=n bi bi−1 Bn 2x i=n + ∑ |bi − bi−1 | + ∑ Bi−1 Bn+1 Bn i=0 i=1   i=n+1 2x i=n+1 bi bi−1 = ∑ |bi − bi−1 | + ∑ Bi−1 . Bn+1 i=0 i=1

Hence the inequality (HW) holds for n + 1. Therefore by induction it holds for any n ≥ 0. From our assumptions on {bn }, we deduce that αn → 0 when n → +∞, and lim xn − xn−1  = 0.

n→∞

2.6 Ishikawa and Mann Iterations in Banach spaces

33

Since xn − T (xn ) ≤ xn − (1 − αn ) T (xn−1 ) + (1 − αn )T (xn−1 ) − T (xn ) + αn T (xn ) ≤ αn x + xn − xn−1  + αn x,

we obtain lim xn − T (xn ) = 0,

n→∞

i.e., {xn } is an a.f.p.s. for T . Next we prove that lim sup T (xn ) − P(x), x − P(x) ≤ 0. n→∞

Indeed assume not. Then there exist D > 0 and a subsequence {xψ (n) } of {xn } such that   T (xψ (n) ) − P(x), x − P(x) ≥ D, n ≥ 1. We may assume that {xψ (n) } weakly converges to y ∈ K. Since {xn } is an a.f.p.s., then T is demiclosed, i.e., y ∈ F(T ) and {T (xψ (n) )} also weakly converges to y. So we must have y − P(x), x − P(x) ≥ D > 0. This is a contradiction with the inequality (NPP). So we have lim sup T (xn ) − P(x), x − P(x) ≤ 0. n→∞

∞

Moreover using the Abel-Dini theorem [129], we conclude that

∑ αn = +∞ which

n=0

implies i=n

∏ (1 − αi ) = 0, n→+∞ lim

(E1)

i=m

for any m ∈ N. Fix ε > 0. Then there exists n0 ≥ 0 such that T (xn ) − P(x), x − P(x) ≤ ε , and αn x − P(x)2 ≤ ε , for any n ≥ n0 . Then for any n ≥ n0 , we have 2 x − P(x)2 xn+1 − P(x)2 = αn+1 2αn+1 (1 − αn+1 ) < T (xn ) − P(x), x − P(x) >

+ (1 − αn+1 )2 T (xn ) − P(x)2 2 ≤ αn+1 x − P(x)2 + 2αn+1 ε + (1 − αn+1 ) T (xn ) − P(x)2

≤ αn+1 3ε + (1 − αn+1 ) T (xn ) − P(x)2 ≤ 3αn+1 ε + (1 − αn+1 ) xn − P(x)2 .

34

2 Fixed Point Theory in Metric Spaces: An Introduction

Hence xm − P(x)2 ≤ 3ε + xn0 − P(x)2

m

∏

(1 − αi ), m > n0 .

i=n0 +1

Using the limit (E1), we get lim sup xn − P(x)2 ≤ 3ε . n→∞

Since ε was taken arbitrarily positive, we conclude that lim xn − P(x) = 0. n→∞ 

Remark 2.6. The proof of Theorem 2.22 shows how difficult such conclusions can be. Moreover, one has to keep in mind that moving from Hilbert spaces to Banach spaces adds more to the difficulty. Halpern’s iteration (HI) finds its roots in the approximation used to find approximate fixed point sequences for a nonexpansive mapping. It is worth mentioning that the Ishikawa process is more general than the Mann process. But research has been concentrated on the latter due probably to the reasons that the formulation of the Mann process is simpler, and that a convergence theorem for the Mann process may possibly lead to a convergence theorem for the Ishikawa process, provided the sequence {βn } satisfies certain appropriate conditions. However, the introduction of the Ishikawa process has value on its own merit. As a matter of fact, the Mann process may fail to converge while the Ishikawa process can still converge for some mappings in a Hilbert space, [45]. Reich [182] shows that if the underlying space X is uniformly convex and has a Fr´echet differentiable norm, T is nonexpansive, and if the sequence {αn } is such that

∞

∑ αn (1 − αn ) = +∞, then the

n=0

sequence generated by the Mann processes converges weakly to a fixed point of T . For a good reference to the theory of successive iterations, one may consult the book by Berinde [24]. Recently, Kozlowski proved the weak convergence of generalized Mann and Ishikawa processes for asymptotic pointwise nonexpansive mappings acting in Banach spaces with Opial property [133] and in uniformly smooth Banach spaces [137].

2.7 Metric Convexity and Convexity Structures If (M, d) is a metric space and if p, q ∈ M then a point r ∈ M is said to be metrically between p and q if p = r = q and d(p, q) = d(p, r) + d(r, q). A metric space is said to be metrically convex if, given any two points p, q ∈ M, there exists at least one point r ∈ M such that r is metrically between p and q.

2.7 Metric Convexity and Convexity Structures

35

The concept of metric convexity is fundamental in the study of abstract metric spaces. However, in its most general form, it fails to satisfy one of the fundamental properties of convexity in the algebraic sense (in a linear space); namely, if A and B are two (metrically) convex subsets of a metric space M, then it need not be the case that A ∩ B is convex. To see this, it is sufficient to consider the ordinary unit circle S in R2 , where the distance between two points is taken to be the length of the shortest arc joining them. The upper half circle and lower half circle of S are each metrically convex but the intersection of these two sets is a set consisting of just two antipodal points. A subset S of a metric space M is called a metric segment joining p, q ∈ S if there exists a closed interval [a, b] in the real line R and an isometry ϕ which maps [a, b] onto S with ϕ (a) = p and ϕ (b) = q. If a metric space M has the property that each two of its points are the endpoints of a metric segment, then clearly M is metrically convex, and it is natural to ask when the converse is true. The answer is surprisingly often. This is a fundamental result in the theory due to Karl Menger [162], one of the pioneers in the study of abstract metric spaces. Theorem 2.23. If M is a complete and metrically convex metric space, then each two points of M are joined by at least one metric segment of M. We shall be concerned almost entirely with complete metric spaces; hence with the stronger notion of convexity assured by Menger’s Theorem whenever the concept arises. For reasons that will be clear later, the usefulness of metric convexity in a fixed point context is hindered by the fact that the family of all convex subsets of a metrically convex metric space need not be stable under intersections. A quick way to remedy this defect is to turn to a more abstract notion of convexity. Definition 2.9. A family C of subsets of a set X is called an (abstract) convexity structure if (1) Both 0/ and X are in C . (2) C is stableunder intersections; that is, if {Dα }α ∈I is any nonempty subfamily Dα ∈ C . of C then α ∈I

(3) C is stable for nested unions; that is, if {Dα }α ∈I is any nonempty subfamily of  Dα ∈ C . C which is totally ordered by set inclusion, then α ∈I

In the study of metric fixed point theory, closed balls turn out to play a fundamental role. A subset A of a bounded metric space M will be said to be admissible of A, and can be written as the intersection of a family of closed balls centered at points of M. The family A (M) of all admissible subsets of M enters into the study of metric fixed point theory in a very natural way, and this is therefore the obvious candidate for the needed underlying convexity structure. However, while this family satisfies assumptions (1) and (2), in general property (3) does not hold. (This fact cannot be rectified even if the collection is enlarged to include all ball intersections rather than just intersections of closed balls). Nonetheless, the family A (M) plays a major

36

2 Fixed Point Theory in Metric Spaces: An Introduction

role in the metric fixed point theory. Thus we shall exchange property (3) with the assumption that the convexity structure under consideration contains all the closed balls of M. The structure A (M) has one very distinctive advantage when compared with metric convexity. Any subset A of a metric space is contained in the set cov(A) :=



{B : B is a closed ball in M and B ⊇ A}.

Moreover cov(A) ∈ A (M). This provides a convenient analog to the concept of a “closed convex hull” in functional analysis, with the family A (M) replacing the family of all closed (algebraically) convex sets. This analogy does not quite work with metric convexity. It is natural to define the convex hull of a subset of A of a convex metric space M to be a set which is closed, convex, contains A, and for which none of its proper subsets has those properties. With this definition, the existence of a unique convex hull for each subset A of a complete convex metric space which has unique metric segments is immediate. Merely take the “convex hull” of A to be the intersection of all closed convex sets which contain A. However, problems arise in the general case since, as we have observed, the intersection of two convex sets need not be a convex set. On the other hand, if M is any compact metric space and if A ⊆ M, then A does have a convex hull, although the convex hull of A need not be unique. This is a classical result proved by Karl Menger in 1931. It rests on the following lemma. Lemma 2.3. Let C1 ⊇ C2 ⊇ ··· be a descending sequence of nonempty closed convex subsets of a compact metric space (M, d). Then

∞ 

Cn is nonempty and convex.

n=1

Proof. The fact that the intersection is nonempty is immediate from compactness. Suppose x, y ∈

∞ 

n=1

Cn with x = y. Then in each of the sets Cn there exists a point zn

such that

1 d(x, zn ) = d(y, zn ) = d(x, y), 2 (where we used the fact that x and y are actually joined by a metric segment which lies in Cn .) By compactness of M the sequence {zn } has a subsequence {znk } which

converges to a point z ∈ M and since each of the sets Cn is closed, z ∈

∞ 

Cn . Since

n=1

the metric d is continuous, 1 d(x, z) = d(y, z) = d(x, y). 2 This proves that 

∞ 

Cn is convex.

n=1

Theorem 2.24. If A is a nonempty subset of a compact and convex metric space M, then A has a convex hull.

2.8 Uniformly Convex Metric Spaces

37

Proof. Let Σ denote the collection of all closed convex subsets of M which contain A. Observe that Σ = 0/ since M ∈ Σ . Since M is compact, for each n ∈ N there is a finite collection Un := {Un(1),Un(2) , · · ·Un(kn ) } of open balls of radius 1/n which covers M. For each K ∈ Σ there is a well defined integer μn (K) (μn (K) ≤ k(n)) such that K intersects μn (K) members of Un . Choose C1 ∈ Σ so that

μ1 (C1 ) = inf{μ1 (C) : C ∈ Σ }. Having defined Cn choose Cn+1 ⊆ Cn so that

μn+1 (Cn+1 ) = inf{μn+1 (C) : C ∈ Σ and C ⊆ Cn }. Then clearly C1 ⊇ C2 ⊇ · · · and A ⊆ C :=

∞ 

Cn . Also, by Lemma 2.3, C is convex.

n=1

Suppose some H ∈ Σ is a proper subset of C. Then there exists a point c ∈ C\H and in turn there exists an integer n for which the open ball U(c; 1/n) ∩ C ⊂ C\H. In particular, some member U of U2n lies in U(c; 1/n); hence

μ2n (H) < μ2n (C) ≤ μ2n (C2n ), contradicting the definition of μ2n (C2n ).

 It is possible to give a much quicker proof of the above theorem using Zorn’s Lemma. (The proof given here is Menger’s original. It predates the discovery of Zorn’s Lemma.) In fact, the following is true. Theorem 2.25. Let M be a convex metric space, and suppose the intersection every descending chain of closed convex subsets of M is itself convex. Then every nonempty subset A of M has a convex hull. Proof. As above, let Σ denote the family of all closed convex subsets of M which contain A, and order Σ by set inclusion . Σ = 0/ since M ∈ Σ , and by assumption every descending chain in (Σ , ⊇) is bounded below by its intersection. By Zorn’s lemma Σ has a minimal element, completing the proof. 

2.8 Uniformly Convex Metric Spaces Let (M, d) be a metric space. Suppose that there exists a family F of metric segments such that any two points x, y in M are endpoints of a unique metric segment [x, y] ∈ F ([x, y] is an isometric image of the real line interval [0, d(x, y)]). In this case, there exists a mapping W : M × M × [0, 1] → M such that for each (x, y, λ ) ∈ M × M × [0, 1] and z ∈ M, we have d (z,W (x, y, λ )) ≤ λ d(z, x) + (1 − λ )d(z, y).

38

2 Fixed Point Theory in Metric Spaces: An Introduction

Throughout let us write W (x, y, λ ) = λ x ⊕ (1 − λ )y, whenever the choice of the mapping W is irrelevant. Moreover, if we have 

1 1 1 1 1 p ⊕ x, p ⊕ y ≤ d(x, y), d 2 2 2 2 2 for all p, x, y in M, then M is said to be a hyperbolic metric space (see [82]). We will say that a subset C of a hyperbolic metric space M is convex if [x, y] ⊂ C whenever x, y are in C. It is not hard to check that balls in hyperbolic metric spaces are convex. Definition 2.10. Let (M, d) be a hyperbolic metric space. We say that M is uniformly convex (in short, UC) if for any a ∈ M, for every r > 0, and for each ε > 0 

1  1 1 δ (r, ε ) = inf 1 − d x ⊕ y, a : d(x, a) ≤ r, d(y, a) ≤ r, d(x, y) ≥ rε > 0. r 2 2 The definition of uniform convexity finds its origin in Banach spaces [50]. To the best of our knowledge, the first attempt to generalize this concept to metric spaces was done in [84]. The reader may also consult [82]. We have the following properties [113]: (a) Let us observe that δ (r, 0) = 0, and δ (r, ε ) is an increasing function of ε for every fixed r. Also, for r1 ≤ r2 there holds  r1  r2  1 − δ r2 , ε ≤ δ (r1 , ε ). 1− r1 r2 (b) If (M, d) is uniformly convex, then (M, d) is strictly convex, i.e., whenever d

1  x ⊕ y, a = d(x, a) = d(y, a) 2 2

1

for any x, y, a ∈ M, then we must have x = y. Obviously, normed linear spaces are hyperbolic spaces. As nonlinear examples, one can consider the Hadamard manifolds [38], the Hilbert open unit ball equipped with the hyperbolic metric [82], and the CAT(0) spaces [122, 120, 126, 156] (see example 2.3). Example 2.2. [82] Let B be the open unit ball of the infinite Hilbert space H. On B, we consider the Poincar´e hyperbolic metric ρ :

ρ (x, y) = inf γ

 1 0

α (γ (t), γ  (t))dt

where the infimum is taken over all piecewise differentiable curves such that γ (0) = x and γ (1) = y, and where

α (x, v) = sup D f (x)(v) f ∈F

2.8 Uniformly Convex Metric Spaces

39

with F = { f : B → B : holomorphic}. Note that (B, ρ ) is a complete metric space (unbounded). Using the Mobius transformations, one can prove that  1/2 (1 − |x|2 )(1 − |y|2 ) ρ (x, y) = Argth 1 − σ (x, y) with σ (x, y) = . |1− < x, y > |2 Using several complicated computations, one may prove that for any x, y in B, and λ ∈ [0, 1], there exists a unique z ∈ B such that

ρ (z, w) ≤ λ ρ (x, w) + (1 − λ )ρ (y, w) for any w ∈ B. In other words, we have z = W (x, y, λ ) = λ x ⊕ (1 − λ )y. In fact, the metric space (B, ρ ) enjoys some geometric properties similar to uniformly convex Banach spaces. Moreover, we have ⎧ ⎪ ⎨ ρ (a, x) ≤ r  1 1  ρ (a, y) ≤ r =⇒ ρ a, x ⊕ y ≤ r(1 − δ (r, ε )) , ⎪ 2 2 ⎩ ρ (x, y) ≥ rε for any a, x, y in B and any positive r and ε , where

 sinh(r(1 + ε /2)) sinh(r(1 − ε /2)) 1/2 1 δ (r, ε ) = 1 − Argth . r cosh(r) It is easy to check that for r > 0 and ε > 0, we have δ (r, ε ) > 0, i.e., the open unit ball (B, ρ ) is uniformly convex. Note that holomorphic mappings are nonexpansive mappings for the Poincar´e distance ρ . Example 2.3. [28] Let (X, d) be a metric space. A geodesic from x to y in X is a mapping c from a closed interval [0, l] ⊂ R to X such that c (0) = x, c (l) = y, and d (c (t) , c (t  )) = |t − t  | for all t,t  ∈ [0, l]. In particular, c is an isometry and d (x, y) = l. The image α of c is called a geodesic (or metric) segment joining x and y. The space (X, d) is said to be a geodesic space if every two points of X are joined by a geodesic and X is said to be uniquely geodesic if there is exactly one geodesic joining x and y for each x, y ∈ X, which will be denoted by [x, y], and called the segment joining x to y. A geodesic triangle Δ (x1 , x2 , x3 ) in a geodesic metric space (X, d) consists of three points x1 , x2 , x3 in X (the vertices of Δ ) and a geodesic segment between each pair of vertices (the edges of Δ ). A comparison triangle for geodesic triangle Δ (x1 , x2 , x3 ) in (X, d) is a triangle Δ (x1 , x2 , x3 ) := Δ (x¯1 , x¯2 , x¯3 ) in R2 such that dR2 (x¯i , x¯ j ) = d (xi , x j ) for i, j ∈ {1, 2, 3}. Such a triangle always exists (see [28]). A geodesic metric space is said to be a CAT (0) space if all geodesic triangles of appropriate size satisfy the following CAT (0) comparison axiom:

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2 Fixed Point Theory in Metric Spaces: An Introduction

Let Δ be a geodesic triangle in X and let Δ ⊂ R2 be a comparison triangle for Δ . Then Δ is said to satisfy the CAT (0) inequality if for all x, y ∈ Δ and all comparison points x, ¯ y¯ ∈ Δ , d (x, y) ≤ d (x, ¯ y) ¯ . Complete CAT (0) spaces are often called Hadamard spaces (see [120]) . If x, y1 , y2 are points of a CAT (0) space and y0 is the midpoint of the segment [y1 , y2 ], i.e., y1 ⊕ y2 , then the CAT (0) inequality implies: y0 = 2 

1 y1 ⊕ y2 1 1 d 2 x, ≤ d 2 (x, y1 ) + d 2 (x, y2 ) − d 2 (y1 , y2 ) . 2 2 2 4 This inequality is the (CN) inequality of Bruhat and Tits [37]. As for the Hilbert space, the (CN) inequality implies that CAT(0) spaces are uniformly convex with  ε2 δ (r, ε ) = 1 − 1 − . 4 Next we give some results of uniformly convex metric spaces which are useful in the study of metric fixed point theory in these metric spaces. We start with the following technical lemma which will help establish a property in uniformly convex metric spaces equivalent to reflexivity in Banach spaces. Lemma 2.4. [113] Assume that (M, d) is uniformly convex. Let {Cn } ⊂ M be a sequence of nonempty, nonincreasing, convex, bounded, and closed sets. Let x ∈ M be such that 0 < d = lim d(x,Cn ) < ∞. n→∞

Let xn ∈ Cn be such that d(x, xn ) → d. Then {xn } is a Cauchy sequence. Recall that a hyperbolic metric space (M, d) is said to have the property (R) [105] if any nonincreasing sequence of nonempty, convex, bounded, and closed sets has a nonempty intersection. Our next result deals with the existence and the uniqueness of the best approximants of convex, closed, and bounded sets in a uniformly convex metric space. This result is of interest by itself as uniform convexity implies the property (R), which reduces to reflexivity in the linear case. Theorem 2.26. Assume that (M, d) is complete and uniformly convex. Let C ⊂ M be nonempty, convex, and closed. Let x ∈ M be such that d(x,C) < ∞. Then there exists a unique best approximant of x in C, i.e., there exists a unique x0 ∈ C such that d(x, x0 ) = d(x,C). The following result gives the analog result to the well-known theorem that states that any uniformly convex Banach space is reflexive. For a reference, the reader may read Theorem 2.1 in [82].

2.8 Uniformly Convex Metric Spaces

41

Theorem 2.27. If (M, d) is complete and uniformly convex, then (M, d) has the property (R). Remark 2.7. Note that any hyperbolic metric space M which satisfies the property (R) is complete. Indeed, let {xn } be a Cauchy sequence in M. Denote

εn = sup{d(xm , xs ) : m, s ≥ n}, n = 1, · · · Our assumption implies that lim εn = 0. In hyperbolic metric spaces, closed balls n→∞

are convex. Therefore the property (R) implies that



B(xn , εn ) = 0. / It is easy to

n≥1

check that this intersection is reduced to one point which is the limit of {xn }. The following technical lemma is useful to establish a metric version of the main results in [201] proved in the setting of Banach spaces. Lemma 2.5. [113] Let (M, d) be a uniformly convex metric space. Assume that there exists R ∈ [0, +∞) such that  1 1  lim sup d(xn , a) ≤ R, lim sup d(yn , a) ≤ R, and lim d a, xn ⊕ yn = R. n→∞ 2 2 n→∞ n→∞ Then lim d(xn , yn ) = 0. n→∞

A metric version of the parallelogram identity goes as follows (see [18, 82, 201]). Theorem 2.28 (Parallelogram Inequality). [113] Let (M, d) be uniformly convex. Fix a ∈ M. For each 0 < r and for each ε > 0 denote

Ψ (r, ε ) = inf

 1 1  1 d 2 (a, x) + d 2 (a, y) − d 2 a, x ⊕ y , 2 2 2 2

1

where the infimum is taken over all x, y ∈ M such that d(a, x) ≤ r, d(a, y) ≤ r, and d(x, y) ≥ rε . Then Ψ (r, ε ) > 0 for any 0 < r and for each ε > 0. Moreover, for a fixed r > 0, we have (a) Ψ (r, 0) = 0 (b) Ψ (r, ε ) is a nondecreasing function of ε ; (c) if lim Ψ (r,tn ) = 0, then lim tn = 0. n→∞

n→∞

The concept of p-uniform convexity was used extensively by Xu [201] (see also [16] p. 310); its nonlinear version for p = 2 is given below [113]. Definition 2.11. We will say that (M, d) is 2-uniformly convex if   Ψ (r, ε ) : r > 0 , ε > 0 > 0. cM = inf r2 ε 2

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2 Fixed Point Theory in Metric Spaces: An Introduction

Note that (M, d) is 2-uniformly convex if and only if   δ (r, ε ) : r > 0 , ε > 0 > 0. inf ε2 Example 2.4. Let M, d) be CAT (0) geodesic metric space. The (CN) inequality implies that r2 ε 2 . Ψ (r, ε ) = 4 1 This clearly implies that any CAT(0) space is 2-uniformly convex with cM = 4 [113]. As an application of the previous technical results, we discuss the existence of fixed points of uniformly Lipschitzian mappings [80]. Definition 2.12. A mapping T : C → C (a subset of M) is said to be uniformly Lipschitzian if there exists a non-negative number k such that d(T n (x), T n (y)) ≤ k d(x, y), for all x and y in C, and n ≥ 1. The smallest such constant k will be denoted by λ (T ). It is well known that if a mapping is uniformly Lipschitzian, then one may find an equivalent distance for which the mapping is nonexpansive (see [82, pages 34–38]). Indeed, let T : C → C be uniformly Lipschitzian. Denote

ρ (x, y) = sup{d(T n x, T n y) : n = 0, 1, 2, . . .} for all x, y ∈ C. Then ρ is an equivalent distance to the distance d and relative to which T is nonexpansive. In this context, it is natural to ask the question; if a set C has the fixed point property (fpp) for nonexpansive mappings with respect to the metric d, then does C also have (fpp) for mappings which are nonexpansive relative to an equivalent metric? This is known as the stability of (fpp). The first result in this direction is due to Goebel and Kirk [80]. Motivated by such questions, many investigated the fixed point property of uniformly Lipschitzian mappings in uniformly convex Banach spaces and hyperbolic metric spaces. Recall that the normal structure coefficient N(M) of the hyperbolic metric space M is defined (see [39]):   diam(C) : C bounded convex subset of M with diam(C) > 0 , N(M) = inf R(C) where diam(C) is the diameter of C, and R(C) is the Chebyshev radius of C. Recall that we have diam(C) = sup{d(x, y) : x, y ∈ C}, and R(C) = inf{sup d(x, y); x ∈ C}. y∈C

2.9 More on Convexity Structures

43

The equation (HE) satisfied in Hilbert spaces has a nonlinear version in uniformly convex hyperbolic metric spaces [113]: Theorem 2.29. Let (M, d) be hyperbolic metric space and let C be a nonempty, closed, and convex subset of M. Assume that M is 2-uniformly convex. Let {xn } be a bounded sequence in C. Then there exists a unique point z ∈ C such that lim sup d 2 (xn , z) + 2cM d 2 (z, x) ≤ lim sup d 2 (xn , x), n→∞

(HEM)

n→∞

for any x ∈ C. The next result is the nonlinear version of Theorem 3 of [201]. Theorem 2.30. [113] Let (M, d) be hyperbolic metric space which is 2-uniformly convex. Let C be a nonempty, closed, convex, and bounded subset of M. Let T : C → C be uniformly Lipschitzian with 

λ (T ) <

1+

1/2  1 + 8cM N(M)2 . 2

Then T has a fixed point in C.

2.9 More on Convexity Structures Kirk’s fixed point theorem involves some kind of compactness and the normal structure property. Both properties are easy to define in metric spaces. The first attempt to extend Kirk’s theorem to metric spaces was done by Takahashi [197]. He considered compact metric spaces in his generalization, which was very restrictive since Kirk’s theorem assumes weak compactness, not strong compactness. Penot [176], on the other hand, defined compactness for convexity structures, which leads to weak compactness in the linear case. Indeed, a convexity structure Σ is said to be compact if property and only if every family of subsets of Σ which has the finite intersection  has a nonempty intersection, i.e., if (Ai )i∈I , with Ai ∈ Σ , then Ai = 0/ provided 

i∈I

Ai = 0/ for any finite subset I f of I. As for the normal structure property, Penot

i∈I f

introduced such a property in convexity structure as follows: Definition 2.13. The convexity structure Σ is said to be normal if and only if for any nonempty and bounded A ∈ Σ not reduced to one point, there exists a ∈ A such that sup{d(a, x) : x ∈ A} < sup{d(x, y) : x, y ∈ A} = diam(A). Penot’s formulation of Kirk’s fixed point theorem becomes

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2 Fixed Point Theory in Metric Spaces: An Introduction

Theorem 2.31. [176] Let (M, d) be a nonempty bounded metric space which possesses a convexity structure which is compact and normal. Then every nonexpansive mapping T : M → M has a fixed point. In the original Kirk’s theorem, the Banach space is supposed to have a normal structure which means that the family of all convex sets is normal. However, this family is a large one, and contains the admissible sets. For example, the Banach space l ∞ is a good example which illustrates the power behind Penot’s formulation. Indeed, l ∞ fails to have the normal structure property but A (l ∞ ) is compact and normal, which implies the following theorem discovered separately by Sine [190] and Soardi [191] in 1979. Theorem 2.32. Let A be a nonempty admissible subset of l ∞ . Then every nonexpansive mapping T : A → A has a fixed point. Remark 2.8. In the original proof of Kirk’s theorem, the weak compactness is used to prove the existence of a minimal invariant set via Zorn’s lemma. Gillespie and Williams [77] showed that a constructive proof may be found which uses only countable compactness. In other words, the convexity structure Σ is assumed to satisfy a countable intersection property, i.e., for any (An )1≤n , with An ∈ Σ , then ∞ 

n=1

An = 0/ provided

m 

An = 0/ for any m ≥ 1. This weakening is very important,

n=1

since in many practical cases we do not have a compactness generated by a topology, but a compactness defined sequentially. The latter usually generates some kind of countable compactness. Note also, that if the convexity structure Σ is uniformly normal, then it is countably compact [12, 105]. This is an amazing metric translation of a well known similar result in Banach spaces due to Maluta [160]. It is natural to ask whether a convexity structure which is countably compact is actually compact. The answer is yes if we are dealing with uniform normal structure [149]. In fact, if A (M) is countably compact and normal, then A (M) is compact. The special role played by l ∞ above is not unique. In fact, it is a special case of a more general theory known as hyperconvex metric spaces [10].

2.10 Common Fixed Points Since the early stages of development of the metric fixed point theory, many asked whether any results that dealt with the existence of the fixed point for one mapping extends to a family of mappings. In this case, we talk about the common fixed point. In particular, we deal with the question of whether the conclusion of Kirk’s fixed point theorem holds for a family of nonexpansive mappings. Since the fixed point set remained invariant when the mappings involved were commutative, it was natural to assume that the given family is commutative. Since the beginning it was clear that in order to obtain a common fixed point result, one has to study the structure of the fixed point set. This opened a new area of study in metric fixed point theory.

2.10 Common Fixed Points

45

Definition 2.14. [108] Let (M, d) be a metric space. A subset A of M is said to be a one-local retract of M if for every family {Bi}i∈I ofclosed balls centered in A with  nonempty intersection, it is the case that A ∩ / Bi = 0. i∈I

It is immediate that each nonexpansive retract of M is a one-local retract (but not conversely). Recall that A ⊂ M is a nonexpansive retract of M if there exists a nonexpansive map r : M → A such that r(a) = a, for every a ∈ A. The following theorem will shed some light on this notion. Theorem 2.33. [108] Let (M, d) be a metric space and N be a nonempty subset of M. The following are equivalent. (i) N is a 1-local retract of M. (ii) N is a nonexpansive retract of N ∪ {x}, for every x ∈ M. (iii) Let H be a metric space and T : N → H be a Lipschitzian map, i.e., there exists α ≥ 0 such that dH (T (a), T (b)) ≤ α dM (a, b), for every a, b ∈ N, where dH and dM stand respectively for the distances on H and M. Then, for every x ∈ M, there exists an extension T ∗ : N ∪ {x} → H, such that dH (T ∗ (a), T ∗ (b)) ≤ α dM (a, b), for every a, b ∈ N ∪ {x}, and T ∗ (a) = T (a) for every a ∈ N. Let (M, d) be a metric space. If we assume that A (M) is compact and normal, then it is not the case that these assumptions remain valid for A (N), where N ⊂ M. Recall that these assumptions are at the core of the Kirk’s fixed point theorem in metric spaces. Theorem 2.34. [108] Let (M, d) be a metric space and N be a nonempty subset one-local retract of M. Assume that A (M) is compact and normal. Then A (N) is compact and normal. The idea of one-local retract was inspired by the beautiful work of Baillon [13]. The following result finds its roots in Baillon’s work. Theorem 2.35. [108] Let (M, d) be a metric space for which A (M) is compact and normal. Let (Mβ )β ∈Γ be a decreasing family of nonempty one-local retracts of M,  Mβ is not empty and is a one-local retract of M. where Γ is totally ordered. Then β ∈Γ

Let us point out that under Kirk’s fixed point theorem, the fixed point set of a nonexpansive mapping is a one-local retract. Indeed let M, d) be a metric space for which A (M) is compact and normal. Let T : M → M be a nonexpansive map. We know that F(T ) is not empty. Let {Bi }i∈I be a family of closed balls centered in F(T ) with a nonempty intersection. Since T is nonexpansive, then S = Bi is T -invariant, i.e., i∈I

46

2 Fixed Point Theory in Metric Spaces: An Introduction

T (S) ⊂ S. The set S belongs to A (M), which implies that A (S) is compact and normal. Therefore Kirk’s fixed point theorem implies that T has a fixed point in S, i.e., S ∩ F(T ) = 0. / This completes the proof of our claim. The next results discuss the existence of fixed points of commutative family of nonexpansive mappings and the structure of their common fixed point set. Theorem 2.36. Let (M, d) be a bounded metric space such that A (M) is compact and normal. Then any commuting family of nonexpansive mappings {T  i }i∈I , Ti :  M → M, has a common fixed point. Moreover if we denote by F {Ti }i∈I the set of   the common fixed points, then F {Ti }i∈I is a one-local retract of M. Proof. Let M and {Ti }i∈I be as assumed in the theorem. Let I f be a nonempty finite subset of I. For any i, j ∈ I f , we have F(Ti ) is nonempty and is a one-local retract of M. Hence, A (F(Ti )) is normal and compact. Since T j commutes with Ti , we have set in T j (F(Ti )) ⊂ F(Ti ). Hence, T j has a fixed point in F(Ti ) and its fixed points  F(Ti ) F(Ti ) is a one-local retract, i.e., F(Ti )∩F(T j ) is a one-local retract. Hence, i∈I f

is not empty and is a one-local retract. Therefore, the conclusion of Theorem 2.36 follows from Theorem 2.35. 

Chapter 3

Modular Function Spaces

In this chapter we recall the foundations of the theory of modular function spaces defined by convex regular function modulars. The results discussed in this chapter will be used throughout the rest of the book. This chapter also presents the reader with an exhaustive list of examples that will frequently reoccur in later parts of the book. In Sect. 3.4 “Generalizations and Special Cases”, we summarize a more general approach, based on [132], which allows nonconvex function modulars and extends the setting to spaces of function with values in Banach spaces. While these results are not necessary for understanding the contents of the remaining chapters, the general framework introduced in this section may help to appreciate the full potential of the theory of modular function spaces.

3.1 Foundations Let Ω be a nonempty set and Σ be a nontrivial σ -algebra of subsets of Ω . Let P be a nontrivial δ -ring of subsets of Ω which means that P is closed with respect to forming of countable intersections, and finite unions and differences. Assume us assume that there exists an further that E ∩ A ∈ P for any E ∈ P and A ∈ Σ . Let  increasing sequence of sets Kn ∈ P such that Ω = Kn . By E we denote the linear space of all simple functions with supports from P. By M∞ we denote the space of all extended measurable functions, that is, all functions f : Ω → [−∞, ∞] such that there exists a sequence {gn } ⊂ E , |gn | ≤ | f | and gn (ω ) → f (ω ) for all ω ∈ Ω . By 1A we denote the characteristic function of the set A. Definition 3.1. Let ρ : M∞ → [0, ∞] be a nontrivial, convex, and even function. We say that ρ is a regular convex function pseudomodular if: (a) ρ (0) = 0; (b) ρ is monotone , that is, | f (ω )| ≤ |g(ω )| for all ω ∈ Ω implies ρ ( f ) ≤ ρ (g), where f , g ∈ M∞ ;

c Springer International Publishing Switzerland 2015  M. A. Khamsi, W. M. Kozlowski, Fixed Point Theory in Modular Function Spaces, DOI 10.1007/978-3-319-14051-3 3

47

48

3 Modular Function Spaces

(c) ρ is orthogonally subadditive , that is, ρ ( f 1A∪B ) ≤ ρ ( f 1A ) + ρ ( f 1B ) for any / where f ∈ M∞ ; A, B ∈ Σ such that A ∩ B = 0, (d) ρ has the Fatou property: | fn (ω )| ↑ | f (ω )| for all ω ∈ Ω implies ρ ( fn ) ↑ ρ ( f ), where f ∈ M∞ ; (e) ρ is order continuous in E , that is, gn ∈ E and |gn (ω )| ↓ 0 implies ρ (gn ) ↓ 0. Similarly as in the case of measure spaces, we say that a set A ∈ Σ is ρ -null if ρ (g1A ) = 0 for every g ∈ E . We say that a property holds ρ -almost everywhere if the exceptional set is ρ -null. As usual we identify any pair of measurable sets whose symmetric difference is ρ -null as well as any pair of measurable functions differing only on a ρ -null set. With this in mind we define M = { f ∈ M∞ : | f (ω )| < ∞ ρ − a.e.}, where each f ∈ M is actually an equivalence class of functions equal ρ -a.e. rather than an individual function. Definition 3.2. Let ρ be a regular convex function pseudomodular. (a) We say that ρ is a regular convex function semimodular if ρ (α f ) = 0 for every α > 0 implies that f = 0 ρ − a.e.; (b) We say that ρ is a regular convex function modular if ρ ( f ) = 0 implies that f = 0 ρ − a.e. The class of all nonzero regular convex function modulars defined on Ω is denoted by ℜ . Remark 3.1. Let us denote ρ ( f , E) = ρ ( f 1E ) for f ∈ M , E ∈ Σ . Also, by convention for α > 0 we will write ρ (α , E) instead of ρ (α 1E ). We will use these notations when convenient. It is easy to prove that ρ ( f , E) is a convex function modular in the sense of Definition 2.1.1 in [132] (more precisely, it is a convex function modular with the Fatou property). Therefore, we can use results of the standard theory of modular function spaces as per the framework defined by Kozlowski in [130, 131, 132, 136] and sketched in Sect. 3.6.1 of this book; see also [166, 167, 169, 170] for the basics of the general modular theory. Remark 3.2. Note that if ρ is a regular convex function modular, then to verify that a set E is ρ -null it suffices to prove that there exists α > 0 such that ρ (α , E) = 0. Definition 3.3. [130, 131, 132] Let ρ be a convex function modular. (a) A modular function space is the vector space Lρ (Ω , Σ ), or briefly Lρ , defined by Lρ = { f ∈ M : ρ (λ f ) → 0 as λ → 0}. (b) The following formula defines a norm in Lρ (frequently called the Luxemburg norm) :  f ρ = inf{α > 0 : ρ ( f /α ) ≤ 1}.

3.1 Foundations

49

Remark 3.3. It is not difficult to prove that  · ρ defines actually a norm such that  f ρ ≤ gρ whenever | f )| ≤ |g| ρ -a.e. It is also straightforward to demonstrate that  fn ρ → 0 if and only if ρ (α fn ) → 0 for every α > 0. While it is interesting to investigate the Luxemburg norm related properties of modular function spaces, we focus our attention on those fixed point results that can be formulated purely in the term of function modulars. The relationship between modulars and respective Luxemburg norms is discussed in Propositions 3.7, 3.12, 3.13 below, see also Example 5.4. In the fixed point theory in modular function spaces, we use often a notion of ρ convergence. It is described, among other important terms, in the following definition. Definition 3.4. Let ρ ∈ ℜ. (a) We say that { fn } is ρ -convergent to f and write fn → f (ρ ) if ρ ( fn − f ) → 0. (b) A sequence { fn } where fn ∈ Lρ is called a ρ -Cauchy sequence if ρ ( fn − fm ) → 0 as n, m → ∞. (c) A set B ⊂ Lρ is called ρ -closed if for any sequence of fn ∈ B, the convergence fn → f (ρ ) implies that f belongs to B. (d) A set B ⊂ Lρ is called ρ -bounded if its ρ -diameter is finite; the ρ -diameter of B is defined as δρ (B) = sup{ρ ( f − g) : f ∈ B, g ∈ B}. (e) A set B ⊂ Lρ is called strongly ρ -bounded if there exists β > 1 such that Mβ (B) = sup{ρ (β ( f − g)) : f ∈ B, g ∈ B} < ∞. (f) A set B ⊂ Lρ is called ρ -compact if for any { fn } in B, there exists a subsequence { fnk } and an f ∈ B such that ρ ( fnk − f ) → 0. (g) A set B ⊂ Lρ is called ρ -a.e. closed if for any { fn } in B, which ρ -a.e. converges to some f , we have f ∈ B. (h) A set B ⊂ Lρ is called ρ -a.e. compact if for any { fn } in B, there exists a subsequence { fnk }, which ρ -a.e. converges to some f ∈ B. (i) Let f ∈ Lρ and B ⊂ Lρ . The ρ -distance between f and B is defined as dρ ( f , B) = inf{ρ ( f − g) : g ∈ B}. Let us start with the following easy but important fact. Proposition 3.1. Assume that a sequence of fn ∈ Lρ ρ -converges to an f ∈ M , that is, ρ ( fn − f ) → 0. Then f ∈ Lρ . 1 Proof. Fix ε > 0 and take any 0 < λ < . Observe that 2 1 1 1 1 ε 1 ρ (λ f ) ≤ ρ (2λ ( f − fn )) + ρ (λ fn ) ≤ ρ ( f − fn ) + ρ (λ fn ) ≤ + ρ (λ fn ), 2 2 2 2 2 2 for n ≥ n0 . Since fn0 ∈ Lρ , it follows that there exists λ0 > 0 such that ρ (λ fn0 ) < ε for λ < λ0 . Hence, ρ (λ f ) ≤ ε for λ < λ0 , which means that f ∈ Lρ . 

50

3 Modular Function Spaces

It is very important to know that the ρ -limit is actually uniquely defined for a given sequence of fn ∈ L(ρ ). Proposition 3.2. Assume that we have two sequences of functions fn ∈ Lρ and gn ∈ Lρ such that ρ ( fn − f ) → 0 and ρ (gn − f ) → 0. Then f = g ρ -a.e. Proof. Since

ρ we have ρ

 f −g 2

 f −g 2

1 1 ≤ ρ ( f − fn ) + ρ (g − gn ) → 0, 2 2

= 0, which implies that f = g ρ -a.e.



Let us note, however, that ρ -convergence does not necessarily imply ρ -Cauchy condition. Also, fn → f (ρ ) does not imply in general λ fn → λ f (ρ ), for λ > 1. The following useful result is a direct consequence of the order continuity property of function modulars, see part (e) of Definition 3.1. Proposition 3.3. If { fn } converges uniformly to f on a set E ∈ P, then for every α > 0, lim ρ (α ( fn − f ), E) = 0. n→∞

The next result follows from the Fatou property of ρ . Proposition 3.4. Let ρ ∈ ℜ and f , fn ∈ M∞ . If fn → f ρ − a.e., then

ρ ( f ) ≤ lim inf ρ ( fn ). n→∞

Proof. Set gn = inf | fk |. Hence, k≥n

lim inf ρ (gn ) ≤ lim inf ρ ( fn ). n→∞

n→∞

Note that, in virtue of the Fatou property, the left-hand side of the above inequality  is equal to ρ ( f ), since |gn | ↑ | f | almost everywhere with respect to ρ .

The next theorem describes an important relationship between the ρ -convergence and the convergence ρ -a.e. Theorem 3.1. Let ρ ∈ ℜ. Assume that { fn } satisfies ρ -Cauchy condition. Then, there exists a subsequence { fnk } of { fn } and a function f ∈ M such that fnk → f ρ -a.e. Proof. Denoting En,m (ε ) = {ω ∈ Ω : | fn (ω ) − fm (ω )| ≥ ε }, we get

ρ (ε , En,m (ε )) ≤ ρ ( fn − fm , En,m (ε )) ≤ ρ ( fn − fm ) → 0. Hence, for any k ∈ N, there exists n0 (k) ∈ N such that if n ≥ n0 (k) and m ≥ n0 (k), then   ρ 2−k , En,m (2−k ) < 2−k .

3.1 Foundations

51

Taking n1 = n0 (1), nk = max{nk−1 , n0 (k)}, we get the sequence gk = fnk . Define Ek = {ω ∈ Ω : |gk (ω ) − gk+1 (ω )| ≥ 2−k }, and observe that ρ (2−k , Ek ) < 2−k . Denote G j =

Em and observe that for every

m= j

j ≥ i ≥ k and ω ∈ Ω \ Gk , there holds |gi (ω ) − g j (ω )| ≤

∞

∞

∑ |gm (ω ) − gm+1 (ω )| ≤ 21−i .

m=i ∞

Define a measurable function h =

∑ 2− j 1G j \G j+1 and calculate

j=1



ρ (h, Gk ) = ρ h,

∞

 (Gn \ Gn+1 )

≤

n=k ∞

∞

n=k

n=k

∑ ρ (hn , Gn \ Gn+1 ) ≤ ∑ ρ (2−n , En ) ≤ 21−k .

Hence, ρ (h, G) = 0, where G =

∞ 

G j . Observe that if ω does not belong to G, then

j=1

there exists k such that for any j ≥ i ≥ k, there holds |gi (ω ) − g j (ω )| < 21−i . Therefore, there exists a real number f (ω ) such that fni (ω ) = gi (ω ) → f (ω ). Clearly, the function f is measurable. It remains to be shown that G is ρ -null. Note that ∞   Dn , where Dn = G ∩ ω ∈ Ω : h(ω ) ≥ n−1 . Therefore, G= n=1

0 = ρ (h, G) ≥ ρ (h, Dn ) ≥ ρ (n−1 , Dn ).  Hence, Dn is ρ -null for every n ∈ N, which implies that G is ρ -null.

We will use the above result to prove the fundamental completeness theorem. Theorem 3.2. Let ρ ∈ ℜ. Lρ is complete with respect to the ρ -convergence. Proof. Let a sequence { fn } ⊂ Lρ be ρ -Cauchy. From Theorem 3.1, there exist a subsequence { fnk } and a function f ∈ M such that fnk → f ρ -a.e. Fix ε > 0 and note that there exists n0 ∈ N such that ρ ( fm − fn ) < ε provided m, n ≥ n0 . Using Proposition 3.4, we get

ρ ( f − fn ) ≤ lim inf ρ ( fnk − fn ) ≤ ε k→∞

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3 Modular Function Spaces

for n ≥ n0 . It remains to be proved that f ∈ Lρ . Let λn → 0. Fix ε > 0 and take k ∈ N such that ρ ( f − fk ) < ε . For n sufficiently large, we obtain

ρ (λn ( f − fk )) ≤ ρ ( f − fk ) < ε , which implies that ρ (λn ( f − fk )) → 0 as n → ∞. Thus, fk − f belongs to Lρ . Since  fk ∈ Lρ and Lρ is a linear space, then f belongs to Lρ as claimed.

Let us follow [132] and prove a modular version of the Egoroff theorem, which plays a critical role in the theory of modular function spaces. Theorem 3.3. Let ρ ∈ ℜ and let fn → f ρ − a.e. where f , fn ∈ M . There exists a nondecreasing sequence of sets Hk ∈ P such that Hk ↑ Ω and { fn } converges uniformly to f on every Hk . Proof. Let us fix α > 0 andrecall that there exists an nondecreasing sequence of sets Kn ∈ P such that Ω = Kn . Let us fix temporarily n ∈ N. We shall prove that there exist sets Fn ∈ P such that Fn ⊂ Kn and

ρ (α , Kn \ Fn ) <

1 2n

(3.1)

and { fm } converges uniformly to f on each Fn . By omitting, if necessary, a ρ null set from Ω , we may assume that the sequence { fn } pointwise converges to f everywhere. Set  ∞   1 m Ek = ω ∈ Kn : | fi (ω ) − f (ω )| < m i=k and observe that E1m ⊂ E2m ⊂ . . . and, since { fn } pointwise converges to f everywhere, that Kn ⊂

∞

Ekm

k=1

for every m ∈ N. Then there exists a positive integer n0 = n0 (m) such that

ρ (α , Kn \ Enm0 (m) ) < Denoting Fn =

∞  m=1

1 . 2m+n

Enm0 (m) ,

3.1 Foundations

53

we have



ρ (α , Kn \ Fn ) = ρ α , Kn \  = ρ α,

Kn \ Enm0 (m)



 Enm0 (m)

m=1



∞ 

∞ 

≤

∞

∑ρ

m=1

m=1



 1 α , Kn \ Enm0 (m) < , n

proving (3.1). It follows from the definition of Fn that to every m ∈ N there corresponds km ∈ N such that for i ≥ km , | fi (ω ) − f (ω )| <

1 , m

for all ω ∈ Fn , which means that { fi } converges uniformly to f on Fn . Let us define now Hk =

k

Fn , H =

n=1

∞

Hk .

k=1

Clearly, {Hk } is a nondecreasing sequence of sets from P such that

ρ (α , Kk \ H) ≤ ρ (α , Kk \ Hk ) ≤ ρ (α , Kk \ Fk ) <

1 2k

and { fm } converges uniformly to f on every Hk . It remains to be proved that Ω \ H is ρ -null. Indeed,

ρ (α , Ω \ H) ≤ ρ (α , Kk \ H) +

∞

∑

n=k+1

ρ (α , Kn \ H) <

∞ 1 1 + ∑ n k 2 2 n=k+1

for every k ∈ N. Thus, ρ (α , Ω \ H) = 0. The proof is complete.



The following useful result is an easy corollary from Theorems 3.1 and 3.3. Proposition 3.5. Let ρ ∈ ℜ. Assume that ρ ( fn − f ) → 0, then there exists a subsequence { fnk } of { fn } such that fnk → f ρ -a.e. Proof. Arguing like in the proof of Theorem 3.1, there exists a subsequence { fnk } of { fn } and a measurable function g such that fnk → g ρ -a.e. We need to show that f = g ρ -a.e. This is not obvious as the modular ρ does not need to satisfy the triangle property. However, by the Egoroff theorem 3.3, there exists an increasing sequence of sets {Hm } from P such that

∞

Hm = Ω and fn converges to g uniformly on

m=1

each Hm . Hence,

 f −g , Hm ≤ ρ ( f − fnk , Hm ) + ρ ( fnk − g, Hm ) → 0, ρ 2

as k → ∞.

Hence, ρ ( f − g, Hm ) = 0 for every m, which implies that f = g ρ -a.e.



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3 Modular Function Spaces

As an immediate consequence of Propositions 3.4 and 3.5, we get the following important result. Proposition 3.6. Let ρ ∈ ℜ. Then ρ -balls Bρ ( f , r) = {g ∈ Lρ : ρ ( f − g) ≤ r} are ρ -closed and ρ -a.e. closed. Let us come back to the question of the relationship between the modular and norm convergence in modular function spaces. Proposition 3.7. Let ρ ∈ ℜ. The following assertions are true: (a) If  f ρ < 1 then ρ ( f ) ≤  f ρ (b)  f ρ ≤ 1 if and only if ρ ( f ) ≤ 1. Proof. Part (a): Let α be any number such that  f ρ < α < 1. Then it follows from the definition of the Luxemburg norm that

 f ≤ 1, ρ α hence

f ρ( f ) = ρ α α



 f ≤ αρ ≤ α. α

Since α was chosen arbitrarily, it follows that ρ ( f ) ≤  f ρ , as claimed. Part (b): Assume first that  f ρ ≤ 1 and fix arbitrarily 0 < λ < 1. Hence λ f ρ < 1. By Part (a) we get ρ (λ f ) ≤ λ f ρ = λ  f ρ . Taking λ → 1 and using the Fatou property, we obtain the desired inequality. Let us prove now the other implication. Assume now that ρ ( f ) ≤ 1. If ρ ( f ) = 1 then by definition,  f ρ ≤ 1, so we may assume that ρ ( f ) < 1. Assume to the contrary that  f ρ > 1. There exists then a sequence of positive numbers {λn } such that λn ↑ 1 and λn f ρ > 1 which, by the definition of the Luxemburg norm, implies that ρ (λn f ) > 1. By passing with n → ∞ and using the Fatou property we get ρ ( f ) ≥ 1, which contradicts our assumption.

 From part (a) of Proposition 3.7, we conclude immediately the following very important fact. Proposition 3.8. Let ρ ∈ ℜ, let fn , f ∈ Lρ . Then  fn − f ρ → 0 implies also the ρ -convergence, that is ρ ( fn − f ) → 0. The next two facts are actually true in any modular space. We reproduce them here for the sake of completeness. Proposition 3.9. Let ρ ∈ ℜ and let f ∈ Lρ . Then  f ρ > 1 implies ρ ( f ) ≥  f ρ . Proof. Let us fix temporarily 1 < α <  f ρ . It follows form the definition of the Luxemburg norm that ρ ( f /α ) > 1 and therefore by the convexity of ρ that 1<ρ

f 1 ≤ ρ ( f ). α α

3.1 Foundations

55

Hence ρ ( f ) ≥ α > 1, which implies that ρ ( f ) ≥  f ρ since α <  f ρ was chosen arbitrarily.

 Proposition 3.10. Let ρ ∈ ℜ, let fn , f ∈ Lρ . Then  fn − f ρ → 0 if and only if ρ (λ ( fn − f )) → 0 for all λ > 0. Proof. If  fn − f ρ → 0 then λ ( fn − f )ρ → 0 for any λ > 0 and hence ρ (λ ( fn − f )) → 0 by Proposition 3.8. Conversely, assume that ρ (λ ( fn − f )) → 0 for all λ > 0 and assume to the contrary that  fn − f ρ does not tend to zero. Passing to a subsequence, if necessary, we can assume that there exists an ε > 0 such that  fn − f ρ > ε . By multiplying both sides of this inequality by a suitable λ > 0, we have λ ( fn − f )ρ > 1. By Proposition 3.9,

ρ (λ ( fn − f )) > λ ( fn − f )ρ > 1, which contradicts the fact that ρ (λ ( fn − f )) → 0.



In the sequel, we will use the following result that describes the relation between the norm and the modular convergence. Proposition 3.11. Let ρ ∈ ℜ. If C ⊂ Lρ is ρ -bounded then C is bounded with respect to the Luxemburg norm. Proof. Assume to the contrary that C is not  · ρ -bounded. There exist therefore hn , h ∈ C such that hn − hρ → ∞. Without loss of generality we can assume that hn − hρ > 1 for all n ∈ N. By Proposition 3.9,

ρ (hn − h) ≥ hn − hρ → ∞,

(3.2)

which contradicts the fact that sup ρ ( f − g) < ∞ by ρ -boundedness of C. 

f ,g∈C

The following definition plays an important role in the theory of modular function spaces. Definition 3.5. Let ρ ∈ ℜ. We say that ρ has the Δ2 property if ρ (2 fn ) → 0 whenever ρ ( fn ) → 0. The next result is an immediate consequence of the above definition and of Proposition 3.7. Proposition 3.12. Let ρ ∈ ℜ. The convergence with respect to the function modular ρ is equivalent to the convergence with respect to the Luxemburg norm  · ρ if and only if ρ has the Δ2 -property. The following comparison of various modular and norm notions is an immediate consequence of the above results.

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3 Modular Function Spaces

Proposition 3.13. Let ρ ∈ ℜ. The following relationships hold for sets C ⊂ Lρ : (a) If C is ρ -closed then C is  · ρ -closed. (b) If C is ρ -bounded then C is  · ρ -bounded. (c) If C is ρ -compact then C is ρ -a.e. compact. (d) If C is  · ρ -compact then C is ρ -compact. (e) Convergence with respect to  · ρ always implies ρ -convergence. (f)  fn − f ρ → 0 if and only if ρ (λ ( fn − f )) → 0 for all λ > 0. (g) If ρ satisfies Δ2 then ·ρ -convergence and ρ -convergence are equivalent in Lρ . (h) If ρ satisfies Δ2 then  · ρ -compactness and ρ -compactness are equivalent in Lρ . A stronger condition than the Δ2 property is sometimes used in the literature, as defined below. Definition 3.6. Let ρ ∈ ℜ. We say that ρ satisfies the Δ2 -type condition if there exists a constant 0 < K < ∞ such that for every f ∈ Lρ we have ρ (2 f ) ≤ K ρ ( f ). Note that if ρ satisfies the Δ2 -type condition then for every f ∈ Lρ , we have ρ ( f ) < ∞. It is also clear that if ρ satisfies the Δ2 -type condition then ρ has the Δ2 -property, and that the converse is not true. In this book we refer mainly to the Δ2 property as defined in Definition 3.5. In some cases, especially in Sect. 5.3.5, we will discuss function modulars satisfying the Δ2 -type condition in the sense of Definition 3.6. It is important to be able to measure the growth of the function modular in the sense of the following definition. Definition 3.7. Let ρ ∈ ℜ. Define the growth function ωρ as follows:   ρ (t f ) : 0 < ρ( f ) < ∞ , ωρ (t) = sup ρ( f ) for all t ≥ 0. Note that in general the growth function ωρ (t) may be equal to infinity for some t > 1, hence it is not very useful in such cases. However, the growth function ωρ has some interesting properties provided ρ satisfies the Δ2 -type condition. Lemma 3.1. [63] Let ρ ∈ R. Assume that ρ is convex and satisfies the Δ2 -type condition. The growth function ωρ has the following properties: (i) ωρ (t) < ∞, for any t ∈ [0, ∞). (ii) ωρ : [0, ∞) → [0, ∞) is a convex, strictly increasing function. Hence, it is continuous. (iii) ωρ (αβ ) ≤ ωρ (α )ωρ (β ), for any α , β ∈ [0, ∞). (iv) ωρ−1 (α )ωρ−1 (β ) ≤ ωρ−1 (αβ ), for any α , β ∈ [0, ∞), where ωρ−1 is the inverse function of ωρ . 1   , for any f ∈ Lρ . (v)  f  ≤ −1 ωρ ρ (1f )

3.1 Foundations

57

Proof. The proofs of (i)–(iv) follow straightforward   from the definition. To prove (v), let us assume, 0 < α < || f ||ρ , hence 1 < ρ αf , which implies 1 < ωρ ρ( f ) and so

ωρ −1

 1 α

 1  1 < . ρ( f ) α

Letting α → || f ||ρ − we obtain || f ||ρ ≤

ωρ −1

1 

1 ρ( f )

.

 Obviously when ρ satisfies the Δ2 -type condition, then the Luxemburg norm convergence and ρ -convergence are equivalent. Using the growth function, we get a better understanding of this result. Lemma 3.2. Let ρ ∈ R. Assume that ρ satisfies the Δ2 -type condition. For every ε > 0, there exists δ > 0 such that ρ ( f ) < δ implies  f  < ε . 1 . Since ω −1 is an increasing function, property ωρ (1/ε ) (v) of Lemma 3.1 implies that  f  < ε whenever ρ ( f ) < δ .



Proof. Let ε > 0. Set δ =

In the sequel we will be using the following notion of the ρ -lower semicontinuity. Definition 3.8. A function λ : C → [0, ∞], where C ⊂ Lρ is nonempty and ρ -closed, is called ρ -lower semicontinuous if for any α > 0, the set Cα = { f ∈ C : λ ( f ) ≤ α } is ρ -closed. Remark 3.4. It can be easily proved that ρ -lower semicontinuity is equivalent to the condition

λ ( f ) ≤ lim inf λ ( fn ) provided f , fn ∈ C, and ρ ( f − fn ) → 0. n→∞

The modular version of the property (R), defined previously for metric spaces, will be essential for the proof of several fixed point theorems in modular function spaces. Following [117], let us formally define this property. Definition 3.9. We say that Lρ has property (R) if and only if every nonincreasing sequence {Cn } of nonempty, ρ -bounded, ρ -closed, convex subsets of Lρ has nonempty intersection.

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3 Modular Function Spaces

In some cases we will be referring to a somewhat stronger notion of the property (R ), defined below. Definition 3.10. We say that Lρ has property (R ) if and only if for every ρ -bounded sequence of fn ∈ Lρ there exists a subsequence { fnk } such that 

clρ (conv{ fnk : k ≥ n})

n∈N

is nonempty and reduced to one point. By clρ (conv(A)), we understand the ρ closure of the smallest convex subset of Lρ containing A. Remark 3.5. It is easy to verify that in any modular function space property (R ) implies property (R). The ρ -lower semicontinuous real-valued convex functions play an important role in modular function spaces because they attain their infimum provided ρ has property (R), as shown in next result. Lemma 3.3. Let us assume that ρ ∈ ℜ has property (R). Let K ⊂ Lρ be nonempty, convex, ρ -closed, and ρ -bounded. If ϕ : K → [0, ∞) is a ρ -lower semicontinuous convex function, then there exists x0 ∈ K such that

ϕ (x0 ) = inf{ϕ (x) : x ∈ K}. Proof. Let m = inf{ϕ (x) : x ∈ K}. The assumptions on ϕ imply m < ∞. For any n ≥ 1, set   1 . Kn = x ∈ K : ϕ (x) ≤ m + n Clearly Kn is not empty and is a convex set because ϕ is a convex function. Also, Kn is ρ -closed since ϕ is ρ -lower semicontinuous. Since ρ satisfies property (R), then K∞ =



Kn = 0. /

n≥1

For x0 ∈ K∞ , there holds

ϕ (x0 ) = inf{ϕ (x) : x ∈ K}, as claimed.



3.2 Space Eρ , Convergence Theorems, and Vitali Property Definition 3.11. Define Lρ0 = { f ∈ Lρ : ρ ( f , ·) is order continuous} and the linear space Eρ = { f ∈ Lρ : λ f ∈ Lρ0 f or every λ > 0}. The position of Eρ with respect to Lρ is characterized in the following theorem.

3.2 Space Eρ , Convergence Theorems, and Vitali Property

59

Theorem 3.4. [130, 131, 132] Let ρ ∈ ℜ. Then Eρ is a  · ρ -closed subspace of Lρ . Proof. First, we will prove that Eρ is a linear subspace of Lρ . Clearly Eρ is a linear space; it suffices, therefore, to prove that Eρ ⊂ Lρ . Let f ∈ Eρ and 0 ≤ λn → 0. Since f ∈ M it follows that there exists a sequence {sm } of P-simple functions such that |sm | ↑ | f |. By the Egoroff theorem, we get a sequence {Hi } such that Hi ∈ P, Hi ↑ Ω , and {sm } converges uniformly to f on every Hk . Let us choose a number λ > 0 such that λn ≤ λ for all natural numbers n, and fix an arbitrary ε > 0. Since f ∈ Eρ , there exists an index k0 ∈ N such that

ε ρ (λ f , Ω \ Hk0 ) < . 3 By the uniform convergence of {sm } on Hk0 we can choose m0 ∈ N such that

ε ρ (2λ (sm0 − f ), Hk0 ) < . 3 Remember that sm0 belongs to E ⊂ Lρ and hence there exists n0 ∈ N such that

ρ (2λn sm0 , Hk0 ) <

ε 3

for n ≥ n0 . Finally, for n ≥ n0 , we have

ρ (λn f ) ≤ ρ (λn f , Ω \ Hk0 ) + ρ (λn f , Hk0 ) ≤ ρ (λ f , Ω \ Hk0 ) + ρ (2λ (sm0 − f ), Hk0 ) + ρ (2λn sm0 , Hk0 ) < ε . This means that the function f is a member of the space Lρ . We will prove now that Eρ is closed. Let fn ∈ Eρ ,  fn − f ρ → 0, f ∈ Lρ . Take a sequence of sets En ∈ Σ with En ↓ 0. / Fix positive numbers ε , α > 0 and choose k0 ∈ N such that

ε ρ (2α ( f − fk0 )) < . 2 Since fk0 ∈ Eρ , it follows that for n sufficiently large,

ε ρ (2α fk0 , En ) < . 2 Thus, for n sufficiently large, we have

ρ (α f , En ) ≤ ρ (2α ( f − fk0 ), En ) + ρ (2α fk0 , En ) ≤ ρ (2α ( f − fk0 )) + ρ (2α fk0 , En ) < ε , which implies that f ∈ Eρ . Therefore, Eρ is a closed subspace of Lρ , as claimed.



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3 Modular Function Spaces

According to Proposition 3.12, the ρ -convergence and the  · ρ -convergence are in general equivalent if only if ρ satisfies Δ2 . However, it is legitimate to ask on what subsets of Lρ such equivalence may hold even ρ does not have Δ2 . We introduce a concept of sets with the Vitali property that will play this role, see [141]. The reference to Giuseppe Vitali is justified by the following version of the Vitali convergence theorem, which was proved in the context of modular function spaces in [132]. Theorem 3.5. [132] Let ρ ∈ ℜ. Let fn ∈ Eρ , f ∈ Lρ , and fn → f ρ -a.e. Then the following conditions are equivalent: (i) f ∈ Eρ and  fn − f ρ → 0. (ii) For every α > 0, the subadditive measures ρ (α fn , ·) are order equicontinuous, that is, if Ek ∈ Σ are such that Ek ↓ 0/ then lim sup ρ (α fn , Ek ) = 0. k→∞ n∈N

/ and Proof. (i) ⇒ (ii). Let us choose a sequence of sets Ek ∈ Σ such that Ek ↓ 0, fix arbitrarily numbers ε > 0 and α > 0. Since  fn − f ρ → 0, it follows that there exists n0 ∈ N such that for all n ≥ n0 ,

ε ρ (2α ( fn − f )) < . 2 From the definition of the space Eρ , it follows that there exists a natural number k0 ∈ N with ρ (α fn , Ek ) < ε (3.3) for k ≥ k0 and n = 1, 2, . . . , n0 − 1. Similarly, since f ∈ Eρ , there exists k1 > k0 such that for any k ≥ k1 ,

ε ρ (2α f , Ek ) < . 2 Thus,

ρ (α fn , Ek ) ≤ ρ (2α ( fn − f )) + ρ (2α f , Ek ) < ε

(3.4)

for all natural k1 > k0 and n ≥ n0 . The inequalities (3.3) and (3.4) give the desired result. (ii) ⇒ (i). Let us fix two numbers ε > 0 and α > 0. Since fn → f ρ -a.e., it follows from the Egoroff theorem (Theorem 3.3) that there exists a sequence of sets Hk ∈ P such that Hk ↑ Ω and { fn } converge uniformly to f on every Hk . By the assumed order equicontinuity of function modulars, we can pick up an index k0 ∈ N such that for all n ∈ N,

ε ρ (2α fn , Ω \ Hk0 ) < . 4 Since { fn } converges uniformly to f on Hk0 ∈ P, one can find n0 ∈ N such that ρ (2α ( fn − f )) <

ε 4

3.2 Space Eρ , Convergence Theorems, and Vitali Property

61

for n ≥ n0 . Let n, m ≥ n0 . We have

ρ (α ( fn − fm )) ≤ ρ (α ( fn − fm ), Ω \ Hk0 ) + ρ (α ( fn − fm ), Hk0 ) ≤ ρ (2α fn , Ω \ Hk0 ) + ρ (2α fm , Ω \ Hk0 ) +ρ (2α ( fn − f )) + ρ (2α ( fm − f )) < ε . Hence the sequence {α fn } satisfies the ρ -Cauchy condition, which implies that { fn } satisfies the Cauchy condition in the sense of the Luxemburg norm  · ρ . Since fn ∈ Eρ , which is complete in the norm sense as the closed subspace of a complete metric space Lρ , it follows that there exists a function g ∈ Eρ such that  fn −gρ → 0. Therefore, in view of Proposition 3.5 there exists a subsequence of { fn } that converges ρ -a.e. to g. On the other hand, fn → f ρ -a.e. Thus, f = g ρ -a.e.  and finally  fn − f ρ → 0, as claimed.

As an immediate consequence of the Vitali convergence theorem, we get a result that is of great importance for applications, namely the following version of the Lebesgue dominated convergence theorem. Theorem 3.6. [132] Let ρ ∈ ℜ. Let fn , f ∈ M , fn → f ρ -a.e. Assume that there exists a function g ∈ Eρ such that | fn (ω )| ≤ |g(ω )| for every n ∈ N. Then  fn − f ρ → 0. Theorem 3.7. [130, 131, 132] Let ρ ∈ ℜ. Then Eρ is the  · ρ closure of the space of all P simple functions E . Proof. We will prove first that cl·ρ (E ) ⊂ Eρ . Take a simple function g ∈ E , a / En ∈ Σ . Denote positive number λ , and a sequence En ↓ 0,

α = sup |λ g(ω )|, ω ∈Ω

and observe that

ρ (λ g, En ) = ρ (λ g, supp(g) ∩ En ) ≤ ρ (α , supp(g) ∩ En ) → 0, because supp(g) ∩ En ∈ P. Hence, E ⊂ Eρ and consequently cl·ρ (E ) ⊂ Eρ , since Eρ is  · ρ -closed. To prove the inverse inclusion, let us choose arbitrarily f ∈ Eρ and ε > 0. Since f ∈ Eρ and Ω is a countable union of sets from P, there exists a set E ∈ P such that, denoting by E  = Ω \ E, we have  f 1E  ρ < ε2 . Let us select a sequence {gn } of P-simple functions such that for all n ∈ N and all ω ∈ E, supp(gn ) ⊂ E, |gn (ω )| ≤ | f (x), gn (ω ) → f (x).

62

3 Modular Function Spaces

By the Lebesgue dominated convergence theorem, gn − f 1E ρ → 0. Hence, gn0 − f 1E ρ <

ε 2

for a certain n0 ∈ N, and gn0 − f ρ ≤ gn0 − f 1E ρ + gn0 1E  ρ +  f 1E  ρ ≤ ε , that is Eρ ⊂ cl·ρ (E ). This completes the proof.



Theorem 3.8. [130, 131, 132] Let ρ ∈ ℜ. Then the following conditions are equivalent: (a) ρ has the Δ2 -property; (b) Lρ0 is a linear subspace of Lρ ; (c) Eρ = Lρ0 = Lρ ; (b)  · ρ -convergence and ρ -convergence are equivalent. Proof. (a) ⇒ (b). Let f ∈ Lρ0 and Ek ↓ 0, / Ek ∈ Σ . Since f ∈ Lρ0 , it follows that ρ ( f , Ek ) → 0. By Δ2 we get that ρ (2 f , Ek ) → 0. Hence, 2 f ∈ Lρ0 and, consequently, Lρ0 is a linear space. (b) ⇒ (c). We have Eρ ⊂ Lρ0 ⊂ Lρ , and Lρ is the smallest linear space containing 0 Lρ . Eρ is the largest space contained in Lρ . Since Lρ0 is linear, then Eρ = Lρ0 = Lρ . (c) ⇒ (a). Suppose to the contrary that there exists a sequence of functions { fn } from Lρ such that ρ ( fn , ·) are order continuous while ρ (2 fn , ·) are not. It follows then that ρ (2 fn , ·) are not uniformly exhaustive, that is, there exists a sequence of disjoint sets {Dk } from Σ and a constant η > 0 such that for every k ∈ N,

ρ (2 fk , Dk ) > η . Observe that ρ ( fk , Dk ) → 0 because

ρ ( fk , Dk ) ≤ sup ρ ( fn , Dk ) → 0, n∈N

which is a consequence of the fact that sup ρ ( fn , ·) is order continuous and hence n∈N

exhaustive. Let { fki } be a subsequence of { fn } such that ∞

∑ ρ ( fki , Dki ) ≤ 1.

i=1

Denoting m

∞

i=1

i=1

sm = ∑ fki 1Dki , f = ∑ fki 1Dki , we get sm ∈ Lρ = Eρ and

ρ ( f − sm ) ≤

∞

∑

i=m+1

ρ ( fki , Dki ) → 0.

3.2 Space Eρ , Convergence Theorems, and Vitali Property

63

The function f is a member of Lρ = Eρ . Indeed, let ε > 0 be chosen arbitrarily. 1 Moreover, let 0 < εk < , εk → 0, λk = 2εk < 1. Then we have 2

ρ (εk f ) ≤ ρ (λk ( f − sm )) + ρ (λk sm ) ≤ ρ ( f − sm ) + ρ (λk sm ).

(3.5)

ε ε and k0 ∈ N such that ρ (λk sm0 ) < 2 2 for natural k ≥ k0 . Hence, ρ (εk f ) < ε for k ≥ k0 , and then f ∈ Lρ = Eρ . Since Lρ = Eρ is linear, it follows that 2 f is a member of Eρ as well. However, it was observed above that ρ (2 f , Dk ) > η > 0. Hence ρ (2 f , ·) is a σ -subadditive measure, which is not exhaustive, i.e., ρ (2 f , ·) must not be order continuous, which means that the function 2 f does not belong to Eρ . This contradiction completes this part of the proof. (a) ⇒ (d). It suffices to prove that ρ ( fn ) → 0 implies ρ (2 fn ) → 0 for fn ∈ Lρ . Assume, therefore, that fn ∈ Lρ and ρ ( fn ) → 0. Let us take an arbitrary subsequence {hn } of { fn }. We can choose a subsequence {gn } of {hn } such that gn → 0 ρ -a.e. As we proved, it follows from (a) that Eρ = Lρ0 = Lρ . It follows from the Vitali convergence theorem that ρ (gn , ·) are order equicontinuous. Hence, ρ (2gn , ·) are also equicontinuous by Δ2 . Using the Vitali convergence theorem again, we get ρ (2gn ) → 0. By the arbitrariness of {hn } we have that ρ (2 fn ) → 0. (d) ⇒ (a). Let fn ∈ Lρ and let ρ ( fn , ·) be order equicontinuous. Assume to the / an ε > 0, and a contrary that there exists a sequence of sets Ek ∈ Σ such that Ek ↓ 0, subsequence {gn } of { fn } such that We may take m0 ∈ N such that ρ ( f − sm0 ) <

ρ (2gk , Ek ) = ρ (2gk 1Ek ) > ε .

(3.6)

On the other hand,

ρ (gk , Ek ) ≤ sup ρ ( fn , Ek ) → 0. n∈N

By (d) then,

ρ (2gk , Ek ) = ρ (2gk 1Ek ) → 0, which contradicts (3.6) and completes the proof.

 Definition 3.12. A set C ⊂ Lρ is said to posses the Vitali property if C ⊂ Eρ , and for any g ∈ Lρ and gn ∈ C with ρ (gn − g) → 0, there exists a subsequence {gnk } of {gn } such that for every α > 0 the subadditive measures ρ (α gnk , ·) are order equicontinuous. Our next result characterizes sets with the Vitali property as those subsets of Eρ on which the ρ -convergence and the  · ρ -convergence are indeed equivalent. Theorem 3.9. Let ρ ∈ ℜ. A set C ⊂ Lρ has the Vitali property if and only the following two conditions are satisfied: (i) C ⊂ Eρ . (ii) If g ∈ Lρ and gn ∈ C with ρ (gn − g) → 0, then gn − gρ → 0.

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3 Modular Function Spaces

Proof. Let us assume first that C ⊂ Lρ has the Vitali property. Hence, by definition, the condition (i) is satisfied. To prove (ii), take a sequence {gn }, and a function g ∈ Lρ such that ρ (gn − g) → 0. By Proposition 3.5, there exists a subsequence {gnk } such that (3.7) gnk → g ρ − a.e. Because C has the Vitali property, there exists a subsequence {gnk } such that for l every α > 0 the submeasures ρ (α gnk , ·) are order equicontinuous. Hence, using l (3.7) and the Vitali theorem (Theorem 3.5), we conclude that gnk − gρ → 0. Since l this reasoning can be repeated for any subsequence of {gn }, every subsequence of {gn } contains a subsequence converging in  · ρ to g, proving (ii). Conversely, let us assume that (i) and (ii) are satisfied. Assume to the contrary that C does not have the Vitali property. Then there exists a sequence {gk } of elements of C, g ∈ Lρ , and α > 0 such that the submeasures ρ (α gn , ·) are not order equicontinuous while ρ (gk − g) → 0. The latter fact implies by (ii) that gn − gρ → 0, which also implies, via closedness of Eρ , that g ∈ Eρ . By Proposition 3.5 we can assume, without loosing generality, that gnk → g ρ -a.e. From Theorem 3.5, we conclude that  gn − gρ cannot tend to zero. Contradiction completes the proof.

Remark 3.6. Combining Proposition 3.13 with Theorem 3.9, we can easily see that a set with the Vitali property is ρ -closed if and only if it is  · ρ -closed. Remark 3.7. Let C ⊂ Lρ be a set with the Vitali property and let a, b ∈ R. Let u : [a, b] → C be a ρ -continuous function, that is, ρ (u(tn ) − u(t)) → 0 provided tn → t. It follows immediately from Theorem 3.9 that u is  · ρ -continuous. As an immediate corollary to Remark 3.7, we obtain the following important result. Remark 3.8. Let Z be a separable linear subspace of (Eρ ,  · ρ ) and let C ⊂ Z have the Vitali property. Assume that the function u : [a, b] → C is ρ -continuous. Then u is the Bochner integrable function with respect the the Lebesgue measure m on [a, b], i.e., u ∈ L1 (Ω , Z, m). Let us discuss the Fatou property in the context of the Vitali property. Proposition 3.14. Let ρ ∈ ℜ. Assume that C ⊂ Eρ has the Vitali property and that

ρ ( fn − f ) → 0, ρ (gn − g) → 0

(3.8)

as n → ∞, with fn , f , gn , g ∈ C. Then,

ρ ( f − g) ≤ lim inf ρ ( fn − gn ). n→∞

(3.9)

Proof. Since C has the Vitali property, it follows from (3.8) that

ρ (( fn − gn ) − ( f − g)) → 0.

(3.10)

3.3 An Equivalent Topology

65

Using (3.10) and the Fatou property of ρ , it is easy to get required inequality (3.9). 

Let us finish this section with few examples of sets with the Vitali property. Example 3.1. If ρ has Δ2 property then every set C ⊂ Lρ has the Vitali property. Example 3.2. Let C ⊂ Eρ . If there exists g ∈ Eρ such that | f (ω ) ≤ |g(ω )| ρ -a.e. for every f ∈ C, then C has the Vitali property. Example 3.3. Let C ⊂ Eρ be  · ρ -conditionally compact. Then C has the Vitali property, see Theorem 3.11 in Sect. 3.4.

3.3 An Equivalent Topology The concept of ρ -a.e. closed, and ρ -a.e. compact sets has been defined using sequences. One of the problems that many authors have found hard to circumvent is whether these notions are related to a topology. In this section we will discuss this problem. In particular, we will construct a topology τ for which the ρ -a.e. compactness is equivalent to the usual compactness for τ . This is crucial when we try to use Zorn’s lemma. Throughout this section, we assume that ρ satisfies the Δ2 -type condition and it is σ -finite, i.e., there exists an increasing sequence of sets Kn ∈ P Kn . such that 0 < ρ (1Kn ) < ∞ and Ω = n≥1

Proposition 3.15. Assume that ρ satisfies the Δ2 -type condition. The functional d : Lρ × Lρ → [0, +∞) defined

 ∞ | f − g| 1 1 1K d( f , g) = ∑ k ρ 1 + | f − g| k k=1 2 ρ (1Kk ) satisfies the following: 1. d( f , g) = 0 if and only if f = g ρ -a.e.; 2. d( f , g) = d(g, f );  ωρ (2)  d( f , h) + d(h, g) ; 3. d( f , g) ≤ 2 for any f , g, and h in Lρ . Proof. (1) and (2) are obvious. To prove (3) we only need to recall the inequality |a + b| |a| |b| ≤ + 1 + |a + b| 1 + |a| 1 + |b| for all positive numbers a, b and use the definition of the growth function ωρ .



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3 Modular Function Spaces

Remark 3.9. The functional d is not a distance because of (3). But there are many mathematical objects that fail the triangle inequality but are very useful tools. That is the case with d. In the next proposition, we discuss the relationship between ρ -a.e. convergence and the convergence for the functional d. Proposition 3.16. Let ρ be a convex, σ -finite modular satisfying the Δ2 -type condition and { fn }n be a sequence in Lρ . If { fn }n is ρ -a.e. convergent to f , then lim d( fn , f ) = 0.

n→∞

Moreover, if lim d( fn , f ) = 0, then there exists a subsequence { fnk }k that converges n→∞ ρ -a.e. to f . Proof. Assume that { fn }n ρ -a.e. converges to f . We will show that lim d( fn , f ) = 0. n→∞

∞

1 Let ε > 0, and choose N ∈ N such that ∑ k < ε . We have 2 k=N+1  | fn − f | 1Kk + ε 1 + | fn − f |

 N | fn − f | 1 1 lim ρ 1Kk + ε . = ∑ k n→∞ 1 + | fn − f | k=1 2 ρ (1Kk )

n→∞

N

1 1 ρ ∑ k n→∞ k=1 2 ρ (1Kk )

lim d( fn , f ) ≤ lim

| fn − f | | fn − f | ρ −a.e. 1K ≤ 1Kk , and 1K −→ 0 as n → ∞, for any k ≥ 1, 1 + | fn − f | k 1 + | fn − f | k Lebesgue convergence theorem will then imply

 | fn − f | lim ρ 1Kk = 0, k ≥ 1. n→∞ 1 + | fn − f | Since

Thus lim d( fn , f ) ≤ ε , for any ε > 0, i.e., lim d( fn , f ) = 0. Conversely, assume now n→∞

n→∞

that lim d( fn , f ) = 0. For any k ≥ 1, we have n→∞

lim ρ

n→∞

| fn − f | 1K 1 + | fn − f | k

 = 0.

Thus there exists a subsequence { fn1 }n of { fn }n such that ρ −a.e.

| fn1 − f | ρ −a.e. 1K −→ 0 1 + | fn1 − f | 1

and so fn1 −→ f in K1 i.e., lim fn1 (x) = f (x) whenever x ∈ K1 \ A1 where A1 ⊂ K1 n→∞

and ρ (1A1 ) = 0. By induction and using a diagonal argument we obtain a subse

quence of { fϕ (n) }n , which converges ρ -a.e. to f.  Next we use the functional d to define the topology τ .

3.4 Compactness and Separability

67

Definition 3.13. Let C be a subset of Lρ . (a) C is said to be d-closed if and only if for any sequence { fn }n in C which d-converges to f , we have f ∈ C. (b) C is d-open if and only if Lρ \C is d-closed. (c) C is said to be d-sequentially compact if for each sequence { fn }n there exists a subsequence { fϕ (n) }n , which d-converges to a point in C. It is easily seen that the family of all d-open subsets of Lρ forms a topology on Lρ . Furthermore from Proposition 3.16, d-sequentially compact sets and ρ -a.e. compact sets are identical. Even though d satisfies (3) instead of the triangular inequality, the usual arguments, which prove that sequential compactness and compactness are identical in metric spaces, hold in this setting. We also have d-sequential compactness and d-compactness identical.

3.4 Compactness and Separability In the context of the previous sections, let us discuss the separability of (Eρ ,  · ρ ). First, we need the following definition. Definition 3.14. The function modular ρ ∈ ℜ is called separable if  f 1(·) ρ ) is a separable set function for each f ∈ E , which means that there exists a countable A ⊂ P such that to every A ∈ P there corresponds a sequence {Ak } of elements of A with ρ (α f , AΔ Ak ) → 0 (3.11) for every α > 0, where Δ denotes the symmetric set difference. We are now ready to prove the following theorem that characterizes separable (Eρ ,  · ρ ) spaces. Theorem 3.10. [132] Let ρ ∈ ℜ. The space (Eρ ,  · ρ ) is a separable Banach space if and only if ρ is separable. Proof. Assume first that (Eρ ,  · ρ ) is separable. Hence the space of all P-simple functions E ⊂ Eρ is separable. Suppose to the contrary that ρ is not a separable function modular. Then there exists a function f ∈ E , a number ε > 0, and an uncountable family of sets B ⊂ P such that  f 1BΔ B ρ > ε for arbitrary B, B ∈ B. Hence,

ε <  f 1BΔ B ρ =  f 1B − f 1B ρ . Put W = { f 1B : B ∈ B}, observe that W is an uncountable subset of E and that g − hρ | > ε for g, h ∈ W , g = h. This fact contradicts the separability of E . Let us prove the converse. For n ∈ N and r ∈ Q, let us denote by {Bn,r } a countable subfamily of sets dense in {Ωn ∩ A : A ∈ P} with respect to the pseudometric (A, B) → r1Ωn ∩(AΔ B) ρ ,

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3 Modular Function Spaces

where A, B ∈ B. Denote

B=



Bn,r .

n∈N, r∈Q

It is enough to prove that {r1B : r ∈ Q, B ∈ B} is dense in {s1A : s ∈ R, A ∈ P}. Fix s ∈ R and ε > 0. By the density of Q in R and by Proposition 3.3, we can choose an r ∈ Q such that ε s1A − r1A ρ < . 3 Since A, Ωn ∈ P and Ωn ↑ Ω , it follows that A \ Ωn ↓ 0/ and that r1A\Ωn ρ < ε3 for n sufficiently large. From the assumption of the separability of ρ we conclude that there exists a set B ∈ Bn,r such that

ε r1A∩Ωn − r1B ρ = r1Ωn ∩(AΔ B) ρ < . 3 Finally, s1A − r1B ρ ≤ s1A − r1A ρ + r1A∩Ωn − r1B ρ + r1A\Ωn ρ < ε . This completes the proof.



Finally, we can combine Theorem 3.10 with Remark 3.8 into the following very useful statement. Proposition 3.17. Let ρ ∈ ℜ be a separable function modular. If u : [a, b] → C is ρ -continuous, where C ⊂ Eρ has the Vitali property, then u ∈ L1 (Ω , Z, m). Next, let us characterize the compact subsets of the space Eρ . Theorem 3.11. [132] Let ρ ∈ ℜ. A set D ⊂ Eρ is conditionally compact in the sense of the Luxemburg norm  · ρ if and only if the following conditions are satisfied: (i) For every α > 0 the set function sup{ρ (α f , ·) : f ∈ D} is order continuous; (ii) For every sequence { fn } of elements from D there exist a subsequence { fnk } and a function f ∈ Eρ such that ρ ( fnk − f ) → 0. Proof. Sufficiency: Let fn ∈ D for n ∈ N. By (ii) we can choose a subsequence { fnk } and a function f ∈ Eρ such that fnk → f ρ -a.e. Combining this with (i) and using the Vitali theorem, we get that  fnk − f ρ → 0. Necessity: To prove (i), let us fix an ε > 0 and sets Ek ↓ 0. / By the conditional compactness of D we may find a finite set { f1 , . . . , fn } of elements of Eρ such that ε to every f ∈ D there corresponds an index i ∈ {1, . . . , n} for which  f − fi ρ < . 2 Since all functions fi belong to Eρ it follows that there exists k0 ∈ N such that ε  fi 1Ek ρ < for k ≥ k0 , 1 ≤ i ≤ n. This completes the proof of (i). To prove (ii), 2 let us choose a sequence { fn } of elements of D. Since D is conditionally compact, it follows that there exists a subsequence { fnk } of { fn } and a function f ∈ Eρ such  that  fnk − f ρ → 0, which implies that ρ ( fnk − f ) → 0.

3.5 Examples

69

3.5 Examples Let us illustrate the notion of modular function spaces with few examples. Example 3.4. L p -space, p ≥ 1, is a modular function space generated by

ρ( f ) =

 R

| f (t)| p dm(t).

Example 3.5. l p -space, p ≥ 1, is a modular function space generated by ∞

ρ ( f ) = ∑ | fi | p . i=1

Example 3.6. Orlicz space is a modular function space generated by

ρ( f ) =

 R

ϕ (| f (t)|)dm(t),

provided ϕ is an Orlicz function, that is, ϕ is a nonnegative, convex function such that ϕ (0) = 0. Example 3.7. Musielak–Orlicz space is a modular function space generated by

ρ( f ) =



φ (t, | f (t)|)dm(t),

R

provided ϕ is nonnegative, convex function in the second variable such that ϕ (x, 0) = 0, and ϕ is a measurable, locally integrable function in the first variable such that ϕ (·, u) > 0 m − a.e. for every u > 0. Example 3.8. Lorentz p-space is a modular function space generated by

ρ ( f ) = sup

 R

τ ∈T

| f (t)| p d μτ (t),

where μ is a σ -finite measure, T is a group of all measure preserving transformations τ : R → R and μτ (E) = μ (τ −1 (E)). Example 3.9. Orlicz–Lorentz space is a modular function space generated by

ρ ( f ) = sup τ ∈T

 R

ϕ (| f (t)|)d μτ (t),

where ϕ is an Orlicz function, μ is a σ -finite measure, T is a group of all measure preserving transformations τ : R → R and μτ (E) = μ (τ −1 (E)). Example 3.10. Let μ be a finite measure on Ω . Denote by (M , μ ) the space of all μ -measurable real-valued functions. Let E ⊂ (M , μ ) be an order continuous Banach function lattice with the Fatou property, that is, a Banach space with a norm  · E such that

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3 Modular Function Spaces

1. If f ∈ (M , μ ), g ∈ E and | f | ≤ |g| μ -a.e., then f ∈ E and  f E ≤ gE , 2. There exists a function in E that is strictly positive μ -a.e., 3. If f , fn ∈ E, | fn | ≤ | f | and fn → 0 μ -a.e., then  fn E → 0, 4. If 0 ≤ fn ∈ E and f ∈ (M , μ ) and fn ↑ f , then  fn E ↑  f E . Let ϕ be an Orlicz function. Define  ϕ ◦ f E , ρ( f ) = ∞,

if ϕ ◦ f ∈ E, otherwise.

It is a straightforward calculation to show that ρ is a regular convex function modular. The corresponding modular function space is called the Calderon–Lozanovskii space defined by the space E and the Orlicz function ϕ and denoted by Eϕ . Note that if E = L1 then Eϕ becomes the Orlicz space Lϕ . Similarly, if E = L p then Eϕ is the Banach space of all measurable functions such that

ρ( f ) =

 Ω

|ϕ ( f (ω ))| p d μ

1

p

< ∞.

For more general discussion of Calderon–Lozanovskii spaces, see for example [43] or [95]. Example 3.11. Let Ω = [0, 1], {Ωn } be a countable disjoint partition of Ω such that m(Ω p ) = 2−p , where m is the Lebesgue measure on [0, 1]. Let P be a δ -ring generated by the sets of the form A ∩ Ω p for all p ∈ N and all measurable sets A ⊂ [0, 1]. For a measurable function f : Ω → R, we put

ρ( f ) =

∞

∑

p=1



1 | f (t)| dm(t) p

Ωp



p

+ sup

p∈N Ω p

| f (t)| p dm(t).

Example 3.12. An extremal flexibility gained by using the apparatus of the modular function spaces can be illustrated as follows: the operator itself is used for the construction of a modular and hence a space in which this operator has required properties. Let us consider for instance the following Uryshon integral operator:  1

T ( f )(x) = 0

k(x, y, | f (y)|)dy + f0 (x),

where f0 is a fixed function and f : [0, 1] → R is Lebesgue measurable. For the kernel k we assume that (a) k : [0, 1] × [0, 1] × R+ → R+ is Lebesgue measurable, (b) k(x, y, 0) = 0, (c) k(x, y, .) is continuous, convex and increasing to +∞,  1

(d) 0

k(x, y,t)dx > 0 for t > 0 and y ∈ (0, 1).

3.6 Generalizations and Special Cases

71

Assume in addition that for almost all t ∈ [0, 1] and measurable functions f , g there holds  1  1  1  k(t, u, |k(u, v, | f (v)|) − k(u, v, |g(v)|)|)dv du ≤ k(t, u, | f (u) − g(u)|)du. 0

0

Setting ρ ( f ) =

0

 1  1 0

0

 k(x, y, | f (y)|)dy dx and using Jensen’s inequality, it is easy

to show that ρ is a nonnegative, even, convex nonlinear functional on the space of measurable functions, Lρ = { f : [0, 1] → R : ∃λ > 0, ρ (λ f ) < ∞}, and that ρ (T ( f ) − T (g)) ≤ ρ ( f − g), that is, T is nonexpansive with respect to ρ . We will come back to this example in Chap. 4 (Example 5.1) and Chap. 5 (Example 7.6).

3.6 Generalizations and Special Cases 3.6.1 General Definition of Function Modular While in this book we will always restrict our considerations to the modular function spaces of scalar-valued measurable functions defined by convex regular function modulars, it is worthwhile to mention a more general framework and, in this context, to point to some interesting applications and possible research directions. The general framework presented in this section follows the book [132] and, as it was mentioned earlier in this chapter, the notions used in this book are special cases of the general theory, hence all the results of the general theory of modular function spaces are applicable to convex regular function modulars defined on spaces of scalar-valued measurable functions. From this perspective, the reader who is not interested in the general theory may skip this section. Let S denote a Banach space equipped with a norm  · S . A function f : Ω → S is called measurable if there exists a sequence of P-measurable simple functions {sn } such that sn (ω ) → f (ω ) for every ω ∈ Ω . It is well known that a function f is measurable if and only if it is separably valued in S and for every open B ⊂ S, f −1 (B) ∈ Σ . Thus, if f is measurable then  f (ω )S is a scalar-valued measurable function of ω . It follows from the properties of scalar measurable functions that there exists a sequence of P-measurable simple functions {sn } such that sn (ω )S ↑  f (ω ))S for every ω ∈ Ω . Let us denote by M (Ω , S) the vector space of all measurable functions acting from Ω to S. Let us recall the definitions of a subadditive and a σ -subadditive measures. Definition 3.15. A set function μ : Σ → [0, ∞) is called a subadditive measure if 1. μ (0) / =0 2. μ (A ∪ B) ≤ μ (A) + μ (B) for A, B ∈ Σ such that A ∩ B = 0/ 3. μ (A) ≤ μ (B) if A, B ∈ Σ with A ⊂ B.

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3 Modular Function Spaces

If, moreover,

μ

∞ 

 En ≤

n=1

∞

∑ μ (En )

n=1

for every sequence of sets En ∈ Σ , then μ is called a σ -subadditive measure. We are now ready to give a general definition of a function modular. Definition 3.16. Let ρ : E × Σ → [0, ∞) satisfy the following conditions: 1. ρ (0, E) = 0 for every E ∈ Σ 2. ρ ( f , E) ≤ ρ (g, E) whenever  f (ω )S ≤ g(ω )S for all ω ∈ E and all f ∈ E , E ∈Σ 3. ρ ( f , ·) : Σ → 0, ∞) is a σ -subadditive measure for every f ∈ E 4. ρ (α , A) → 0 as α ↓ 0 for every A ∈ P 5. If ρ (α , A) = 0 for an α > 0 then sup ρ (β , A) = 0 β >0

6. For every α > 0, ρ (α , ·) is order continuous on P. The definition of ρ is then extended to all f ∈ M (Ω , S) and all sets E ∈ Σ by defining that

ρ ( f , E) = sup{ρ (g, E) : g ∈ E , g(ω )S ≤  f (ω )S , ω ∈ E}. As a convention, we set ρ ( f ) = ρ ( f , Ω ). As already mentioned in Sect. 3.1, it is a straightforward exercise to prove that if S = R and if in addition ρ is convex in its first argument and it has the Fatou property, that is ρ ( fn ) ↑ ρ ( f ) whenever  fn (ω )S ↑  f (ω )S for every ω ∈ Ω , then ρ is a regular convex function modular in the sense of Definition 3.2. This general definition allows us to investigate a wide range of special cases and applications. To the end of this section we introduce few examples of such applications and refer the reader to the corresponding literature. Since these constructs and results will not be used in this book, we limit our discussion to giving some basic facts without proofs. The wide range of applications of the general theory of modular function spaces suggests that extending the fixed point theory to this more general framework will produce results of high interest to researchers from many areas of mathematics and beyond. Since there exist only very few fix point results for nonconvex function modulars or for spaces of functions with values in Banach spaces, this looks like a promising new research direction.

3.6.2 Nonlinear Operator Valued Measures The relationship between the theory of modular function spaces and the theory of nonlinear operator valued measures, discussed below, shows how close the results

3.6 Generalizations and Special Cases

73

presented in this book are to the theory of abstract integration and representation of nonlinear operators. From this point of view, it is not surprising that several applications of fixed point theory in modular function spaces point to problems of nonlinear operator theory and related theory of integral and differential equations. Let F be a Banach space. By N(S, F), we denote the space of all mappings U : S → F such that U(0) = 0 and U are uniformly continuous on bounded subsets of S. A set function μ : P → N(S, F) is called an operator valued measure if μ satisfies the following conditions: 1. μ (0) = 0 2. μ is countably additive in the pointwise sense 3. lim svδ (μ , α , E) = 0 for every E ∈ P and α > 0 δ →0

4. μ (·)r is order continuous for every r ∈ S. Remember that  

 n    svδ (μ , α , E) = sup  ∑ (μ (Ei )ri − μ (Ei )ri ) F

i=1

where the supremum is taken over all finite sequences of sets Ei , and over elements  ri , ri in S such that n

Ei ⊂ E, Ei ∈ P, rS , r S ≤ α , r − r S ≤ δ , 1 ≤ i ≤ n, n ∈ N.

i=1

Recall also the following definition:

μ (E)r = sup{μ (A)rF : A ⊂ E, A ∈ P}. For a P-simple function, f =

n

∑ ri 1Ei , where ri ∈ S and Ei ∈ P. We define the

i=1

integral with respect to the operator valued measure μ over a set E ∈ Σ by  E

n

f d μ = ∑ μ (E ∩ Ei )ri . i=1

The domain of the integration is then extended to the space M (μ ) of all measurable functions f such that there exists a sequence of simple functions fn converging  everywhere to f for which the integrals (·)

fn d μ are uniformly countably additive

on Σ . Let us assume also that there exists α > 0 such that if f ∈ E such that  f (ω )S ≤ α in E, then



E



E

f d μ = 0 for every

gd μ = 0 for all g ∈ E . Define



     ρ ( f , E) = sup  gd μ  : g ∈ E , g(ω )S ≤  f (ω )S , ω ∈ E . E

F

74

3 Modular Function Spaces

It is not difficult to show that ρ defined by (3.6.2) is a function modular in the sense of Definition 3.16. The integral introduced above may be regarded as a nonlinear operator defined in M (μ ) with values in Banach space F. It may be proved that Lρ0 ⊂ M (μ ) and that the integral is continuous in Eρ , see [131]. Hence the apparatus of the theory of modular function spaces can be applied to spaces of functions integrable with respect to operator valued measures. The linear operator valued measures were investigated by Dobrakov in a series of papers starting from [60] and [61]. Integration of bounded functions with respect to nonlinear operator valued measures was considered by Batt [15] and Friedman and Tong [75] in connection with the representation of orthogonally additive operators. This integration was extended to the larger class M (μ ) by Kozlowski and Szczypinski in [145, 146]. For the exposition of this theory we refer the reader to the paper by Szczypinski [189]. Example 3.13. Let S and F be two Banach spaces and let ϕ : S → F be a locally uniformly continuous function with ϕ (0) = 0. Assume that m is a positive finite measure on Σ . Let us define an operator valued measure μ (A)r = m(A)ϕ (r), for A ∈ Σ and r ∈ S. As shown in [189], if ϕ ◦ f : Ω → F is integrable in the sense of Bochner, then f is integrable with respect to μ and  E

f dμ =

 E

(ϕ ◦ f )dm.

Example 3.14. Let F be a Banach space, S = R and let m be a countably additive vector valued measure m : Σ → F. Define an operator valued measure μ (E)r = rm(E) for r ∈ R and E ∈ Σ . The integral of f : Ω → R with respect to μ is equal to the integral introduced by Dunford and Schwartz [70]. The following two examples show that some nonlinear operators important for the theory of integral and differential equations can be represented as integrals with respect to respective operator valued measures. These examples resonate with examples presented in this book, see Example 3.12. Example 3.15. Let m be a positive σ -additive measure on Σ . Let S = R and let F = L1 (Ω ) and the function k : Ω × Ω × R → R satisfy the following conditions: 1. k(ω , σ , 0) ≡ 0 2. k(ω , σ , r) is continuous in r almost everywhere in Ω × Ω 3. k(ω , σ , r) is measurable in (ω , σ ) for every fixed r 4. To every ε > 0, there exists δ > 0 such that if m(A) < δ then  ! ! ! ! ! k(ω , σ , r)dm(σ )!dm(ω ) < ε Ω

A

3.6 Generalizations and Special Cases

75

uniformly in r. Define μ : Σ → N(R, L1 (Ω )) by

μ (E)r =

 E

k(·, σ , r)dm(σ ) ∈ L1 (Ω ).

It is easy to verify that μ is an operator valued measure and that for f ∈ M (μ ), we have    f d μ (ω ) = k(ω , σ , f (σ ))dm(σ ). E

E

Hence the integral of f with respect to μ is actually the Urysohn integral operator. Example 3.16. Let m be a positive σ -additive measure on Σ . Let S = R and F = L1 (Ω ) and let H : Ω × R → R be such that H(·, r) ∈ L1 (Ω ) for every r ∈ R, H(ω , ·) is continuous almost everywhere in ω , and H(·, 0) ≡ 0. Define μ as

μ (A)r = H(·, r1A ) for A ∈ Σ . Then μ is an operator valued measure such that for any f ∈ M (μ ),   f d μ (ω ) = H(ω , f (ω )), A

that is taking the integral with respect to μ is equivalent to applying the Nemytskii operator defined by H.

3.6.3 Nonlinear Fourier Transform It is well known that the linear Fourier transform F ( f )(x) =

 R

e−tx f (t)dm(t)

(3.12)

is the linear isometry from the Hilbert space L2 onto itself. The situation is much more complicated with the nonlinear version of the Fourier transform defined as 

F( f )(x) =

R

e−tx ϕ ( f (t))dm(t),

(3.13)

where ϕ : R → R is an increasing, continuous, odd function such that lim ϕ (u) = +∞, and lim ϕ (u) = −∞.

u→+∞

u→−∞

The major question is how to define the proper domain of F, that is, such a subspace E of the space M of all m-measurable functions that T : E → L2 and is continuous. This task can be achieved by using the tools and methods of modular function

76

3 Modular Function Spaces

spaces. Indeed, let us define

ρ ( f , A) = sup{T (g)L2 : g ∈ E , g(ω )S ≤ 1A (ω ) f (ω )S }.

(3.14)

It is not difficult to prove that the formula (3.14) defines a function modular as well as that T : Eρ → L2 and that T is continuous (see e.g., [132]). We will show that Eρ = E ψ where ψ is an Orlicz function defined by ψ (u) = (ϕ (u))2 , and E ψ = { f ∈ M : ρ (λ f ) < ∞, f or all λ > 0}. Let f ∈ E ψ . Let α > 0 and Ak ∈ P (that is Ak are of finite measure) be such that / To prove that f ∈ Eρ we need to show that ρ (α f , Ak ) → 0. Since for every Ak ↓ 0. constant α > 0 and every g ∈ E the function ϕ (α g(·)) belongs to L2 and the linear Fourier transform F : L2 → L2 is an isometry, then we get the following sequence of equalities:

ρ (α f , Ak ) = sup

!2

  !  1  2 ! ! e−itx ϕ (α g(t))dm(t)! dm(x) : g ∈ E , |g| ≤ | f | ! R

Ak

 !2 1 

  !  2 ! ! = sup !F 1Ak (·)ϕ (α g(·)) (x)! dm(x) : g ∈ E , |g| ≤ | f |

 R     = sup F 1Ak (·)ϕ (α g(·))  2 : g ∈ E , |g| ≤ | f | L 

= sup 1Ak (·)ϕ (α g(·))L2 : g ∈ E , |g| ≤ | f |

  1  2 = sup ϕ 2 (α g(t))dm(t) : g ∈ E , |g| ≤ | f | ≤

 Ak

Ak

ϕ (α f (t))dm(t) 2



1 2

= Ak

ψ (α f (t))dm(t)

1 2

→ 0.

Hence, E ψ ⊂ Eϕ . In order to prove the inverse inclusion, let us take f ∈ Eϕ and hence note that ρ (α f , Ak ) → 0. For every simple function gk ∈ E such that |gk | ≤ | f 1Ak |, we get !2  ! 1 2 ! ! e−itx ϕ (α gk (t))dm(t)! dm(x) ≤ ρ (α f , Ak ) → 0, ! R

Ak

  which means that F ϕ (gk (·)) → 0. Since E ⊂ E ϕ , it follows that the function

ϕ (α gk (·)) is a member of the space L2 . The inverse Fourier transform acts continuously from L2 to L2 and therefore {ϕ (α gk (·))} converges to zero in L2 , that is,  R

ϕ 2 (α gk (t))dm(t) → 0.

(3.15)

3.6 Generalizations and Special Cases

77

We claim now that f ∈ E ψ . Indeed, assume to the contrary that there exists α > 0 such that  ϕ 2 (α f (t))dm(t) = ∞. R

There exist then ε > 0 and a sequence Ak ↓ 0/ such that for every k,  Ak

ϕ 2 (α f (t))dm(t) > ε .

Hence we may choose a sequence of simple functions gk so that |gk | ≤ | f 1Ak | and 

ε ϕ 2 (α gk (t))dm(t) > , 2 R

which contradicts (3.15). The contradiction completes the proof of the equality Eρ = E ψ .

Chapter 4

Geometry of Modular Function Spaces

In this chapter, we discuss the general geometrical methods for consideration of fixed point properties, similarly, in the case of Banach space setting. The most common approach in the Banach space fixed point theory for nonexpansive mappings is to assume the uniform convexity of the norm, which implies the reflexivity, which in turn guarantees the weak compactness of the closed bounded sets. As we will see further, the notion of a uniform convexity of function modulars in conjunction with the property (R) being the modular equivalence of the Banach space reflexivity equips us with powerful tools for proving the fixed point property in modular function spaces. Also, this theory provides a set of powerful techniques for proving existence of common fixed points for commutative families of mappings acting in modular function spaces, and for investigating the topological properties of the set of common fixed points.

4.1 Uniform Convexity in Modular Function Spaces Let us start by reminding ourselves that the modulus of convexity of a Banach space X is defined as     x+y  : x, y ∈ X, x ≤ 1, y ≤ 1, x − y ≥ ε .  δ (ε ) = inf 1 −  2  A Banach space X is called uniformly convex if δ (ε ) > 0 for every ε > 0. We will investigate notions of the modular equivalents of uniform convexity of ρ . As demonstrated below, one concept of uniform convexity in normed spaces generates several different types of uniform convexity in modular function spaces. This is due primarily to the fact that, in general, modulars are not homogeneous. Definition 4.1. Let ρ ∈ ℜ. We define the following uniform convexity type properties of the function modular ρ :

c Springer International Publishing Switzerland 2015  M. A. Khamsi, W. M. Kozlowski, Fixed Point Theory in Modular Function Spaces, DOI 10.1007/978-3-319-14051-3 4

79

80

4 Geometry of Modular Function Spaces

(a) Let r > 0, ε > 0. Define   D1 (r, ε ) = ( f , g) : f , g ∈ Lρ , ρ ( f ) ≤ r, ρ (g) ≤ r, ρ ( f − g) ≥ ε r . Let



  f +g 1 : ( f , g) ∈ D1 (r, ε ) , if D1 (r, ε ) = 0, δ1 (r, ε ) = inf 1 − ρ / r 2

/ We say that ρ satisfies (UC1) if for every and δ1 (r, ε ) = 1 i f D1 (r, ε ) = 0. / for r > 0, ε > 0, we have δ1 (r, ε ) > 0. Note, that for every r > 0, D1 (r, ε ) = 0, ε > 0 small enough. (b) We say that ρ satisfies (UUC1) if for every s ≥ 0, ε > 0, there exists η1 (s, ε ) > 0 depending on s and ε such that

δ1 (r, ε ) > η1 (s, ε ) > 0 for r > s. (c) Let r > 0, ε > 0. Define 

  f −g ≥ εr . D2 (r, ε ) = ( f , g) : f , g ∈ Lρ , ρ ( f ) ≤ r, ρ (g) ≤ r, ρ 2 Let



1 δ2 (r, ε ) = inf 1 − ρ r

f +g 2



 : ( f , g) ∈ D2 (r, ε ) , if D2 (r, ε ) = 0, /

/ We say that ρ satisfies (UC2) if for every and δ2 (r, ε ) = 1 i f D2 (r, ε ) = 0. / for r > 0, ε > 0, we have δ2 (r, ε ) > 0. Note, that for every r > 0, D2 (r, ε ) = 0, ε > 0 small enough. (d) We say that ρ satisfies (UUC2) if for every s ≥ 0, ε > 0 there exists η2 (s, ε ) > 0 depending on s and ε such that

δ2 (r, ε ) > η2 (s, ε ) > 0 for r > s. (e) We say that ρ is strictly convex, (SC), if for every f , g ∈ Lρ such that ρ ( f ) = ρ (g) and 

ρ ( f ) + ρ (g) f +g = ρ 2 2 we have f = g. Remark 4.1. Let us observe that for i = 1, 2, δi (r, 0) = 0 , and δi (r, ε ) is an increasing function of ε for every fixed r. Note also that 

 δ1 (r, ε ) = inf δ (r, h) : h ∈ Lρ , ρ (h) ≥ rε ,

  h ≥ rε , δ2 (r, ε ) = inf δ (r, h) : h ∈ Lρ , ρ 2 



4.1 Uniform Convexity in Modular Function Spaces

where

81



   h 1 : f ∈ Lρ , ρ ( f ) ≤ r, ρ ( f + h) ≤ r . δ (r, h) = inf 1 − ρ f + r 2

Proposition 4.1. The following conditions characterize relationship between the above defined notions: (a) (UUCi) implies (UCi) f or i = 1, 2; (b) δ1 (r, ε ) ≤ δ2 (r, ε ); (c) (UC1) implies (UC2); (d) (UC2) implies (SC); (e) (UUC1) implies (UUC2); Proof. The proofs of these implications follow immediately from Definition 4.1.

 Remark 4.2. Observe that, denoting ρα (u) = αρ (u), and the corresponding moduli of convexity by δρα ,i , where i = 1, 2, we have r  δρα ,i (r, ε ) = δρ ,i ,ε , α and

δρ ,i (r, ε ) = δρα ,i (rα , ε ). Hence, ρ is (UCx), where (UCx) is any of the conditions from Definition 4.1, if and 1 only if there exists α > 0 such that ρα is (UCx). In particular, taking α = , it is r enough to prove any of the conditions defining (UCx) with r = 1. Example 4.1. It is known that in Orlicz spaces, the Luxemburg norm is uniformly convex if and only if ϕ is uniformly convex and Δ2 property holds; this result can be traced to early papers by Luxemburg [157], Milnes [163], Akimovic [4], and Kaminska [101]. It is also known that, under suitable assumptions, (UC2) in Orlicz spaces is equivalent to the very convexity of the Orlicz function [117, 44]. Remember that the function ϕ is called very convex if or every ε > 0 and any x0 > 0, there exists δ > 0 such that 

1 ε (x − y) ≥ (ϕ (x) + ϕ (y)) ≥ εϕ (x0 ), ϕ 2 2 implies

ϕ

 1 1 (x + y) ≤ (1 − δ ) (ϕ (x) + ϕ (y)) . 2 2

Typical examples of Orlicz functions that are are very convex and do not satisfy Δ2 2 condition are: ϕ1 (t) = e|t| − |t| − 1 and ϕ2 (t) = et − 1, [163, 147]. Therefore, these are the examples of Orlicz spaces that are not uniformly convex in the norm sense, and hence the classical Kirk theorem cannot be applied. However, these spaces are

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4 Geometry of Modular Function Spaces

uniformly convex in the modular sense, and respective modular fixed point results can be applied. Let us recall a notion of bounded away sequences of real numbers. Definition 4.2. A sequence {tn } ⊂ (0, 1) is called bounded away from 0 if there exists 0 < a < 1 such that tn ≥ a for every n ∈ N. Similarly, {tn } ⊂ (0, 1) is called bounded away from 1 if there exists 0 < b < 1 such that tn ≤ b for every n ∈ N. We need to prove a series of technical results that will play an essential role in proving our fixed point theorems. First, let us mention the following result whose proof is elementary. Note that for t = 12 , this result follows directly from Definition 4.1. Lemma 4.1. Let ρ ∈ ℜ be (UUC1) and let t ∈ (0, 1). Then for every s > 0, ε > 0 there exists η1t (s, ε ) > 0 depending only on s and ε such that

δ1t (r, ε ) > η1t (s, ε ) > 0 f or any r > s. The next lemma introduces a useful technique which is used extensively for investigating convergence to fixed points in the (UUC1) modular function spaces. It was introduced in [115] for the case tn = 12 and extended to more general case in [54], see, for example, [188] for the Banach space equivalent. Lemma 4.2. Let ρ ∈ ℜ be (UUC1) and let {tn } ⊂ (0, 1) be bounded away from 0 and 1. If there exists R > 0 such that lim sup ρ ( fn ) ≤ R, lim sup ρ (gn ) ≤ R, n→∞

n→∞

lim ρ (tn fn + (1 − tn )gn ) = R,

n→∞

then, lim ρ ( fn − gn ) = 0.

n→∞

Proof. Assume to the contrary that this is not the case and fix an arbitrary γ > 0. Passing to a subsequence if necessary, we may assume that there exists an ε > 0 such that ρ ( fn ) ≤ R + γ , ρ (gn ) ≤ R + γ (4.1) while

ρ ( fn − gn ) ≥ (R + γ )ε .

(4.2)

Since {tn } is bounded away from 0 and 1, there exists 0 < a < b < 1 such that a ≤ tn ≤ b for all natural n. Passing to a subsequence if necessary, we can assume that there exists t0 ∈ [a, b] with tn → t0 . For every t ∈ [01] and f , g ∈ D1 (R + γ , ε ), let us define λ f ,g (t) = ρ (t f + (1 − t)g). Observe that the function λ f ,g : [0, 1] → [0, R + γ ] is a convex function. Hence, the function

λ (t) = sup{λ f ,g (t) : f , g ∈ D1 (R + γ , ε )}

4.1 Uniform Convexity in Modular Function Spaces

83

is also convex on [0, 1], and consequently it is a continuous function on [a, b]. Noting that 1 δ1t (R + γ , ε ) = 1 − λ f ,g (t), r t we conclude that δ1 (R + γ , ε ) is a continuous function of t ∈ [a, b]. Hence lim δ tn (R + γ , ε ) = δ1t0 (R + γ , ε ). n→∞ 1

(4.3)

By (4.1) and (4.2)

δ1tn (R + γ , ε ) ≤ 1 −

1 ρ (tn fn + (1 − tn )gn ). R+γ

(4.4)

t

From (4.3), we deduce that the left hand side of (4.4) tends to δ10 (R + γ , ε ) as n → ∞ γ while the right hand side tends to R+ γ in view of (4.2). Hence

δ1t0 (R + γ , ε ) ≤

γ . R+γ

(4.5)

t

By (UUC1) and Lemma 4.1, there exists η10 (R, ε ) > 0 satisfying 0 < η1t0 (R, ε ) ≤ δ1t0 (R + γ , ε ).

(4.6)

Combining (4.5) with (4.6), we get t

0 < η10 (R, ε ) ≤

γ . R+γ

 Letting γ → 0, we get a contradiction that completes the proof.

Lemma 4.3. Let ρ satisfies (UC2). Let f , g ∈ Lρ and r > 0 be such that f = g, ρ ( f ) ≤ r and ρ (g) ≤ r. Then, 

f +g < r. ρ 2 h > 0. By the definition of δ2 , it follows Proof. Set h = f −g. Since f = g, then ρ 2 from 

   f −g h h 1 =ρ = ρ r ρ 2 2 r 2

that

ρ

f +g 2



"

≤ r 1 − δ2

Note that

δ2

1 r, ρ r

1 r, ρ r

# h . 2

 h >0 2

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4 Geometry of Modular Function Spaces

 h 1 > 0. Hence because ρ satisfies (UC2) and ρ r 2 

f +g < r, ρ 2 as claimed.



Lemma 4.4. Let ρ satisfies (UC2). Then, for every r > 0, δ2 (r, 1) = 1. Proof. Fix r > 0. Let (x, y) ∈ D2 (r, 1), i.e., ρ (x) ≤ r, ρ (y) ≤ r, and 

x−y ≥ r. ρ 2 Observe that r≤ρ

x−y

ρ (x) + ρ (y) ≤ r. 2 

x−y = r. ρ 2

2

Hence

≤

Setting z = −y, we get

r=ρ

x−z 2



=ρ

x+y 2

 ≤ r.

By applying Lemma 4.3 to f = x and g = z, we obtain that x = z, i.e., x = −y, which implies that 

x+y = 0. ρ 2 Hence, by the definition of δ2 , we conclude that δ2 (r, 1) = 1 as claimed.



In the next theorem, we investigate relationship between the uniform convexity of function modulars and the unique best approximant property which generalizes well known properties of Hilbert and uniformly convex Banach spaces. For other results on best approximation in modular function spaces see [142, 143, 118, 119]. Theorem 4.1. Assume ρ ∈ ℜ is (UUC2). Let C ⊂ Lρ be nonempty, convex, and ρ closed. Let f ∈ Lρ be such that d = dρ ( f ,C) < ∞. There exists then a unique best ρ -approximant of f in C, that is, a unique g0 ∈ C such that

ρ ( f − g0 ) = dρ ( f ,C). Proof. The uniqueness follows from the strict convexity of ρ ; remember that (UUC2) implies (SC). Let us prove the existence of the ρ -approximant. Since C is ρ -closed, we may assume without loss of any generality that d := dρ ( f ,C) > 0.

4.1 Uniform Convexity in Modular Function Spaces

85

Clearly there exists a sequence { fn } ∈ C such that 

1 . ρ ( f − fn ) ≤ d 1 + n   1 fn is ρ -Cauchy. Assume to the contrary that this is not the case. We claim that 2 There exist an ε0 > 0 and a subsequence { fnk } of { fn } such that 

f nk − f n p ≥ ε0 , ρ 2 for any p, k ≥ 1. Since ρ is (UUC2) then ρ is (UC2). Hence  

f nk + f n p ε0 ≤ 1 − δ2 d(k, p), d(k, p), ρ f− 2 d(k, p) 

1 d. For p, k ≥ 1, we have d(k, p) ≤ 2d. Hence where d(k, p) = 1 + min(n p , nk ) 

 ε0 ε0  . ≥ δ2 d(k, p), δ2 d(k, p), d(k, p) 2d Since ρ is (UUC2) then there exists η > 0 such that  ε  0 ≥η δ2 r, 2d for any r >

d d . Since d(k, p) ≥ d > , we get 3 3 

fn + fn p ≤ (1 − η ) d(k, p), ρ f− k 2

fn + fn p ∈ C. Using the definition of d, we for any k, p ≥ 1. By the convexity of C, k 2 get 

f nk + f n p d≤ρ f− ≤ (1 − η ) d(k, p), 2 for any k, p ≥1. Ifwe let k, p go to infinity, we get d ≤ (1 − η)d, which  is impos1 1 fn is ρ -Cauchy. By the completeness of Lρ , fn ρ -converges sible. Hence 2 2   fm + fn fm + g and ∈ C and ρ -converges to to a g ∈ Lρ . Fix m ≥ 1. Since 2 2 fm + g ∈ C. Letting m → ∞, we get 2g ∈ C. By C is ρ -closed, then we have 2

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4 Geometry of Modular Function Spaces

Propositions 3.4 and 3.5, passing to a subsequence if necessary, we get  

fn fn + fm ≤ lim inf lim inf ρ f − . ρ ( f − 2g) ≤ lim inf ρ f − g − n→∞ n→∞ m→∞ 2 2 Since ρ is convex, we get 

fn + fm ρ ( f − fn ) + ρ ( f − fm ) ≤ lim inf lim inf ≤ d. lim inf lim inf ρ f − n→∞ m→∞ n→∞ m→∞ 2 2 Hence, ρ ( f −2g) ≤ d. Since 2g ∈ C, we get d ≤ ρ ( f −2g). Therefore, ρ ( f −2g) = d.  In other words, g0 = 2g is the ρ -approximant of f in C.

We will use Theorem 4.1 in the proof of our next result to establish a relationship between the modular uniform convexity and the property (R), which is a modular equivalent of the Milman–Pettis theorem, which states that the uniform convexity of a Banach space implies its reflexivity. Theorem 4.2. Let ρ ∈ ℜ be (UUC2), then Lρ has property (R). Proof. Let {Cn } be a nonincreasing sequence of nonempty, ρ -bounded, ρ -closed, convex subsets of Lρ . According to Definition 3.9 we need to demonstrate that {Cn } has nonempty intersection. Fix any f ∈ C1 . By the ρ -boundedness of C1 , there exists a finite constant M > 0 such that for any n ≥ 1, ρ ( f − g) < M for any g ∈ Cn ⊂ C1 . Hence, sup dρ ( f ,Cn ) < ∞. n≥1

Using the proximinality of ρ -closed convex subsets of Lρ (Theorem 4.1), for every n ≥ 1 there exists fn ∈ Cn such that ρ ( f − fn ) = dρ ( f ,Cn ). It is easy to show that {dρ ( f ,Cn )} is nondecreasing and bounded. Hence lim dρ ( f ,Cn ) = d exists. n→∞

Assume that d = 0, then dρ ( f ,Cn ) = 0, for any n ≥ 1. Since {Cn } are ρ -closed, we  get f ∈ Cn for any n ≥ 1, which implies Cn = 0. / Therefore, we can assume d > 0. n≥1   1 fn is ρ -Cauchy. Indeed if we assume the contrary, In this case we claim that 2 then there exists an ε0 > 0 and a subsequence { fnk } of { fn } such that 

f nk − f n p ≥ ε0 , ρ 2 for any p, k ≥ 1. Since ρ is (UUC2) it follows that ρ is (UC2). Hence  

 ε  f nk + f n p 0 d, ≤ 1 − δ2 d, ρ f− 2 d

4.1 Uniform Convexity in Modular Function Spaces

87

for any p, k ≥ 1. So

dρ ( f ,Cmax(n p ,nk ) ) ≤ ρ

fn + fn p f− k 2



  ε  0 ≤ 1 − δ2 d, d, d

for any p, k ≥ 1. If we let p, k → ∞, we will get  ε   0 d, d ≤ 1 − δ2 d, d  ε  0 which is a contradiction because δ2 d, > 0 by (UC2). Hence the sequence d   1 fn is ρ -Cauchy, and therefore it ρ -converges to some g ∈ Lρ . Let us prove that 2 fk + f p ∈ Cn , for any p, k ≥ n. Fix any 2g ∈ Cn , for any n ≥ 1. Indeed, we have 2   fk + f p fk ρ -converges to + g as p → ∞, and Cn is ρ -closed, then k ≥ n. Since 2 2 fk + g ∈ Cn , for any k ≥ n. If we let k → ∞, we get 2g ∈ Cn , for any n ≥ 1. Hence 2  Cn = 0, / which completes the proof.

 n≥1

In fact, the conclusion of Theorem 4.2 may be improved to have an intersection property for any family of subsets. Theorem 4.3. Let ρ ∈ ℜ be (UC2). Let {Cα }α ∈Γ be a decreasing family of nonempty, convex, ρ -closed subsets of Lρ , where (Γ , ≺) is upward directed. Assume that there  exists f ∈ Lρ such that sup dρ ( f ,Cα ) < ∞. Then, Cα = 0. / α ∈Γ

α ∈Γ

Proof. Set d = sup dρ ( f ,Cα ). Without loss of generality, we may assume d > 0. α ∈Γ

For Any n ≥ 1, there exists αn ∈ Γ such that 

1 < dρ ( f ,Cαn ) ≤ d. d 1− n . In particular we have Since (Γ , ≺) is upward directed, we may assume αn ≺ αn+1 Cαn = 0. / Clearly C0 Cαn+1 ⊂ Cαn for any n ≥ 1. Since ρ is (UC2), we get C0 = n≥1

is ρ -closed and dρ ( f ,C0 ) = sup dρ ( f ,Cαn ) = d. n≥1

Again using the property (UC2) satisfied by ρ , there exists unique g0 ∈ C0 such that dρ ( f ,C0 ) = ρ ( f − g0 ). Let us prove that g0 ∈ Cα for any α ∈ Γ . Fix α ∈ Γ . If for some n ≥ 1 we have α ≺ αn , then obviously we have g0 ∈ Cαn ⊂ Cα . Therefore, let us assume that α ≺ αn , for any n ≥ 1. Since Γ is upward directed, there exists βn ∈ Γ such that αn ≺ βn and α ≺ βn for any n ≥ 1. We can also assume that βn ≺ βn+1 for

88

4 Geometry of Modular Function Spaces

any n ≥ 1. Again we have C1 = C1 ⊂ C0 . Moreover we have



Cβn = 0. / Since Cβn ⊂ Cαn , for any n ≥ 1, we get

n≥1

d = dρ ( f ,C0 ) ≤ dρ ( f ,C1 ) = sup dρ ( f ,Cβn ) ≤ d. n≥1

Hence, dρ ( f ,C1 ) = d which implies the existence of a unique point g1 ∈ C1 such that dρ ( f ,C1 ) = ρ ( f − g1 ) = d. Since ρ is uniformly convex, it must be (SC). Hence, g0 = g1 . In particular, we have g0 ∈ Cβn , for any n ≥ 1. Since α ≺ βn , we conclude that Cβn ⊂ Cα , for any n ≥ 1, which implies g0 ∈ Cα . Since α was taking arbitrary   Cα , which implies Cα = 0. /

 in Γ , we get g0 ∈ α ∈Γ

α ∈Γ

Since ρ is convex, ρ -closed balls are convex. Theorem 4.3 implies the following. Corollary 4.1. Let ρ ∈ ℜ and C ⊂ Lρ be nonempty, convex, ρ -closed, and ρ bounded. Assume ρ is (UC2). Then A (C) is compact.

4.2 Parallelogram Inequality and Minimizing Sequence Property We will establish now a modular version of the parallelogram inequality for uniformly convex modular function spaces . See [201] and [18] for the norm version of this inequality. Lemma 4.5. For each 0 < s < r and ε > 0, set 

 f +g 1 2 1 , Ψ (r, s, ε ) = inf ρ ( f ) + ρ 2 (g) − ρ 2 2 2 2 where the infimum is taken over all f , g ∈ Lρ such that ρ ( f ) ≤ r, ρ (g) ≤ r, max (ρ ( f ), ρ (g)) ≥ s, and ρ ( f − g) ≥ rε . If ρ ∈ ℜ is (UUC1), then Ψ (r, s, ε ) > 0 for any 0 < s < r and ε > 0. Moreover, for a fixed r, s > 0, we have (a)Ψ (r, s, 0) = 0; (b)Ψ (r, s, ε ) is a nondecreasing function of ε ; (c)if lim Ψ (r, s,tn ) = 0, then lim tn = 0. n→∞

n→∞

Proof. It is easy to see that Ψ (r, s, ε ) ≥ 0 for any 0 < s < r and ε > 0. We want to show that Ψ (r, s, ε ) > 0. Assume to the contrary that there exist 0 < s < r and ε > 0 such that Ψ (r, s, ε ) = 0. Then there exist { fn } and {gn } such that  f +g  1 1 2 n n =0 ρ ( fn ) + ρ 2 (gn ) − ρ 2 n→∞ 2 2 2 lim

(4.7)

4.2 Parallelogram Inequality and Minimizing Sequence Property



89



and ρ ( fn ) ≤ r, ρ (gn ) ≤ r, max ρ ( fn ), ρ (gn ) ≥ s, and ρ ( fn − gn ) ≥ rε . Since ρ is convex, we get

ρ2

 f + g   ρ ( f ) + ρ (g ) 2 ρ 2 ( f ) + ρ 2 (g ) n n n n n n ≤ , ≤ 2 2 2

where we used the inequality 2ab ≤ a2 + b2 , for any a, b ∈ R. Hence  ρ ( f ) − ρ (g ) 2 1  f +g  1 n n n n , ≤ ρ 2 ( fn ) + ρ 2 (gn ) − ρ 2 2 2 2 2   which implies lim ρ ( fn ) − ρ (gn ) = 0. Without loss of any generality, we may n→∞

assume that lim ρ ( fn ) = R exists. This also implies that lim ρ (gn ) = R. By (4.7) n→∞ n→∞ we get then  f +g  n n = R. lim ρ (gn ) = lim ρ 2 n→∞ n→∞ 2 Observe that   R = lim max ρ ( fn ), ρ (gn ) ≥ s > 0. n→∞

By Lemma 4.2 then ρ ( fn − gn ) → 0 contradicting the fact that ρ ( fn − gn ) ≥ rε > 0. The proofs of (a), (b), and (c) are straightforward.

 Let us introduce a notion of a ρ -type , a powerful technical tool that will be used in the proofs of our fixed point results. Definition 4.3. Let C ⊂ Lρ be convex and ρ -bounded. A function τ : C → [0, ∞] is called a ρ -type (or shortly a type) if there exists a sequence {gk } of elements of C such that for any f ∈ C there holds

τ ( f ) = lim sup ρ (gk − f ). k→∞

The following lemma establishes a crucial minimizing sequence property of uniformly convex modular function spaces. It will be used in conjunction with the parallelogram property in the proof of the main fixed point result for nonexpansive mappings acting in modular function spaces. Lemma 4.6. Assume that ρ ∈ ℜ is (UUC1). Let C be a ρ -closed ρ -bounded convex nonempty subset of Lρ . Let τ be a ρ -type defined on C. Then, any minimizing sequence of τ is ρ -convergent. Its ρ -limit is independent of the minimizing sequence. Proof. Let { fn } ⊂ C be such that τ ( f ) = lim sup ρ ( fn − f ), τ0 = inf{τ (h) : h ∈ C}. n→∞

Let {gk } be a minimizing sequence of τ . Since C is ρ -bounded, there exists R > 0 such that ρ ( f − g) ≤ R for any f , g ∈ C. The rest of the proof is split into two cases. Case 1: Assume that τ0 > 0. Let us choose a σ > 0 such that τ0 − σ > 0. Let us fix

90

4 Geometry of Modular Function Spaces

gm and gk and select a subsequence {hn } of { fn } such that 

 gm + gk gm + gk = lim ρ − hn . 0 < τ0 ≤ τ n→∞ 2 2 Then, for n sufficiently large, we have

 gm + gk − hn ≤ max (ρ (gm − hn ), ρ (gk − hn )) . 0 < τ0 − σ ≤ ρ 2 Using (4.7) from Lemma 4.5, we have



 gm + gk 1 1 1 − hn ≤ ρ 2 (gm −hn )+ ρ 2 (gk −hn )−Ψ R, τ0 − σ , ρ (gm −gk ) , ρ2 2 2 2 R and after passing with n to infinity, we get 

 1 2 1 2 1 2 gk + gm ≤ τ (gk ) + τ (gm ) − Ψ R, τ0 − σ , ρ (gk − gm ) . τ 2 2 2 R Hence,

τ02

 1 2 1 2 1 ≤ τ (gk ) + τ (gm ) − Ψ R, τ0 − σ , ρ (gk − gm ) , 2 2 R

for any k, m ≥ 1, which implies that

 1 1 1 Ψ R, τ0 − σ , ρ (gk − gm ) ≤ τ 2 (gk ) + τ 2 (gm ) − τ02 . R 2 2

 1 Hence, lim Ψ R, τ0 − σ , ρ (gk − gm ) = 0. The properties of Ψ imply that k,m→∞ R {gk } is ρ -Cauchy. Since Lρ is ρ -complete and C is ρ -closed, then {gk } is ρ convergent to a g ∈ C. Let us prove that any other minimizing sequence also ρ converges to g. Indeed, let {un } ∈ C be any minimizing sequence of τ . Using the same argument as previously, we have 

 1 2 1 2 1 2 2 gn + un ≤ τ (gn ) + τ (un ) − Ψ R, τ0 − σ , ρ (gn − un ) , τ0 ≤ τ 2 2 2 R for some σ > 0 such that τ0 − σ > 0 and

 1 1 1 Ψ R, τ0 − σ , ρ (gn − un ) ≤ τ 2 (gn ) + τ 2 (un ) − τ02 , R 2 2 for any n ≥ 1. As before, we get lim ρ (gn − un ) = 0. Since ρ is convex, we have n→∞

ρ

u−g 3



1 1 1 ≤ ρ (u − un ) + ρ (un − gn ) + ρ (gn − g), 3 3 3

4.3 Uniform Noncompact Convexity in Modular Function Spaces

91

where u is the ρ -limit of {un }. Clearly, our assumptions imply that ρ

u−g 3

 = 0,

hence u = g. This completes the proof of the lemma for Case 1. Case 2: Assume that τ0 = 0. Let K=



convρ ({ fk ; k ≥ n}) ,

n≥1

which is nonempty in view of property (R): recall (UUC1) implies (R) by Theorem 4.2. Let f∞ ∈ K, h ∈ C, ε > 0. By definition of τ , there exists n0 > 0 such that for every n > n0 ρ ( fn − h) ≤ τ (h) + ε . Therefore, fn ∈ Bρ (h, τ (h) + ε ) for n > n0 . This fact implies K ⊂ convρ ({ fn : n ≥ n0 }) ⊂ Bρ (h, τ (h) + ε ). Hence, f∞ ∈ Bρ (h, τ (h) + ε ). Since this is true for every ε > 0, f∞ ∈ Bρ (h, τ (h)), that is, ρ ( f∞ − h) ≤ τ (h). (4.8) Let {gk } be a minimizing sequence of τ . Using (4.8) with h = gk , we get

ρ ( f∞ − gk ) ≤ τ (gk ) → τ0 = 0 as k → ∞, which means that {gk } is ρ -convergent to f∞ . Since this limit is independent of the  sequence {gk }, the proof of Case 2 and Lemma 4.6 is complete.

4.3 Uniform Noncompact Convexity in Modular Function Spaces Compactness plays an essential role in many fundamental theorems. This is the case with Schauder fixed point theorem for example. However, there are major problems when dealing with mappings, which are not compact. In order to investigate noncompactness, Kuratowski [150] was the first one to introduce the concept of “measure on noncompactness” in connection with certain problems in General Topology. In this section, we extend this concept to modular function spaces. Definition 4.4. Let ρ ∈ R. Let A be a nonempty ρ -bounded subset of Lρ . Define the Kuratowski measure of noncompactness of A by   ε > 0 : A can be covered by a f inite number , α (A) = inf o f sets with ρ − diameter smaller than ε

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4 Geometry of Modular Function Spaces

and the Hausdorff measure of noncompactness of A by   ε > 0 : A can be covered by a f inite number . χ (A) = inf o f ρ − balls o f radius smaller than ε for any subset A of Lρ . We will make the obvious convention that the infimum over an empty set is infinite. One can easily notice that χ (A) ≤ α (A) ≤ δρ (A), for every ρ -bounded A ⊂ Lρ . Proposition 4.2. The following properties hold. (1) If A ⊂ B, then χ (A) ≤ χ (B) and α (A) ≤ α (B). ¯ = χ (A) and α (A) ¯ = α (A). (2) χ (A) ¯ (3) If α (A) = 0, then A is ρ -sequentially compact. nonempty ρ -closed ρ -bounded subset of (4) Let {An } be a decreasing sequence of  An is a nonempty ρ -compact set. Lρ . Assume that lim α (An ) = 0, then n→∞

n≥1

ρ -bounded con(5) Let {An } be a decreasing sequence of nonempty ρ -closed  An is nonempty and vex subset of Lρ . Assume that lim χ (An ) = 0. Then n→∞ n≥1   χ An = 0. n≥1

Recall that A¯ is the smallest ρ -closed subset of Lρ which contains A. Proof. The proof of (1) is obvious. In order to prove (2), let us first notice that ¯ Let ε > 0, then there exists Bρ (x1 , r),..,Bρ (xn , r) such that χ (A) ≤ χ (A). A ⊂ Bρ (x1 , r) ∪ .. ∪ Bρ (xn , r), where r ≤ χ (A) + ε . The Fatou property implies that ρ -balls are ρ -closed and therefore A¯ ⊂ Bρ (x1 , r) ∪ .. ∪ Bρ (xn , r), ¯ ≤ r ≤ χ (A) + ε . Since ε was taken arbitrarily positive, we which implies χ (A) ¯ get χ (A) ≤ χ (A). In order to prove the second claim of (2), we will only need ¯ Indeed set δρ (A) = d. Then for any f ∈ A, we have to show that δρ (A) = δρ (A). ¯ A ⊂ Bρ ( f , d). Since ρ -balls are ρ -closed, we get A¯ ⊂ Bρ ( f , d). So for any g ∈ A, ¯ we have ρ ( f − g) ≤ d. Hence A ⊂ Bρ (g, d), for all g ∈ A. Again using the fact that ¯ i.e. ρ -balls are ρ -closed, we get A¯ ⊂ Bρ (g, d). Hence ρ ( f − g) ≤ d, for any f , g ∈ A, ¯ ¯ ¯ δρ (A) ≤ d = δρ (A). Since δρ (A) ≤ δρ (A) is obvious, we deduce δρ (A) = δρ (A). ¯ Let us prove (3). Assume that α (A) = 0. Let { fn } be any sequence of elements in ¯ Let ε > 0. Using the definition of α , we can find a subsequence { fn } such that A. ρ ( fn − fm ) ≤ ε , for all n , m ∈ N. The diagonal argument will show that { fn } has a ρ -Cauchy subsequence which is ρ -convergent since Lρ is ρ -complete. In order to prove (4), let {An } be a decreasing sequence of ρ -closed ρ -bounded

4.3 Uniform Noncompact Convexity in Modular Function Spaces

93

nonempty subsets of Lρ such that lim α (An ) = 0. Choose fn ∈ An and let ε > 0. n→∞

Then, there exists n0 ≥ 1 such that α (An ) < ε , for n ≥ n0 . Therefore, there exist D1 , .., Dk such that Di , An0 ⊂ 1≤i≤k



with δρ (Di ) ≤ ε , for i = 1, · · · , k. So fn ∈

Di , for all n ≥ n0 . Hence one Di

1≤i≤k

contains infinitely many fn . So there exists a subsequence { fn } of { fn } such that ρ ( fn − fm ) ≤ ε , for all n , m . Again the diagonal argument will imply that { fn } has a ρ -Cauchy subsequence which is therefore ρ -convergent to f ∈ Lρ . Since {An } are  ρ -closed and decreasing, we deduce that f ∈ An for all n ≥ 1. Hence An is not n≥1

empty and α (∩An ) = 0 holds from (1). Using (3), we conclude that



An is ρ -

n≥1

compact. Finally, let us prove (5). Let {An } be a decreasing sequence of ρ -closed ρ -bounded nonempty convex subsets of Lρ such that lim χ (An ) = 0. Let ε > 0, then there exists n→∞

n0 ∈ N such that χ (An0 ) < ε . Let fn ∈ An , for any n ≥ 1. Then there exist infinitely many elements from { fn } which belong to a ρ -ball Bρ ( f , ε ), for some f ∈ Lρ . This implies that there exists a subsequence { fn } such that

 1 ( f  − fm ) ≤ ρ ( fn − f ) + ρ ( f − fm ) ≤ 2 ε , ρ 2 n  , m . As noted before, we secure the existence of a subsequence { f  } for every n n  1  f  is ρ -Cauchy. Let h ∈ Lρ be its ρ -limit. Fix n ≥ 1. Then such that 2n  1 1 ( f  + fm ) is ρ -convergent to fn + h. Since An is ρ -closed and con2 n 2   m ≥n vex, we deduce that 12 fn + h ∈ An . Since (An ) is decreasing, we get 12 fm + h ∈ An , for any m ≥ n . Again since ( 12 fm +h)m ≥n is ρ -convergent to h+h = 2h, we obtain that 2h ∈ An . This clearly implies that 2h ∈ ∩An . Using (3) we get χ (∩An ) = 0. The proof of Proposition 4.2 is complete.



Remark 4.3. Since in general ρ is not subadditive, there is no reason to have α (A) = 0 whenever A is ρ -compact. In fact, it can be shown that ρ satisfies the Δ2 -condition if and only if α (A) = 0 whenever A is ρ -compact. For more on α and χ , one can consult [17]. Similarly as Goebel and Sekowski [83] did for Banach spaces, we can define a new scaling for modular spaces using the measures of noncompactness α and χ .

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4 Geometry of Modular Function Spaces

Definition 4.5. The ρ -modulus of noncompact convexity Δ χ (resp. Δα ) is defined as ⎫ ⎧ ⎨ c > 0 : f or any ρ − bounded convex A ⊂ Lρ ⎬ (II) Δ χ (r, ε ) = sup ρ − bounded and f ∈ Lρ such that A ⊂ Bρ ( f , r) , ⎭ ⎩ with χ (A) ≥ rε , then dρ ( f , A) ≤ r(1 − c) for every r > 0 and ε > 0. For the definition of Δα , one will replace χ (A) by α (A) in (II). Define the characteristic of noncompact convexity of Lρ by

 εχ (r, Lρ ) = sup ε > 0 : Δ χ (r, ε ) = 0 ,

 εα (r, Lρ ) = sup ε > 0 : Δα (r, ε ) = 0 ,

and for every r > 0.

Since χ ≤ α , we get Δ χ ≤ Δα and εα ≤ εχ . In any Banach space X, it can be easily proved that εχ (X) = 0 if and only if εα (X) = 0. In modular spaces, it is not the case in general. Definition 4.6. The modular function space Lρ is said to be α (resp. χ )-uniformly ρ -noncompact convex if and only if εα (r, Lρ ) = 0 (resp. εχ (r, Lρ ) = 0), for every r > 0. Clearly if Lρ is χ -uniformly ρ -noncompact convex, then Lρ is α -uniformly ρ noncompact convex. Let ρ ∈ R. In Definition 4.1, we introduced the ρ -modulus of uniform convexity of Lρ as   1 δ1 (r, ε ) = inf 1 − ρ ( f + h) , r for any r > 0 and ε > 0, where the infimum is taken over all f , h ∈ Lρ such that

ρ ( f ) ≤ r, ρ ( f + h) ≤ r, and ρ (h) ≥ rε . Let us discuss the connection between ρ -uniform convexity and α -uniform ρ noncompact convexity. Let A ⊂ Lρ be nonempty ρ -bounded and convex such that A ⊂ Bρ ( f , r) with α (A) ≥ rε for some f ∈ Lρ , r > 0 and ε > 0. Let ζ < 1 then, by definition of α , one can find h1 , h2 ∈ A such that ρ (h1 − h2 ) ≥ rζ ε . Therefore, we have 

( f − h1 ) + ( f − h2 ) ≤ r(1 − δ1 (r, ζ ε )). ρ 2 Since

h1 + h2 ∈ A, we deduce that 2 dρ ( f , A) ≤ r(1 − δ1 (r, ζ ε )).

4.3 Uniform Noncompact Convexity in Modular Function Spaces

95

Which clearly implies that

δ1 (r, ζ ε )) ≤ Δα (r, ε ), for every r > 0, ε > 0 and ζ < 1. Goebel and Sekowski [83] proved that whenever the characteristic of uniform noncompact convexity of any Banach space is less than 1, then the space is reflexive. In what follows, we investigate the validity of this result in modular function spaces. First we investigate the link between proximinality and the ρ -modulus of noncompact convexity. More exactly, given f ∈ Lρ , we consider the minimization problem of finding h ∈ C such that

ρ ( f − h) = inf{ρ ( f − g) : g ∈ C}, where C ⊂ Lρ is not empty. Such a h is called a best approximant. Problems of finding best approximants are important in approximation theory [168, 118] and probability theory [52]. Theorem 4.4. Let ρ ∈ ℜ. Assume that Lρ is α -uniformly ρ -noncompact convex. Then for any nonempty C ρ -bounded ρ -closed convex subset of Lρ and f ∈ Lρ such that dρ ( f ,C) < ∞, the set Pρ ( f ,C) = {g ∈ C : dρ ( f ,C) = ρ ( f − g) } is nonempty ρ -compact and convex. Proof. We assume without any loss of generality that d = dρ ( f ,C) > 0. Consider the sets Cn = C ∩ Bρ ( f , d + 1n ) for n ≥ 1. Clearly {Cn } is a decreasing sequence of ρ -closed, ρ -bounded, nonempty, convex subsets of C. Assume that inf α (Cn ) = lim α (Cn ) > 0. Then, since Lρ is α -uniformly ρ -noncompact convex, n≥1

n→∞

there exists Δ > 0 such that



1 , dρ ( f ,Cn ) ≤ (1 − Δ ) d + n

for every n ≥ 1. Since dρ ( f ,C) ≤ dρ ( f ,Cn ), we get 

1 , d ≤ (1 − Δ ) d + n for every n ≥ 1. This contradicts the fact that d > 0. Hence inf α (Cn )= lim α (Cn ) = 0, 

n≥1

n→∞

and by Proposition 4.2, we deduce that Cn is a nonempty ρ -compact convex subset of C. Clearly  Cn . Pρ ( f ,C) = n≥1

The proof of Theorem 4.4 is hence complete.



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4 Geometry of Modular Function Spaces

It is not clear whether the conclusion of Theorem 4.4 remains true if we assume that εα (Lρ ) < 1. Note also how Theorem 4.4 relates to Theorem 4.1. In a similar manner, our next result is parallel to Theorem 4.2. Theorem 4.5. Let ρ ∈ ℜ. Assume Lρ is α -uniformly ρ -noncompact convex. Then Lρ has the Property (R). Proof. Let (Cn )n≥0 be a decreasing sequence of ρ -bounded ρ -closed nonempty convex subsets of Lρ . Fix f ∈ C0 , then r = sup dρ ( f ,Cn ) ≤ δρ (C0 ) < ∞. n≥1

Define Kn = Cn ∩ Bρ ( f , r). Clearly {Kn } is a decreasing sequence of nonempty ρ -closed convex subsets of Lρ . Using the same argument as in the proof of Theorem 4.4, one can show that lim α (Kn ) = 0. So, ∩Kn is a nonempty ρ -compact n→∞ convex subset of Lρ . Therefore ∩Cn is nonempty, which completes the proof of Theorem 4.5.

 One may ask if under the assumptions of Theorem 4.5, we have the intersection property for any family of subsets. The answer to this question is in the affirmative. Theorem 4.6. Let ρ ∈ ℜ. Assume Lρ is α -uniformly ρ -noncompact convex. Then for any decreasing family (Aβ )β ∈Γ of ρ -bounded, ρ -closed nonempty convex sub Aβ is nonempty. sets of Lρ and any directed set (Γ , ), we have β ∈Γ

Proof. Let (Aβ )β ∈Γ be a decreasing family of ρ -bounded, ρ -closed nonempty convex subsets of Lρ . We can assume without any loss of generality that Γ has a minimum, say β0 . Let f ∈ Aβ0 , then r = sup {dρ ( f , Aβ ) : β ∈ Γ } ≤ δρ (Aβ0 ) < ∞. Set Kβ = Aβ ∩ Bρ ( f , r), for any β ∈ Γ . Then (Kβ )β ∈Γ is a decreasing family of nonempty ρ -closed convex subsets of Lρ . Using the same argument as in the proof of Theorem 4.5, we get inf{α (Kβ ) : β ∈ Γ } = 0. Therefore for every n ≥ 1, there exists βn ∈ Γ such that 1 α (Kβn ) ≤ . n Clearly one can choose {βn } ⊂ Γ such that {Kβn } is decreasing. Using Proposition 4.2, we obtain that K∞ = ∩Kβn is a nonempty ρ -compact subset. We can assume that there exists β  ∈ Γ such that βn ≤ β  , for every n ≥ 1. Otherwise, we would have K∞ =

 n≥1

Kβn =

 β ∈Γ

Kβ .

4.3 Uniform Noncompact Convexity in Modular Function Spaces

97

Since (Kβ ) is decreasing, we deduce that Kβ ⊂ Kβ  ⊂ K∞ , for every β ≥ β  . Put δ = δρ (Kβ  ). As α (Kβ  ) = 0, one can find {Ai }1≤i≤k such that Kβ  ⊂



Ai with δρ (Ai ) ≤

1≤i≤k

δ . 2

For any β ≥ β  choose fβ ∈ Kβ . It is not hard to show that there exists i0 ∈ {1, 2, .., k} such that for every β ≥ β  there exists γ ≥ β such that fγ ∈ Ai0 . Indeed, assume to the contrary that for every i ∈ {1, .., k} there exists βi ≥ β  such that fγ does not belong to Ai , for all γ ≥ βi . Since Γ is a directed set, there exists β ∈ Γ such that β ≥ βi for every i ∈ {1, .., k}. So fβ does not belong to any Ai . Contradicting the fact that fβ ∈ Kβ ⊂ Kβ  ⊂ A1 ∪ .. ∪ Ak . Therefore, there exists such i0 ∈ {1, .., k}. Set Cβ = { fγ ; γ ≥ β and fγ ∈ Ci0 }. Then for every β ≥ β  , Cβ is nonempty. Moreover, we have for every ζ ≥ β , there exists γ ≥ ζ such that fγ ∈ Aβ . Since Cβ ⊂ Ai , we get δρ (Cβ ) ≤ δ2 . Let Lβ = convρ (Cβ ), for all β ≥ β  , where convρ (A) is the smallest ρ -closed convex subset, which contains A. Then δρ (Lβ ) ≤ δ2 and {Lβ }β ≥β  is a decreasing family of nonempty ρ -closed convex sets with Lβ ⊂ Kβ , for every β ≥ β  . Using α (Kβ ) = 0 we deduce that α (Lβ ) = 0, for every β ≥ β  . We may repeat the same construction to get another decreasing family (Lβ2 ) of ρ -closed convex subsets with Lβ2 ⊂ Lβ

and δρ (Lβ2 ) ≤ δ4 , since α (Lβ  ) = 0, for all β ≥ β  . So by induction, we obtain a decreasing family {Lβn }β ≥β  of ρ -closed convex nonempty subsets with Lβn ⊂ Lβn−1 and δρ (Lβn ) ≤

δ , 2n

for all β ≥ β  and all n ≥ 2. Since Lρ is ρ -complete, we get 

Lβn = 0, / f or any β ≥ β  .

n≥1

Clearly ∩Lβn is reduced to one point, say hβ for all β ≥ β  . Since Lβn ⊂ Lγn for any β ≥ γ ≥ β  and any n ≥ 1, we obtain hβ = hγ for every β ≥ γ ≥ β  . Therefore, hβ  ∈ Lβn ⊂ Kβ for all n ≥ 1 and all β ≥ β  , which clearly implies 0/ =

 β ≥β 



Kβ ⊂

This completes the proof of Theorem 4.6.

β ∈Γ



Aβ .

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4 Geometry of Modular Function Spaces

4.4 Opial and Kadec–Klee Properties The original definitions of Opial and Kadec–Klee properties are closely linked to the weak topology in Banach spaces, as presented in Sect. 2.5.4. In recent years, many authors tried to extend these definitions to other topologies. The investigation of these two properties in modular function spaces finds its root in [109]. Definition 4.7. Let ρ ∈ ℜ. 1. We will say that Lρ satisfies the ρ -a.e.-Opial property if for every { fn } ⊂ Lρ ρ -a.e. convergent to 0, such that there exists β > 1 for which sup ρ (β fn ) = M < ∞, n

then we have lim inf ρ ( fn ) < lim inf ρ ( fn + f ), n→∞

n→∞

for every f ∈ Eρ not equal to 0. 2. We will say that Lρ satisfies the ρ -a.e.-uniform Opial property if for every ε > 0 there exists η > 0 such that for every { fn } ⊂ Lρ ρ -a.e. convergent to 0 such that there exists β > 1 for which sup ρ (β fn ) = M < ∞ n

then, for every f ∈ Eρ such that ρ ( f ) ≥ ε , we have lim inf ρ ( fn ) + η ≤ lim inf ρ ( fn + f ). n→∞

n→∞

These definitions can be strengthen to the following one, which will play an important role in this book. Definition 4.8. We say that Lρ satisfies the ρ -a.e. strong Opial property or shortly strong Opial property if for every { fn } ∈ Lρ which is ρ -a.e. convergent to 0 such that there exists a β > 1 for which sup ρ (β fn ) < ∞, n

the following equality holds for any g ∈ Eρ lim inf ρ ( fn + g) = lim inf ρ ( fn ) + ρ (g). n→∞

n→∞

(4.9)

4.4 Opial and Kadec–Klee Properties

99

Clearly, the ρ -a.e.-uniform Opial property implies ρ -a.e.-Opial property. Also, it is not difficult to prove that the ρ -a.e. strong Opial property implies ρ -a.e.-uniform Opial property. Remark 4.4. If we consider the Luxemburg norm on Lρ , then we can define a variant of the Opial properties in this case. Indeed, we will say that (Lρ , .ρ ) satisfies the ρ -a.e.-uniform Opial property if 

inf lim inf  fn + f ρ − 1 > 0, n→∞

where the infimum is taken over all { fn } ⊂ Lρ ρ -a.e. convergent to 0 such that lim inf  fn ρ ≥ 1, and  f ρ ≥ c, for any c > 0. n→∞

Let us give the definition of the Kadec–Klee property in modular function spaces. Definition 4.9. We will say that Lρ satisfies ρ -a.e.-Kadec-Klee property if for some ε ∈ (0, 2] and every r > 0, there exists η > 0 such that ρ ( f ) ≤ r(1 − η ) for every { fn } ⊂ Eρ ρ -a.e. convergent to f ∈ Eρ for which   (i) there exists β > 1 such that sup ρ β [ fn − f ] = M < ∞, n

(ii) ρ ( fn ) ≤ r,for every ≥ 1,    n fn − fm 1 fn = inf ρ : n = m > r ε . (iii) sep 2 2 We will say that Lρ satisfies ρ -a.e.-uniform Kadec–Klee property if the above still holds for every ε ∈ (0, 2]. Remark 4.5. If we consider the Luxemburg norm on Lρ , then we can define a variant of the Kadec–Klee property in this case. Indeed, we will say that (Lρ ,  · ρ ) satisfies the ρ -a.e.-uniform Kadec–Klee property if for any ε ∈ (0, 2], we have 

inf 1 −  f ρ > 0, where the infimum is taken over all { fn } ⊂ Lρ ρ -a.e. convergent to f , such that  fn ρ ≤ 1, for any n ≥ 1, and sep{ fn } = inf  fn − fm ρ : n = m > ε . In the next theorem, we show that a large class of modular function spaces enjoy the ρ -a.e strong Opial property. Recall that a function modular ρ is said to be orthogo/ Note that all nally additive if ρ ( f , A ∪ B) = ρ ( f , A) + ρ ( f , B) whenever A ∩ B = 0. Orlicz and Musielak–Orlicz modulars are orthogonally additive. Let us begin with the following technical result.

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4 Geometry of Modular Function Spaces

Lemma 4.7. [27] Let ρ ∈ ℜ. Let ε > 0 and k > 1 be such that k ε < 1. Then for every f , g ∈ Lρ such that ρ (k f ) < ∞ and ρ (1/ε (k − 1)g) < ∞, we have |ρ ( f + g) − ρ ( f )| ≤ ε |ρ (k f ) − kρ ( f )| + 2 ρ (Cε g), where Cε = 1/ε (k − 1). Proof. Put

α = 1 − kε , β = ε , γ = (k − 1)ε . Then, we have α + β + γ = 1 and f + g = α f + β k f + γ Cε g. Since ρ is convex, we get

ρ ( f + g) ≤ α ρ ( f ) + β ρ (k f ) + γρ (Cε g). Hence

ρ ( f + g) − ρ ( f ) ≤ ε (ρ (k f ) − kρ ( f )) + (k − 1)ερ (Cε g) which implies

ρ ( f + g) − ρ ( f ) ≤ ε (ρ (k f ) − kρ ( f )) + ρ (Cε g). On the other hand, if we put

α=

1 ε ε (k − 1) , β= , γ= , 1 + kε 1 + kε 1 + kε

then f = α ( f + g) + β k f + γ (−Cε g). Hence

ρ ( f ) − ρ ( f + g) ≤ ε [ρ (k f ) − kρ ( f )] + ε (k − 1)ρ (−Cε g). This will imply

ρ ( f ) − ρ ( f + g) ≤ ε [ρ (k f ) − kρ ( f )] + ρ (Cε g). The proof is therefore complete.



Theorem 4.7. [109] Let ρ ∈ ℜ. Assume that ρ is orthogonally additive. Let { fn } ⊂ Lρ be ρ -a.e. convergent to 0. Assume there exists β > 1 such that sup ρ (β fn ) = M < ∞. n

Let g ∈ Eρ . Then

  lim ρ ( fn + g) − ρ ( fn ) − ρ (g) = 0,

n→∞

which implies lim inf ρ ( fn + g) = lim inf ρ ( fn ) + ρ (g), n→∞

n→∞

i.e., Lρ posses the ρ -a.e. strong Opial property.

4.4 Opial and Kadec–Klee Properties

101

Proof. Since { fn } converges ρ -a.e. to 0, then by Egoroff’s Theorem, there exists  an increasing sequence of sets Hk ∈ P such that Ω = Hk and { fn } converges uniformly to f on every Hk . On the other hand we have |ρ ( fn + g) − ρ ( fn ) − ρ (g)| ≤ |ρ ( fn + g, Hm ) − ρ ( fn , Hm ) − ρ (g, Hm )| + |ρ ( fn + g, Hmc ) − ρ ( fn , Hmc ) − ρ (g, Hmc )| , where Ac denotes the complement of the subset A. Using Lemma 4.7, we get |ρ ( fn + g, Hm ) − ρ (g, Hm )| ≤ ε (ρ (kg, Hm ) − kρ (g, Hm )) + 2ρ (Cε fn , Hm ), for every ε > 0 such that ε k < 1. Since { fn } converges uniformly to 0 on every Hm we have lim sup |ρ ( fn + g, Hm ) − ρ ( fn , Hm ) − ρ (g, Hm )| ≤ ερ (kg). n→∞

Using the same ideas, we get lim sup |ρ ( fn + g, Hmc ) − ρ ( fn , Hmc ) − ρ (g, Hmc )| ≤ ε lim sup ρ (k fn ) + 2ρ (Cε g, Hmc ) n→∞

n→∞

+ρ (g, Hmc ). Hence lim sup |ρ ( fn + g, Hmc ) − ρ ( fn , Hmc ) − ρ (g, Hmc )| ≤ ε sup ρ (k fn ) + 2ρ (Cε g, Hmc ) n→∞

n

+ρ (g, Hmc ). Therefore lim sup |ρ ( fn +g)−ρ ( fn )−ρ (g)| ≤ ε ρ (kg)+ε sup ρ (k fn )+2ρ (Cε g, Hmc )+ ρ (g, Hmc ). n→∞

n

Let m go to infinity. Using the fact that g ∈ Eρ , we get lim sup |ρ ( fn + g) − ρ ( fn ) − ρ (g)| ≤ ε ρ (kg) + ε sup ρ (k fn ). n→∞

n

Finally, we let ε go to 0 to get lim sup |ρ ( fn + g) − ρ ( fn ) − ρ (g)| ≤ 0. n→∞

The proof is therefore complete.



Remark 4.6. Observe that the Opial property in the norm sense, [173], does not necessarily hold for several classical Banach function spaces. For instance, the norm Opial property does not hold for L p spaces for 1 ≤ p = 2 while the modular strong Opial property holds in L p for all p ≥ 1.

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4 Geometry of Modular Function Spaces

Note also that if ρ satisfies the Δ2 -condition, then one may assume f ∈ Lρ in the definition of the strong Opial property. As an implication of Theorem 4.7, one can get the following result in L p spaces. Corollary 4.2. Let ρ ∈ ℜ. Let p ≥ 1 and { fn } be a sequence of L p -uniformly bounded functions on a measure space. Assume that { fn } converges almost everywhere to f ∈ L p . Then lim inf || fn || p = lim inf || fn − f || p + || f || p . n→∞

n→∞

When p = 1, the conclusion of Corollary 4.2 gives the main result of [25, 152]. Let us add that when p < 1 the conclusion of Corollary 4.2 is still true. This will not be a simple deduction from Theorem 4.7 since the function ϕ (t) = t p is not convex. A technical assumption [27] can be added to get a more general result (see also [151]). Theorem 4.8. Let ρ ∈ ℜ. Assume that ρ has the ρ -a.e. strong Opial property. Let C be a convex, strongly ρ -bounded and ρ -a.e. compact nonempty subset of the modular function space Lρ . Let { fn }n be a sequence of elements of C. Let Φ : C → [0, +∞) be a ρ -type defined by Φ (g) = lim sup ρ ( fn − g). Then, for any sequence {gm } in C n→∞

which ρ -a.e. converges to g ∈ C we have

Φ (g) ≤ lim inf Φ (gm ), m→∞

i.e., ρ -types are lower semicontinuous in the sense of convergence ρ -a.e. Moreover, Φ attains its minimum in C. Proof. Since C is ρ -a.e. compact, there exists a subsequence { fφ (n) } of { fn } such ρ −a.e

that fφ (n) −→ f ∈ C and lim ρ ( fφ (n) − g) = lim sup ρ ( fn − g). Hence n→∞

n→∞

Φ (gm ) = lim sup ρ ( fn − gm ) ≥ lim sup ρ ( fφ (n) − gm ) ≥ lim inf ρ ( fφ (n) − gm ). n→∞

n→∞

n→∞

Equality (4.9) of the strong Opial property implies that lim inf ρ ( fφ (n) − gm ) = lim inf ρ ( fφ (n) − f ) + ρ ( f − gm ). n→∞

n→∞

Thus, Φ (gm ) ≥ lim inf ρ ( fφ (n) − f ) + ρ ( f − gm ), for any m ≤ 1. Hence n→∞

lim inf Φ (gm ) ≥ lim inf ρ ( fφ (n) − f ) + lim inf ρ ( f − gm ). m→∞

n→∞

m→∞

Using again (4.9), we get lim inf ρ ( f − gm ) = lim inf ρ (gm − g) + ρ (g − f ), m→∞

m→∞

4.4 Opial and Kadec–Klee Properties

103

which implies lim inf Φ (gm ) ≥ lim inf ρ ( fφ (n) − f ) + lim inf ρ (gm − g) + ρ (g − f ). m→∞

n→∞

m→∞

(4.10)

On the other hand,

Φ (g) = lim sup ρ ( fn − g) = lim ρ ( fφ (n) − g) = lim inf ρ ( fφ (n) − g), n→∞

n→∞

n→∞

which implies

Φ (g) = lim inf ρ ( fφ (n) − f ) + ρ ( f − g). n→∞

(4.11)

From (4.10) and (4.11), it is clear that

Φ (g) ≤ lim inf Φ (gm ). m→∞

To prove that Φ attains its minimum in C, let Φ0 = inf{Φ ( f ) : f ∈ C} which is finite due to the ρ -boundedness of C. There exists then a sequence of gn ∈ C such that Φ0 = lim Φ (gn ). By the ρ -a.e. compactness of C, passing to a subsequence if n→∞ necessary, there exists g ∈ C such that gn → g ρ -a.e. Using the ρ -a.e. lower semicontinuity of Φ we have

Φ0 ≤ Φ (g) ≤ lim Φ (gn ) = Φ0 , n→∞

which completes the proof of Theorem 4.8. 

An easy consequence of Theorem 4.7 and Theorem 4.8 is the ρ -a.e. lower semicontinuity of the ρ -types for orthogonally additive function modulars for C satisfying hypothesis of Theorem 4.7. In the next theorem, we discuss the Kadec–Klee property in modular function spaces. Theorem 4.9. Let ρ ∈ ℜ. Assume that ρ is orthogonally additive. Let { fn } ⊂ Lρ be ρ -a.e. convergent to 0. Assume there exists β > 1 such that sup ρ (β fn ) = M < ∞. n

Then the modular function space Lρ has the ρ -a.e.-uniform Kadec–Klee property. Proof. Let ε > 0, r > 0 and { fn } be as in Definition 4.9. Assume { fn } is ρ -a.e. convergent to f . Theorem 4.7 implies that ρ has the strong Opial property. Hence lim inf ρ ( fn − f ) + ρ ( f ) = lim inf ρ ( fn ). n→∞

n→∞

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4 Geometry of Modular Function Spaces

Our assumption on { fn } implies that lim inf ρ ( f − fn ) ≥ r n→∞

ε . 2

 ε . ρ( f ) ≤ r 1 − 2

Therefore The proof is complete. 

We will now discuss the Opial and Kadec–Klee properties in modular function spaces seen as a Banach space endowed with the Luxemburg norm. Most of the results obtained here are found in [100]. Lemma 4.8. [100] Let ρ ∈ ℜ. Assume that ρ is an orthogonally additive modular function with the Δ2 -type condition. Let { fn } be a sequence in Lρ which ρ -a.e. converges to 0. Let f ∈ Lρ . Then to every ε > 0, there exist a subsequence { fnk }, a sequence {gk } in Lρ and a function g ∈ Lρ such that (i) lim  fnk − gk  = 0, k→∞

/ for any k ≥ 1, (ii) supp gk ∩ supp g = 0, (iii)  f − g < ε . Proof. Using the Egoroff theorem , we are assured of the existence of a sequence of  sets Ωk ∈ P such that Ωk = Ω , Ωk ∩ Ωl = 0, / whenever k = l, and { fn } converges uniformly to the null function on every Ωk . Using the orthogonal additivity of ρ and the Δ2 -condition, we get ρ ( f ) < ∞, and

ρ( f ) =

∑ ρ ( f , Ωk ).

k≥1

for any f ∈ Lρ . Let ε > 0, and consider δ > 0 given by Lemma 3.2. Let us fix f ∈ Lρ . Then there exists k0 ≥ 1 such that

∑ ρ ( f , Ωk ) < δ .

k>k0

Let {εn } be a real sequence in (0, 1) such that lim εn = 0. For every εn , consider the n→∞

corresponding δn given by Lemma 3.2. Using the properties of the function modular  k=k 0  ρ , we know that ρ α , Ωk → 0, as α decreases to 0. Thus there exists α0 > 0 k=1

such that

 k=k 0  δ1 ρ α0 , Ωk < . 2 k=1

4.4 Opial and Kadec–Klee Properties

105

Since { fn } converges uniformly to 0 on

k=k 0

Ωk , there exists n1 ≥ 1 such that for

k=1

any n ≥ n1 , we have | fn (x)| ≤ α0 , for any x ∈

k=k 0

Ωk . If we consider the function

k=1

fn1 , then there exists k1 > k0 such that

∑ ρ ( f n1 , Ω k ) <

k>k1

Define the function g1 = fn1 1Ak

0 ,k1

δ1 . 2

, where Ak0 ,k1 =



Ωk . So g1 ∈ Lρ and

k0
k0 k0       ρ ( fn1 − g1 ) = ρ fn1 , Ωk ∪ Ω k = ρ f n1 , Ω k + ρ f n1 , Ωk k=1

k>k1

k=1

k>k1

which implies k0   ρ ( fn1 − g1 ) ≤ ρ α0 , Ωk +

∑ ρ ( f n1 , Ω k ) ≤

k>k1

k=1

δ1 δ1 + = δ1 . 2 2

Hence  fn1 − g1  ≤ ε1 . Suppose that we have constructed k0 < k1 < · · · < kl , n1 < n2 < · · · < nl such that ρ ( fni − gi ) ≤ δi , where gi = fni 1Ak

0 ,k1

,

for i ∈ [1, l]. Using the properties of ρ , there exists αl+1 > 0 such that k=k   δ l l+1 . ρ αl+1 , Ωk < 2 k=1

Since { fn } converges uniformly to 0 on

k=k l

Ωk , we can find nl+1 > nl such that

k=1

| fnl+1 (x)| ≤ αl+1 , for any x ∈

k=k l

Ωk . Also, we can find kl+1 > kl such that

k=1

∑

ρ ( fnl+1 , Ωk ) <

k>kl+1

Set gl+1 = fnl+1 1

δl+1 . 2



kl
Ωk

.

106

4 Geometry of Modular Function Spaces

Then, we have ρ ( fnl+1 − gl+1 ) ≤ δl+1 . Therefore, we have  fnl+1 − gl+1  ≤ εl+1 . By induction, we constructed a subsequence { fnk } and a sequence {gk } such that lim  fnk − gk  = 0. Next, we define the function k→∞

g= f 1

Ωk

.

1≤k≤k0

/ for any k ≥ 1. Moreover, we have It is obvious that supp gk ∩ supp g = 0,

ρ ( f − g) ≤

∑ ρ ( f , Ωk ) < δ ,

k>k0

 which implies  f − gρ < ε .

As a consequence to the above technical results, we obtain the following theorem. Theorem 4.10. Let ρ ∈ ℜ. Assume that ρ is a orthogonally additive function modular satisfying the Δ2 -type condition. Let ε > 0. Let { fn } be in Lρ such that (1)  fn ρ ≤ 1, for any n ≥ 1, (2) sep( fn ) = inf{ fn − fm ρ : n = m} > ε , (3) { fn } ρ -a.e. converges to f ∈ Lρ . Then  f ρ ≤

ωρ−1

1 1 1 − 1/2ωρ (1/ε )

 < 1.

In other words, (Lρ ,  · ρ ) has the uniform Kadec–Klee property with respect to the ρ -a.e. convergence. Proof. Fix ε > 0. Let { fn } be a sequence such that (1), (2), and (3) hold. Let γ > 0 and define ε f fn , g= , and ε  = . gn = γ +1 γ +1 γ +1 Then gn ρ < 1, for anyn ≥ 1. Hence ρ (gn ) ≤ gn ρ < 1. Since gn − gm ρ > ε  ,

we get ρ (gn − gm )/ε  > 1, for any n = m. Using the definition of the growth function, we have



 gn − gm 1 ≤ ωρ ρ ρ (gn − gm ),  ε ε which implies

1 < ωρ

for any n = m. Hence

1 ε



ρ (gn − gm ),

1

 < ρ (gn − gm ), 1 ωρ ε

4.4 Opial and Kadec–Klee Properties

107

for any n = m. Fix m ≥ 1. Using Theorem 4.7, we get 1

 ≤ lim inf ρ (gn − gm ) = lim inf ρ (gn − g) + ρ (g − gm ). n→∞ n→∞ 1 ωρ ε If we let m → ∞, we get 1

 ≤ 2 lim inf ρ (gn − g). n→∞ 1 ωρ ε On the other hand, we also have lim inf ρ (gn ) = lim inf ρ (gn − g) + ρ (g). n→∞

n→∞

Hence 1

0 ≤ ρ (g) = lim inf ρ (gn ) − lim inf ρ (gn − g) ≤ 1 − n→∞

n→∞

2ωρ

1 ε

.

Using property (v) of Lemma 3.1 and the monotonicity of ωρ−1 , we get  f ρ ≤ (1 + γ )

ωρ−1



1 1 1−1/2ωρ ((1+γ )/ε )

.

Since γ is arbitrary and both functions ωρ and ωρ−1 are continuous, we deduce  f ρ ≤

ωρ−1



1 1 1−1/2ωρ (1/ε )

 = η (ε ).

In order to finish the proof of Theorem 4.10, weneed to check that η (ε ) < 1. Since 1 > 0. Since ρ satisfies the Δ2 -type ωρ (1) = 1, then η (ε ) < 1 if and only if ωρ ε condition, then there K > 0 such that ρ (2 f ) ≤ K ρ ( f ), for any f ∈ Lρ . This 

exists 1 1 ≥ . Since ε ∈ (0, 2], then we have will imply that ωρ 2 K



 1 1 1 ≥ > 0. ≥ ωρ ωρ ε 2 K

 The last result of this section deals with the Opial property.

108

4 Geometry of Modular Function Spaces

Theorem 4.11. Let ρ ∈ ℜ. Assume that ρ is an orthogonally additive modular function satisfying the Δ2 -type condition. Let c > 0. Let { fn } be in Lρ such that { fn } ρ -a.e. converges to 0, and lim inf  fn ρ ≥ 1. Then n→∞

lim inf  fn + f ρ ≥ n→∞

ωρ−1

1 1+ ωρ (1/c)

 > 1,

for any f ∈ Lρ such that  f ρ ≥ c. In other words, (Lρ ,  · ρ ) has the uniform Opial property with respect to the ρ -a.e. convergence. Proof. Fix c > 0. Let { fn } be in Lρ such that { fn } ρ -a.e. converges to 0, and lim inf  fn ρ ≥ 1. Let f ∈ Lρ be such that  f ρ ≥ c. Let ε ∈ (0, c). Lemma 4.8 will n→∞

force the existence of a subsequence { fnk } of { fn }, a sequence {gn } and a function / for g ∈ Lρ such that lim  fnk − gk ρ = 0,  f − gρ < ε , and supp gk ∩ supp g = 0, k→∞

any k ≥ 1. We may also assume that lim inf  fn + f ρ = lim inf  fnk + f ρ . n→∞

k→∞

Hence lim inf gk ρ ≥ 1, and gρ ≥ c − ε . k→∞

1 1 Fix t ∈ (0, 1). Define the functions h = g and hk = gk , for any k ≥ 1. Then we t t have gρ 1 > gρ ≥ c − ε , and lim inf hk ρ ≥ > 1. hρ = k→∞ t t So, for k large enough, we may assume hk ρ > 1. Set 

1 −1 . δε = ωρ 1 + ωρ (1/(c − ε )) Since hk and h have disjoint support, the additivity of ρ implies





 hk + h hk h =ρ +ρ ρ δε δε δε Using the properties of ωρ , we get





 hk h h 1 1 +ρ ≥ ρ ρ (hk ) + ρ δε δε ωρ (δε ) ωρ (δε /(c − ε )) c−ε 1 1 + ωρ (δε ) ωρ (δε )ωρ (1/(c − ε )) " # 1 1 1+ =1 = ωρ (δε ) ωρ (1/(c − ε ))

>

4.4 Opial and Kadec–Klee Properties

109

Hence lim inf gk +gρ ≥ t δε , which implies lim inf gk +gρ ≥ δε , by letting t → 1. k→∞

k→∞

Therefore, we have lim inf  fn + f ρ = lim  fnk + f ρ n→∞

k→∞

≥ lim inf gk + gρ − lim  fnk − gk ρ −  f − gρ k→∞

k→∞

≥ δε − ε . Since ε was arbitrary and the functions ωρ and ωρ−1 are continuous, we get lim inf  fn + f ρ ≥ n→∞

ωρ−1

 1 . 1+ ωρ (1/c)

In order to complete the proof of Theorem 4.11, we need to show that 

1 > 1. ωρ−1 1 + ωρ (1/c) Clearly this will happen if and only if ωρ (1/c) > 0. Using the same ideas as those  applied at the end of the proof of Theorem 4.10, we can show that ωρ (1/c) > 0.

Remark 4.7. The lower bound found in'Theorem 4.11 is optimal. Indeed, assume Lρ = L p ([0, 1]), 1 ≤ p < ∞, and ρ ( f ) = [0,1] | f (x)| p dx. Then the ρ -a.e. is the classical a.e. convergence, which is equivalent to convergence locally in measure . In this case, the best lower bound is (c p + 1)1/p which is the exact value that we obtained in Theorem 4.11 since ωρ (t) = t p .

Chapter 5

Fixed Point Existence Theorems in Modular Function Spaces

This chapter presents a series of fixed point existence theorems for nonlinear mappings acting in modular functions spaces. We cover a range of different types of mappings including ρ -contractions and their pointwise asymptotic versions and ρ nonexpansive mappings and pointwise asymptotic ρ -nonexpansive mappings, under various assumptions on the function modular ρ and on the sets these mappings are defined in. In addition, we will discuss some examples and applications to the theory of differential equations.

5.1 Definitions Let us introduce definitions of modular contractions, pointwise contractions, asymptotic pointwise mappings, and associated notions. Frequently, they are collectively referenced in the literature as generalized contractions and generalized nonexpansive mappings, respectively. Definition 5.1. Let ρ ∈ ℜ and C ⊂ Lρ be nonempty, and ρ -closed. A mapping T : C → C is called a ρ -contraction or shortly contraction if there exists a number c ∈ [0, 1) such that

ρ (T ( f ) − T (g)) ≤ cρ ( f − g),

for all f , g ∈ C.

Definition 5.2. Let ρ ∈ ℜ and C ⊂ Lρ be nonempty, and ρ -closed. A mapping T : C → C is called a pointwise ρ -contraction or shortly pointwise contraction if there exists α : C → [0, 1) such that

ρ (T ( f ) − T (g)) ≤ α ( f )ρ ( f − g),

for all f , g ∈ C.

Definition 5.3. Let ρ ∈ ℜ and C ⊂ Lρ be nonempty, and ρ -closed. A mapping T : C → C is called an asymptotic pointwise mapping if there exists a sequence

c Springer International Publishing Switzerland 2015  M. A. Khamsi, W. M. Kozlowski, Fixed Point Theory in Modular Function Spaces, DOI 10.1007/978-3-319-14051-3 5

111

112

5 Fixed Point Existence Theorems in Modular Function Spaces

of mappings αn : C → [0, ∞) such that

ρ (T n ( f ) − T n (g)) ≤ αn ( f )ρ ( f − g),

for all f , g ∈ Lρ .

(a) If αn ( f ) = 1 for every f ∈ Lρ and n ∈ N, then T is called ρ -nonexpansive or shortly nonexpansive. (b) If {αn } converges pointwise to α : C → [0, 1), then T is called asymptotic pointwise ρ -contraction or shortly asymptotic pointwise contraction. (c) If lim sup αn ( f ) ≤ 1 for any f ∈ Lρ , then T is called asymptotic pointwise ρ n→∞

nonexpansive or shortly asymptotic pointwise nonexpansive. (d) If αn is a constant function for every n, and lim sup αn ≤ 1 for any f ∈ Lρ , then n→∞

T is called asymptotically ρ -nonexpansive or shortly asymptotically nonexpansive. Remark 5.1. It follows immediately from the definition that every ρ -contraction is a pointwise ρ -contraction, and every pointwise ρ -contraction is an asymptotic pointwise ρ -contraction. Remark 5.2. It follows immediately from the definition that every ρ -nonexpansive mapping is asymptotically ρ -nonexpansive, and every asymptotically ρ -nonexpansive mapping is an asymptotic pointwise ρ -nonexpansive mapping. Let us look at some examples which illustrate the role of the above defined notions. We start with a typical application of the methods of modular function spaces to the theory of nonlinear integral equations. Example 5.1. As suggested in the introduction, an operator itself may be used for the construction of a function modular and hence a space in which this operator has required properties like ρ -nonexpansiveness or ρ -contraction. Let us consider for instance the following, important for the theory of integral equations, Urysohn operator:  1

T ( f )(x) = 0

k(x, y, | f (y)|)dy + f0 (x),

where f0 is a fixed function and f : [0, 1] → R is Lebesgue measurable. For the kernel k we assume that (a) k : [0, 1] × [0, 1] × R+ → R+ is Lebesgue measurable (b) k(x, y, 0) = 0 (c) k(x, y, ·) is continuous, convex and increasing to +∞  1

(d) 0

k(x, y,t)dx > 0 for t > 0 and y ∈ (0, 1).

Assume, in addition, that for almost all t ∈ [0, 1] and all measurable functions f , g there exists a constant K > 0 such that   1  1  1 k(t, u, |k(u, v, | f (v)|)−k(u, v, |g(v)|)|)dv du ≤ K k(t, u, | f (u)−g(u)|)du. 0

0

0

5.1 Definitions

Setting ρ ( f ) =

113

 1  0

0

1

 k(x, y, | f (y)|)dy dx and using Jensen’s inequality it is easy

to show that ρ is a function modular and that ρ (T ( f ) − T (g)) ≤ K ρ ( f − g) on Lρ , that is, T is K-Lipschitzian with respect to ρ . Let us summarize what we have done: given an integral operator we have constructed a modular function space in which this operator is Lipschitzian with the constant K. Obviously, if K < 1 then T becomes a ρ -contraction, or if K ≤ 1, ρ -nonexpansive mapping. As we will see, applying relevant fixed point theorems one can solve the corresponding Urysohn integral equation. The next example, taken from [79], illustrates the role of the asymptotic assumptions. Please remember that the Hilbert space l 2 is a modular function space too. Example 5.2. Let C denote the unit ball in l 2 and let T be defined as follows: T : (x1 , x2 , x3 , . . .) → (0, x12 , A2 x2 , A3 x3 , . . .) 1 where Ai ∈ (0, 1) are numbers such that ∏∞ i=2 Ai = 2 . Then T is Lipschitzian with constant 2 but it is not nonexpansive. Moreover, i

T i (x) − T i (y) ≤ 2 ∏ A j x − y j=2

for i = 2, 3, . . ., which implies that i

lim ki = 2 lim ∏ A j = 1,

i→∞

i→∞

j=2

that is, T is asymptotically nonexpansive. The following simple finite-dimensional example gives another view on asymptotic pointwise contractions and asymptotic pointwise nonexpansive mappings. Example 5.3. Let a be any real √ number from (0, 1) and C = [a, 1]. Let T : [a, 1] → R be defined simply as T (x) = x. Note that if x, y ∈ C with x ≤ y then √ 1 √ | x − y| ≤ √ |x − y|, 2 x which makes T pointwise nonexpansive (and actually nonexpansive) but only if a ≥ 14 . We know that T has an obvious fixed point at x = 1. However, if we started with a < 14 we would not be able to use fixed point results like Banach contraction principle or Kirk’s theorem to prove its existence. On the other hand, since in the same situation 1 1 |T n (x) − T n (y)| ≤ n x 2n −1 |x − y|, 2 it is easy to see that T is an asymptotic pointwise contraction in C = [a, 1] for any a ∈ (0, 1) and hence, by Theorem 5.3, it will have a unique fixed point in C which

114

5 Fixed Point Existence Theorems in Modular Function Spaces

can be found by taking the limit of orbits. An easy calculation will show that this limit is indeed equal to 1.

5.2 Contractions in Modular Function Spaces 5.2.1 Banach Contraction Principle in Modular Function Spaces As we already discussed in Sect. 2.1 the Banach contraction principle (Theorem 2.1) is an extremely important tool used by many to guarantee the existence and uniqueness of solutions seen as fixed point of self-mappings defined on metric spaces. It also provides a constructive method to find those fixed points. It was natural to try to extend such celebrated theorem to the context of modular function spaces. That is exactly what we will do in our next result. Theorem 5.1. Let ρ ∈ ℜ. Let C ⊂ Lρ be nonempty, ρ -closed and ρ -bounded. Let ¯ T : C → C be a ρ -contraction.  Then, T has a unique fixed point f ∈ C. Moreover, for any f ∈ C, ρ T n ( f ) − f¯ → 0 as n → ∞, where T n is the n-th iterate of T . Proof. Since T is ρ -contraction, there exists α < 1 such that

ρ (T ( f ) − T (g)) ≤ α ρ ( f − g),

for all f , g ∈ C.

Let us fix f0 ∈ C. Since C is ρ -bounded, hence

δρ (C) = sup{ρ ( f − g) : f , g ∈ C} < ∞. Observe that

    ρ T n+k ( f0 ) − T n ( f0 ) ≤ α ρ T n+k−1 ( f0 ) − T n−1 ( f0 )   ≤ α n ρ T k ( f0 ) − f0 ≤ α n δρ (C) → 0,

for any n, k ≥ 1. Since α < 1 and δρ (C) < ∞, we conclude that {T n ( f0 )} is ρ ρ ρ ¯ Cauchy. The ρ -completeness  n  of L (Theorem 3.2) implies the existence of f ∈ L ¯ ¯ such that lim ρ T ( f0 ) − f = 0. Because C is ρ -closed, we get f ∈ C. Since n→∞

ρ

f¯ − T ( f¯) 2



    ≤ ρ f¯ − T n ( f0 ) + ρ T n ( f0 ) − T ( f¯)     ≤ ρ f¯ − T n ( f0 ) + α ρ T n−1 ( f0 ) − f¯ → 0, as n → ∞.

Therefore, T ( f¯) = f¯, which means that f¯ is a fixed point of T . To prove the uniqueness part observe that if T ( f1 ) = f1 and T ( f2 ) = f2 , then

ρ ( f1 − f2 ) = ρ (T ( f1 ) − T ( f2 )) ≤ α ρ ( f1 − f2 ).

(5.1)

5.2 Contractions in Modular Function Spaces

115

Since α < 1 and the right-hand side is finite, equality (5.1) can hold only if f1 = f2 . 

Next, we discuss the extension of Theorem 5.1 to the case of pointwise ρ -contractions or asymptotic pointwise ρ -contractions. Theorem 5.2. Let us assume that ρ ∈ ℜ. Let C ⊂ Lρ be nonempty, ρ -closed, and ρ -bounded. Let T : C → C be a pointwise ρ -contraction or asymptotic pointwise ρ -contraction. Then T has at most one fixed point f0 ∈ C. Moreover, if f0 is a fixed point of T , then the orbit {T n ( f )} ρ -converges to f0 for any f ∈ C. Proof. Since every pointwise ρ-contraction is an asymptotic pointwise ρ-contraction, we can assume that T is an asymptotic pointwise ρ -contraction, i.e., there exists a sequence of mappings αn : C → [0, ∞) such that

ρ (T n ( f ) − T n (g)) ≤ αn ( f ) ρ ( f − g),

for all f , g ∈ C.

where {αn } converges pointwise to α : C → [0, 1). Let f1 , f2 ∈ C be two fixed points of T . Then we have

ρ ( f1 − f2 ) = ρ (T n ( f1 ) − T n ( f2 )) ≤ αn ( f1 ) ρ ( f1 − f2 ), for any n ≥ 1. If we let n → ∞, we will get

ρ ( f1 − f2 ) ≤ α ( f1 ) ρ ( f1 − f2 ). Since α ( f1 ) < 1 and C is ρ -bounded, we conclude that ρ ( f1 − f2 ) = 0, i.e. f1 = f2 . This proves that T has at most one fixed point. To prove the convergence, assume that f0 is the fixed point of T . Fix an arbitrary f ∈ C. Let us prove that {T n ( f )} ρ -converges to f0 . Indeed we have     ρ T n+m ( f ) − f0 = ρ T n+m ( f ) − T n ( f0 ) ≤ αn ( f0 ) ρ (T m ( f ) − f0 ) , for any n, m ≥ 1. Hence   lim sup ρ T n+m ( f ) − f0 ≤ lim sup αn ( f0 ) ρ (T m ( f ) − f0 ) . m→∞



m→∞



Since lim sup ρ T n+m ( f ) − f0 = lim sup ρ (T m ( f ) − f0 ), we get m→∞

m→∞

lim sup ρ (T m ( f ) − f0 ) ≤ αn ( f0 ) lim sup ρ (T m ( f ) − f0 ) , m→∞

m→∞

for any n ≥ 1. If we let n → ∞, we obtain lim sup ρ (T m ( f ) − f0 ) ≤ α ( f0 ) lim sup ρ (T m ( f ) − f0 ) . m→∞

m→∞

116

5 Fixed Point Existence Theorems in Modular Function Spaces

Since α (x0 ) < 1, we get lim sup ρ (T n ( f ) − f0 ) = 0, which implies the desired conn→∞

clusion lim ρ (T n ( f ) − f0 ) = 0.

n→∞



5.2.2 Case of Uniformly Continuous Function Modulars We will start our discussion with the existence of fixed point results in the case of uniformly continuous function modulars. Definition 5.4. Let ρ ∈ ℜ. We will say that a function modular ρ is uniformly continuous if for every ε > 0 and L > 0, there exists δ > 0 such that |ρ (g) − ρ (h + g)| ≤ ε , if ρ (h) ≤ δ and ρ (g) ≤ L, f , g ∈ Lρ . Remark 5.3. Let us mention that uniform continuity holds for a large class of function modulars. For instance, it can be proved that in Orlicz spaces over a finite atomless measure [44] or in sequence Orlicz spaces [101] the uniform continuity of the Orlicz modular is equivalent to the Δ2 property. Lemma 5.1. Let ρ ∈ ℜ be uniformly continuous. Let K ⊂ Lρ be nonempty, convex, ρ -closed, and ρ -bounded. Then, any ρ -type τ : K → [0, ∞) is ρ -lower semicontinuous in K. Proof. Let τ be a ρ -type defined on K. Then there exists { fn } in K such that

τ ( f ) = lim sup ρ ( f − fn ). n→∞

Since K is ρ -bounded, then τ ( f ) ≤ δρ (K) < ∞. Let us fix arbitrarily α > 0. Define Cα = { f ∈ K : τ (x) ≤ α }. We need to prove that Cα is ρ -closed. Without loss of generality we can assume that Cα is nonempty. Let {gk } be a sequence in Cα such that lim ρ (g0 − gk ) = 0, with g0 ∈ K. We need to prove that g0 ∈ Cα . Since K is k→∞

ρ -bounded, then L = sup ρ (g0 − fn ) < ∞. Let us fix an arbitrary ε > 0. By uniform n≥1

continuity of ρ there exists δ > 0 such that |ρ (g) − ρ (h + g)| ≤ ε i f ρ (h) ≤ δ and ρ (g) ≤ L.

(5.2)

Since lim ρ (g0 − gk ) = 0, there exists p ≥ 1 such that ρ (g0 − gk ) ≤ δ , for k ≥ p. k→∞

Using (5.2) with h = g0 − g p and g = fn − g0 we have |ρ ( fn − g0 ) − ρ ( fn − g p )| ≤ ε , f or every n ≥ 1.

5.2 Contractions in Modular Function Spaces

117

By the definition of τ , we have then that τ (g0 ) ≤ α + 2ε because g p ∈ Cα . Since ε was chosen arbitrarily we have finally τ (g0 ) ≤ α , i.e., g0 ∈ Cα , as claimed. 

We will utilize Lemma 5.1 to prove the following fixed point theorem. Theorem 5.3. Assume that ρ ∈ ℜ is uniformly continuous and has property (R). Let K ⊂ Lρ be nonempty, convex, ρ -closed, and ρ -bounded. Let T : K → K be an asymptotic pointwise ρ -contraction or asymptotic pointwise ρ -contraction. Then, T has a unique fixed point f0 ∈ K. Moreover, the orbit {T n ( f )} ρ -converges to f0 for any f ∈ K. Proof. Since every pointwise ρ -contraction is an asymptotic pointwise ρ -contraction we can assume that T is an asymptotic pointwise ρ -contraction. In view of Theorem 5.2, it is enough to show that T has a fixed point. Let us fix f ∈ K and define the ρ -type τ (g) = lim sup ρ (T n ( f ) − g), n→∞

for g ∈ K. By Lemma 5.1 the ρ -type τ is ρ -lower semicontinuous in K. Since ρ has property (R), Lemma 3.3 ensures us the existence of f0 ∈ K such that

τ ( f0 ) = inf{τ (g) : g ∈ K}. Let us prove that τ ( f0 ) = 0. Indeed, for any n, m ≥ 1 we have

ρ (T n+m ( f ) − T m ( f0 )) ≤ αm ( f0 ) ρ (T n ( f ) − f0 ). If we let n go to infinity, we get

τ (T m ( f0 )) ≤ αm ( f0 ) τ ( f0 ), which implies

τ ( f0 ) = inf{τ (g) : g ∈ K} ≤ τ (T m ( f0 )) ≤ αm ( f0 ) τ ( f0 ). Passing with m to infinity, we get τ ( f0 ) ≤ α ( f0 ) τ ( f0 ), which forces τ ( f0 ) = 0, as α ( f0 ) < 1. Hence, lim sup ρ (T n ( f ) − f0 ) = 0 holds, i.e., n→∞

lim ρ (T n ( f ) − f0 ) = 0.

n→∞

The ρ -continuity of T implies f0 is a fixed point of T .



Remark 5.4. The conclusions of Theorem 5.3 were originally proved in [114] using slightly different techniques.

118

5 Fixed Point Existence Theorems in Modular Function Spaces

5.2.3 Case of Modular Function Spaces with Strong Opial Property A typical method of proof for the fixed point theorems is to construct a fixed point by finding an element on which a specific type function attains its minimum. As a matter of fact, that is exactly how we proved our Theorem 5.3. To be able to proceed with this method, one has to know that such an element indeed exists. In case of Theorem 5.3 the assumption of uniform continuity of ρ and property (R) ensured such an existence because in this case ρ -types are ρ -lower semicontinuous which was proved in Lemma 5.1. In the case of ρ with the strong Opial property, we cannot guarantee anymore that the ρ -type is ρ -lower semicontinuous. However, in view of Theorem 4.8, ρ -types are ρ -a.e. lower semicontinuous, provided C are strongly ρ -bounded, which ensures that they attain their minimum in C. Theorem 5.4. Let ρ ∈ ℜ. Assume that Lρ has the ρ -a.e. strong Opial property. Let K ⊂ Eρ be a nonempty, ρ -a.e. compact, strongly ρ -bounded, convex set. Then, any T : K → K asymptotic pointwise ρ -contraction or pointwise ρ -contraction has a unique fixed point f0 ∈ K. Moreover, the orbit {T n ( f )} ρ -converges to f0 , for any f ∈ K. Proof. Since every pointwise ρ -contraction is an asymptotic pointwise ρ -contraction we can assume that T is an asymptotic pointwise ρ -contraction. In view of Theorem 5.2 it is enough to show that T has a fixed point. Let us fix then f ∈ K and define the ρ -type by τ (h) = lim sup ρ (T n ( f ) − h), n→∞

for h ∈ K. By Theorem 4.8 there exists f0 ∈ K such that

τ ( f0 ) = inf{τ (h) : h ∈ K}. Using the same argument as in the proof of Theorem 5.3, we will get τ ( f0 ) = 0. Hence, lim ρ (T n ( f ) − f0 )) = 0. The ρ -continuity of T forces f0 to be a fixed point n→∞ of T . 

Remark 5.5. The conclusions of Theorem 5.4 were proven by Khamsi and Kozlowski in [114] using different methods.

5.2.4 Quasi-Contraction Mappings in Modular Function Spaces ´ c [49] introduced the conAs a generalization to Banach contraction principle, Ciri´ cept of quasi-contraction mappings. In this section, we investigate this kind of mappings in modular function spaces.

5.2 Contractions in Modular Function Spaces

119

´ c definition, we introduce the concept of quasi-contractions in modSimilarly to Ciri´ ular spaces. Definition 5.5. Let ρ ∈ R. Let C be a nonempty subset of Lρ . A mapping T : C → C is said to be a quasi ρ -contraction if there exists k < 1 such that for any f , g ∈ C  ρ (T ( f ) − T (g)) ≤ k max ρ ( f − g); ρ ( f − T ( f )); ρ (g − T (g));  ρ ( f − T (g)); ρ (g − T ( f )) . In the sequel we prove an existence fixed point theorem for such mappings. First, let T and C be as in the above definition. For any f ∈ C define the orbit O( f ) = { f , T ( f ), T 2 ( f ), · · · }, and its ρ -diameter by

δρ ( f ) = diam(O( f )) = sup{ρ (T n ( f ) − T m ( f )) : n, m ∈ N} . Lemma 5.2. Let ρ ∈ ℜ. Let C be a nonempty subset of Lρ and T : C → C be a quasi ρ -contraction. Let f ∈ C be such that δρ ( f ) < ∞. Then for any n ≥ 1, we have

δρ (T n ( f )) ≤ kn δρ ( f ) , where k is the constant associated with the quasi-contraction definition of T . Moreover, we have ρ (T n ( f ) − T n+m ( f )) ≤ kn δρ ( f ) for any n, m ∈ N. Proof. Let n, m ∈ N. We have  ρ (T n ( f ) − T m (g)) ≤ k max ρ (T n−1 ( f ) − T m−1 (g)); ρ (T n−1 ( f ) − T n ( f ));

 ρ (T m (y) − T m−1 (g)); ρ (T n−1 ( f ) − T m (g)); ρ (T n ( f )−T m−1 (g))

for any f , g ∈ C. This obviously implies that

δρ (T n ( f )) ≤ kδρ (T n−1 ( f )) for any n ≥ 1. Hence for any n ≥ 1, we have

δρ (T n ( f )) ≤ kn δρ ( f ) . Moreover, for any n, m ∈ N, we have

ρ (T n ( f ) − T n+m ( f )) ≤ δρ (T n ( f )) ≤ kn δρ ( f ) .



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5 Fixed Point Existence Theorems in Modular Function Spaces

The next lemma will be helpful to prove the main result of this section. Lemma 5.3. Let ρ ∈ ℜ. Let C be a ρ -closed nonempty subset of Lρ and T : C → C a quasi ρ -contraction. Let f ∈ C such that δρ ( f ) < ∞. Then {T n ( f )} ρ -converges to h ∈ C. Moreover, we have

ρ (T n ( f ) − h) ≤ kn δρ ( f ), for any n ≥ 1. Proof. Lemma 5.2 implies that {T n ( f )} is ρ -Cauchy. Since Lρ is ρ -complete and C is ρ -closed, then there exists h ∈ C such that {T n ( f )} ρ -converges to h. Since

ρ (T n ( f ) − T n+m ( f )) ≤ kn δρ ( f ), for any n, m ∈ N, the Fatou property (once we let m → ∞) will imply

ρ (T n ( f ) − h) ≤ kn δρ ( f ) . 

Next we prove that h is in fact a fixed point of T and it is unique provided some extra assumptions. Theorem 5.5. Let C, T , and f be as in Lemma 5.3. Assume ρ (h − T (h)) < ∞ and ρ ( f − T (h)) < ∞. Then the ρ -limit h of {T n ( f )} is a fixed point of T . Moreover, if g is any fixed point of T in C such that ρ (h − g) < ∞, then we have h = g. Proof. We have  ρ (T ( f ) − T (h)) ≤ k max ρ ( f − h); ρ ( f − T ( f )); ρ (T (h) − h);  ρ (T ( f ) − h); ρ ( f − T (h)) . From the previous results, we get   ρ (T ( f ) − T (h)) ≤ k max δρ ( f ); ρ (h − T (h)); ρ ( f − T (h)) . Assume that for n ≥ 1 we have

  ρ (T n ( f ) − T (h)) ≤ max kn δρ ( f ); kρ (h − T (h)); kn ρ ( f − T (h)) .

Then

 ρ (T n+1 ( f ) − T (h)) ≤ k max ρ (T n ( f ) − h); ρ (T n ( f ) − T n+1 ( f ));

 ρ (h − T (h)); ρ (T n+1 ( f ) − h); ρ (T n ( f ) − T (h)) .

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121

Hence

  ρ (T n+1 ( f ) − T (h)) ≤ k max kn δρ ( f ); ρ (h − T (h)); ρ (T n ( f ) − T (h)) .

Using our previous assumption, we get   ρ (T n+1 ( f ) − T (h)) ≤ max kn+1 δρ ( f ); kρ (h − T (h)); kn+1 ρ (x − T (h)) . So by induction, we have   ρ (T n ( f ) − T (h)) ≤ max kn δρ ( f ); kρ (h − T (h)); kn ρ ( f − T (h)) , for any n ≥ 1. Therefore we have lim sup ρ (T n ( f ) − T (h)) ≤ kρ (h − T (h)) . n→∞

Using the Fatou property, we get

ρ (h − T (h)) ≤ lim inf ρ (T n ( f ) − T (h)) ≤ kρ (h − T (h)) . n→∞

Since k < 1, we get ρ (h − T (h)) = 0 implying T (h) = h. Let g be another fixed point of T such that ρ (h − g) < ∞. Then we have

ρ (h − g) = ρ (T (h) − T (g)) ≤ kρ (h − g) which implies h = g. This completes the proof of Theorem 5.5.



5.3 Nonexpansive and Pointwise Asymptotic Nonexpansive Mappings 5.3.1 Case of Uniformly Convex Function Modulars Similarly as in Banach and metric spaces, the fixed point existence theorems for the ρ -nonexpansive mappings (and for their pointwise and pointwise asymptotic generalizations) acting in modular function spaces are much harder to obtain than for ρ -contractions. Let us first consider the following question: Are the ρ -nonexpansive mappings really different from the mappings nonexpansive with respect to the Luxemburg norm associated with the modular ρ ? First we will show the following simple result.

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5 Fixed Point Existence Theorems in Modular Function Spaces

Proposition 5.1. Let ρ ∈ ℜ. If for every λ > 0

ρ (λ (T ( f ) − T (g))) ≤ ρ (λ ( f − g))

(5.3)

then, T ( f ) − T (g)ρ ≤  f − gρ . Proof. Assume to the contrary that there exist f , g ∈ Lρ and α > 0 such that  f − gρ < α < T ( f ) − T (g)ρ .  

  f −g f −g   < 1. < 1, which by Proposition 3.7 part (a) implies that ρ Then,  α ρ α It also implies that    T ( f ) − T (g)    , 1<  α ρ

which, by Proposition 3.7 part (b), yields 1 < ρ

λ = α −1 , we obtain

 T ( f ) − T (g) . Finally, setting α

ρ (λ ( f − g)) < 1 < ρ (λ (T ( f ) − T (g))) . This contradiction completes the proof of Proposition 5.1.



In view of Proposition 5.1, we need to ask whether the inequality (5.3) needs to hold for every λ > 0 in order to ensure the norm nonexpansiveness? If we knew that it sufficed to assume it merely for λ = 1, then there would be no real reason to consider ρ -nonexpansiveness. The answer to this question can be found in the following example of a mapping which is ρ -nonexpansive but it is not .ρ -nonexpansive. Example 5.4. Let X = (0, ∞) and Σ be the σ -algebra of all Lebesgue measurable subsets of X. Let P denote the δ -ring of subsets of finite measure. Define a function modular by  1 ∞ ρ( f ) = 2 | f (x)|x+1 dm(x). e 0 Let B be the set of all measurable functions f : (0, ∞) → R such that 0 ≤ f (x) ≤ 1/2. Consider the map  f (x − 1), for x ≥ 1, T ( f )(x) = 0, for x ∈ [0, 1]. Clearly, we have T (B) ⊂ B. For every f , g ∈ B and λ ≤ 1, we have

ρ (λ (T ( f ) − T (g))) ≤ λ ρ (λ ( f − g)) , which implies that T is ρ -nonexpansive. On the other hand, if we take f = 1[0,1] , then T ( f )ρ > e ≥  f ρ ,

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123

which clearly implies that T is not .ρ -nonexpansive. Note that T is linear. For historical reasons, let us quote without proof an early example of a fixed point theorem for ρ -nonexpansive mappings acting in modular function spaces, see Theorem 2.13 [116]. Theorem 5.6. Let ρ ∈ ℜ satisfy the regular growth condition , that is, ωρ (t) < 1 for all t ∈ [0, 1). Let C ⊂ Lρ be a convex, ρ − bounded and ρ -a.e. compact such that C − C ⊂ Lρ0 . Assume in addition that for every sequence fn ∈ C such that fn → f ρ -a.e. with f ∈ C, and for every sequence of sets Gk ↓ 0, /

 lim sup ρ ( fn − f , Gk ) = 0. k→∞

n∈N

If T : C → C is ρ -nonexpansive, then T has a fixed point. The above theorem sheds an interesting light on the following example by Alspach, [8]. Example 5.5. Define the operator  min{2, 2 f (x)}, T ( f )(x) = max{0, 2 f (2x − 1) − 2},

for x ∈ [0, 1/2], for x ∈ (1/2, 1]

on C, a convex subset of L1 [0, 1], defined by   C = f ∈ L1 [0, 1] : 0 ≤ f (x) ≤ 2 a.e. and

1

 f (x)dx = 1 .

0

Let Lρ = L1 [0, 1]. The operator T is an L1 -isometry on C hence is ρ -nonexpansive, but it does not have any fixed points. It is easy to see that all assumptions of Theorem 5.6 are satisfied except for the ρ -a.e. compactness of C. To see that C is not ρ -a.e. compact take a sequence fn = 2gn where gn = 1An and An = [2−1 − 2−n , 1 − 2−n ]. Obviously fn ∈ C but fn → 0 ρ -a.e. while the zero function does not belong to C. Note that Theorem 5.6 gives an intrinsic reason why T , defined on B = { f ∈ L1 [0, 1] : 0 ≤ f (x) ≤ 2 a.e.}, must have a fixed point while T : C → C does not have to. Moreover, we do not refer to any geometrical properties of L1 or its subspaces. A more advanced fixed point results for ρ -nonexpansive mappings started showing up in the early 2000s, see for example [63, 64, 65]. It was only after a proper modular function space geometry was established that it became possible to obtain an elegant modular fixed point theory for ρ -nonexpansive mappings, summarized below in a generalized version which covers also the asymptotic pointwise case, Theorem 5.7. On a high level, the working of this theory can be summarized as follows: 1. The uniform convexity property implies the unique best approximant property (Theorem 4.1).

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5 Fixed Point Existence Theorems in Modular Function Spaces

2. The uniform convexity property via the unique best approximant property implies the property (R) (Theorem 4.2). 3. The uniform convexity property implies the parallelogram property (Lemma 4.5). 4. The parallelogram property implies the minimizing sequence property for type functions when the minimum is strictly positive (Lemma 4.6, Case 1). 5. The property (R) implies the minimizing sequence property for type functions when the minimum is equal to zero (Lemma 4.6, Case 2). 6. The minimizing sequence property for type functions implies the fixed point property for asymptotic pointwise noenxpansive mappings; the modular limit of a minimizing sequence for a type function defined by an orbit is an obvious candidate for a fixed point. As Theorem 5.7 below shows, this is indeed the case. Theorem 5.7. Assume ρ ∈ ℜ is (UUC1). Let C be a ρ -closed ρ -bounded convex nonempty subset of Lρ . Then, any T : C → C pointwise asymptotically nonexpansive mapping has a fixed point. Moreover, the set of all fixed points F(T ) is ρ -closed and convex. Proof. Let f ∈ C. Define the ρ -type

τ (h) = lim sup ρ (T n ( f ) − h), for any h ∈ C. n→∞

Define τ0 = inf{τ (h) : h ∈ C} and note that τ0 < ∞ due to the ρ -boundedness of C. Let {gn } ⊂ C be a minimizing sequence of τ and g ∈ C its ρ -limit which exists in view of Lemma 4.6. We will prove that g is a fixed point of T . First notice that

τ (T m (h)) ≤ αm (h)τ (h), for any h ∈ C and m ≥ 1. In particular, we have

τ (T m (gn )) ≤ αm (gn )τ (gn ), for any n, m ≥ 1. By induction, we will build an increasing sequence {mk } such that 1 αmk +m (gk ) ≤ 1 + , k for k, m ≥ 1. Since T is asymptotic pointwise nonexpansive, we have lim sup αm (g1 ) ≤ 1. m→∞

Therefore, there exists m1 ≥ 1 such that for any m ≥ 1 we have 1 αm1 +m (g1 ) ≤ 1 + . 1

(5.4)

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125

Since lim sup αm (g2 ) ≤ 1, there exists m2 > m1 such that for any m ≥ 1, we have m→∞

1 αm2 +m (g2 ) ≤ 1 + . 2 Assume now that mk has been built. Since lim sup αm (gk+1 ) ≤ 1, m→∞

then there exists mk+1 > mk such that for any m ≥ mk+1 , we have

αm (gk+1 ) ≤ 1 +

1 , k+1

which completes our induction claim and proves (5.4). Let n ≥ 1, k ≥ 1 and p ≥ 0 be integers. From (5.4), it follows that 

  1 ρ T n+mk +p ( f ) − T mk +p (gk ) ≤ 1 + ρ (T n ( f ) − gk ) . (5.5) k By taking lim sup of both sides of (5.5), we obtain n→∞



τ0 ≤ τ T

mk +p

 1 (gk ) ≤ 1 + τ (gk ). k 

(5.6)

Passing with k → ∞ in (5.6) gives us   lim τ T mk +p (gk = τ0 , k→∞

which forces {T mk +p (gk )} to be a minimizing sequence of τ , for any p ≥ 0. Lemma 4.6 implies {T mk +p (gk )} is ρ -convergent to g, for any p ≥ 0. In particular, taking p = 1, we conclude that   ρ T mk +1 (gk ) − g → 0. (5.7) From

  ρ T mk +1 (gk ) − T (g) ≤ α1 (g)ρ (T mk (gk ) − g) ,

we get the following

  ρ T mk +1 (gk ) − T (g) → 0.

(5.8)

Since the ρ -limit of any ρ -convergent sequence is unique then (5.7) together with (5.8) imply that T (g) = g. To prove that F(T ) is ρ -closed, let fn ∈ F(T ) and ρ ( fn − f ) → 0. Observe that 

1 (T ( f ) − f ) ≤ ρ (T ( f ) − T ( fn )) + ρ (T ( fn ) − fn ) + ρ ( fn − f ) ρ 3 ≤ α1 ( f )ρ ( fn − f ) + ρ ( fn − f ),

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5 Fixed Point Existence Theorems in Modular Function Spaces

for any n ≥ 1. If we let n → ∞, we get 

1 (T ( f ) − f ) = 0. ρ 3 Hence f ∈ F(T ) proving F(T ) is ρ -closed. It remains to prove that F(T ) is convex. f +g ∈ F(T ), for any f , g ∈ F(T ). Note that to this end it suffices to show that h = 2 Without loss of generality, we can assume that f = g. Let k > 0 be an integer. We have ρ ( f − T k (h)) = ρ (T k ( f ) − T k (h)) ≤ αk ( f )ρ ( f − h) , and

ρ (g − T k (h)) = ρ (T k (g) − T k (h)) ≤ αk (g)ρ (g − h) . 

f −g Since ρ ( f − h) = ρ (g − h) = ρ , and 2 

1 f −g 1 ≤ ρ ( f − T k (h)) + ρ (g − T k (h)), ρ 2 2 2 we conclude that

lim ρ ( f − T (h)) = lim ρ (g − T (h)) = ρ k

k

k→∞

k→∞

 f −g . 2

Similarly we have 

1 h + T k (h) 1 ≤ ρ ( f − h) + ρ ( f − T k (h)), ρ f− 2 2 2 

1 h + T k (h) 1 ≤ ρ (g − h) + ρ (g − T k (h)). ρ g− 2 2 2

and

Since

ρ

f −g 2



1 ≤ ρ 2

 

1 h + T k (h) h + T k (h) + ρ g− , f− 2 2 2

we conclude that   

f −g h + T k (h) h + T k (h) = lim ρ g − =ρ . lim ρ f − k→∞ k→∞ 2 2 2 Therefore we have

lim ρ ( f − T (h)) = lim ρ k

k→∞

k→∞

h + T k (h) f− 2

 = ρ ( f − h).

5.3 Nonexpansive and Pointwise Asymptotic Nonexpansive Mappings

127

Lemma 4.2 applied to Ak = f − T k (h) and Bk = T k (h) − g implies that ρ (Ak − Bk ) → 0. Hence 

Ak − Bk k ≤ lim ρ (Ak − Bk ) = 0. lim ρ (h − T (h)) = lim ρ k→∞ k→∞ k→∞ 2 Clearly we will get lim ρ (h − T k+m (h)) = 0, for any integer m ≥ 0. Since k→∞

ρ (T m (h) − T k+m (h)) ≤ αm (h)ρ (h − T k (h)) we get lim ρ (T m (h) − T k+m (h)) = 0. Finally using the inequality k→∞

ρ

h − T m (h) 2



1 1 ≤ ρ (h − T k+m (h)) + ρ (T m (h) − T k+m (h)), 2 2

and by letting k → ∞, we get T m (h) = h, for any integer m ≥ 0. Taking m = 1 we get h ∈ F(T ), which completes the proof. 

Remark 5.6. Observe that the statement of the above theorem is parallel to the celebrated Browder and G¨ohde fixed point theorems (see [32, 85]) but formulated purely in terms of function modulars without any reference to norms. Also, note that Theorem 5.7 extends outside nonexpansiveness and assumes merely asymptotic pointwise ρ -nonexpansiveness of the mapping T . Therefore, Theorem 5.7 can be actually understood as parallel not only to the Browder and G¨ohde fixed point theorems for nonexpansive mappings in Banach spaces but also to the result of Kirk and Xu [128] for asymptotic pointwise nonexpansive mappings in Banach spaces. In the next section, we will discuss the analogue to Kirk’s fixed point theorem [120] in modular function spaces.

5.3.2 Normal Structure Property in Modular Function Spaces Because of the important role played by the normal structure property during the first 20 years (since 1965) of the fixed point theory, it was clear from the beginning that an analogue property in modular function spaces must be defined and investigated [116]. First, we have to introduce some basic definitions. Definition 5.6. Let C be a ρ -bounded subset of Lρ . (1) The quantity rρ ( f ,C) = sup {ρ ( f − g) : g ∈ C} will be called the ρ -Chebyshev radius of C with respect to f .

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5 Fixed Point Existence Theorems in Modular Function Spaces

(2) The ρ -Chebyshev radius of C is defined by Rρ (C) = inf {rρ ( f ,C) : f ∈ C}. (3) The ρ -Chebyshev center of C is defined by Cρ (C) = { f ∈ C : rρ ( f ,C) = Rρ (C)}. Note that Rρ (C) ≤ rρ ( f ,C) ≤ δρ (C), for any f ∈ C and any ρ -bounded nonempty subset C of Lρ . Observe that Cρ (C) may be empty in general. Definition 5.7. Let C be a ρ -bounded subset of Lρ . (1) We say that A is a ρ -admissible subset of C if A =



Bρ ( fi , ri ) ∩C, where fi ∈ C,

i∈I

ri ≥ 0, I is an arbitrary index set and Bρ ( f , r) = {g ∈ Lρ ; ρ ( f − g) ≤ r} is the ρ -closed ball centered at f with radius r. The family of all admissible subsets of C will be denoted by Aρ (C). If D is a subset of C, we write coC (D) =



Bρ ( f , rρ ( f , D)) ∩C.

f ∈D

Note that coC (D) ∈ Aρ (C) and is the smallest ρ -admissible subset of C which contains D. (2) Aρ (C) is said to be compact if any family {Aα }α ∈Γ of elements of Aρ (C), has  a nonempty intersection provided Aα = 0, / for any finite subset F ⊂ Γ . α ∈F

(3) We will say that Aρ (C) is countably compact or satisfies the property (R) if any sequence {An }n≥1 of elements of Aρ (C), which are nonempty and decreasing, has a nonempty intersection. (4) Aρ (C) is said to be normal if each ρ -admissible subset A of C, not reduced to a single point, we have Rρ (A) < δρ (A). (5) We will say that Aρ (C) is uniformly normal if there exists c ∈ (0, 1) such that for any nonempty A ∈ Aρ (C), not reduced to one point, we have Rρ (A) ≤ c δρ (A). Define the ρ -uniform normal structure coefficient of C as   Rρ (A) ˜ : A ∈ Aρ (C), A = 0/ and δρ (A) > 0 . N(C) = sup δρ (A)

Next we give the modular version of Kirk’s fixed point theorem [120]. Theorem 5.8. Let C be a nonempty ρ -closed ρ -bounded subset of Lρ . Assume that the family Aρ (C) is normal and compact. Let T : C → C be ρ -nonexpansive. Then T has a fixed point. Proof. Since Aρ (C) is compact there exists a minimal nonempty A ∈ Aρ (C) such that T (A) ⊂ A. It is easy to check that coC (T (A)) = A. Indeed, we know that coC (T (A)) is a nonempty ρ -admissible subset of C. Since T (A) ⊂ A, then coC (T (A)) ⊂ A. Hence T (coC (T (A))) ⊂ T (A) ⊂ coC (T (A)). The minimality of A

5.3 Nonexpansive and Pointwise Asymptotic Nonexpansive Mappings

129

will force A = coC (T (A)) as claimed. Let us prove that δρ (A) = 0, i.e., A is reduced to one point. Suppose that δρ (A) = 0. For any f ∈ A, we have rρ ( f , A) ≤ δρ (A) ≤ δρ (C) < ∞,   and A ⊂ Bρ ( f , rρ ( f , A)). Hence T (A) ⊂ Bρ T ( f ), rρ ( f , A) , for any f ∈ A, since T is ρ -nonexpansive. Thus   coC (T (A)) ⊂ Bρ T ( f ), rρ ( f , A) . So rρ (T ( f ), A) ≤ rρ ( f , A), for any f ∈ A. Next we fix f0 ∈ A and define A0 = { f ∈ A : rρ ( f , (A) ≤ rρ ( f0 , A)}. Clearly A0 is not empty since f0 ∈ A0 . Moreover, we have    A0 = Bρ f , rρ ( fo , A) ∩ A ∈ Aρ (C). f ∈A

Since rρ (T ( f ), A) ≤ rρ ( f , A), for any f ∈ A, we get T (A0 ) ⊂ A0 . The minimality of A implies A0 = A. In particular we have rρ ( f , A) = rρ ( f0 , A), for any f ∈ A. Hence δρ (A) = sup rρ ( f , A) = rρ ( f0 , A). Since f0 was taken arbitrary in A, we get f ∈A

Rρ (A) = δρ (A). This is a contradiction with the assumption Aρ (C) is normal. Thus, we must have δρ (A) = 0, i.e., A is reduced to one point which is fixed by T . 

Next, we give a constructive result discovered for Banach spaces by Kirk [124] which relaxes the compactness assumption in the above theorem. The main ingredient in Kirk’s constructive proof is a technical lemma due to Gillespie and Williams [77]. The next lemma is the modular version of this technical result. Lemma 5.4. Let C be a ρ -bounded subset of Lρ . Let T : C → C be a ρ -nonexpansive mapping. Assume that Aρ (C) is normal. Let A ∈ Aρ (C) be nonempty and T invariant, i.e., T (A) ⊂ A. Then there exists a nonempty A0 ∈ Aρ (C) such that A0 is T -invariant, A0 ⊂ A and

δρ (A0 ) ≤

δρ (A) + Rρ (A) . 2

Proof. Set r = 12 (δρ (A) + Rρ (A)). We assume that δρ (A) > 0, otherwise we can take the set A = A0 . Since Aρ (C) is normal, we have Rρ (A) < δρ (A). Hence Rρ (A) < r, which implies the existence of f ∈ A such that rρ ( f , A) < r. Therefore, the set D = { f ∈ A : A ⊂ Bρ ( f , r)} =

 g∈A

Bρ (g, r) ∩ A

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5 Fixed Point Existence Theorems in Modular Function Spaces

is a nonempty ρ -admissible subset of C. Note that there is no reason for D to be T -invariant. Consider the family F = {M ∈ Aρ (C) : D ⊂ M and T (M) ⊂ M}. 

Note that F is nonempty, since C ∈ F . Set L =

M. The set L is a ρ -admissible

M∈F

subset of C which contains D. Using the definition of F , we deduce that L is T -invariant. Consider B = D ∪ T (L), and observe that coC (B) = L. Indeed, since B ⊂ T (L) ⊂ L and L ∈ Aρ (C), we have coC (B) ⊂ L. From this we obtain T (coC (B)) ⊂ T (L) ⊂ B ⊂ coC (B). Hence coC (B) ∈ F , and L ⊂ coC (B). This gives the desired equality. Define A0 = { f ∈ L : L ⊂ Bρ ( f , r)} =



Bρ (g, r) ∩ L.

g∈L

We claim that A0 is the desired set. Observe that A0 is nonempty since it contains D (by definition of D) and is a ρ -admissible subset of C. On the other hand, it is clear that δρ (A0 ) ≤ r. To complete the proof, we have to show that A0 is T -invariant. Let f ∈ A0 . By definition of A0 , we have L ⊂ Bρ ( f , r). Since T is ρ -nonexpansive, we have T (L) ⊂ Bρ (T ( f ), r). For any g ∈ D, there holds L ⊂ Bρ (g, r). But T ( f ) ∈ L, so T ( f ) ∈ Bρ (g, r), which implies g ∈ Bρ (T ( f ), r). Hence D ⊂ Bρ (T ( f ), r) holds. Since B = D ∪ T (L), we get B ⊂ Bρ (T ( f ), r). Therefore, we must have coC (B) = L ⊂ Bρ (T ( f ), r). By the definition of A0 , it follows that T ( f ) ∈ A0 . In other words, A0 is T -invariant. Finally, note that since L ⊂ A, then A0 ⊂ A. 

Next we give the modular analogue of Kirk’s fixed point theorem [124]. Theorem 5.9. Let C be a ρ -bounded and ρ -closed nonempty subset of Lρ . Assume that Aρ (C) is normal and countably compact. If T : C → C is ρ -nonexpansive, then T has a fixed point. Proof. Let F = {D ∈ Aρ (C) : D = 0/ and T (D) ⊂ D}. The family F is not empty since C ∈ F . Define δ˜ : F → [0, +∞) by

δ˜ (D) = inf {δρ (B) : B ∈ F and B ⊂ D}. Set D1 = C. By definition of δ˜ (D1 ), there exists D2 ∈ F such that D2 ⊂ D1 and δρ (D2 ) < δ˜ (D1 ) + 1. Assume D1 , D2 , · · · , Dn , for n ≥ 1, are constructed. Again by

5.3 Nonexpansive and Pointwise Asymptotic Nonexpansive Mappings

131

definition of δ˜ (Dn ), there exists Dn+1 ∈ F such that δρ (Dn+1 ) < δ˜ (Dn ) + 1n and  Dn is not empty. Dn+1 ⊂ Dn . Since Aρ (C) is countably compact, then D∞ = n≥1

Clearly we have D∞ ∈ F . Assume that D∞ is not reduced to one point. Using Lemma 5.4, there exists D∗ ∈ F and D∗ ⊂ D∞ such that

δρ (D∗ ) ≤

Rρ (D∞ ) + δρ (D∞ ) . 2

(5.9)

Since D∗ ⊂ Dn , we get 1 1 δρ (D∗ ) ≤ δρ (D∞ ) ≤ δρ (Dn+1 ) ≤ δ˜ (Dn ) + ≤ δρ (D∗ ) + , n n for any n ≥ 1. If we let n → ∞, we get δρ (D∗ ) = δρ (D∞ ). Then the inequality (5.9) implies δρ (D∞ ) ≤ Rρ (D∞ ). This is in contradiction with the assumption that Aρ (C) is normal. Hence, D∞ is reduced to one point which is a fixed point of T since D∞ is T -invariant.

 As a corollary we get the following result. Corollary 5.1. Let ρ ∈ ℜ. Let C be a ρ -bounded and ρ -closed nonempty subset of Lρ . Assume that C is ρ -a.e. compact and Aρ (C) is normal. If T : C → C is ρ nonexpansive, then T has a fixed point. Proof. Recall that ρ -balls are ρ -a.e. closed because of Proposition 3.4. Hence, any element of Aρ (C) is ρ -a.e. closed subset of C. Since C is ρ -a.e. sequentially compact, we deduce that Aρ (C) is countably compact. Therefore, the conclusion of Corollary 5.1 follows from Theorem 5.9.

 In the next result we relax the assumption of ρ -a.e. compactness. Theorem 5.10. Let C be a ρ -bounded and ρ -closed nonempty subset of Lρ . Assume that Aρ (C) is uniformly normal. If T : C → C is ρ -nonexpansive, then T has a fixed point. Proof. Without loss of generality we may assume that C is not reduced to one point. Set C1 = C. Using Lemma 5.4, there exists C2 ∈ Aρ (C) such that C2 ⊂ C1 , T (C2 ) ⊂ C2 and δρ (C1 ) + Rρ (C1 ) . δρ (C2 ) ≤ 2 If we proceed by induction we will construct a sequence {Cn } of nonempty elements of Aρ (C) such that Cn+1 ⊂ Cn , T (Cn ) ⊂ Cn and

δρ (Cn+1 ) ≤

δρ (Cn ) + Rρ (Cn ) , n = 1, 2, · · · 2

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5 Fixed Point Existence Theorems in Modular Function Spaces

Since Aρ (C) is uniformly normal, there exists c ∈ (0, 1) such that Rρ (A) ≤ c δρ (A), for any A ∈ Aρ (C). Hence

δρ (Cn+1 ) ≤

Thus

δρ (Cn ) ≤

1+c δρ (Cn ), n = 1, 2, · · · 2

1+c 2

n

δρ (C), n = 1, 2, · · ·

Let fn ∈ Cn , for any n ≥ 1. Since

ρ ( fn+h − fn ) ≤ δρ (Cn ) ≤

1+c 2

n

δρ (C), n, h = 1, 2, · · ·

and 1 + c < 2, we conclude that { fn } is ρ -Cauchy. Since Lρ is ρ -complete, we deduce that { fn } is ρ -convergent to f ∈ Lρ . Since C is ρ -closed, we conclude that  f ∈ C. In fact, we have f ∈ Cn , for any n ≥ 1. Hence C∞ = Cn is not empty. n≥1

Observe that

δρ (C∞ ) ≤ δρ (Cn ) ≤

1+c 2

n

δρ (C), n = 1, 2, · · ·

This will force δρ (C∞ ) ≤ 0, i.e., C∞ is reduced to one point, i.e., C∞ = { f }. Since each Cn is T -invariant, for any n ≥ 1, we conclude that C∞ is also T -invariant, i.e., T ( f ) = f . This completes the proof of Theorem 5.10. 

Next, we discuss the normal structure property when Lρ is uniformly ρ -noncompact convex modular space. Theorem 5.11. Let ρ ∈ ℜ has property (R’). Assume that the characteristic of noncompact convexity of Lρ satisfies εα (r, Lρ ) < 1, for some r > 0. Then for any nonempty, ρ -closed, ρ -bounded convex K ⊂ Lρ such that δρ (K) ≤ r, there exists f ∈ K for which sup{ρ ( f − g) : g ∈ K} < δρ (K). Proof. Assume this is not the case. Then there exists a nonempty, ρ -closed, ρ bounded convex, not reduced to one point, set K ⊂ Lρ such that δρ (K) ≤ r and sup{ρ ( f − g) : g ∈ K} = δρ (K) = d > 0, for all f ∈ K. Fix f1 ∈ K and let f2 ∈ K such that 

1 ρ ( f2 − f1 ) ≥ d 1 − 3 . 2

5.3 Nonexpansive and Pointwise Asymptotic Nonexpansive Mappings

133

Assume that ( f1 , .., fn ) have been constructed, then let fn+1 ∈ K be such that   

1 1 n . ρ fn+1 − ∑ fi ≥ d 1 − n i=1 (n + 1)3 Hence there exists a sequence { fn } ⊂ K such that   

1 n 1 ≤ ρ fn+1 − ∑ fi ≤ d. d 1− (n + 1)3 n i=1 Using the convexity of ρ , we can get 

1 ≤ ρ ( f − fn+1 ) ≤ d, d 1− (n + 1)2 for every f =

∑

(5.10)

αi fi , with αi ≥ 0 and ∑ αi = 1. Indeed put α = max{αi }, then

1≤i≤n

 1 αi 1 1 ∑ fi = nα f + ∑ n − nα fi . n 1≤i≤n 1≤i≤n

we have

Since

1 αi 1 + ( − ) = 1, we obtain nα ∑ n nα

 1 αi 1 1 n ∑ fi − fn+1 = nα ( f − fn+1 ) + ∑ n − nα ( fi − fn+1 ). n i=1 1≤i≤n

Therefore,

d 1− and then

which implies

1 (n + 1)3

d 1−

 ≤

 1 αi 1 ρ ( f − fn+1 ) + ∑ − ρ ( fi − fn+1 ), nα nα 1≤i≤n n

1 (n + 1)3

 ≤



1 1 ρ ( f − fn+1 ) + 1 − d, nα nα

nα d 1− (n + 1)3

 ≤ ρ ( f − fn+1 ).

Since nα ≤ n + 1, we obtain the inequality (5.10). In particular, we have 

1 ρ ( fn − fm ) ≥ d 1 − 2 , m

(5.11)

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5 Fixed Point Existence Theorems in Modular Function Spaces

for every m > n ≥ 1. Since Lρ satisfies the property (R’), there exists a subsequence { fn } of { fn } such that    convρ { fi : i ≥ n} = {h}. n≥1



Denote Cn = convρ { fi :

 i ≥ n} . Using (5.11), it is not hard to show that

α (Cn ) ≥ d, for every n ≥ 1. Let f ∈ K, then Cn ⊂ Bρ ( f , d) since δρ (K) = d. Because εα (r, Lρ ) < 1, one can find Δ > 0 such that dρ ( f ,Cn ) ≤ (1 − Δ )d. Theorem 4.4 implies that Cn ∩ Bρ ( f , (1 − Δ )d) = Kn , is nonempty, for all n ≥ 1. Since the property (R’) implies the property (R) and (Kn ) is a decreasing sequence of ρ -closed ρ -bounded nonempty convex subsets, we obtain   Kn = Cn ∩ Bρ ( f , (1 − Δ )d) n

n

is nonempty. This clearly implies that

ρ ( f − h) ≤ (1 − Δ )d. Since this is true for any f ∈ K, we get sup{ρ ( f − h) : f ∈ K} ≤ (1 − Δ )d, contradicting our assumption on K. This completes the proof of Theorem 5.11. 

Remark 5.7. As a consequence of this theorem, whenever Lρ has property (R’), and the characteristic of noncompact convexity of Lρ satisfies εα (r, Lρ ) < 1, for some r > 0, then for any nonempty ρ -closed, ρ -bounded convex subset K of Lρ , and δρ (K) ≤ r, we know that Aρ (K) is normal and countably compact. Under these assumptions, Theorem 5.9 will imply that any T : K → K ρ -nonexpansive has a fixed point.

5.3.3 Case of Uniformly Lipschitzian Mappings In this section, we study the existence of fixed points for a more general class of mappings: uniformly Lipschitzian mappings. Fixed point theorems for this class

5.3 Nonexpansive and Pointwise Asymptotic Nonexpansive Mappings

135

of mappings in Banach spaces have been studied in [42, 62] and in metric spaces in [154] (for further information about this subject, see Chapter VIII of [11] and references therein). The main tool used in this section is the coefficient of normal structure property in modular sense. The following technical lemma will be useful to prove the main theorem of this section. Lemma 5.5. Let ρ ∈ ℜ be a function modular satisfying the Δ2 -type condition. Assume that ρ has the ρ -a.e. strong Opial property. Let K be a strongly ρ -bounded and ρ -a.e sequentially compact subset of Lρ . Let { fn }n and {gn }n be two sequences in K. Then, there exists g ∈ Ωae ({gn }) ∩ K such that lim sup ρ (g − fn ) ≤ lim sup lim sup ρ (g j − fn ), n→∞

n→∞

j→∞

where Ωae ({gn }) is the set of all ρ -a.e. limit of subsequences of {gn }. Proof. Let { fn }n and {gn }n be two sequences in K. Define θ : K → [0, +∞] by

θ (h) = lim sup ρ (h − fn ). n→∞

Since K is ρ -a.e. sequentially compact and ρ -bounded, there exist a subsequence {gϕ (n) } of {gn } and a subsequence { fψ (n) } of { fn } such that ⎧ ρ −a.e ⎪ ⎪ ⎨ gϕ (n) −→ g, ρ −a.e fψ (n) −→ f , ⎪ ⎪ ⎩ lim ρ ( fψ (n) − g) = lim sup ρ ( fn − g). n→∞

n→∞

Clearly we have g ∈ Ωae ({gn }) ∩ K and f ∈ K. Next we prove that

θ (g) ≤ lim sup θ (g j ). j→∞

Indeed, the ρ -a.e. strong Opial property implies

θ (g j ) = lim sup ρ ( fn − g j ) n→∞

≥ lim inf ρ ( fψ (n) − g j ) n→∞

= lim inf ρ ( fψ (n) − f ) + ρ ( f − g j ), n→∞

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5 Fixed Point Existence Theorems in Modular Function Spaces

for any j ≥ 1. Thus lim sup θ (g j ) ≥ lim inf ρ ( fψ (n) − f ) + lim sup ρ ( f − g j ) n→∞

j→∞

j→∞

≥ lim inf ρ ( fψ (n) − f ) + lim inf ρ ( f − gφ ( j) ) n→∞

j→∞

= lim inf ρ ( fψ (n) − f ) + lim inf ρ (gφ ( j) − g) + ρ ( f − g). n→∞

j→∞

On the other hand, we have

θ (g) = lim sup ρ ( fn − g) = lim ρ ( fψ (n) − g) = lim inf ρ ( fψ (n) − f ) + ρ ( f − g). n→∞

n→∞

n→∞

Therefore, we have θ (g) ≤ lim sup θ (g j ). j→∞



The following lemma was inspired by [42] where a similar result is proved in reflexive Banach spaces (see also [154] for a version in metric spaces with additional properties). Lemma 5.6. Let ρ ∈ ℜ be a function modular satisfying the Δ2 -type condition. Assume that ρ satisfies the ρ -a.e. strong Opial property. Let C be a strongly ρ bounded and ρ -a.e compact subset of Lρ . Let { fn } be a sequence in C and c a ˜ constant such that c > N(C). Then there exists f ∈ C such that (i) lim sup ρ ( f − f j ) ≤ c δρ ({ fn }), j→∞

(ii) ρ ( f − g) ≤ lim sup ρ ( fn − g), for any g ∈ C. n→∞

Proof. Let { fn } be a sequence in C. Set A=

∞ 

Am ⊂ C,

m=1

  where Am = coC { f j : j ≥ m} ∈ Aρ (C), for any m ≥ 1. Since ρ satisfies the Fatou property, any ρ -ball is ρ -a.e. closed. Using the ρ -a.e compactness of C, then A = 0. / It is easy to check that δρ (Am ) ≤ δρ ({ fn }) and Am ⊂ Bρ (g, sup ρ (g − f j )), for any m ≥ 1 and g ∈ C. Hence for any f ∈ A and g ∈ C, we have

j≥m

ρ (g − f ) ≤ rρ (g, A) ≤ rρ (g, An ) ≤ sup ρ (g − f j ), j≥n

for any n ≥ 1. In fact we have rρ (g, An ) = sup ρ (g − f j ), for any n ≥ 1. Therefore, j≥n

we have

ρ (g − f ) ≤ lim sup ρ (g − fn ), n→∞

5.3 Nonexpansive and Pointwise Asymptotic Nonexpansive Mappings

137

i.e., (ii) holds for any f ∈ A. Next we prove that there exists f ∈ A such that (i) holds. Without loss of generality we may assume that δρ ({ fn }) > 0. Choose ε > 0 ˜ δρ ({ fn }) + ε ≤ c δρ ({ fn }). By the definition of Rρ (Am ), there exists such that N(C) gm ∈ Am such that ˜ ˜ δρ (Am ) + ε ≤ N(C) δρ ({ fn }) + ε ≤ c δρ ({ fn }), rρ (gm , Am ) < Rρ (Am ) + ε ≤ N(C) for any m ≥ 1. Since sup ρ (gm − f j ) = rρ (gm , Am ), we have j≥m

lim sup ρ (gm − f j ) ≤ c δρ ({ fn }),

(5.12)

j→∞

for any m ≥ 1. Using Lemma 5.5, there exists f ∈ Ωae ({gn }) ∩C such that lim sup ρ ( f − f j ) ≤ lim sup lim sup ρ (gn − f j ). n→∞

j→∞

(5.13)

j→∞

We will check that f ∈ A. Indeed, for all i, n integers such that i ≥ n we have gi ∈ Ai ⊂ An . Thus, Ωae ({gn }) ∩C ⊂ An for any n ≥ 1, which implies f ∈ A. Using the inequalities (5.12) and (5.13), we get lim sup ρ ( f − f j ) ≤ c δρ ({ fn }), j→∞

which completes the proof of Lemma 5.6. 

Using the two lemmas above, we are ready to prove the main theorem of this section. Theorem 5.12. [63] Let ρ ∈ ℜ be a function modular satisfying the Δ2 -type condition. Assume that ρ satisfies the ρ -a.e. strong Opial property. Let K be a strongly ˜ ρ -bounded and ρ -a.e compact convex subset of Lρ . Suppose that N(K) < 1. Let T : K → K be a k-uniformly Lipschitzian mapping, i.e.,

ρ (T n ( f ) − T n (g)) ≤ k ρ ( f − g), −1/2 . Then T has a fixed point. ˜ for any f , g ∈ K. Suppose that k < (N(K))

Proof. If k ≤ 1, then T is ρ -nonexpansive, so Theorem 5.10 shows that T has a ˜ fixed point. Therefore, we may assume that k > 1. Choose a constant c ∈ (N(K), 1) −1/2 . Fix f0 ∈ K. By Lemma 5.6, we can inductively construct a such that 1 < k < c sequence { fn } ⊂ K such that (1) lim sup ρ (T n ( fm ) − fm+1 ) ≤ c δ ({T n ( fm )}n ), n→∞

(2) ρ ( fm+1 − g) ≤ lim sup ρ (T n ( fm ) − g), n→∞

138

5 Fixed Point Existence Theorems in Modular Function Spaces

for all g ∈ K and m ≥ 0. Denote Dm = lim sup ρ (T n ( fm ) − fm+1 ). For n, h ∈ N, we n→∞

have

ρ (T n+h ( fm ) − T n ( fm )) ≤ k ρ ( fm − T h ( fm )) ≤ k lim sup ρ (T i ( fm−1 ) − T h ( fm )) i→∞

≤ k lim sup ρ (T i−h ( fm−1 ) − fm ) 2

i→∞

≤ k Dm−1 . 2

Since Dm = lim sup ρ (T n ( fm ) − fm+1 ) ≤ c δ ({T n ( fm )}n ), we obtain n→∞

Dm ≤ c k2 Dm−1 ≤ (c k2 )m D0 , m = 1, 2, · · · Thus, for any j ∈ N, we have   ρ ( f j+1 − f j ) ≤ ωρ (2) ρ ( f j+1 − T n ( f j )) + ρ ( f j − T n ( f j ))   ≤ ωρ (2) ρ ( f j+1 − T n ( f j )) + lim sup ρ (T m ( f j−1 ) − T n ( f j )) m→∞   n ≤ ωρ (2) ρ ( f j+1 − T ( f j )) + k lim sup ρ (T m−n ( f j−1 ) − f j ) m→∞   n ≤ ωρ (2) ρ ( f j+1 − T ( f j )) + k D j−1 , where ωρ (t) is the growth function associated to the modular ρ (see Definition 3.7). Taking lim sup as n → ∞, we obtain

ρ ( f j+1 − f j ) ≤ ωρ (2)(D j + k D j−1 ) ≤ ωρ (2)((c k2 ) j + k (c k2 ) j−1 )D0 ≤ ωρ (2)(c k2 + k)(c k2 ) j−1 D0 ≤ A(c k2 ) j , where A = ωρ (2) c ck+1 k D0 and j ∈ N, which implies 1 1 . ≤ 2 j A(c k ) ρ ( f j+1 − f j ) Using Lemma 3.1, we obtain

ωρ −1





 j



1 1 1 1 −1 −1 , ≤ ωρ −1 ≤ ω ω ρ ρ A c k2 A(c k2 ) j ρ ( f j+1 − f j )

5.3 Nonexpansive and Pointwise Asymptotic Nonexpansive Mappings

139

for any j ∈ N. Therefore, we have || f j+1 − f j ||ρ ≤

ωρ

−1



1 1

ρ ( f j+1 − f j )

 ≤B 

ωρ −1

1 

1 c k2

j ,

  for any j ∈ N. Since ωρ (1) = 1 and ck2 < 1, we conclude that ωρ−1 ck12 > 1. Hence { fn } is a Cauchy sequence in (Lρ , ||.||ρ ). Since (Lρ , ||.||ρ ) is complete, there exists f ∈ Lρ such that || fn − f ||ρ → 0. Since under Δ2 -condition norm convergence and modular convergence are identical, { fn } is ρ -convergent to f . Thus, Proposition 3.5 implies the existence of a subsequence of { fn } ρ -a.e convergent to f . Note that f belongs to K since K is ρ -a.e closed. Next we prove that f is a fixed point of T . Indeed,   ρ ( f − T ( f )) ≤ ωρ (3) ρ ( f − f j+1 ) + ρ ( f j+1 − T n ( f j )) + ρ (T n ( f j ) − T ( f ))   ≤ ωρ (3) ρ ( f − f j+1 ) + ρ ( f j+1 − T n ( f j )) + k ρ (T n−1 ( f j ) − f )  ≤ ωρ (3) ρ ( f − f j+1 ) + ρ ( f j+1 − T n ( f j ))    n−1 +k ωρ (2) ρ (T ( f j ) − f j+1 ) + ρ ( f j+1 − f ) , for any n ≥ 1 and j ∈ N. Taking lim sup as n → ∞, we have     ρ ( f − T f ) ≤ ωρ (3) ρ ( f − f j+1 ) + D j + k ωρ (2) D j + ρ ( f j+1 − f ) , for any j ∈ N. Taking the limit as j → ∞, we obtain ρ ( f − T f ) = 0, i.e., T ( f ) = f . 

The following lemma gives a relationship between the coefficient of ρ -normal structure and the ρ -modulus of uniform convexity. Lemma 5.7. Let ρ ∈ ℜ staisfy the Δ2 -type condition. Let C be a ρ -bounded nonempty convex subset of Lρ not reduced to one point. Then,

 ˜ N(C) ≤ 1 − inf δ2 (d, ε ) : d ∈ (0, δρ (C)] ,

for any ε ∈ 0,

 1 . ωρ (2)

Proof. Let B ∈ Aρ (C) be nonempty and not reduced to one point, i.e., δρ (B) > 0. Since C is convex and the ρ -balls are convex, we deduce that B is convex. Denote d = δρ (B) and r = Rρ (B). Let ε ∈ (0, 1). There exist f , g ∈ B such that

140

5 Fixed Point Existence Theorems in Modular Function Spaces

ρ ( f − g) ≥ εδ (B). Hence

ρ

f −g 2

 ≥

dε ρ ( f − g) ≥ . ωρ (2) ωρ (2)

Let h ∈ B. We know that ρ (h − f ) ≤ d , ρ (h − g) ≤ d and 

dε (h − f ) − (h − g) ≥ . ρ 2 ωρ (2) 

ε , we have By the definition of δ2 d, ωρ (2)  



(h − f ) + (h − g) f +g ε =ρ ≤ d 1 − δ2 d, , ρ h− 2 2 ω (2) for all h ∈ B. Thus,



Rρ (B) ε r = ≤ 1 − δ2 d, . δρ (B) d ω (2)

Therefore, 

  ε ˜ : 0 < d ≤ δρ (C) N(C) ≤ sup 1 − δ2 d, ω (2) 

  ε : 0 < d ≤ δρ (C) . ≤ 1 − inf 1 − δ2 d, ω (2) 

Remark 5.8. If we use the modulus of uniform convexity 

  f +g 1 : ( f , g) ∈ D1 (r, ε ) , δ1 (r, ε ) = inf 1 − ρ r 2   where D1 (r, ε ) = ( f , g) : f , g ∈ Lρ , ρ ( f ) ≤ r, ρ (g) ≤ r, ρ ( f − g) ≥ ε r , then we do not need ρ to satisfy the Δ2 -type condition. The estimate found in Lemma 5.7 becomes 

˜ N(C) ≤ 1 − inf δ2 (d, ε ) : d ∈ (0, δρ (C)] , for any ε ∈ (0, 1). Next we give an improvement to the conclusion of Theorem 5.12 in the case of asymptotically regular mappings. First we need the following definition.

5.3 Nonexpansive and Pointwise Asymptotic Nonexpansive Mappings

141

Definition 5.8. Let C be a nonempty subset of Lρ and T : C → C. We denote by |T | the ρ -Lipschitz constant of T , i.e.,   ρ T ( f ) − T (g) : f = g, f , g ∈ C |T | = sup ρ ( f − g) and s(T ) = lim inf |T n |. We say that T is asymptotically regular if n→∞

lim ρ (T n+1 ( f ) − T n ( f )) = 0,

n→∞

for any f ∈ C. The following lemma will be needed to prove the next theorem. Lemma 5.8. Let ρ ∈ ℜ satisfy the Δ2 -type condition. Let { fn } and {gn } be two sequences in Lρ . Assume lim ρ (gn ) = 0. Then n→∞

lim sup ρ ( fn + gn ) = lim sup ρ ( fn ). n→∞

n→∞

Proof. Using the properties of ρ and ωρ , we get

 g  fn n , +ρ ρ ( fn + gn ) ≤ ρ 1−ε ε for any ε ∈ (0, 1). Thus

ρ ( fn + gn ) ≤ ωρ

1 1−ε



 1 ρ ( fn ) + ωρ ρ (gn ) ε

which implies

lim sup ρ ( fn + gn ) ≤ ωρ n→∞

1 1−ε

 lim sup ρ ( fn ). n→∞

Since ε is arbitrary and

ωρ

1 1−ε

 → 1 as ε → 0+

we obtain lim sup ρ ( fn + gn ) ≤ lim sup ρ ( fn ). n→∞

n→∞

Furthermore, the same argument proves lim sup ρ ( fn ) = lim sup ρ ( fn + gn − gn ) ≤ lim sup ρ ( fn + gn ). n→∞

n→∞

n→∞

142

5 Fixed Point Existence Theorems in Modular Function Spaces

Putting the inequalities together we get lim sup ρ ( fn + gn ) = lim sup ρ ( fn ) as n→∞

n→∞

claimed. 

Theorem 5.13. [64] Let ρ ∈ ℜ be a function modular satisfying the Δ2 -type condition. Assume that ρ has the ρ -a.e. strong Opial property. Let C be a strongly ρ bounded, ρ -a.e. compact subset of Lρ . Let T : C → C be an asymptotically regular mapping such that s(T ) < 2. Then, T has a fixed point. Proof. Using the definition of s(T ), there exists a subsequence {T nk } such that s(T ) = lim |T nk | = lim inf |T n |. Define the function r : C → [0, +∞) by k→∞

n→∞

 r( f ) = inf r > 0 : ∃g ∈ C such that lim inf ρ ( f − T nk (g)) ≤ r . k→∞

1 + s(T ) . Then our assumption 2 on s(T )implies s(T ) < b < 2. Since ωρ (1) = 1, there exists  ε ∈ (0, 1) such that 1 1 γ 1 . Then choose γ ∈ (0, 1) such that ωρ . Set < < ωρ ε b−1 ε b−1

  1 1 1 , . δ = min γ + (1 − b)ωρ 2 2 ε Note that r( f ) ≤ δρ (C), for any f ∈ C. Set b =

Note that 0 < δ < 1. Now, we choose a number μ ∈ (0, 1) such that ) (   γ − δ + ωρ ε1 (1 − b) δ  ,   . μ < min ωρ 1−1 ε ωρ ε1 b Finally denote (

δ γ −δ  1  , b(1 + μ ) −   α = max 1 + μ − ωρ 1−ε ωρ ε1

) .

Then 0 < α < 1. Let f ∈ C. Assume that r( f ) = 0. For any m ≥ 1, there exists gm ∈ C such that 1 lim inf ρ ( f − T nk (gm )) ≤ . k→∞ m Then we have  ρ ( f − T ( f )) ≤ ωρ (3) ρ ( f − T nk (gm )) + ρ (T nk (gm ) − T nk +1 (gm )) +ρ (T nk +1 (gm ) − T ( f )  ≤ ωρ (3) (1 + |T |) ρ ( f − T nk (gm )) + ρ (T nk (gm ) − T nk +1 (gm )) .

5.3 Nonexpansive and Pointwise Asymptotic Nonexpansive Mappings

143

If we take the lim inf when nk → +∞, we get  1 ρ ( f − T ( f )) ≤ ωρ (3) (1 + |T |) + 0 , m where we used the asymptotic regularity of T . Now if we let m → +∞, we get ρ ( f − T ( f )) = 0, i.e., T ( f ) = f . Next assume that r( f ) > 0. Since γ < 1, we can find a positive integer k0 ≥ 1 such that |T nk0 | < b and ρ ( f − T nk0 ( f )) > γ r( f ). Otherwise, we will have lim inf ρ ( f − T n ( f )) ≤ lim inf ρ ( f − T nk ( f )) ≤ γ r( f ). n→∞

k→∞

Hence r( f ) ≤ γ r( f ) < r( f ), which is a contradiction since r( f ) > 0. Therefore there exists k0 ≥ 1 such that |T nk0 | < b and ρ ( f − T nk0 ( f )) > γ r( f ). Since μ > 0, we can also find g ∈ C such that lim inf ρ ( f − T nk (g)) ≤ r( f )(1 + μ ). k→∞

Consider a subsequence {nk } of {nk } such that {T nk (g)} is ρ -a.e convergent in C, say to h, and ρ ( f − T nk (g)) = lim inf ρ ( f − T nk (g)). lim  k →∞

k→∞

Using Lemma 5.8 and the asymptotic regularity of T , we get lim sup ρ (T nk0 ( f ) − T nk (g)) ≤ |T nk0 | lim sup ρ ( f − T nk −nk0 (g)) k →∞

k →∞

= |T

nk0

= |T

nk0

| lim sup ρ ( f−T nk (g)+T nk (g)−T nk −nk0 (g)) k →∞

| lim sup ρ ( f − T nk (g)). k →∞

We claim that lim inf ρ (T nk (g) − h) ≤ α r( f ).  k →∞

Indeed if

ρ ( f − h) ≥

ωρ

δ  1  r( f ), 1−ε

(5.14)

144

5 Fixed Point Existence Theorems in Modular Function Spaces

then using the ρ -a.e. strong Opial property, we get inf ρ (T nk (g) − f ) − ρ ( f − h) lim inf ρ (T nk (g) − h) = lim   k →∞

k →∞

≤ (1 + μ )r( f ) − " = (1 + μ ) −

ωρ

≤ α r( f ).

ωρ

δ  1  r( f )

1−ε # δ  1  r( f ) 1−ε

Otherwise, assume that

ρ ( f − h) <

ωρ

δ  1  r( f ). 1−ε

Using again the ρ -a.e. strong Opial property, we get inf ρ (T nk (g) − T nk0 ( f )) − ρ (T nk0 ( f ) − h). lim inf ρ (T nk (g) − h) = lim   k →∞

k →∞

The properties of the modular ρ and the growth function ωρ give

 1 1 nk0 nk0 ε (T ( f ) − h) + ρ (T ( f ) − f ) = ρ (1 − ε )( f − h) ε 1−ε



 1 1 nk0 ρ (T ( f ) − h) + ωρ ρ (h − f ). ≤ ωρ ε 1−ε Hence

ρ (T

nk0

  ωρ 1−1 ε 1 nk0   ρ (T ( f ) − f ) −   ρ (h − f ). ( f ) − h) ≥ ωρ ε1 ωρ ε1

Since inf ρ (T nk (g) − T nk0 ( f )) − ρ (T nk0 ( f ) − h) lim inf ρ (T nk (g) − h) = lim   k →∞

k →∞ nk0

≤ |T

| lim inf ρ (T nk −nk0 (g) − f ) − ρ (T nk0 ( f ) − h)  k →∞

≤ b(1 + μ )r( f ) − ρ (T nk0 ( f ) − h),

5.3 Nonexpansive and Pointwise Asymptotic Nonexpansive Mappings

145

we get lim inf ρ (T  k →∞

nk 

  ωρ 1−1 ε 1 nk0   ρ (h − f ) (g)−h) ≤ b(1 + μ )r( f )−  1  ρ (T f − f ) + ωρ ε ωρ ε1 ≤ b(1 + μ )r( f ) −

ωρ

γ  1  r( f ) +

≤ α r( f ),

ε

ωρ

δ  1  r( f ) ε

which proves the inequality (5.14). The latter implies, by the definition of the function r, that r(h) ≤ α r( f ). Moreover   

h− f h− f ≤ ωρ (2)ρ ρ (h − f ) = ρ 2 2 2   nk  nk  inf ρ (h − T (g)) + lim inf ρ (T (g) − f ) ≤ ωρ (2) lim k →∞ k →∞   ≤ ωρ (2) α r( f ) + (1 + μ )r( f ) = Ar( f ), where A = ωρ (2)(α + 1 + μ ). Set h0 = f and h1 = h. Assume that h0 , h1 , · · · , hn−1 are defined, we consider hn as the corresponding element to hn−1 in the above construction. Thus, r(hn ) ≤ α r(hn−1 ) ≤ · · · ≤ α n r(h0 ), and ρ (hn+1 − hn ) ≤ Ar(hn ). Hence, we have constructed {hn } such that

ρ (hn+1 − hn ) ≤ K α n ≤ β n , where K = Ar( f ), which implies 1 1 . ≤ K αn ρ (hn+1 − hn ) Using the properties of ωρ , we get 



n



1 1 1 1 −1 −1 −1 −1 . ≤ω ω ω ≤ω K α K αn ρ (hn+1 − hn ) Since ||hn+1 − hn ||ρ ≤

ω −1



1 1 ρ (hn+1 −hn )

≤

ω −1

1  1   n , ω −1 α1 K

and α < 1, we conclude that {hn } is a Cauchy sequence in (Lρ , ||.||ρ ). Therefore there exists h ∈ Lρ such that lim ||hn − h||ρ = 0, because (Lρ , ||.||ρ ) is complete. n→∞

Hence, {hn } is ρ -convergent to h which implies the existence of a subsequence {hϕ (n) }n of {hϕ (n) }n which converges to h ρ -a.e. Since C is ρ -a.e. closed, we get h ∈ C. We finish the proof of Theorem 5.13 by showing that r(h) = 0. Indeed, let

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5 Fixed Point Existence Theorems in Modular Function Spaces

ε > 0. We choose n large enough such that ε ε r(hn ) < , and ρ (hn − h) < . 2 2 ε There exists g ∈ C such that lim inf ρ (T nk (g) − hn ) ≤ . Hence k→∞ 2 

n T k (g) − h lim inf ρ ≤ lim inf ρ (T nk (g) − hn ) + ρ (hn − h) ≤ ε , k→∞ k→∞ 2 which implies lim inf ρ (T nk (g) − h) ≤ ωρ (2)ε . k→∞

Therefore we have r(h) ≤ ωρ (2)ε . Since ε was taken arbitrarily positive, we get r(h) = 0. This implies, as we proved earlier, that T (h) = h, i.e., T has a fixed point. 

5.3.4 Common Fixed Point Theorems in Modular Function Spaces The purpose of this section is to give an outline of a common fixed point theory for nonexpansive mappings defined on some subsets of modular function spaces. One frequently used approach to prove the existence of a common fixed point is the investigation of the fixed point set of one mapping. In this section, we lay down the work done in [7] extending the ideas developed in [108] in metric spaces to modular function spaces. Definition 5.9. Let ρ ∈ ℜ and C ⊂ Lρ be nonempty. A nonempty subset D of C is said to be a one-local retract of C if for every family {Bi : i ∈ I} of ρ -balls centered in D such that, if   Bi = 0/ C i∈I

holds then the following is true as well   D / Bi = 0. i∈I

Recall that D ⊂ C is a ρ -nonexpansive retract of C if there exists a ρ -nonexpansive map R : C → D such that R( f ) = f , for every f ∈ D. It is immediate that each ρ nonexpansive retract of Lρ is a one-local retract (but not conversely). The following result will shed some light on the interest generated around this concept. Theorem 5.14. Let ρ ∈ ℜ and C ⊂ Lρ be nonempty, ρ -closed, and ρ -bounded. Assume that A (C) is compact and normal. Then for any ρ -nonexpansive mapping T : C → C, the fixed point set F(T ) is a nonempty one-local retract of C.

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147

Proof. Theorem 5.8 shows that F(T ) is nonempty. Let us complete the proof by showing it is a one-local retract of C. Let {Bρ ( fi , ri )}i∈I be any family of ρ -closed balls such that fi ∈ F(T ), for any i ∈ I, and   C0 = C





= 0. /

Bρ ( fi , ri )

i∈I

/ Since { fi }i∈I ⊂ F(T ), and T is ρ Let us prove that F(T ) ∩ (∩i∈I Bρ ( fi , ri )) = 0. nonexpansive, then T (C0 ) ⊂ C0 . Clearly, C0 ∈ A (C) and is nonempty. Then we have A (C0 ) ⊂ A (C). Therefore, A (C0 ) is compact and ρ -normal. Theorem 5.8 implies that T has a fixed point in C0 which will give us   F(T )





= 0. /

Bρ ( fi , ri )

i∈I

 Next we discuss some properties of one-local retract subsets. Theorem 5.15. Let ρ ∈ ℜ and C ⊂ Lρ be nonempty. Let D be a nonempty subset of C. The following are equivalent. (i) D is a one-local retract of C. (ii) D is a ρ -nonexpansive retract of D ∪ { f }, for every f ∈ C. Proof. Let us prove (i) ⇒ (ii). Let f ∈ C. We may assume that f ∈ D. In order to construct a ρ -nonexpansive retract R : D ∪ { f } → D, we only need to find R( f ) ∈ D such that ρ (R( f ) − g) ≤ ρ ( f − g), for every g ∈ D. Since f ∈



Bρ (g, ρ ( f − g)) and f ∈ C, then

g∈D

 C∩





Bρ (g, ρ ( f − g))

= 0. /

g∈D

Since D is a one-local retract of C, we get  D0 = D ∩





Bρ (g, ρ ( f − g))

= 0. /

g∈D

Any point in D0 will work as R( f ). Next, we prove that (ii) ⇒ (i). In order to prove that D is a one-local retract of C, let {Bρ ( fi , ri )}i∈I be any family of ρ -closed balls such that fi ∈ D, for any i ∈ I, and   C0 = C ∩

 i∈I

Bρ ( fi , ri )

= 0. /

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5 Fixed Point Existence Theorems in Modular Function Spaces

Let us prove that D ∩ (∩i∈I Bρ ( fi , ri )) = 0. / Let f ∈ C0 . If f ∈ D, we have nothing to prove. Assume otherwise that f ∈ D. Property (ii) implies the existence of a ρ nonexpansive retract R : D ∪ { f } → C. It is easy to check that / R( f ) ∈ D ∩ (∩i∈I Bρ ( fi , ri )) = 0, which completes the proof of Theorem 5.15. 

The following technical lemma will be useful for the next results. Lemma 5.9. Let ρ ∈ ℜ and C ⊂ Lρ be nonempty, and ρ -bounded. Let D be a nonempty one-local retract of C. Set coC (D) = C ∩



{A : A ∈ A (C) and D ⊂ A}.

Then (i) rρ ( f , D) = rρ ( f , coC (D)), for any f ∈ C (ii) Rρ (coC (D)) = Rρ (D) (iii) δρ (coC (D)) = δρ (D) Proof. Let us first prove (i). Fix f ∈ C. Since D ⊂ coC (D), we get rρ ( f , D) ≤ rρ ( f , coC (D)). Set r = rρ ( f , D). We have D ⊂ Bρ ( f , r) ∈ A (C). From the definition of coC (D) it follows that coC (D) ⊂ Bρ ( f , r). Hence rρ ( f , coC (D)) ≤ r = rρ ( f , D), which implies rρ ( f , D) = rρ ( f , coC (D)). Next, let us prove (ii). Let f ∈ D. We have f ∈ coC (D). Using (i), we get rρ ( f , D) = rρ ( f , coC (D)) ≥ Rρ (coC (D)). Hence, Rρ (D) ≥ Rρ (coC (D)). Let f ∈ coC (D) and set r = rρ ( f , coC (D)). Since we have D ⊂ coC (D) ⊂ Bρ ( f , r) then f∈



Bρ (g, r),

g∈D

which implies that C∩(



Bρ (g, r)) = 0. /

g∈D

Since D is a one-local retract of C, we get / D0 = D ∩ (∩g∈D Bρ (g, r)) = 0. Let g ∈ D0 . Observe that rρ (g, D) ≤ r. Hence, Rρ (D) ≤ r. Since f was arbitrary chosen in coC (D), we get Rρ (D) ≤ Rρ (coC (D)), which implies Rρ (D) = Rρ (coC (D)).

5.3 Nonexpansive and Pointwise Asymptotic Nonexpansive Mappings

149

Finally, let us prove (iii). Since D ⊂ coC (D), we get δρ (D) ≤ δρ (coC (D)). Next set d = δρ (D). Then, for any f ∈ D, we have D ⊂ Bρ ( f , d). Hence coC (D) ⊂ Bρ ( f , d). This implies that  Bρ (g, d). f∈ g∈coC (D))

Since f was arbitrarily chosen in D, we get 

D⊂

Bρ (g, d).

g∈coC (D))

The definition of coC (D) implies coC (D) ⊂



Bρ (g, d).

g∈coC (D))

Therefore, for any f , g ∈ coC (D), we have ρ ( f − g) ≤ d. δρ (coC (D)) ≤ d=δρ (D), which implies δρ (D) = δρ (coC (D)). 

Hence

As an application of this lemma we get the following result. Theorem 5.16. Let ρ ∈ ℜ and C ⊂ Lρ be nonempty, ρ -closed, and ρ -bounded. Assume that A (C) is compact and ρ -normal. If D is a nonempty one-local retract of C, then A (D) is compact and ρ -normal. Proof. Using the definition of one-local retract, it is easy to see that A (D) is compact. Let us show that A (D) is ρ -normal. Let A0 ∈ A (D) nonempty and not reduced to one point. Set 

coC (A0 ) = C ∩ ( {A : A ∈ A (C) and A0 ⊂ A}). From Lemma 5.9 we have Rρ (coC (A0 )) = Rρ (A0 ), and δρ (coC (A0 )) = δρ (A0 ). Since coC (A0 ) ∈ A (C), then we must have Rρ (coC (A0 )) < δρ (coC (A0 )) because A (C) is ρ -normal. Therefore, Rρ (A0 ) < δρ (A0 ), which completes the proof of our claim.

 The next result is quite amazing. Its metric counterpart has found many applications. As we will see toward the end this section, we already have few quite important applications of Theorem 5.17 (Theorems 5.18, 5.19, 5.20, and Corollary 5.2) and the future research will for sure extend this list. Most of the ideas in the proof below actually go back to the pioneering work of Baillon [13].

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5 Fixed Point Existence Theorems in Modular Function Spaces

Theorem 5.17. Let ρ ∈ ℜ and C ⊂ Lρ be nonempty, ρ -closed, and ρ -bounded. Assume that A (C) is compact and ρ -normal. Let (Cβ )β ∈Γ be a decreasing family of one-local retracts of C, where (Γ , ≺) is totally ordered. Then 

Cβ = 0/

β ∈Γ

and is a one-local retract of C. 

Proof. First, let us prove that

Cβ is not empty. Consider the family

β ∈Γ

)

( F=

∏ Aβ

β ∈Γ

∏ Cβ

F is not empty since

∏ Aβ ⊂ ∏ Bβ

β ∈Γ

β ∈Γ

: Aβ ∈ A (Cβ ) and (Aβ ) is decreasing

.

∈ F . F will be ordered by inclusion, i.e.,

β ∈Γ

if and only if Aβ ⊂ Bβ for any β ∈ Γ . From Theorem 5.16, we

know that A (Cβ ) is compact, for every β ∈ Γ . Therefore, F satisfies the hypothesis of Zorn’s lemma. Hence for every D ∈ F , there exists a minimal element A ∈ F such that A ⊂ D. We claim that if A = ∏ Aβ is minimal, then there exists β0 ∈ Γ β ∈Γ

such that δ (Aβ ) = 0 for every β  β0 . Assume not, i.e., δ (Aβ ) > 0 for every β ∈ Γ . Fix β ∈ Γ . For every K ⊂ C, set 

coβ (K) = Consider A =

Bρ ( f , rρ ( f , K)).

f ∈Cβ

∏ Aα where

α ∈Γ

⎧  ⎨ Aα = coβ (Aβ ) ∩ Aα if α ≤ β ⎩

Aα = Aα

if α ≥ β .

The family (Aα ≥β ) is decreasing since A ∈ F . Let α ≤ γ ≤ β . Then Aγ ⊂ Aα since Aγ ⊂ Aα and Aβ = coβ (Aβ ) ∩ Aβ . Hence the family (Aα ) is decreasing. On the other hand if α ≺ β , then coβ (Aβ ) ∩ Aα ∈ A (Cα ) since Cβ ⊂ Cα . Hence Aα ∈ A (Cα ). Therefore, we have A ∈ F . Since A is minimal, then A = A . Hence Aα = coβ (Aβ )



Aα , for every α ≺ β .

Let f ∈ Cβ and α ≺ β . Since Aβ ⊂ Aα , then rρ ( f , Aβ ) ≤ rρ ( f , Aα ). Because coβ (Aβ ) =

 g∈Cβ

Bρ (g, rρ (g, Aβ ))

5.3 Nonexpansive and Pointwise Asymptotic Nonexpansive Mappings

151

we have coβ (Aβ ) ⊂ Bρ (g, rρ (g, Aβ )) which implies rρ (g, Aβ ) ≤ rρ (g, Aα ). Since Aα ⊂ coβ (Aβ ), then rρ (g, Aβ ) ≤ rρ (g, Aα ) ≤ rρ (g, coβ (Aβ )) ≤ rρ (g, Aβ ). Therefore, we have rρ (g, Aα ) = rρ (g, Aβ ) for every g ∈ Cβ . Using the definition of the ρ -Chebyshev radius Rρ , we get Rρ (Aα ) ≤ Rρ (Aβ ). Let f ∈ Aα and set s = rρ ( f , Aα ). Then f ∈ coβ (Aβ ) since Aα ⊂ coβ (Aβ ). Hence, f ∈(



Bρ (g, s))



coβ (Aβ ).

g∈Aβ

Since Cβ is a one-local retract of C, then ⎛ Sβ = Cβ ∩ ⎝



⎞ Bρ (g, s)⎠ ∩ coβ (Aβ ) = 0. /

g∈Aβ

Since Aβ = Cβ ∩ coβ (Aβ ), then we have Sβ = Aβ ∩

 

 Bρ (g, s) .

g∈Aβ

Let h ∈ Sβ , then h ∈



Bρ (g, s). Hence, rρ (h, Aβ ) ≤ s which implies

g∈Aβ

Rρ (Aβ ) ≤ s = rρ ( f , Aα ), for every f ∈ Aα . Hence Rρ (Aβ ) ≤ Rρ (Aα ). Therefore we have Rρ (Aβ ) = Rρ (Aα ), for every α , β ∈ Γ . Since δρ (Aβ ) > 0 for every β ∈ Γ , set Aβ to be the ρ -Chebyshev center of Aβ , i.e., Aβ = Cρ (Aβ ), for every β ∈ Γ . Since Rρ (Aβ ) = Rρ (Aα ), for every α , β ∈ Γ , then the family (Aβ ) is decreasing. Indeed, let α ≺ β and f ∈ Aβ . Then we have rρ ( f , Aβ ) = Rρ (Aβ ). Since we proved that rρ (g, Aβ ) = rρ (g, Aα ), for every g ∈ Cβ , then rρ ( f , Aα ) = rρ ( f , Aβ ) = Rρ (Aβ ) = Rρ (Aα ), which implies that f ∈ Aα . Therefore, we have A =

∏ Aβ ∈ F . Since A ⊂ A and

β ∈Γ

A is minimal, we get A = A . Therefore, we have Cρ (Aβ ) = Aβ for every β ∈ Γ . This contradicts the fact that A (Cβ ) is normal for every β ∈ Γ . Hence, there exists β0 ∈ Γ such that δ (Aβ ) = 0, for every β  β0 .

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5 Fixed Point Existence Theorems in Modular Function Spaces

The proof of our claim is therefore complete. Then we have Aβ = { f }, for every  β  β0 . This clearly implies that f ∈ Cβ = 0. / In order to complete the proof, we need to show that S =



β ∈Γ

Cβ is a one-local retract of C. Let (Bi )i∈I be a family

β ∈Γ

of ρ -balls centered in S such that



Bi = 0. / Set Dβ = (

i∈I



Bi ) ∩Cβ , for any β ∈ Γ .

i∈I

Since Cβ is a one-local retract of C, and the family (Bi ) is centered in Cβ , then Dβ is not empty and Dβ ∈ A (Cβ ). Therefore, D = ∏ Dβ ∈ F . Let A = ∏ Aβ ⊂ D β ∈Γ

be a minimal element of F . The above proof shows that 0/ =

 β ∈Γ

Aβ ⊂



β ∈Γ

Dβ .

β ∈Γ

The proof of Theorem 5.17 is therefore complete. 

The next theorem is useful to prove the main common fixed point result of this section. Theorem 5.18. Let ρ ∈ ℜ and C ⊂ Lρ be nonempty, ρ -closed, and ρ -bounded. Assume that A (C) is compact and ρ -normal. Let (Cβ )β ∈Γ be a family of one-local  Cβ is not empty and is a retracts of C such that for any finite subset I of Γ , β ∈I

one-local retract of C. Proof. Consider the family F ofsubsets I ⊂ Γ such that for any finite subset J ⊂ Γ (empty or not), we know that Cα is a nonempty one-local retract of C. Note α ∈I∪J

that F is not empty since any finite subset of Γ is in F . Using Theorem 5.17, we can show that F satisfies the hypothesis of Zorn’s lemma. Hence F has a maximal element I ⊂ Γ . Assume I = Γ . Let α ∈ Γ \ I. Obviously we have I ∪ {α } ∈ F . This isa clear contradiction with the maximality of I. Therefore we have I = Γ ∈ F , i.e., Cβ is not empty and is a one-local retract of C. β ∈Γ



In order to show that any family of commutative ρ -nonexpansive mappings have a common fixed point, we need to establish that it is the case for any finite family. Theorem 5.19. Let ρ ∈ ℜ and C ⊂ Lρ be nonempty, ρ -closed, and ρ -bounded. Assume that A (C) is compact and ρ -normal. Then for any finite family F = {T1 , T2 , · · · , Tn } of commutative ρ -nonexpansive mappings defined on C has / Moreover, the set of common a common fixed point, i.e., F(T1 ) ∩ · · · ∩ F(Tn ) = 0. fixed point set, denoted F(F ) = F(T1 ) ∩ · · · ∩ F(Tn ), is a one-local retract of C.

5.3 Nonexpansive and Pointwise Asymptotic Nonexpansive Mappings

153

Proof. Let us first prove Theorem 5.19 for two mappings T1 and T2 . Using Theorem 5.14, we know that F(T1 ) is a nonempty one-local retract of C. Since T1 and T2 are commutative, then T2 (F(T1 )) ⊂ F(T1 ). Theorems 5.8 and 5.16 show that the restriction of T2 to F(T1 ) has a fixed point. Again Theorem 5.14 implies that the common fixed point set F(T1 ) ∩ F(T2 ) is a nonempty one-local retract of C. Using the same argument shows that the conclusion of Theorem 5.19 is valid for any finite number of mappings. 

We are now ready to prove the main result of this section. Theorem 5.20. Let ρ ∈ ℜ and C ⊂ Lρ be nonempty, ρ -closed, and ρ -bounded. Assume that A (C) is compact and ρ -normal. Then for any family F = {Ti : i ∈ I} of ρ -nonexpansive mappings defined on C has a common fixed point, commutative   i.e., F(Ti ) = 0. / Moreover the set of common fixed point set, denoted F(F ) = F(Ti ), i∈I

i∈I

is a one-local retract of C. Proof. Let Γ = {β : β be a nonempty finite subset of I}. Theorem 5.19 implies that for every β ∈ Γ , the set Fβ of common fixed point set of the mappings Ti , i ∈ β , is a nonempty one-local retract of C. Clearly the family (Fβ )β ∈Γ is decreas ing and satisfies the assumptions of Theorem 5.18. Therefore, we have that Fβ β ∈Γ

is nonempty and is a one-local retract of C. The proof of Theorem 5.20 is complete. 

Using Corollary 4.1 we get the following result. Corollary 5.2. Let ρ ∈ ℜ and C ⊂ Lρ be nonempty, convex, ρ -closed, and ρ bounded. Assume ρ is (UUC2). Then for any family F = {Ti : i ∈ I} of commutative ρ -nonexpansive mappings defined on C has a common fixed point, i.e., F(F ) =



F(Ti ) = 0. /

i∈I

Moreover the set of common fixed points is a one-local retract of C.

5.3.5 Asymptotic Nonexpansive Mappings in Modular Function Spaces Satisfying Δ2 -type Condition Throughout this section, we assume that ρ has strong Opial property, and satisfies exists an increasing sequence of sets the Δ2 -type condition and σ -finite, i.e., there Kn . Kn ∈ P such that 0 < ρ (1Kn ) < ∞ and Ω = n≥1

154

5 Fixed Point Existence Theorems in Modular Function Spaces

Let C be a convex, strongly ρ -bounded and ρ -a.e. compact nonempty subset of the modular function space Lρ . Recall that the mapping T : C → C is said to be a asymptotic ρ -nonexpansive mapping if there exists a sequence of positive integers {kn }n which converge to 1 such that for every n ∈ N and f , g ∈ C we have

ρ (T n ( f ) − T n (g)) ≤ kn ρ ( f − g). Denote F the family of all subsets K of C satisfying the following property: K is nonempty, convex and ρ -a.e. closed such that f ∈ K implies Ωa.e. ( f ) ⊂ K

(5.15)

where Ωa.e. ( f ) is the set of all ρ -a.e. limits of subsequences of {T n ( f )}. Note that Ωa.e. ( f ) is not empty since C is ρ -a.e. compact, and F is nonempty since C ∈ F . Ordering F by inclusion, there exists a nonempty minimal element H in F which satisfies (5.15) since C is ρ -a.e. compact for the topology generated by d (see Proposition 3.15). The following lemma is the counterpart in modular function spaces of Lemma 2.1 in [202] for Banach spaces. Lemma 5.10. Under the above assumptions, for each f ∈ H define the functional r f (g) = lim sup ρ (T n ( f ) − g), n→∞

for any g ∈ Lρ . Then the functional r f (·) is constant on H and this constant is independent of f in H. Proof. Let t > 0 and f ∈ H. Set Ht ( f ) = { g ∈ H : r f (g) ≤ t } . It is easily seen that Ht ( f ) is convex because ρ is convex. We claim that Ht ( f ) is ρ -a.e. closed. Indeed, assume that {gm } ∈ Ht ( f ) ρ -a.e. converges to g ∈ H. Using Theorem 4.8, we get   r f (g) = lim sup ρ (T n ( f ) − g) ≤ lim inf lim sup ρ (T n ( f ) − gm ) = lim inf r f (gm ), n→∞

m→∞

n→∞

m→∞

which implies r f (g) ≤ t, i.e., g ∈ Ht ( f ). Hence Ht ( f ) is ρ -a.e. closed. Since H is ρ -a.e. compact, we deduce that Ht ( f ) is ρ -a.e. compact. Next, we claim that Ht ( f ) satisfies property (5.15). Indeed, let g ∈ Ht ( f ) and h ∈ Ωa.e. (g). We need to check that h ∈ Ht ( f ). By definition of Ωa.e. (g), there exists an increasing sequence of

5.3 Nonexpansive and Pointwise Asymptotic Nonexpansive Mappings

155

integers {ni }i such that T ni (g)→h ρ -a.e. Theorem 4.8 implies that   r f (h) = lim sup ρ (T n ( f ) − h) ≤ lim inf lim sup ρ (T n ( f ) − T ni (g)) i→∞ n→∞ n→∞       ni ≤ lim inf r f T (g) ≤ lim sup r f T ni (g) ≤ lim sup r f T m (g) i→∞ m→∞ i→∞   n m ≤ lim sup lim sup ρ (T ( f ) − T (g)) m→∞ n→∞   ≤ lim sup km lim sup ρ (T n−m ( f ) − g) m→∞ n→∞   ≤ lim sup lim sup ρ (T n ( f ) − g) ≤ t. m→∞

n→∞

Hence h ∈ Ht ( f ) as claimed. The minimality of H implies that Ht ( f ) is empty or equal to H. From this, it is clear that rt (.) is constant on H. In order to complete the proof of this lemma, we need to prove that r f is independent of f . Let f , g ∈ H. Since C is ρ -a.e. compact, there exists a subsequence {T ni (g)}i of {T n (g)}n which ρ -a.e. converges to h ∈ C. Since H satisfies property (5.15), we have h ∈ H. Theorem 4.8 implies ρ (T n ( f ) − h) ≤ lim inf ρ (T n ( f ) − T ni (g)). i→∞

Hence, r f = r f (h) = lim sup ρ (T n ( f ) − h) n→∞   ≤ lim sup lim inf ρ (T n ( f ) − T ni (g)) i→∞ n→∞   ≤ lim sup lim sup ρ (T n ( f ) − T m (g)) n→∞

m→∞

≤ lim sup ρ ( f − T m (g)) = rg ( f ) = rg , m→∞

which implies rg = r f .

 Keeping in mind that if ρ satisfies the Δ2 -type condition, then the ρ -convergence and the Luxemburg norm convergence coincide, and we have the following result as Lemma 5.11. Lemma 5.11. Let ρ be a convex function modular satisfying the Δ2 -type condition. Let S be a nonempty, norm compact subset of Lρ with diamρ (S) > 0. Then there exists f ∈ conv(S) such that sup{ρ (g − f ) : g ∈ S} < diamρ (S). Proof. Since S is compact and ρ is norm continuous, there exist f0 , f1 ∈ S such that ρ ( f0 − f1 ) = diamρ (S). Let S0 be a maximal subset of S such that f0 , f1 ∈ S0 and

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5 Fixed Point Existence Theorems in Modular Function Spaces

for any f , g ∈ S0 , f = g, we have ρ ( f − g) = diamρ (S). Since S is compact, S0 must be finite. Write S0 = { f0 , f1 , f2 , ....., fn } and define h=

f0 + f1 + · · · + fn . n+1

Since S is compact, there exists g0 ∈ S such that

ρ (g0 − h) = sup{ρ (g − h) : g ∈ S}. On the other hand, using the convexity of ρ , we get     k=n k=n 1 1 g0 − ∑ fk ρ (g0 − h) = ρ ∑ k=0 n + 1 k=0 n + 1  k=n 1 ≤ ∑ ρ (g0 − fk ) ≤ diamρ (S). k=0 n + 1 If ρ (g0 − h) = diamρ (S), then we must have ρ (g0 − fk ) = diamρ (S), for k = 0, 1, ...., n. This contradicts the maximality of S0 . Hence, sup{ρ (g − h) : g ∈ S} = ρ (g0 − h) < diamρ (S). 

Theorem 5.21. Let ρ be a convex σ -finite function modular with strong Opial property satisfying the Δ2 -type condition and C be a strongly ρ -bounded, ρ -a.e. compact subset of Lρ . Let T : C → C be an asymptotic ρ -nonexpansive mapping. Let H be a convex subset of C such that: (i) If f ∈ H then Ωρ −a.e ( f ) ⊂ H. (ii) For each f ∈ H, any subsequence {T ni ( f )}i of {T n ( f )}n , has a ρ -convergent subsequence. Then T has a fixed point. Proof. Consider the family F of nonempty ρ -a.e. compact subset of H, which satisfies property (5.15). F is not empty since H ∈ F . By the previous results, F has a minimal element. Let K be a minimal element of F . Assume that K has more than one point, i.e., δρ (K) > 0. Let f ∈ K. Set S = Ω||·||ρ ( f ) = {g ∈ H : T ni ( f ) || · ||ρ -converges to g for some ni ↑ ∞}. It is easy to see that S ⊂ K. We claim that S = T (S). Indeed, let g ∈ S. Then there exists a sequence {T ni ( f )}i which ||.||ρ -converges to g. Since T is continuous, we have T ni +1 ( f ) → T (g) in norm. By the definition of S, we get T (g) ∈ S, i.e., T (S) ⊂ S. Let us show the other inclusion, i.e., S ⊂ T (S). Let g ∈ S. Again by the definition of S, there exists a sequence {T ni ( f )}i which || · ||ρ -converges to g. The sequence {T ni −1 ( f )}i has a norm convergent subsequence, say {T nφ (i) −1 ( f )}i . Let

5.3 Nonexpansive and Pointwise Asymptotic Nonexpansive Mappings

157

h be its || · ||ρ -limit. Since T is continuous, we get T (h) = T (lim T nφ (i) −1 ( f )) = lim T nφ (i) ( f ) = g. i→∞

i→∞

Hence g ∈ T (S), i.e., S ⊂ T (S). Therefore, T (S) = S is true. Next, notice that the assumption (ii) implies that S is norm compact. Lemma 5.11 implies the existence of f0 ∈ conv(S) ⊂ K such that sup{ρ (g − f0 ) : g ∈ S} < δρ (S).

(A)

Let r = sup{ρ (g − f0 ) : g ∈ S}. Set D = {h ∈ K; sup ρ (g − h) ≤ r}. Since f0 ∈ D g∈S

and ρ is convex, D is a nonempty convex subset of K. We claim that D = K. Indeed, let us first show that D is ρ -a.e. compact. By the assumption (ii), it is enough to show that D is ρ -a.e. closed. Let {hn }n be a sequence in D such that hn →h ∈ Lρ ρ -a.e. Fix g ∈ S. Since g − hn →g − h ρ -a.e. Theorem 4.8 implies

ρ (g − h) ≤ lim inf ρ (g − hn ), n→∞

which yields   ρ (g − h) ≤ lim inf sup{ρ ( f − hn ) : f ∈ S} ≤ r. n→∞

Hence, sup{ρ (h − g) : g ∈ S} ≤ r, i.e., h ∈ D. Next, we check that D satisfies property (5.15)). Indeed, let f ∈ D and g ∈ Ωa.e ( f ). Then there exists a sequence of natural numbers ni such that T ni ( f )→g ρ -a.e.. Using Theorem 4.8 we obtain

ρ (g − h) ≤ lim inf ρ (T ni ( f ) − h) ≤ lim sup ρ (T n ( f ) − h), n→∞

n→∞

for any h ∈ S. Since T (S) = S, there exists a sequence {un }n in S such that h = T n (un ), for any n ≥ 1. Hence,

ρ (g − h) ≤ lim sup ρ (T n ( f ) − T n (un )) n→∞

≤ lim sup kn ρ ( f − un ) n→∞

≤ lim sup ρ ( f − un ) n→∞

≤ sup{ρ ( f − u); u ∈ S} ≤ r. So sup{ρ (g − h) : h ∈ S} ≤ r which implies g ∈ D. Thus D satisfies property (5.15) and by minimality of K, we obtain D = K. On the other hand,

δρ (D) ≤ r < δρ (S) ≤ δρ (K),

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5 Fixed Point Existence Theorems in Modular Function Spaces

which is a contradiction. Therefore, K is reduced to one point. Property (5.15) forces this point to be a fixed point of T . 

Now we are ready to state and prove the main result of this section. Theorem 5.22. [65] Let ρ be a convex, σ -finite function modular with strong Opial property satisfying the Δ2 -type condition. Let C be a convex, strongly ρ -bounded and ρ -a.e. compact, nonempty subset of Lρ . Let T : C → C be asymptotic ρ nonexpansive. Then T has a fixed point. Proof. Let F be the family of nonempty convex subsets of C, which satisfy the property (5.15). F is not empty since C ∈ F . By Zorn’s lemma, F has a minimal element H. Let us show that H satisfies the hypothesis of Theorem 5.21. It suffices to check that H satisfies property (ii). Let r be the function defined on H as in Lemma 5.10. If r = 0 we have lim T n ( f ) = g, for any f , g ∈ H, which implies n→∞ (ii). Otherwise, assume that r > 0. Let f ∈ H be such that there exists a sequence {T ni ( f )}i , which has no norm convergent subsequence. Thus, there exists ε > 0 and a subsequence {T n(k) ( f )}k such that 

sep({T n(k) ( f )}k ) = inf{ρ (T n(k) ( f ) − T n(k ) ( f )) : k = k } ≥ ε . ρ −a.e

Since H is ρ -a.e. compact, there exists f∞ ∈ H such that T n(k) ( f ) −→ f∞ ∈ H as k → ∞. Without loss of generality, we may assume the existence of lim ρ (T n(k) ( f ) − f∞ ) = l.

k→∞

ε Since lim sup ρ (T n ( f ) − f ) = r, we choose η > 0 such that η < , and an integer 2 n→∞ n0 ≥ 1, such that for all n ≥ n0 we have ρ (T n ( f ) − f ) < r + η . Fix n ≥ n0 . There exists k0 ≥ 1 such that for all k ≥ k0 , we have n(k) ≥ n + n0 and   ρ (T n f − T n(k) f ) = ρ T n ( f ) − T n+(n(k)−n) ( f )   ≤ kn ρ f − T n(k)−n ( f ) < kn (r + η ). ρ −a.e

Note that if fn −→ f and sep{ fn }n ≥ ε , then by Theorem 4.8, we have

ε ≤ lim inf lim inf ρ ( fn − fm ) ≤ 2 lim inf ρ ( fn − f ). m→∞

n→∞

n→∞

Using again Theorem 4.8, we get lim inf ρ ( fn ) = lim inf ρ ( fn − f ) + ρ ( f ) ≥ n→∞

n→∞

ε + ρ ( f ). 2

5.3 Nonexpansive and Pointwise Asymptotic Nonexpansive Mappings

159

In particular, since {T n(k) f − T n f }k is ρ -a.e. convergent to f∞ − T n f as k → ∞ and satisfies sep({T n(k) f − T n f }k ) ≥ ε , we get

ε ρ (T n f − f∞ ) ≤ lim inf ρ (T n(k) f − T n f ) − . k→∞ 2 Hence,

ε ρ ( f∞ − T n f ) ≤ r + η − , 2

which implies r = lim sup ρ ( f∞ − T n f ) ≤ r + η − n→∞

ε < r. 2

This contradiction completes the proof of Theorem 5.22. 

Consider Lρ = L p (Ω , μ ), p ≥ 1, for a σ -finite measure μ . Let C be a convex, bounded and closed nonempty subset of L p , for 1 < p < ∞. Let T : C → C be an asymptotic nonexpansive mapping. Using the uniform convexity of L p when p > 1, one can see that T has a fixed point. However the result does not hold for p = 1 (even for nonexpansive mappings, see [8]). Since L1 is a modular function space, Theorem 5.22 implies the existence of fixed point if p = 1 when C is ρ -a.e. compact. Thus we can state. Corollary 5.3. Let C be a convex, bounded, and closed nonempty subset of L1 (Ω , μ ), where μ is σ -finite measure. Assume that C is compact for the topology of the convergence local in measure. Let T : C → C be an asymptotic nonexpansive mapping. Then, T has a fixed point. Proof. Under the above hypothesis, ρ -a.e. compact sets and compact sets in the topology of convergence local in measure are identical.



5.3.6 KKM and Ky Fan Theorems in Modular Function Spaces Among the results equivalent to the Brouwer’s fixed point theorem, the theorem of Knaster–Kuratowski–Mazurkiewicz occupies a special place [67]. Let ρ ∈ ℜ and let C ⊂ Lρ be nonempty. The set of all subsets of C is denoted as 2C . The notation conv(A) describes the convex hull of A, while convρ (A) describes the smallest ρ -closed convex subset of Lρ which contains A. Recall that a family {Aα : Aα ∈ 2Lρ , α ∈ Γ } is said to have the finite intersection property if the intersection of each finite subfamily is not empty. Definition 5.10. Let ρ ∈ ℜ and let C ⊂ Lρ be nonempty. A multivalued mapping G : C → 2Lρ is called a Knaster–Kuratowski–Mazurkiewicz mapping (in short KKM-mapping) if G( fi ) conv({ f1 , . . . , fn }) ⊂ 1≤i≤n

for any f1 , . . . , fn ∈ C.

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5 Fixed Point Existence Theorems in Modular Function Spaces

Theorem 5.23. Let ρ ∈ ℜ. Let C ⊂ Lρ be nonempty and G : C → 2Lρ be a KKMmapping such that for any f ∈ C, G( f ) is nonempty and finitely ρ -closed. Then, the family {G( f ) : f ∈ C} has the finite intersection property. Proof. Assume this is not the case, i.e., there exist f1 , . . . , fn ∈ C such that 

G( fi ) = 0. /

1≤i≤n

Define L = convρ ({ fi }) in Lρ . Our assumptions imply that L ∩ G( fi ) is ρ -closed for every i = 1, 2, . . . , n. Since the norm convergence implies the modular convergence, we deduce that L ∩ G( fi ) is closed for the Luxemburg norm .ρ for any i ∈ {1, . . . , n}. Thus for every f ∈ L, there exists i0 such that f does not belong to  L ∩ G( fi0 ) since L ∩ G( fi ) = 0. / Hence, 1≤i≤n



 d f , L ∩ G( fi0 ) = inf{ f − gρ : g ∈ L ∩ G( fi0 )} > 0, because L ∩ G( fi0 ) is closed. We use the function    α ( f ) = ∑ d f , L G( fi ) > 0, 1≤i≤n

where f ∈ K = conv{ f1 , . . . , fn } to define the map T : K → K by T(f) =

   1 d f , L G( fi ) fi . ∑ α ( f ) 1≤i≤n

Clearly, T is a continuous map. Since K is a compact convex subset of the Banach space (Lρ ,  f ρ ), Brouwer’s theorem implies the existence of a fixed point f0 ∈ K of T , i.e., T ( f0 ) = f0 . Set  

  I = i : d f0 , L G( fi ) = 0 . Clearly, f0 = Hence, f0 ∈



   1 d f0 , L G( fi ) fi . ∑ α ( f0 ) i∈I

G( fi ) and f0 ∈ conv({ fi : i ∈ I}) as this contradicts the assumption

i∈I

  conv { fi : i ∈ I} ⊂ G( fi ) . i∈I



As an immediate consequence, we obtain the following result.

5.3 Nonexpansive and Pointwise Asymptotic Nonexpansive Mappings

161

Theorem 5.24. Let ρ ∈ ℜ. Let C ⊂ Lρ be nonempty and G : C → 2Lρ be a KKMmapping such that for any f ∈ C, G( f ) is nonempty and ρ -closed. Assume that there exists f0 ∈ C such that G( f0 ) is ρ -compact. Then, we have 

G( f ) = 0/ .

f ∈C

Notice that the ρ -compactness of G( f0 ) may be weakened, i.e., we can still reach the conclusion if one involves an auxiliary multivalued map and a suitable topology on Lρ , as shown in the next theorem whose proof is immediate. Theorem 5.25. Let ρ ∈ ℜ. Let C ⊂ Lρ be nonempty and G : C → 2Lρ be a KKMmapping such that for any f ∈ C, G( f ) is nonempty and finitely ρ -closed. Assume there is a multivalued map K : C → 2Lρ such that G( f ) ⊂ K( f ) for every f ∈ C and 

K( f ) =

f ∈C



G( f ) .

f ∈C

If there is some topology τ on Lρ such that each K( f ) is τ -compact, then 

G( f ) = 0/ .

f ∈C

Before we state an analog to Ky Fan fixed point result [67], we need the following definition. Definition 5.11. Let ρ ∈ ℜ. Let C ⊂ Lρ be a nonempty ρ-closed subset. Let T :C→Lρ be any map. T is called ρ -continuous if {T ( fn )} ρ -converges to T ( f ) whenever { fn } ρ -converges to f . Also T is called strongly ρ -continuous if T is ρ -continuous and lim inf ρ (g − T ( fn )) = ρ (g − T ( f )), n→∞

for any sequence { fn } ⊂ C, which ρ -converges to f and for any g ∈ C. In general, it is unknown for what type of function modular ρ , ρ -continuity implies strong ρ -continuity. However, the Δ2 -property is enough to provide this implication. Lemma 5.12 is needed to prove the modular version of the Ky Fan fixed point theorem. Lemma 5.12. Let ρ ∈ ℜ. Let K ⊂ Lρ be nonempty convex and ρ -compact. Let T : K → Lρ be strongly ρ -continuous. Then, there exists f0 ∈ K such that   ρ ( f0 − T ( f0 )) = inf ρ f − T ( f0 ) . f ∈K

162

5 Fixed Point Existence Theorems in Modular Function Spaces

Proof. Consider the map G : K → 2Lρ defined by

 G(g) = f ∈ K : ρ ( f − T ( f )) ≤ ρ (g − T ( f )) . Since T is strongly ρ -continuous, then for any sequence { fn } ⊂ G(g) which ρ converges to f , we have

ρ ( f − T ( f )) ≤ lim inf ρ ( fn − T ( fn )) ≤ lim inf ρ (g − T ( fn )) = ρ (g − T ( f )), n→∞

n→∞

on the basis of the Fatou property and the continuity of T . Clearly, this implies that G(g) is ρ -closed for any g ∈ K. Next, we show that G is a KKM-mapping. Assume not. Then, there exists {g1 , . . . , gn } ⊂ K and f ∈ conv({gi }) such that G(gi ). This clearly implies f ∈ 1≤i≤n

ρ (gi − T ( f )) < ρ ( f − T ( f )) , for i = 1, . . . , n . Let ε > 0 be such that ρ (gi − T ( f )) ≤ ρ ( f − T ( f )) − ε , for i = 1, 2, . . . , n. Since ρ is convex, for any g ∈ conv({gi }), we have

ρ (g − T ( f )) ≤ ρ ( f − T ( f )) − ε . As f ∈ conv({gi }), so we get

ρ ( f − T ( f )) ≤ ρ ( f − T ( f )) − ε . Contradiction. Therefore, G is a KKM-mapping. By the ρ -compactness of K, we deduce that G(g) is compact for any g ∈ K. Theorem 5.24 implies the existence of G(g). Hence, f0 ∈ g∈K

ρ ( f0 − T ( f0 )) ≤ ρ (g − T ( f0 )) for any g ∈ K. In particular, we have   ρ ( f0 − T ( f0 )) = inf ρ g − T ( f0 ) . g∈K



We are now ready to state Ky Fan Fixed Point Theorem in modular function spaces. Theorem 5.26. Let ρ ∈ ℜ. Let K ⊂ Lρ be nonempty convex and ρ -compact. Let T : K → Lρ be strongly ρ -continuous. Assume that for any f ∈ K, with f = T ( f ), there exists α ∈ (0, 1) such that     / (5.16) K ∩ Bρ f , αρ ( f − T ( f )) ∩ Bρ T ( f ), (1 − α )ρ ( f − T ( f )) = 0. Then, T has a fixed point, i.e., T (g) = g for some g ∈ K.

5.4 Applications to Differential Equations

163

Proof. From the previous lemma, there exists f0 ∈ K such that   ρ ( f0 − T ( f0 )) = inf ρ g − T ( f0 ) . g∈K

(5.17)

We claim that f0 is a fixed point of T . Assume it is not, i.e., f0 = T ( f0 ). Then, our assumption on K implies the existence of α ∈ (0, 1) such that     K0 = K ∩ Bρ f0 , αρ ( f0 − T ( f0 )) ∩ Bρ T ( f0 ), (1 − α )ρ ( f0 − T ( f0 )) = 0/ . Let g ∈ K0 . Then, ρ (g−T ( f0 )) ≤ (1− α )ρ ( f0 −T ( f0 )). This implies a contradiction with (5.17).

 Note that the condition (5.16) is satisfied if T (K) ⊂ K, which implies the following important result. Theorem 5.27. Let ρ ∈ ℜ. Let K ⊂ Lρ be nonempty convex and ρ -compact. Let T : K → K be strongly ρ -continuous. Then, T has a fixed point, i.e., T (g) = g for some g ∈ K.

5.4 Applications to Differential Equations Let ρ ∈ ℜ. To the end of this section we consider the following initial value problem for an unknown function u : [0, A] → C, where C ⊂ Eρ : ( u(0) = f (5.18) u (t) + (I − T )u(t) = 0, where f ∈ C and A > 0 are fixed and T : C → C is ρ -nonexpansive. Let us introduce the following convenient notations that are used throughout this section. For any t > 0 we define K(t) = 1 − e−t =

 t

es−t ds.

(5.19)

0

For any function v : [0, A] → Lρ , where A > 0, and any t ∈ [0, A], we define  t

S(v)(t) = 0

es−t v(s)ds.

(5.20)

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5 Fixed Point Existence Theorems in Modular Function Spaces

Similarly, we denote n−1

Sτ (v)(t) =

∑ (ti+1 − ti )eti −t v(ti ),

(5.21)

i=0

for any τ = {t0 , . . . ,tn }, a subdivision of the interval [0, A]. Let us start with the following technical lemma. Lemma 5.13. Let ρ ∈ ℜ be separable. Let x, y : [0, A] → Lρ be two Bochnerintegrable  · ρ -bounded functions, where A > 0. Then for every t ∈ [0, A] we have  t   ρ e−t y(t) + es−t x(s)ds ≤ e−t ρ (y(t)) + K(t) sup ρ (x(s)). 0

(5.22)

s∈[0,t]

Proof. Without any loss of generality we can assume that sup ρ (x(s)) < ∞. Let s∈[0,t]

τ = {t0 , . . . ,tn } be a subdivision of the interval [0, A]. Define n−1

Zτ (t) = e−t y(t) + Sτ (x)(t) = e−t y(t) + ∑ (ti+1 − ti )eti −t x(ti ). i=0

It follows from the Lebesgue dominated convergence theorem for Bochner integrals, that for every t ∈ [0, A], Zτ (t) converges in .ρ to Z(t) defined by Z(t) = e−t y(t) + S(x)(t) = e−t y(t) +

 t

es−t x(s)ds,

0

whenever |τ | = sup |ti+1 − ti | → 0. Using the Fatou property of ρ and Proposition 0≤i≤n

3.6, we can easily see that

ρ (Z(t)) ≤ lim inf ρ (Zτ (t)). |τ |→0

(5.23)

Since, n−1

e−t + ∑ (ti+1 − ti )eti −t ≤ 1, i=0

and ρ is convex, it follows that

ρ (Zτ (t)) ≤ e−t ρ (y(t)) +

  n−1 ∑ (ti+1 − ti )eti −t sup ρ (x(s)). i=0

(5.24)

s∈[0,t]

By combining (5.24) with (5.23) and letting |τ | → 0 we get the desired inequality (5.22). 

We are now ready to prove our main result.

5.4 Applications to Differential Equations

165

Theorem 5.28. Let ρ ∈ ℜ be separable. Let C ⊂ Eρ be a nonempty, convex, ρ bounded, ρ -closed set with the Vitali property. Let T : C → C be a ρ -nonexpansive mapping. Let us fix f ∈ C and A > 0 and define the sequence of functions un : [0, A] → C by the following inductive formula: ( u0 (t) = f (5.25) ' un+1 (t) = e−t f + 0t es−t T (un (s))ds. Then for every t ∈ [0, A] there exists u(t) ∈ C such that

ρ (un (t) − u(t)) → 0

(5.26)

and the function u : [0, A] → C defined by (5.26) is a solution of the initial value problem (5.18). Moreover,

ρ ( f − un (t)) ≤ K n+1 (A)δρ (C).

(5.27)

Proof. We show that un (t) ∈ C for every n ∈ N and every t ∈ [0, A]. Indeed, for a given subdivision τ of [0, A], define   uτn+1 (t) = e−t f + Sτ T (un (ti )) (t). Since C is a convex set, it is easy to prove by induction that uτn (t) ∈ C. Note that, by the properties of the Bochner integral, uτn (t) − un (t)ρ → 0. Since C is  · ρ -closed it is also ρ -closed. Hence, un (t) ∈ C as claimed. Now we prove that

ρ (un+p (t) − un (t)) ≤ K n+1 (A)δρ (C)

(5.28)

for all t ∈ [0, A] and every n, p ∈ N. We prove (5.28) by induction with respect to n. For n = 0 we have u p (t) − u0 (t) =

 t 0

es−t T (u p−1 (s))ds.

Applying Lemma 5.13 with y(t) = 0 and x(t) = T (u p−1 (s)) − f we have

ρ (u p (t) − u0 (t)) ≤ K(t) sup ρ (T (u p−1 (s)) − f ) ≤ K(A)δρ (C), s∈[0,t]

because both f , and T (u p−1 (s)) belong to C. Now, assume that (5.28) holds for an n ∈ N. Then,  t   es−t T (un+p (s)) − T (un (s)) ds. un+1+p (t) − un+1 (t) = 0

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5 Fixed Point Existence Theorems in Modular Function Spaces

Applying Lemma 5.13 again, with y(t) = 0 and x(t) = T (un+p (s)) − T (un (s)) we have   ρ (un+1+p (t) − un+1 (t)) ≤ K(A) sup ρ T (un+p (s)) − T (un (s)) s∈[0, t]

≤ K(A) sup ρ (un+p (s) − un (s)) s∈[0, t]

≤K

n+1

(A) δρ (C),

where we used the ρ -nonexpansiveness of T and the inductive assumption. It follows directly from (5.28) that for every t ∈ [0, A], the sequence {un (t)} is a ρ -Cauchy sequence, the latter implies that un (t) − u(t)ρ → 0.

(5.29)

Remember that T is ρ -nonexpansive and hence it is ρ -continuous in C, which, via the Vitali property, implies that T is  · ρ -continuous in C. Using this fact, it is easy to prove by induction that un is a  · ρ -continuous mapping from [0, A] to C. Consequently, functions λn : [0, A] → C defined by λn (s) = T (un (s)) are  · ρ -continuous and bounded. Hence, it follows from the Lebesgue dominated convergence theorem for Bochner integrals that lim Sτ (T (uk (t)) −

|τ |→0

 t 0

es−t T (uk (s))dsρ = 0,

(5.30)

for every k ∈ N. Using the ρ -nonexpansiveness of T and the fact that ρ (un (t)−u(t)) → 0, we have the following:    n−1  ρ Sτ (T (u(t)) − Sτ (T (un (t)) ≤ ∑ (ti+1 − ti )eti −t ρ T (u(t)) − T (un (t)) i=0

≤ A eA−t ρ (u(t) − un (t)) → 0, which, via the Vitali property, implies that lim Sτ (T (u(t)) − Sτ (T (un (t))ρ = 0.

n→∞

(5.31)

For every t ∈ [0, A] we have now Sτ (T (u(t)) − u(t) + e−t f ρ ≤ Sτ (T (u(t)) − Sτ (T (un (t))ρ +Sτ (T (un (t)) −  t

+ 0

 t 0

es−t T (un (s))dsρ

es−t T (un (s))ds − u(t) + e−t f ρ .

5.4 Applications to Differential Equations

Observing that

 t 0

167

es−t T (un (s))ds = un+1 − e−t f

and using (5.31), (5.30), and (5.29) we obtain that lim Sτ (T (u(t)) − u(t) + e−t f ρ = 0.

|τ |→0

Hence, the function s → es−t T (u(s) is Bochner-integrable on [0, A] and  t 0

es−t T (u(s))ds = u(t) − e−t f .

(5.32)

Using (5.32) it can be easily proved, via the standard calculus methods, that u is differentiable and is a solution of the initial value problem (5.18). The proof of Theorem 5.28 is now complete. 

Remark 5.9. It is clear that the solution of the the initial value problem (5.18) can be extended to a solution u(t) defined on [0, +∞) such that its restriction to the interval [0, A] is the ρ -limit of the sequence {un } defined as in the proof of Theorem 5.28. Remark 5.10. Theorem 5.28 extends results of [106] proven for norm continuous, nonexpansive mappings acting in Musielak–Orlicz spaces with the Δ2 property. Example 5.6. Let us recall an example of a modular generated by the Urysohn nonlinear integral operator:  1

T ( f )(x) = 0

k(x, y, | f (y)|)dy + f0 (x),

where f0 is an arbitrarily chosen function and the unknown f : [0, 1] → R is Lebesgue measurable. For the kernel k we assume that, (a) k : [0, 1] × [0, 1] × R+ → R+ is Lebesgue measurable, (b) k(x, y, 0) = 0, (c) k(x, y, .) is continuous, convex and increasing to +∞,  1

(d) 0

k(x, y,t)dx > 0 for t > 0 and y ∈ (0, 1).

In addition, assume that for almost all t ∈ [0, 1] and measurable functions f , g there holds  1  1  1  k(t, u, |k(u, v, | f (v)|) − k(u, v, |g(v)|)|)dv du ≤ k(t, u, | f (u) − g(u)|)du. 0

0

Setting ρ ( f ) =

0

 1  1 0

0

 k(x, y, | f (y)|)dy dx and using Jensen’s inequality it is easy

to show that ρ is a convex function modular, and that T is nonexpansive with respect

168

5 Fixed Point Existence Theorems in Modular Function Spaces

to ρ . Let us fix an r > 0 and set C = { f ∈ Eρ : ρ ( f − f0 ) ≤ r}. It is easy to see that T : C → C. If we assume additionally that there exists a constant M > 0 and a Bochner-integrable function h : [0, 1] × [0, 1] → [0, ∞) such that for every u ≥ 0 and x, y ∈ [0, 1] k(x, y, 2u) ≤ M k(x, y, u) + h(x, y), (5.33) then the modular ρ has the property Δ2 in the sense of Definition 3.5. Using Theorem 5.28 it is then easy to see that the corresponding initial value problem ( u(0) = f0 (5.34) u (t) + (I − T )u(t) = 0, has a solution in C that can be calculated as the ρ -limit of the sequence {un } as defined in Theorem 5.28. Still in the context of Theorem 5.28, let us introduce the following notation: for f ∈ C we write u f (t) ≡ f for all t ≥ 0. Let us ask a question for which f ∈ C, u f are the solutions of (5.18). The answer is provided in our next result. Theorem 5.29. Under the assumptions of Theorem 5.28, u f is a solution of (5.18) if and only if f ∈ F(T ). Proof. It is obvious that for every fixed point f ∈ F(T ), u f is a solution of (5.18). Conversely, let us assume that u f is a solution of (5.18). Arguing like in the proof of Theorem 5.28, we obtain that

ρ ( f − un (t)) ≤ K n+1 (A)δρ (C)

(5.35)

for any n ∈ N, any A > 0 and any t ∈ [0, A]. Noting that e−t f + K(t)T ( f ) − un+1 (t) =

 t 0

  es−t T ( f ) − T (un (s)) ds

and applying Lemma 5.13 we obtain     ρ e−t f + K(t)T ( f ) − un+1 (t) ≤ K(t) sup ρ T ( f ) − T (un (s) . s∈[0,t]

Using the fact that T is ρ -nonexpansive and inequality (5.35) we get   ρ e−t f + K(t)T ( f ) − un+1 (t) ≤ K n+2 (A)δρ (C), hence, {un (t)} ρ -converges to e−t f + K(t)T ( f ) for any t ≥ 0. The uniqueness of the ρ -limit implies that f = u f (t) = e−t f + K(t)T ( f ), which yields T ( f ) = f , as claimed. 

5.4 Applications to Differential Equations

169

Remark 5.11. Theorem 5.29 establishes a connection between the theory of differential equations in modular function spaces and the fixed point theory for nonlinear mappings acting in modular function spaces. It is clear that, for example, that applying it in conjunction with our fixed point theorem 5.7 we obtain a strong characterization of stationary points of the system (5.18). Remark 5.12. There is a solid body of work on the subject of ordinary and partial differential equations in Orlicz and Musielak–Orlicz spaces, especially in the context of Orlicz–Sobolev and Musielak–Orlicz–Sobolev spaces, see for instance [48, 86, 87, 93, 196], and more recent works often related to applications to modeling of “smart fluids,” see e.g., [6, 22, 53, 88, 89, 92, 187] and the papers referenced there. Frequently in these applications, a nonlinear extension of modular function spaces are used leading to the concepts of the modular metric spaces, see e.g., [3, 46, 47]. Typically these results use the classical techniques of differential equations with values in Banach spaces. As outlined above, our approach is to use the modular notions, like the ρ -nonexpansiveness, whenever this is practical. Our results generalize the results of Khamsi from [106] for norm continuous, nonexpansive mappings acting in Musielak–Orlicz spaces Lϕ with ϕ satisfying the Musielak–Orlicz version of the Δ2 condition, in relation to the problem of existence of ρ -nonexpansive semigroups of mappings, which, in their turn, extended classical Banach space results of [51, 183].

Chapter 6

Fixed Point Construction Processes

Assume ρ ∈ ℜ is (UUC1). Let C be a ρ -closed, ρ -bounded convex nonempty subset of Lρ . Let T : C → C be a pointwise asymptotically nonexpansive mapping. According to Theorem 5.7 the mapping T has a fixed point. The proof of this important theorem is of the existential nature and does not describe any algorithm for constructing a fixed point of an asymptotic pointwise ρ -nonexpansive mapping. This chapter aims at filling this gap. Therefore, we will define iterative processes for the fixed point construction in modular function spaces and we will prove their convergence. These algorithms will be based on classical iterative methods introduced originally by Mann in [161] and Ishikawa in [97]. The results of this section draw mostly on the research exposed in [54].

6.1 Preliminaries Denoting an (x) = max(αn (x), 1), we observe that without loss of generality we can assume that T is asymptotic ρ -pointwise nonexpansive if

ρ (T n ( f ) − T n (g)) ≤ an ( f )ρ ( f − g), lim an ( f ) = 1, an ( f ) ≥ 1,

n→∞

for all f , g ∈ C, n ∈ N,

for all f ∈ C, n ∈ N.

Define bn ( f ) = an ( f ) − 1. In view of the above, we have lim bn ( f ) = 0.

n→∞

The above notation will be consistently used throughout this section. By T (C) we will denote the class of all asymptotic pointwise nonexpansive mappings T : C → C.

c Springer International Publishing Switzerland 2015  M. A. Khamsi, W. M. Kozlowski, Fixed Point Theory in Modular Function Spaces, DOI 10.1007/978-3-319-14051-3 6

171

172

6 Fixed Point Construction Processes

In this section, we will impose some restrictions on the behavior of an and bn . Definition 6.1. Define Tr (C) as a class of all T ∈ T (C) such that an is a bounded function for every n ≥ 1 and

∞

∑ bn (x) < ∞ for every x ∈ C.

n=1

Remark 6.1. The restrictions described in Definition 6.1 are typical for controlling the convergence of iterative processes for asymptotically nonexpansive mappings, see, for example, [133]. Let us start with the following technical result. Lemma 6.1. Let ρ ∈ ℜ. Let C ⊂ Lρ be a convex set, and let T ∈ Tr (C). If {xk } is a ρ -approximate fixed point sequence for T, that is, ρ (T (xk ) − xk ) → 0 as k → ∞, then for every fixed m ∈ N there holds 

m T (xk ) − xk → 0, as k → ∞. ρ m Proof. It follows from the fact that an is a bounded function for every n ≥ 1 that there exists a finite constant M > 0 such that m−1

∑ sup{a j (x) : x ∈ C}

≤ M.

j=1

Using the convexity of ρ and the ρ -nonexpansiveness of T we get   

m T (xk ) − xk 1 m−1 j+1 =ρ ρ ∑ (T (xk ) − T j (xk )) m m j=0   m−1 1 m−1 j+1 j ≤ ∑ ρ (T (xk )−T (xk )) ≤ ρ (T (xk )−xk ) ∑ a j (xn ) + 1 m j=0 j=1 ≤

1 (M + 1)ρ (T (xk ) − xk ) → 0, m

as k → ∞.

 Corollary 6.1. If, under the hypothesis of Lemma 6.1, ρ satisfies additionally the Δ2 condition, then ρ (T m (xk ) − xk ) → 0 as k → ∞.

6.2 Demiclosedness Principle The following modular version of the Demiclosedness Principle will be used in the proof of our convergence Theorem 6.2. Our proof, the Demiclosedness Principle, uses the parallelogram inequality valid in the modular spaces with the (UUC1)

6.2 Demiclosedness Principle

173

property (Lemma 4.5). For the discussion of the Demiclosedness Principle in the Banach space context, see Sect. 2.5.3. Theorem 6.1 (Demiclosedness Principle). [54] Let ρ ∈ ℜ. Assume that (a) ρ is (UUC1), (b) ρ has Strong Opial Property, (c) ρ has Δ2 property and is uniformly continuous. Let C⊂Lρ be nonempty, convex, strongly ρ-bounded, and ρ -closed, and let T ∈ Tr(C). Let {xn } ⊂ C, and x ∈ C. If xn → x ρ − a.e. and ρ (T (xn ) − xn ) → 0 then x ∈ F(T ). Proof. Let us recall that by definition of the uniform continuity of ρ to every ε > 0 and L > 0, there exists δ > 0 such that |ρ (g) − ρ (g + h)| ≤ ε ,

(6.1)

provided ρ (h) < δ and ρ (g) ≤ L. Fix any m ∈ N. Noting that ρ (xn −x) ≤ M < ∞ due to the strong ρ -boundedness of C and that ρ (T m (xn ) − xn ) → 0 by Corollary 6.1, it follows then from (6.1) with g = xn − x and h = T m (xn ) − xn that |ρ (xn − x) − ρ (xn − x + T m (xn ) − xn )| → 0,

as n → ∞.

Hence, lim sup ρ (xn − x) = lim sup ρ (T m (xn ) − x) . n→∞

(6.2)

n→∞

Define the ρ -type ϕ by

ϕ (x) = lim sup ρ (xn − x). n→∞

By (6.2) we get

ϕ (x) = lim sup ρ (T m (xn ) − x) . n→∞

Hence, for every y ∈ C there holds

ϕ (T m (y)) = lim sup ρ (T m (xn ) − T m (y))

(6.3)

n→∞

≤ am (y) lim sup ρ (xn − y) = am (y)ϕ (y) n→∞

Using (6.3) with y = x and by passing with m to infinity we conclude that lim sup ϕ (T m (x)) ≤ ϕ (x).

(6.4)

m→∞

Since ρ satisfies the Strong Opial property, it also satisfies the Opial property. Since xn → x ρ − a.e., it follows via the Opial property that for any y = x

ϕ (x) = lim sup ρ (xn − x) < lim sup ρ (xn − y) = ϕ (y), n→∞

n→∞

174

6 Fixed Point Construction Processes

which implies that

ϕ (x) = inf{ϕ (y) : y ∈ C}.

(6.5)

Combining (6.4) with (6.5), we have

ϕ (x) ≤ lim sup ϕ (T m (x)) ≤ ϕ (x), m→∞

that is, lim sup ϕ (T m (x)) = ϕ (x). m→∞

We claim that lim ρ (T m (x) − x) = 0.

m→∞

(6.6)

Assume to the contrary that (6.6) does not hold, that is

ρ (T m (x) − x) does not tend to zero. (6.7) 

m T (x) − x does not tend to zero. By By Δ2 , then it follows from (6.7) that ρ 2 passing to a subsequence if necessary we can assume that there exists 0 < t < M such that 

m T (x) − x > t > 0, for m ∈ N, ρ 2 which implies that t ρ (xn − x) + ρ (xn − T m (x)) > , 2

for every m, n ∈ N.

Hence, t max {ρ (xn − x), ρ (xn − T m (x))} ≥ , 4

for every m, n ∈ N.

Applying the parallelogram inequality from Lemma 4.5, 

 1 z+y 1 1 ≤ ρ 2 (z) + ρ 2 (y) − Ψ r, s, ρ (z − y) , ρ2 2 2 2 r where ρ (z) ≤ r, ρ (y) ≤ r and max{ρ (z), ρ (y)} ≥ s for 0 < s < r, with r = M, s = 4t , z = xn − x, y = T m (x), we get 

 1 t 1 x + T m (x) 1 ≤ ρ 2 (xn −x)+ ρ 2 (xn−T m (x))−Ψ M, , ρ (x−T m (x)) . ρ 2 xn − 2 2 2 4 M Note that by (6.5)

ϕ 2 (x) ≤ ϕ 2

x + T m (x) 2





x + T m (x) = lim sup ρ 2 xn − . 2 n→∞

6.3 Generalized Mann Iteration Process

175

Combining the last two formulas we obtain

 t 1 1 1 2 2 m m ϕ (x)≤ lim sup ρ (xn −x)+ lim sup ρ (xn−T (x))−Ψ M, , ρ (x − T (x)) , 2 n→∞ 2 n→∞ 4 M 2

which implies

 t 1 1 1 0 ≤ Ψ M, , ρ (x − T m (x)) ≤ ϕ 2 (T m (x)) − ϕ 2 (x). 4 M 2 2 Hence,

 t 1 m 0 ≤ lim sup Ψ M, , ρ (x − T (x)) 4 M m→∞ 1 1 2 ≤ lim sup ϕ (T m (x)) − ϕ 2 (x) ≤ 0. 2 m→∞ 2 Using the properties of Ψ , we conclude that ρ (x − T m (x)) tends to zero itself, which contradicts our assumption (6.7). Hence, ρ (x − T m (x)) → 0 as m → ∞ as claimed in (6.6). Clearly, then ρ (x − T m+1 (x)) → 0 as m → ∞, that is, T m+1 (x) → x(ρ ) while T m+1 (x) → T (x)(ρ ) as well as by the ρ -continuity of T . By the uniqueness of the ρ -limit, we obtain T (x) = x, that is, x ∈ F(T ), which completes the proof. 

6.3 Generalized Mann Iteration Process Let us define our main iterative process for the construction of fixed points in modular function spaces. Definition 6.2. Let T ∈ Tr (C) and let {nk } be an increasing sequence of natural numbers. Let {tk } ⊂ (0, 1) be bounded away from 0 and 1. The generalized Mann iteration process generated by the mapping T , the sequence {tk }, and the sequence {nk }, denoted by gM(T, {tk }, {nk }) is defined by the following iterative formula: xk+1 = tk T nk (xk ) + (1 − tk )xk , where x1 ∈ C is chosen arbitrarily. Definition 6.3. We say that a generalized Mann iteration process gM(T, {tk }, {nk }) is well-defined if (6.8) lim sup ank (xk ) = 1. k→∞

Remark 6.2. Observe that by the definition of asymptotic pointwise nonexpansiveness, lim ak (x) = 1 for every x ∈ C. Hence, we can always select a subsequence k→∞

{ank } such that (6.8) holds. In other words, by a suitable choice of {nk } we can always make gM(T, {tk }, {nk }) well-defined.

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6 Fixed Point Construction Processes

Let us quote an elementary lemma about real numbers, see for example [36]. Lemma 6.2. Suppose {rk } is a bounded sequence of real numbers and {dk,n } is a doubly-index sequence of real numbers which satisfy lim sup lim sup dk,n ≤ 0, and rk+n ≤ rk + dk,n n→∞

k→∞

for each k, n ≥ 1. Then {rk } converges to an r ∈ R. The following result provides an important technique which will be used in this chapter. Lemma 6.3. Let ρ ∈ ℜ be (UUC1). Let C ⊂ Lρ be a ρ -closed, ρ -bounded, and convex set. Let T ∈ Tr (C) and let {nk } ⊂ N. Assume that a sequence {tk } ⊂ (0, 1) is bounded away from 0 and 1. Let w be a fixed point of T and gM(T, {tk }, {nk }) be a generalized Mann process. Then, there exists r ∈ R such that lim ρ (xk − w) = r. k→∞

Proof. Let w ∈ F(T ). Since

ρ (xk+1 − w) ≤ tk ρ (T nk (xk ) − w) + (1 − tk )ρ (xk − w) = tk ρ (T nk (xk ) − T nk (w)) + (1 − tk )ρ (xk − w) ≤ tk (1 + bnk (w))ρ (xk − w) + (1 − tk )ρ (xk − w) = tk bnk (w)ρ (xk − w) + ρ (xk − w) ≤ bnk (w)diamρ (C) + ρ (xk − w), it follows that for every n ∈ N,

ρ (xk+n − w) ≤ ρ (xk − w) + diamρ (C)

k+n−1

∑

bni (w).

i=k

Denote r p = ρ (x p − w) for every p ∈ N and dk,n = diamρ (C)

k+n−1

∑

bni (w). Observe

i=k

that since T ∈ Tr (C), it follows that lim sup lim sup dk,n = 0. By Lemma 6.2, there k→∞

n→∞

exists an r ∈ R such that lim ρ (xk − w) = r as claimed. k→∞



The next result will be essential for proving the convergence theorems for iterative process. Lemma 6.4. Let ρ ∈ ℜ be (UUC1). Let C ⊂ Lρ be a ρ -closed, ρ -bounded, and convex set, and T ∈ Tr (C). Assume that a sequence {tk } ⊂ (0, 1) is bounded away from 0 and 1. Let {nk } ⊂ N and gM(T, {tk }, {nk }) be a generalized Mann iteration process. Then (6.9) lim ρ (T nk (xk ) − xk ) = 0, k→∞

and lim ρ (xk+1 − xk ) = 0.

k→∞

(6.10)

6.3 Generalized Mann Iteration Process

177

Proof. By Theorem 5.7, T has at least one fixed point w ∈ C. In view of Lemma 6.3, there exists r ∈ R such that (6.11) lim ρ (xk − w) = r. k→∞

Note that lim sup ρ (T nk (xk ) − w) = lim sup ρ (T nk (xk ) − T nk (w)) k→∞

(6.12)

k→∞

≤ lim sup ank (w)ρ (xk − w) ≤ r, k→∞

and that lim ρ (tk (T nk (xk ) − w) + (1 − tk )(xk − w)) = lim ρ (xk+1 − w) = r.

k→∞

k→∞

Set fk = T nk (xk ) − w, gk = xk − w, and note that lim sup ρ (gk ) ≤ r by (6.11), and lim sup ρ ( fk ) ≤ r by (6.12). Observe also that

k→∞

k→∞

lim ρ (tk fk + (1 − tk )gk ) = lim ρ (tk T nk (xk ) + (1 − tk )xk − w) = lim ρ (xk+1 − w) = r.

k→∞

k→∞

k→∞

Hence, it follows from Lemma 4.2 that lim ρ (T nk (xk ) − xk ) = lim ρ ( fk − gk ) = 0,

k→∞

k→∞

which by the construction of the sequence {xk } is equivalent to lim ρ (xk+1 − xk ) = 0,

k→∞

as claimed.



In the next lemma, we prove that under suitable assumptions, the sequence {xk } becomes an approximate fixed point sequence, which will provide an important step in the proof of the generalized Mann iteration process convergence. First, we need to define the following notions. Definition 6.4. A strictly increasing sequence {ni } ⊂ N is called quasiperiodic if the sequence {ni+1 − ni } is bounded, or equivalently if there exists a number p ∈ N such that any block of p consecutive natural numbers must contain a term of the sequence {ni }. The smallest of such numbers p will be called a quasiperiod of {ni }. Lemma 6.5. Let ρ ∈ ℜ be (UUC1) satisfying Δ2 . Let C ⊂ Lρ be a ρ -closed, ρ bounded, and convex set, and T ∈ Tr (C). Let {tk } ⊂ (0, 1) be bounded away from 0 and 1. Let {nk } ⊂ N be such that the generalized Mann process gM(T, {tk }, {nk }) is well-defined. If, in addition, the set of indices J = { j : n j+1 = 1 + n j } is quasiperiodic, then {xk } is an approximate fixed point sequence, that is, lim ρ (T (xk ) − xk ) = 0.

k→∞

(6.13)

178

6 Fixed Point Construction Processes

Proof. Let p ∈ N be a quasiperiod of J . Observe that it is enough to prove that ρ (T (xk ) − xk ) → 0 as k → ∞ through J . Indeed, from this fact it follows that if we fix ε > 0 then ρ (T (xk ) − xk ) < ε for sufficiently large k ∈ J . By the quasiperiodicity of J , to every positive integer k there exists jk ∈ J such that |k − jk | ≤ p. Assume that k − p ≤ jk ≤ k (the proof for the other case is identical). Since T is ρ -Lipschitzian with the Lipschitz constant ε M = sup{a1 (x) : x ∈ C}, there exist a 0 < δ < such that 3

ρ (T (x) − T (y)) < ε i f ρ (x − y) < δ . Note that by (6.10) and by Δ2 , ρ (p(xk+1 − xk )) <

δ for k sufficiently large. This p

implies that

ρ (xk − x jk ) ≤

 δ 1 ρ (p(xk − xk−1 )) + ... + ρ (p(x jk +1 − x jk )) ≤ p = δ , p p

and therefore 

1 xk − T (xk ) 1 1 ε ε ≤ ρ (xk −x jk )+ ρ (x jk−T (x jk ))+ ρ (T (x jk )−T (xk ))≤ δ + + < ε , ρ 3 3 3 3 3 3

which demonstrates that

ρ

xk − T (xk ) 3

 →0

as k → ∞. By Δ2 we get ρ (T (xk ) − xk ) → ∞. To prove that ρ (T (xk ) − xk ) → 0 as k → ∞ through J , observe that, since nk+1 = nk +1 for such k, there holds 

1 xk − T (xk ) 1 ≤ ρ (xk − xk+1 ) + ρ (xk+1 − T nk+1 (xk+1 )) ρ 4 4 4 1 1 + ρ (T nk+1 (xk+1 ) − T nk+1 (xk )) + ρ (T T nk (xk ) − T (xk )) 4 4 1 1 nk+1 (xk+1 )) ≤ ρ (xk − xk+1 ) + ρ (xk+1 − T 4 4 1 1 + ank+1 (xk+1 )ρ (xk − xk+1 ) + M ρ (T nk (xk ) − xk ) 4 4 which tends to zero in view of (6.9), (6.10), and (6.8).

 We are now ready to prove our main convergence result, the proof follows [54].

6.3 Generalized Mann Iteration Process

179

Theorem 6.2. Let ρ ∈ ℜ. Assume that (i) ρ is (UUC1), (ii) ρ has Strong Opial Property, (iii) ρ has Δ2 property and is uniformly continuous. Let C ⊂ Lρ be nonempty, ρ − a.e. compact, convex, strongly ρ -bounded, and ρ closed, and let T ∈ Tr (C). Assume that a sequence {tk } ⊂ (0, 1) is bounded away from 0 and 1. Let {nk } ⊂ N and gM(T, {tk }, {nk }) be a well-defined generalized Mann iteration process. Assume, in addition, that the set of indices J = { j : n j+1 = 1 + n j } is quasiperiodic. Then, there exists x ∈ F(T ) such that xn → x ρ -a.e. Proof. Observe that by Theorem 5.7 the set of fixed points F(T ) is nonempty, and ρ -closed. Also Note that by Theorem 4.8, it follows from the Strong Opial property of ρ that any ρ -type attains its minimum in C. By Lemma 6.5 the sequence {xk } is an approximate fixed point sequence, that is

ρ (T (xk ) − xk ) → 0 as k → ∞. Consider y, z ∈ C, two ρ -a.e. cluster points of {xk }. There exits then {yk }, {zk } subsequences of {xk } such that yk → y ρ -a.e., and zk → z ρ -a.e. By Theorem 6.1, y ∈ F(T ) and z ∈ F(T ). By Lemma 6.3, there exist ry , rz ∈ R such that ry = lim ρ (xk − y), rz = lim ρ (xk − z). k→

k→

We claim that y = z. Assume to the contrary that y = z. Then by the Strong Opial property we have ry = lim inf ρ (yk − y) < lim inf ρ (yk − z) k→∞

k→∞

= lim inf ρ (zk − z) < lim inf ρ (zk − y) = ry . k→∞

k→∞

The contradiction implies that y = z. Therefore, {xk } has at most one ρ -a.e. cluster point. Since C is ρ -a.e. compact, it follows that the sequence {xk } has exactly one ρ -a.e. cluster point, which means that ρ (xk ) → x ρ -a.e. Using Theorem 6.1 again, we get x ∈ F(T ) as claimed. 

Remark 6.3. It is easy to see that we can always construct a sequence {nk } with the quasiperiodic properties specified in the assumptions of Theorem 6.2. When constructing concrete implementations of this algorithm, the difficulty will be to ensure that the constructed sequence {nk } is not “too sparse” in the sense that the generalized Mann process gM(T, {tk }, {nk }) remains well-defined. The similar, quasiperiodic type assumptions are common in the asymptotic fixed point theory, see, for example, [36, 133, 144].

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6 Fixed Point Construction Processes

If T is ρ-nonexpansive then obviously it is also asymptotic pointwise ρ-nonexpansive with an (x) = 1 for any x ∈ Ca and any natural n. Hence, one can take nk = k as the defining sequence for the generalized Mann process and consequently J = { j : n j+1 = 1 + n j } is always quasiperiodic. Thus, we have the following result for ρ -nonexpansive mappings. Theorem 6.3. Let ρ ∈ ℜ. Assume that (i) ρ is (UUC1), (ii) ρ has Strong Opial Property, (iii) ρ has Δ2 property and is uniformly continuous. Let C ⊂ Lρ be nonempty, ρ − a.e. compact, convex, strongly ρ -bounded, and ρ closed, and let T : C → C be ρ -nonexpansive. Assume that a sequence {tk } ⊂ (0, 1) is bounded away from 0 and 1. Let gM(T, {tk }, k}) be generalized Mann iteration process, which must be well-defined because of ρ -nonexpansiveness of T . Then, there exists x ∈ F(T ) such that xn → x ρ -a.e.

6.4 Generalized Ishikawa Iteration Process The two-step Ishikawa iteration process is a generalization of the one-step Mann process. The Ishikawa iteration process provides more flexibility in defining the algorithm parameters which is important from the numerical implementation perspective. Definition 6.5. Let T ∈ Tr (C) and let {nk } be an increasing sequence of natural numbers. Let {tk } ⊂ (0, 1) be bounded away from 0 and 1, and {sk } ⊂ (0, 1) be bounded away from 1. The generalized Ishikawa iteration process generated by the mapping T , the sequences {tk }, {sk }, and the sequence {nk }, denoted by gI(T, {tk }, {sk }, {nk }) is defined by the following iterative formula:   xk+1 = tk T nk sk T nk (xk ) + (1 − sk )xk + (1 − tk )xk , where x1 ∈ C is chosen arbitrarily. Definition 6.6. We say that a generalized Ishikawa iteration process gI(T, {tk }, {sk }, {nk }) is well-defined if lim sup ank (xk ) = 1.

(6.14)

k→∞

Remark 6.4. Observe that, by the definition of asymptotic pointwise nonexpansiveness, lim ak (x) = 1 for every x ∈ C. Hence, we can always select a subsequence k→∞

{ank } such that (6.14) holds. In other words, by a suitable choice of {nk } we can always make gI(T, {tk }, {sk }, {nk }) well-defined.

6.4 Generalized Ishikawa Iteration Process

181

Lemma 6.6. Let ρ ∈ ℜ be (UUC1). Let C ⊂ Lρ be a ρ -closed, ρ -bounded, and convex set. Let T ∈ Tr (C) and let {nk } ⊂ N. Let {tk } ⊂ (0, 1) be bounded away from 0 and 1, and {sk } ⊂ (0, 1) be bounded away from 1. Let w ∈ F(T ) and gI(T, {tk }, {sk }, {nk }) be a generalized Ishikawa process. There exists then an r ∈ R such that lim ρ (xk − w) = r. k→∞

Proof. Define Tk : C → C by   Tk (x) = tk T nk sk T nk (x) + (1 − sk )x + (1 − tk )x, x ∈ C. It is easy to see that xk+1 = Tk (xk ) and that F(T ) ⊂ F(Tk ). Moreover, a straight calculation shows that each Tk satisfies

ρ (Tk (x) − Tk (y)) ≤ Ak (x)ρ (x − y), where Ak (x) = 1 + tk ank (Mk (x))(1 + sk ank (x) − sk ) − tk ,

(6.15)

and Mk (x) = sk T nk (x) + (1 − sk )x. Note that Ak (x) ≥ 1 which follows directly from the fact that ank (x) ≥ 1 and from (6.15). Using (6.15) and the fact that Mk (w) = w we have Bk (w) = Ak (w) − 1 = tk (1 + sk ank (w))(ank (w) − 1) ≤ (1 + ank (w))bnk (w). Fix any M > 1. Since lim ank (w) = 1, it follows that there exists a k0 ≥ 1 such that k→∞

for k > k0 , ank (w) ≤ M. Therefore, by the same argument as in the proof of Lemma 6.3, we deduce that for k > k0 and n > 1

ρ (xk+n − w) ≤ ρ (xk − w) + diamρ (C)

k+n−1

∑

Bi (w)

i=k

≤ ρ (xk − w) + diamρ (C)(1 + M)

k+n−1

∑

bni (w).

i=k

Arguing like in the proof of Lemma 6.3, we conclude that there exists an r ∈ R such  that lim ρ (xk − w) = r.

k→∞

Lemma 6.7. Let ρ ∈ ℜ be (UUC1). Let C ⊂ Lρ be a ρ -closed, ρ -bounded, and convex set. Let T ∈ Tr (C) and let {nk } ⊂ N. Let {tk } ⊂ (0, 1) be bounded away from 0 and 1, and {sk } ⊂ (0, 1) be bounded away from 1. Let gI(T, {tk }, {sk }, {nk }) be a generalized Ishikawa process. Define yk = sk T nk (xk ) + (1 − sk )xk . Then lim ρ (T nk (yk ) − xk ) = 0,

k→∞

(6.16)

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6 Fixed Point Construction Processes

or equivalently lim ρ (xk+1 − xk ) = 0,

k→∞

(6.17)

Proof. By Theorem 5.7, F(T ) = 0. / Let us fix w ∈ F(T ). By Lemma 6.6, the limit lim ρ (xk − w) exists. Let us denote it by r. Since w ∈ F(T ), T ∈ Tr (C), and k→∞

lim ρ (xk − w) = r, by Lemma 6.6 we have the following     lim sup ρ T nk (yk ) − w = lim sup ρ T nk (yk ) − T nk (w) k→∞ k→∞   ≤ lim sup ank (w)ρ (yk − w)= lim sup ank (w)ρ sk T nk (xk ) + (1 − sk )xk − w

k→∞

k→∞

k→∞

≤ lim sup[sk ank (w)ρ (T (xk ) − w) + (1 − sk )ank (w)ρ (xk − w)] nk

k→∞

≤ lim sup[sk a2nk (w)ρ (xk − w) + (1 − sk )ank (w)ρ (xk − w)] ≤ r. k→∞

Note that

  lim ρ tk (T nk (yk ) − w) + (1 − tk )(xk − w) k→∞   = lim ρ tk T nk (yk ) + (1 − tk )xk − w = lim ρ (xk+1 − w) = r. k→∞

k→∞

Applying Lemma 4.2 with uk = T nk (yk ) − w and vk = xk − w, we obtain the desired equality lim ρ (T nk (yk ) − xk ) = 0, while (6.17) follows from (6.16) via the construck→∞

tion formulas for xk+1 and yk . 

Lemma 6.8. Let ρ ∈ ℜ be (UUC1) satisfying Δ2 . Let C ⊂ Lρ be a ρ -closed, ρ bounded, and convex set. Let T ∈ Tr (C) and let {nk } ⊂ N. Let {tk } ⊂ (0, 1) be bounded away from 0 and 1, and {sk } ⊂ (0, 1) be bounded away from 1. Let gI(T, {tk }, {sk }, {nk }) be a well-defined generalized Ishikawa process. Then   (6.18) lim ρ T nk (xk ) − xk = 0. k→∞

Proof. Let yk = sk T nk (xk ) + (1 − sk )xk . Hence T nk (xk ) − xk =

 1  nk T (xk ) − yk . 1 − sk

Since {sk } ⊂ (0, 1) is bounded away from 1, there exists 0 < s < 1 such that sk ≤ s for every k ≥ 1. Hence,

ρ (T nk (xk ) − xk ) = ρ



 1   1 (T nk (xk ) − yk ) . (T nk (xk ) − yk ) ≤ ρ 1 − sk 1−s

6.4 Generalized Ishikawa Iteration Process

183

  The right-hand side of this inequality tends to zero because ρ T nk (xk ) − yk → 0 by Lemma 6.7 and the fact that ρ satisfies Δ2 . 

Lemma 6.9. Let ρ ∈ ℜ be (UUC1) satisfying Δ2 . Let C ⊂ Lρ be a ρ -closed, ρ -bounded, and convex set, and T ∈ Tr (C). Let {tk } ⊂ (0, 1) be bounded away from 0 and 1 and {sk } ⊂ (0, 1) be bounded away from 1. Let {nk } ⊂ N be such that the generalized Ishikawa process gI(T, {tk }, {sk }, {nk }) is well-defined. If, in addition, the set J = { j : n j+1 = 1 + n j } is quasiperiodic, then {xk } is an approximate fixed point sequence, i.e., lim ρ (T (xk ) − xk ) = 0.

k→∞

Proof. The proof is analogous to that of Lemma 6.5 with (6.17) used instead of (6.10) and (6.18) replacing (6.9). 

Theorem 6.4. Let ρ ∈ ℜ. Assume that (i) ρ is (UUC1), (ii) ρ has Strong Opial Property, (iii) ρ has Δ2 property and is uniformly continuous. Let C ⊂ Lρ be a nonempty, ρ − a.e. compact, convex, strongly ρ -bounded, and ρ -closed set, and let T ∈ Tr (C). Let T ∈ Tr (C). Let {tk } ⊂ (0, 1) be bounded away from 0 and 1, and {sk } ⊂ (0, 1) be bounded away from 1. Let {nk } be such that the generalized Ishikawa process gI(T, {tk }, {sk }, {nk }) is well-defined. If, in addition, the set J = { j : n j+1 = 1 + n j } is quasiperiodic, then {xk } generated by gI(T, {tk }, {sk }, {nk }) converges ρ -a.e. to a fixed point x ∈ F(T ). Proof. The proof is analogous to that of Theorem 6.2 with Lemma 6.5 replaced by Lemma 6.9, and Lemma 6.3 replaced by Lemma 6.6.

 Similarly as with the Generalized Mann process, we can argue that if T is ρ -nonexpansive then obviously it is also asymptotic pointwise ρ -nonexpansive with an (x) = 1 for any x ∈ C and any natural n. Hence, one can take nk = k as the defining sequence for the generalized Ishikawa process and consequently J = { j : n j+1 = 1 + n j } is always quasiperiodic. Thus, we have the following result for ρ -nonexpansive mappings. Theorem 6.5. Let ρ ∈ ℜ. Assume that (i) ρ is (UUC1), (ii) ρ has Strong Opial Property, (iii) ρ has Δ2 property and is uniformly continuous. Let C ⊂ Lρ be nonempty, ρ − a.e. compact, convex, strongly ρ -bounded, and ρ -closed, and let T : C → C be ρ -nonexpansive. Let {tk } ⊂ (0, 1) be bounded away from 0 and 1, and {sk } ⊂ (0, 1) be bounded away from 1. Then {xk } generated by gI(T, {tk }, {sk }, {k}) converges ρ -a.e. to a fixed point x ∈ F(T ).

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6 Fixed Point Construction Processes

6.5 Strong Convergence It is interesting that, assuming C is ρ -compact (which is a strong form of compactness in modular function spaces), instead of ρ -a.e. compactness (which is a weak form of compactness), both generalized Mann and Ishikawa processes converge strongly to a fixed point of T even without assuming the Opial property or any quasiperiodicity assumptions. The next result proves exactly that. Theorem 6.6. Let ρ ∈ ℜ satisfy conditions (UUC1) and Δ2 . Let C ⊂ Lρ be a ρ compact, ρ -bounded, and convex set, and let T ∈ Tr (C) . Let {tk } ⊂ (0, 1) be bounded away from 0 and 1, and {sk } ⊂ (0, 1) be bounded away from 1. Let {nk } be such that the generalized Mann process gM(T, {tk }, {nk }) (respectively, Ishikawa process gI(T, {tk }, {sk }, {nk })) is well-defined. Then, there exists a fixed point x ∈ F(T ) such that the sequence {xk } generated by gM(T, {tk }, {nk }) (respectively, gI(T, {tk }, {sk }, {nk })) converges strongly to a fixed point of T , that is, lim ρ (xk − x) = 0. k→∞

Proof. By the ρ -compactness of C we can select a subsequence {x pk } of {xk } such that there exists x ∈ C with lim ρ (T (x pk ) − x) = 0.

k→∞

Note that

ρ

x pk − x 2



 1   1  ≤ ρ x pk − T (x pk ) + ρ T (x pk − x , 2 2

(6.19)

(6.20)

which tends to zero by Lemma 6.4 (respectively Lemma 6.9) and by (6.19). By Δ2 if follows from (6.20) that

ρ (x pk − x) → 0 as k → ∞.

(6.21)

Observe that by the convexity of ρ and by ρ -nonexpansiveness of T , we have 

 1  1  1  T (x) − x ≤ ρ T (x) − T (x pk ) + ρ T (x pk ) − x pk + ρ (x pk − x) ρ 3 3 3 3  1 1  1 ≤ ρ (x − x pk ) + ρ T (x pk ) − x pk + ρ (x pk − x), 3 3 3 which tends to zero by (6.21) and by Lemma 6.4 (respectively Lemma 6.9). From there we get ρ (T (x) − x) = 0 which implies that x ∈ F(T ). Applying Lemma 6.3 (respectively Lemma 6.6) we conclude that lim ρ (xk − x) exists. By (6.21) this limit k→∞

must be equal to zero which implies that lim ρ (xk − x) = 0, as claimed. k→∞ 

Remark 6.5. Observe that in view of the Δ2 assumption, the ρ -compactness of the set C assumed in Theorem 6.6 is equivalent to the compactness in the sense of the Luxemburg norm defined by ρ .

Chapter 7

Semigroups of Nonlinear Mappings in Modular Function Spaces

Let us recall that a family {Tt }t≥0 of mappings forms a semigroup if T0 (x) = x, and Ts+t = Ts (Tt (x)). Such a situation is quite typical in mathematics and applications. For instance, in the theory of dynamical systems, the modular function space Lρ would define the state space and the mapping (t, x) → Tt (x) would represent the evolution function of a dynamical system. The question about the existence of common fixed points, and about the structure of the set of common fixed points, can be interpreted as a question whether there exist points that are fixed during the state space transformation Tt at any given point of time t, and if yes—what the structure of a set of such points may look like. In the setting of this chapter, the state space may be an infinite dimensional. Therefore, it is natural to apply these result to not only to deterministic dynamical systems but also to stochastic dynamical systems. Because of wide body of potential applications, the theory of semigroups of nonlinear mappings in modular function spaces, initiated in the 1992 paper by Khamsi [106], has become recently a subject of an intensive development, see [9, 55, 56, 135, 139, 140, 141]. There is, however, a lot of space for future research in this area.

7.1 Definitions Let us start with the modular definitions of Lipschitzian—in the modular sense— mappings, and of associated definitions of semigroups of nonlinear mappings. Definition 7.1. Let ρ ∈ ℜ and let C ⊂ Lρ be nonempty and ρ -closed. A mapping T : C → C is called a ρ -Lipschitzian if there exists a constant 0 < L such that

ρ (T ( f ) − T (g)) ≤ Lρ ( f − g),

for any f , g ∈ Lρ .

Definition 7.2. A one-parameter family F = {Tt : t ≥ 0} of mappings from C into itself is said to be a ρ -Lipschitzian (respectively ρ -nonexpansive) semigroup on C if F satisfies the following conditions:

c Springer International Publishing Switzerland 2015  M. A. Khamsi, W. M. Kozlowski, Fixed Point Theory in Modular Function Spaces, DOI 10.1007/978-3-319-14051-3 7

185

186

7 Semigroups of Nonlinear Mappings in Modular Function Spaces

(i) T0 (x) = x for x ∈ C; (ii) Tt+s (x) = Tt (Ts (x)) for x ∈ C and t, s ≥ 0; (iii) for each t ≥ 0, Tt is ρ -Lipschitzian (respectively ρ -nonexpansive). Definition 7.3. A one-parameter family F = {Tt : t ≥ 0} of mappings from C into itself is said to be a ρ -contractive semigroup on C if F satisfies the following conditions: (i) T0 (x) = x for x ∈ C; (ii) Tt+s (x) = Tt (Ts (x)) for x ∈ C and t, s ≥ 0; (iii) for each t ≥ 0, Tt is a ρ -contraction with a constant 0 < Lt < 1 such that lim sup Lt < 1. t→∞

Definition 7.4. A semigroup F = {Tt : t ≥ 0} is called strongly continuous if for every z ∈ C, the following function   Λz (t) = ρ Tt (z)) − z (7.1) is continuous at every t ∈ [0, ∞). Definition 7.5. A semigroup F = {Tt : t ≥ 0} is called continuous if for every z ∈ C, the mapping t −→ Tt (z) is ρ -continuous at every t ∈ [0, ∞), i.e.,  ρ Ttn (z)−Tt (z) → 0 as tn → t.

7.2 Fixed Point Existence for Semigroups of Nonexpansive Mappings Following [135], we will prove the existence of common fixed points for contractive and nonexpansive semigroups. Theorem 7.1. Let ρ ∈ ℜ. Assume that Lρ has the ρ -a.e. strong Opial property. Let C ⊂ Eρ be a nonempty, strongly ρ -bounded, ρ -a.e. compact convex set. Let F be a ρ -contractive semigroup onC. Then, F has a unique common fixed point z ∈ C and  for each u ∈ C, ρ Tt (u) − z → 0 as t → ∞. Proof. First, let us prove the uniqueness. Assume that z, w ∈ F . Then we have

ρ (z − w) = ρ (Tt (z) − Tt (w)) ≤ Lt ρ (z − w), for any t ≥ 0. If we let t → ∞, we will get

ρ (z − w) ≤ Lρ (z − w), where L = lim sup Lt . t→∞

7.2 Fixed Point Existence for Semigroups of Nonexpansive Mappings

187

Since L < 1, we conclude that ρ (z − w) = 0 or equivalently that z = w, hence there must exist at most one common fixed point for F . To prove the existence of the common fixed point, let us fix any x ∈ C and define the ρ -type τ by τ (u) = lim sup ρ (Tt (x) − u), (7.2) t→∞

for u ∈ C. By Theorem 4.8, there exists z ∈ C such that

τ (z) = inf{τ (y) : y ∈ C}. Let us prove that τ (z) = 0. To this end, take s,t ≥ 0 and observe that by the nonexpansiveness of Tt we have

ρ (Ts+t (x) − Tt (z)) ≤ Lt ρ (Ts (x) − z), and then letting s → ∞

τ (Tt (z)) ≤ Lt τ (z)

which implies the following

τ (z) = inf{τ (y) : y ∈ C} ≤ τ (Tt (z)) ≤ Lt τ (z). Passing with t → ∞ we get

τ (z) ≤ Lτ (z).

Since 0 < L < 1 it follows that τ (z) = 0. Hence, by (7.2), lim ρ (Tt (x) − z) = 0.

(7.3)

t→∞

Since Ts is a ρ -contraction for any given s ≥ 0, it follows that lim ρ (Tt (x) − Ts (z)) = lim ρ (Ts+t (x) − Ts (z)) ≤ lim ρ (Tt (x) − z) = 0.

t→∞

t→∞

t→∞

(7.4)

By the uniqueness of the ρ -limit, we conclude from (7.3) and (7.4) that Ts (z) = z for any s ≥ 0, i.e z ∈ F . To prove the convergence of orbits, let us fix any u ∈ C. We shall prove that {Tt (u)} converges to z. Indeed, we have

ρ (Tt+s (u) − z) = ρ (Tt+s (u) − Tt (z)) ≤ Lt ρ (Ts (u) − z), for any t, s ≥ 0. Hence lim sup ρ (Tt+s (u) − z) ≤ lim sup Lt ρ (Ts (u) − z). s→∞

s→∞

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7 Semigroups of Nonlinear Mappings in Modular Function Spaces

Since lim sup ρ (Tt+s (u) − z) = lim sup ρ (Ts (u) − z), we get s→∞

t→∞

lim sup ρ (Ts (u) − z) ≤ Lt lim sup ρ (Ts (u) − z), s→∞

s→∞

for any t ≥ 0. If we let t → ∞, we obtain lim sup ρ (Ts (u) − z) ≤ L lim sup ρ (Ts (u) − z). s→∞

s→∞

Since L < 1, we get lim sup ρ (Ts (u) − z) = 0. s→∞

Clearly, we can derive the same equality where lim sup is replaced by lim inf which implies the desired conclusion: lim ρ (Ts (u) − z) = 0. s→∞ 

Let us prove now the existence of common fixed points for ρ -nonexpansive semigroups. Theorem 7.2. Assume ρ ∈ ℜ is (UUC1). Let C be a ρ -closed ρ -bounded convex nonempty subset. Let F be a ρ -nonexpansive semigroup on C. Then, the set F(F ) of common fixed points is nonempty, ρ -closed, and convex. Proof. Let us fix an x ∈ C and define the ρ -type

τ (y) = lim sup ρ (Tt (x) − y), t→∞

where y ∈ C. Let τ0 = inf{τ (y) : y ∈ C} and let {zn } be a minimizing sequence for τ , i.e., lim τ (zn ) = τ0 . n→∞

By Lemma 4.6, there exists a z ∈ C such that lim ρ (zn − z) = 0.

n→∞

We are going to prove that z ∈ F(F ). Noting that

ρ (Ts+t (x) − Tt (y)) ≤ ρ (Ts (x) − t) and passing with s → ∞ we get

τ (Tt (y)) ≤ τ (y). In particular, for any n ≥ 1 we have

τ (Tt (zn )) ≤ τ (zn ).

(7.5)

7.2 Fixed Point Existence for Semigroups of Nonexpansive Mappings

189

Let us fix any sequence tk → ∞ and note that for every s > 0 the sequence {Ttk +s (zk )} is a minimizing sequence for τ . Indeed, using (7.5) we obtain

τ0 ≤ τ (Ttk +s (zk )) ≤ τ (zk ) → τ0 . Hence, Lemma 4.6 implies that lim ρ (Ttk +s (zk ) − z) = 0,

(7.6)

lim ρ (Ttk (zk ) − z) = 0.

(7.7)

k→∞

and in particular for s = 0, k→∞

Using (7.7) we get

ρ (Ttk +s (zk ) − Ts (z)) ≤ ρ (Ttk (zk ) − z) → 0. Since the ρ -limit is unique, (7.6) and (7.2) give Ts (z) = z, i.e., z ∈ F(F ) as claimed. Let us prove that F(F ) is ρ -closed. Let xn ∈ F(F ) and ρ (xn − x) → 0. Observe that for every t ≥ 0,

 1 (Tt (x) − x) ≤ ρ (Tt (x) − Tt (xn )) + ρ (Tt (xn ) − xn ) + ρ (xn − x) ρ 3 ≤ ρ (xn − x) + ρ (xn − x) → 0. Hence, x ∈ F(Tt ) proving that x ∈ F(F ), and consequently, that F(F ) is ρ -closed. To prove convexity of F(F ), we need to demonstrate that w=

u+v ∈ F(F ) 2

provided u, v ∈ F(F ). Indeed, let t > 0. Define x = Tt (w) − u, y = Tt (w) − v. Note that x+y = Tt (w) − w. 2 Define 

v−u r=ρ 2 and observe that

ρ (x) = ρ (Tt (w) − u) = ρ (Tt (w) − Tt (u)) ≤ ρ (w − u) = ρ

v−u 2

 = r.

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7 Semigroups of Nonlinear Mappings in Modular Function Spaces

Similarly, ρ (y) ≤ r. Hence, x, y ∈ D2 (r, 1) and therefore 

1 x+y 1 = 1 − ρ (Tt (w) − w). δ2 (r, 1) ≤ 1 − ρ r 2 r Using the assumed (UUC1) and Proposition 4.1, we conclude that ρ satisfies (UC2). Hence by Lemma 4.4, δ2 (r, 1) = 1, which yields 1 ρ (Tt (w) − w) ≤ 1 − δ2 (r, 1) = 0. r Therefore, Tt (w) = w, i.e., w ∈ F(F ), completing the proof. 

Remark 7.1. Recently, Al-Mezel, Al-Roqi, and Khamsi, provided in [7] a partial generalization of the second theorem showing, under suitable assumptions, the existence of common fixed points of a commutative family of ρ -nonexpansive mappings defined on a ρ -closed ρ -bounded convex nonempty subset of Lρ .

7.3 Characterization of the Set of Common Fixed Points In this section, we will show how, under suitable assumptions, the set of common fixed points of a ρ -nonexpansive semigroup can be represented as the set of fixed points of just one ρ -nonexpansive mapping or—in a different context—by an intersection of the fixed point sets of two suitably chosen ρ -nonexpansive mappings. We will use these representation theorems to show how to construct common fixed points of semigroups of nonlinear mappings. The idea of such representations of semigroups in Banach spaces was developed by Suzuki, [192, 193] and then extended to the setting of modular function spaces by Kozlowski [140] and by Alsulami and Kozlowski [9]. Let 0 < α < β , and α ≤ s ≤ β . Define inductively a sequence {An (s)} of subsets of [α , β ] by A1 (s) = {s}, An+1 (s) =



{|α − t|, |β − t|} f or n ∈ N.

t∈An (s)

Set A(s) =

∞

An (s).

n=1

We will use the following three results concerning real numbers. Lemma 7.1. [193] If α /β is an irrational number, then for every s ∈ [α , β ], the closure of A(s) is equal to [0, β ].

7.3 Characterization of the Set of Common Fixed Points

191

Lemma 7.2. [193] If α /β is a rational number, then for every s ∈ [α , β ], the set A(s) is finite. Lemma 7.3. [192] Let t be a nonnegative real number and let {βn } be a sequence in (0, ∞) converging to 0. Define sequences {δn } in (0, ∞) and {kn } in N ∪ {0} as follows: 1. δ1 = t; 2. kn = [δn /βn ], for n ∈ N; 3. δn+1 = δn − kn βn , for n ∈ N. Then for all n ∈ N the following hold: 1. 0 ≤ δn+1 < βn ; 2. δn → 0; n

3.

∑ k j β j + δn+1 = 1;

j=1 ∞

4.

∑ k j β j = t.

j=1

We are now ready to prove our main theorem about representation of a set of common fixed points by a set of fixed points of one nonexpansive mapping. Theorem 7.3. [140] Let F = {Tt : t ≥ 0} be a strongly continuous nonexpansive semigroup on a ρ -bounded subset C of a modular function space Lρ , where ρ ∈ ℜ. Let α and β be positive real numbers such that α /β is an irrational number. Let λ ∈ (0, 1) be arbitrary. Then   F(F ) = F λ Tα + (1 − λ )Tβ . Proof. Note first that the mapping λ Tα + (1 − λ )Tβ is nonexpansive with respect to the convex modular ρ . Without loosing generality we can assume that α < β . It suffices to prove   F λ Tα + (1 − λ )Tβ ⊂ F(F ), as the opposite direction inclusion is trivial. Also, this inclusion is trivial when / We can assume, therefore, that there exists w ∈ C such F(λ Tα + (1 − λ )Tβ ) = 0. that λ Tα (w) + (1 − λ )Tβ (w) = w. (7.8) Since F is a strongly continuous semigroup it follows that the function Λw is a continuous real-valued function on the interval [0, β ] and hence it attains its maximum at a number τ ∈ [0, β ] which means that  

   ρ Tτ (w) − w = max ρ Tt (w) − w : t ∈ [0, β ] .

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7 Semigroups of Nonlinear Mappings in Modular Function Spaces

Since τ ∈ A(τ ) ⊂ [0, β ], we have  

   ρ Tτ (w) − w = max ρ Ts (w) − w : s ∈ A(τ ) . Let us prove by induction that for every n ∈ N and any s ∈ An (τ ),     ρ Tτ (w) − w = ρ Ts (w) − w .

(7.9)

(7.10)

Since A1 (τ ) = {τ }, (7.10) is true for n = 1. Let us make an inductive assumption that (7.10) holds for n = k. Fix arbitrary t ∈ Ak (τ ). By the inductive assumption we have     ρ Tτ (w) − w = ρ Tt (w) − w . (7.11) Substituting (7.8) into the right hand side of (7.11) and using the convexity of ρ , and then using nonexpansiveness, we obtain the following       ρ Tτ (w) − w ≤ λ ρ Tt (w) − Tα w + (1 − λ )ρ Tt (w) − Tβ w     ≤ λ ρ T|α −t| (w) − w + (1 − λ )ρ T|β −t| (w) − w   ≤ ρ Tτ (w) − w , where the last inequality comes from the fact that |α − t| and |β − t| belong to Ak+1 (τ ) ⊂ A(τ ) and from (7.9). From (7.3) it follows easily that       ρ Tτ (w) − w = ρ T|α −t| (w) + ρ T|β −t| (w) . By arbitrariness of t ∈ Ak (τ ), we conclude that (7.10) holds for k + 1 and hence by induction it also holds for all natural n. From (7.10) it follows that for any s ∈ A(τ ),     ρ Tτ (w) − w = ρ Ts (w) − w . (7.12) By Lemma 7.1, A(τ ) is dense in [0, β ]. Hence, by the continuity of Λw we deduce from (7.12) that     ρ Tτ (w) − w = ρ Ts (w) − w , for every s ∈ [0, β ]. Substituting s = 0 and remembering that T0 (w) = w we have   ρ Tτ (w) − w = 0,   and consequently ρ Ts (w) − w = 0 for every s ∈ [0, β ] implying Ts (w) = w for any s ∈ [0, β ]. Let t be any positive real number, hence t = nβ + s for a n ∈ N ∪ {0} and

7.3 Characterization of the Set of Common Fixed Points

193

s ∈ [0, β ]. Therefore Tt (w) = Tβn ◦ Ts (w) = Tβn (w) = w, which means that w ∈ F(F ), as claimed. 

Please note that we did not assume in Theorem 7.3 that the common fixed points actually exist. This theorem, however, reduces the question of existence of common fixed points to the question of existence of fixed points for each ρ -nonexpansive mapping. We have, therefore, the following result which corresponds to the Banach space results from [34, 193]. Theorem 7.4. Let F = {Tt : t ≥ 0} be a strongly continuous nonexpansive semigroup on a ρ -closed, ρ -bounded, and convex subset C of a modular function space Lρ , where ρ ∈ ℜ. Assume that every ρ -nonexpansive mapping on C has a fixed point. Then the set of common fixed points of F is nonempty. Remark 7.2. Please note that our Theorem 7.4 combined with the existence theorem for ρ -nonexpansive mappings (Theorem 5.7 ) gives us an alternative proof of the existence part of Theorem 7.2. As we saw, the strong continuity assumption of the semigroup was of critical importance. Let us give an important example when this condition is satisfied. It is easy to see that if a semigroup F is continuous and the modular ρ is uniformly continuous then F is strongly continuous. Consequently, we have the following result. Theorem 7.5. Let F = {Tt : t ≥ 0} be a continuous nonexpansive semigroup on a subset C of a modular function space Lρ , where ρ ∈ ℜ is uniformly continuous. Let α and β be positive real numbers such that α /β is an irrational number. Let λ ∈ (0, 1) be arbitrary. Then   F(F ) = F λ Tα + (1 − λ )Tβ . A natural question arises: what representation of common fixed point sets can be achieved without assuming that the semigroup F is strongly continuous or without the uniform continuity of ρ ? It turns out that we can indeed characterize finite intersections of sets of fixed points. Let us limit ourselves to the case of two mappings. The proof of the general case follows the same pattern. Theorem 7.6. Let F = {Tt : t ≥ 0} be a nonexpansive semigroup on a subset C of a modular function space Lρ , where ρ ∈ ℜ. Let α and β be positive real numbers such that α /β is a rational number. Let λ ∈ (0, 1) be arbitrary. Then   F(Tα ) ∩ F(Tβ ) = F λ Tα + (1 − λ )Tβ .

194

7 Semigroups of Nonlinear Mappings in Modular Function Spaces

Proof. Assume that α < β . It suffices to prove that   F λ Tα + (1 − λ )Tβ ⊂ F(Tα ) ∩ F(Tβ ).   Fix an arbitrary w ∈ F λ Tα + (1 − λ )Tβ . By Lemma 7.2, the set A(0) is finite. Therefore, there exists τ ∈ A(0) such that  

   ρ Tτ (w) − w = max ρ Ts (w) − w : s ∈ A(0) . Arguing like in the proof of Theorem 7.3, we conclude that     ρ Tτ (w) − w = ρ Ts (w) − w , for every s ∈ A(0). Since 0 ∈ A(0) and T0 (w) = w, it follows that   ρ Tτ (w) − w = 0. Using this and the fact that both α ∈ A(0) and β ∈ A(0), we have       ρ Tα (w) − w = ρ Tβ (w) − w = ρ Tτ (w) − w = 0, which implies that Tα (w) = Tβ (w) = w. The proof is complete. 

Let us prove the following result which relates to Bruck’s theorem in Banach spaces, see [34]. Theorem 7.7. [9] Let ρ ∈ ℜ be a strictly convex function modular. Let C ⊂ Lρ and let T and S be two ρ -nonexpansive mappings from C into Lρ with a common fixed point. Then for each λ ∈ (0, 1), a mapping U : C → Lρ defined by U(x) = λ S(x) + (1 − λ )T (x), for x ∈ Lρ is ρ -nonexpansive and F(U) = F(S) ∩ F(T ). Proof. A straightforward calculation shows that the mapping U is ρ -nonexpansive. It is also clear that F(S) ∩ F(T ) ⊂ F(U). Therefore, to complete the proof we need only to prove the converse inclusion.

7.3 Characterization of the Set of Common Fixed Points

To this end, let us fix x ∈ F(U) and w ∈ F(S) ∩ F(T ). Let us calculate:   ρ (x − w) = ρ λ S(x) + (1 − λ )T (x) − w     ≤ λ ρ S(x) − w + (1 − λ )ρ T (x) − w     = λ ρ S(x) − S(w) + (1 − λ )ρ T (x) − T (w)

195

(7.13)

≤ λ ρ (x − w) + (1 − λ )ρ (x − w). In particular, (7.13) yields the following     ρ (x − w) = ρ S(x) − w = ρ T (x) − w .

(7.14)

Indeed, let us assume to the contrary that   ρ S(x) − w < ρ (x − w).

(7.15)

Combining (7.13) with (7.15) we have     ρ (x − w) ≤ λ ρ S(x) − w + (1 − λ )ρ T (x) − w < λ ρ (x − w) + (1 − λ )ρ (x − w) = ρ (x − w),

(7.16)

which is impossible. Since the same reasoning can be applied assuming that   ρ T (x) − w < ρ (x − w), we conclude that the claim (7.14) holds. Set f = S(x) − w and g = T (x) − w and observe that (7.14) implies that ρ ( f ) = ρ (g). Straight calculation shows that     (7.17) ρ λ f + (1 − λ )g = ρ U(x) − w . On the other hand, it follows from (7.14) and from the assumption x ∈ F(U) that     λ ρ ( f ) + (1 − λ )ρ (g) = λ ρ S(x) − w + (1 − λ )ρ T (x) − w = λ ρ (x − w) + (1 − λ )ρ (x − w) = ρ (x − w) = ρ (U(x) − w). Comparing (7.17) to (7.18), we obtain immediately   ρ λ f + (1 − λ )g = λ ρ ( f ) + (1 − λ )ρ (g),

(7.18)

(7.19)

which by the strict convexity of ρ implies that f = g, and consequently that S(x) = T (x). Compute x = U(x) = λ S(x) + (1 − λ )T (x) = λ S(x) + (1 − λ )S(x) = S(x),

(7.20)

196

7 Semigroups of Nonlinear Mappings in Modular Function Spaces

and hence x ∈ F(S). Similarly, we can prove that x ∈ F(T ). Hence, x ∈ F(S) ∩ F(T ) as claimed. 

Our next result, interesting by itself, will be used in the proof of our representation result, Theorem 7.9. Theorem 7.8. [9] Let ρ ∈ ℜ and let F = {Tt : t ≥ 0} be a continuous semigroup of mappings on a subset C of Lρ . Let {αn } be a sequence of nonnegative numbers converging to α ∈ [0, ∞) such that αn = α for all n ∈ N. Then we have the following representation of the set of all common fixed points of F , ∞ 

F(F ) =

F(Tαn ).

(7.21)

n=1

Proof. We only need to prove that

∞ 

F(Tαn ) ⊂ F(F ) as the other direction is

n=1

trivial. Let z ∈ C be such that Tαn (z) = z for every n ∈ N. Observe first that if {tn } is a sequence of nonnegative real numbers such that Ttn (z) = z and tn → t, where t ∈ [0, ∞), then Tt (z) = z. Indeed,  T (z) − z  1   1   t ≤ ρ Tt (z) − Ttn (z) + ρ Ttn (z) − z ρ 2 2 2  1  = ρ Tt (z) − Ttn (z) → 0, 2 by the continuity of F . Hence Tt (z) = z as claimed. The above observation implies, in particular, that Tα (z) = z. Let us define βn = |αn − α | > 0, where n ∈ N. It follows from the assumptions that each βn is a positive real number and that βn → 0. Note that max{αn , α } = min{αn , α } + βn . Hence, denoting mn = min{αn , α } and Mn = max{αn , α }, we have Tβn (z) = Tβn ◦ Tmn (z) = Tβn +mn (z) = TMn (z) = z. Fix any t > 0. By Lemma 7.3, there exists a sequence {kn } in N ∪ {0} such that ∞

t = ∑ ki βi . i=1

Denoting n

sn = ∑ ki βi , i=1

7.3 Characterization of the Set of Common Fixed Points

197

we obtain for each n ∈ N with sn > 0 k

Tsn (z) = Tβknn ◦ Tβ n−1 ◦ · · · ◦ Tβk2 ◦ Tβk1 (z) = z. n−1

2

1

Since T0 (z) = z, it follows that Tsn (z) = z for every n ∈ N. Because sn → t and F is a continuous semigroup, we conclude, as previously observed, that Tt (z) = z which concludes the proof of the theorem. 

The following technical result about real numbers (Lemma 3 in [192]) will be used in the proof of our next representation theorem. /Q . Lemma 7.4. [192] Let α and β be positive real numbers satisfying α /β ∈ Define sequences {αn } in (0, ∞) and {kn } in N as follows: 1. α1 = max{α , β }; 2. α2 = min{α , β }; 3. kn = [αn /αn+1 ], for all n ∈ N; 4. αn+2 = αn − kn αn+1 , for all n ∈ N. Then the following hold: 1. 0 < αn+1 < αn , for all n ∈ N; 2. kn ∈ N, for all n ∈ N; / Q, for all n ∈ N; 3. αn /αn+1 ∈ 4. {αn } converges to 0. In our next result, we will represent the set of all common fixed points of a semigroup by an intersection of just two mappings from this semigroup. In contrast to Theorem 7.6 we will assume only that the semigroup is continuous, i.e., we will not require the strong continuity. Theorem 7.9. [9] Let ρ ∈ ℜ and let F = {Tt : t ≥ 0} be a continuous semigroup of mappings on a subset C of Lρ . Let α > 0 and β > 0 be two real numbers such that α /β ∈ / Q. Then F(F ) = F(Tα ) ∩ F(Tβ ). Proof. We only need to prove that F(Tα ) ∩ F(Tβ ) ⊂ F(F )

(7.22)

as the converse inclusion is obvious. To this end, let us fix z ∈ C such that z ∈ F(Tα ) ∩ F(Tβ ). Let {αn } in (0, ∞) and {kn } in N be two sequences defined as in Lemma 7.4. We will show that Tαn (z) = Tαn+1 (z) = z, for every n ∈ N. Observe that Tα1 (z) = z and Tα2 (z) = z.

(7.23)

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7 Semigroups of Nonlinear Mappings in Modular Function Spaces

Thus, Eq. (7.23) holds for n = 1. Now, suppose that Tαn (z) = Tαn+1 (z) = z. Then, we have n (z) = Tαn+2 +kn αn+1 (z) = Tαn (z) = Tαn+1 (z) = z. Tαn+2 (z) = Tαn+2 ◦ Tαkn+1

Hence, by induction, Tαn (z) = z for every n ∈ N and consequently F(Tα ) ∩ F(Tβ ) ⊂

∞ 

  F Tαn .

(7.24)

n=1

Since, by construction, {αn } is a sequence of positive numbers converging to 0 ∈ [0, ∞), it follows from Theorem 7.8 that F(F ) =

∞ 

  F Tαn .

(7.25)

n=1

Combining (7.24) with (7.25) we obtain the desired inclusion (7.22) which completes the proof. 

The next result is an immediate consequence of the fact that the uniform convexity (UUC1) implies the strict convexity (SC) of ρ , and of Theorems 7.2, 7.7, and 7.9. Theorem 7.10. [9] Let ρ ∈ ℜ be (UUC1), and let F = {Tt : t ≥ 0} be a continuous semigroup of ρ -nonexpansive mappings on a ρ -closed, ρ -bounded, convex, / Q. nonempty subset of Lρ . Let α > 0 and β > 0 be two real numbers such that α /β ∈ Fix an arbitrary λ ∈ (0, 1). Then F(F ) = 0/ and   (7.26) F(F ) = F λ Tα + (1 − λ )Tβ . Remark 7.3. Note that we already obtained the conclusion of Theorem 7.10 in the case when the assumption of the uniform convexity was replaced by that the strong continuity of the semigroup F , see Theorem 7.5.

7.4 Convergence of Mann Iteration Processes We concluded the previous section with Theorem 7.10 which says that, under suitable assumptions, the set of all common fixed points of a continuous semigroup of ρ -nonexpansive mappings is nonempty and can be represented as the set of all fixed points of just one ρ -nonexpansive mapping. In this section, we will demonstrate how this result can be applied to the construction of such a common fixed point. This idea can be summarized as follows: using the results of the previous sections, we can reduce a problem of constructing a common fixed point for a semigroup of mappings to a problem of constructing a fixed point for just one ρ -nonexpansive mapping.

7.4 Convergence of Mann Iteration Processes

199

In the current section, we will prove the convergence of the Mann iterative process to a common fixed point of a continuous semigroup. Let us start with the definition of the Mann process, see [161] for the original definition. Compare also with Sect. 2.6 of this book. Definition 7.6. Let ρ ∈ ℜ, C ⊂ Lρ , and let T be a ρ -nonexpansive self-mapping on C. Let σ ∈ (0, 1). The Mann iteration process generated by the mapping T and the constant σ , denoted by M(T, σ ) , is defined by the following iterative formula: xk+1 = σ T (xk ) + (1 − σ )xk , where x1 ∈ C is chosen arbitrarily. Let us start with an important technical result which corresponds to Lemma 6.3. The latter one was proved in a more complex settings of asymptotic ρ -nonexpansive mappings. Lemma 7.5. Let ρ ∈ ℜ be (UUC1), C ⊂ Lρ be a ρ -closed, ρ -bounded, and convex set. Let T : C → C be ρ -nonexpansive and let σ ∈ (0, 1). Denote by {xk } a sequence of elements of C generated by a Mann process M(T, σ ). Assume that w is a fixed point of T . Then there exists r ∈ R such that lim ρ (xk − w) = r.

k→∞

Proof. Since

ρ (xk+1 − w) ≤ σ ρ (T (xk ) − w) + (1 − σ )ρ (xk − w) = σ ρ (T (xk ) − T (w)) + (1 − σ )ρ (xk − w) ≤ σ ρ (xk − w) + (1 − σ )ρ (xk − w) = ρ (xk − w), it follows that {ρ (xk − w)} is a nonincreasing sequence of nonnegative numbers, hence it is convergent to a number r ∈ R. 

Lemma 7.6. Let ρ ∈ ℜ be (UUC1), C ⊂ Lρ be a ρ -closed, ρ -bounded, and convex set. Let T : C → C be ρ -nonexpansive and let σ ∈ (0, 1). Denote by {xk } a sequence of elements of C generated by a Mann process M(T, σ ). Then lim ρ (T (xk ) − xk ) = 0,

k→∞

and lim ρ (xk+1 − xk ) = 0.

k→∞

200

7 Semigroups of Nonlinear Mappings in Modular Function Spaces

Proof. By Theorem 5.7, T has at least one fixed point w ∈ C. In view of Lemma 7.5, there exists r ∈ R such that lim ρ (xk − w) = r.

(7.27)

k→∞

Note that lim sup ρ (T (xk ) − w) = lim sup ρ (T (xk ) − T (w)) ≤ lim sup ρ (xk − w) ≤ r, (7.28) k→∞

k→∞

k→∞

and that lim ρ (σ (T (xk ) − w) + (1 − σ )(xk − w)) = lim ρ (xk+1 − w) = r.

k→∞

k→∞

Set fk = T (xk ) − w, gk = xk − w, and note that lim sup ρ (gk ) ≤ r by (7.27), and lim sup ρ ( fk ) ≤ r by (7.28). Observe also that

k→∞

k→∞

lim ρ (σ fk + (1 − σ )gk ) = lim ρ (σ T (xk ) + (1 − σ )xk − w)

k→∞

k→∞

= lim ρ (xk+1 − w) = r. k→∞

Hence, it follows from Lemma 4.2 that lim ρ (T (xk ) − xk ) = lim ρ ( fk − gk ) = 0,

k→∞

k→∞

which by the construction of the sequence {xk } is equivalent to lim ρ (xk+1 − xk ) = 0,

k→∞

as claimed. 

We are now ready to prove the following version of the Mann process convergence theorem for a single ρ -nonexpansive mapping. Theorem 7.11. [9] Let ρ ∈ ℜ. Assume that 1. ρ is (UUC1), 2. ρ has strong Opial property, 3. ρ has Δ2 property and is uniformly continuous. Let C ⊂ Lρ be nonempty, ρ −a.e. compact, convex, strongly ρ -bounded and ρ -closed. Let T : C → C be ρ -nonexpansive and σ ∈ (0, 1). Denote by {xk } a sequence of elements of C generated by a Mann process M(T, σ ). Then there exists x ∈ F(T ) such that xn → x ρ -a.e.

7.4 Convergence of Mann Iteration Processes

201

Proof. Observe that by Theorem 5.7 the set of fixed points F(T ) is nonempty, convex, and ρ -closed. By Lemma 7.6, the sequence {xk } is an approximate fixed point sequence, that is, ρ (T (xk ) − xk ) → 0 as k → ∞. Consider y, z ∈ C, two ρ -a.e. cluster points of {xk }. There exits then {yk }, {zk } subsequences of {xk } such that yk → y ρ -a.e., and zk → z ρ -a.e. By Theorem 6.1, y ∈ F(T ) and z ∈ F(T ). By Lemma 7.5, there exist ry , rz ∈ R such that ry = lim ρ (xk − y), rz = lim ρ (xk − z). k→

k→

We claim that y = z. Assume to the contrary that y = z. Then by the strong Opial property we have ry = lim inf ρ (yk − y) < lim inf ρ (yk − z) k→∞

k→∞

= lim inf ρ (zk − z) < lim inf ρ (zk − y) = ry . k→∞

k→∞

The contradiction implies that y = z. Therefore, {xk } has at most one ρ -a.e. cluster point. Since C is ρ -a.e. compact it follows that the sequence {xk } has exactly one ρ -a.e. cluster point, which means that ρ (xk ) → x ρ -a.e. Using Theorem 6.1 again, we get x ∈ F(T ) as claimed.

 Let us combine now Theorem 7.11 with Theorem 7.10 to demonstrate the convergence of an iterative algorithm to a common fixed point of a semigroup of nonlinear mappings in modular function spaces. Theorem 7.12. [9] Let ρ ∈ ℜ. Assume that 1. ρ is (UUC1), 2. ρ has strong Opial property, 3. ρ has Δ2 property and is uniformly continuous. Let C ⊂ Lρ be nonempty, ρ − a.e. compact, convex, strongly ρ -bounded and ρ closed. Let F = {Tt : t ≥ 0} be a continuous semigroup of ρ -nonexpansive map/ Q. pings on C. Assume that α > 0 and β > 0 are two real numbers such that α /β ∈ Fix λ , κ ∈ (0, 1) such that κ + λ < 1. Define a sequence {xn } in C by x1 ∈ C and xn+1 = κ Tα (xn ) + λ Tβ (xn ) + (1 − κ − λ )xn ,

(7.29)

for natural n ≥ 2. Then {xn } ρ -a.e. converges to a common fixed point of the semigroup F . Proof. Define a mapping S by S=

κ λ Tα + T κ +λ κ +λ β

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7 Semigroups of Nonlinear Mappings in Modular Function Spaces

and observe that S : C → C is ρ -nonexpansive. Fix any x1 ∈ C and let {xn } be generated by the Mann process M(S, σ ), where σ = κ + λ : xn+1 = σ S(xn ) + (1 − σ )xn , which is exactly the sequence defined by (7.29). By Theorem 7.11, there exists x ∈ F(S) such that xn → x ρ -a.e. By Theorem 7.10, F(F ) = F(S), hence x ∈ F(F ). The proof is complete. 

7.5 Convergence of Ishikawa Iteration Processes Definition 7.7. Let ρ ∈ ℜ, C ⊂ Lρ , and let T be a ρ -nonexpansive self-mapping on C. Let σ , τ ∈ (0, 1). The Ishikawa iteration process generated by the mapping T and the constants σ and τ , denoted by I(T, σ , τ ) , is defined by the following iterative formula: xk+1 = σ T (τ T (xk ) + (1 − τ )xk ) + (1 − σ )xk , where x1 ∈ C is chosen arbitrarily. Lemma 7.7. Let ρ ∈ ℜ be (UUC1). Let C ⊂ Lρ be a ρ -closed, ρ -bounded, and convex set. Let T : C → C be ρ -nonexpansive and let σ , τ ∈ (0, 1). Denote by {xk } a sequence of elements of C generated by the Ishikawa process I(T, σ , τ ). Assume that w is a fixed point of T . Then, there exists then an r ∈ R such that lim ρ (xk − w) = r. k→∞

Proof. Using a similar calculation to the one used in the proof of Lemma 7.5, it is not difficult to prove that {xk } is a bounded nonincreasing sequence of nonnegative numbers, hence it is convergent to a number r ∈ R. 

Lemma 7.8. Let ρ ∈ ℜ be (UUC1). Let C ⊂ Lρ be a ρ -closed, ρ -bounded, and convex set. Let T : C → C be ρ -nonexpansive and Let σ , τ ∈ (0, 1). Denote by {xk } a sequence of elements of C generated by the Ishikawa process I(T, σ , τ ). Define yk = τ T (xk ) + (1 − τ )xk . Then lim ρ (T (yk ) − xk ) = 0,

(7.30)

lim ρ (xk+1 − xk ) = 0,

(7.31)

k→∞

or equivalently k→∞

7.5 Convergence of Ishikawa Iteration Processes

203

Proof. By Theorem 5.7, F(T )=0. / Let us fix w ∈ F(T ). By Lemma 7.7, lim ρ (xk−w) k→∞

exists. Let us denote it by r. Since w ∈ F(T ), T ∈ Tr (C), and lim ρ (xk − w) = r by k→∞

Lemma 7.7, we have the following: lim sup ρ (T (yk ) − w) = lim sup ρ (T (yk ) − T (w)) k→∞

k→∞

≤ lim sup ρ (yk − w) k→∞

= lim sup ρ (τ T (xk ) + (1 − τ )xk − w) k→∞   ≤ lim sup τρ (T (xk ) − w) + (1 − τ )ρ (xk − w) k→∞   ≤ lim sup τρ (xk − w) + (1 − τ )ρ (xk − w) ≤ r. k→∞

Note that lim ρ (σ (T (yk ) − w) + (1 − σ )(xk − w))

k→∞

= lim ρ (σ T (yk ) + (1 − σ )xk − w) = lim ρ (xk+1 − w) = r. k→∞

k→∞

Applying Lemma 4.2 with uk = T (yk ) − w and vk = xk − w, we obtain the desired equality lim ρ (T (yk ) − xk ) = 0, while (7.31) follows from (7.30) via the construck→∞

tion formulas for xk+1 and yk . 

Lemma 7.9. Let ρ ∈ ℜ be (UUC1) satisfying Δ2 . Let C ⊂ Lρ be a ρ -closed, ρ -bounded, and convex set. Let T : C → C be ρ -nonexpansive and let σ , τ ∈ (0, 1). Denote by {xk } a sequence of elements of C generated by the Ishikawa process I(T, σ , τ ). Then lim ρ (T (xk ) − xk ) = 0. k→∞

Proof. Let yk = τ T (xk ) + (1 − τ )xk . Hence, T (xk ) − xk =

1 (T (xk ) − yk ). 1−τ

Since τ ∈ (0, 1), there exists 0 < s < 1 such that τ ≤ s. Hence,

ρ (T (xk ) − xk ) = ρ

 1  1   (T (xk ) − yk ) . (T (xk ) − yk ) ≤ ρ 1−τ 1−s

The right-hand side of this inequality tends to zero because ρ (T (xk ) − yk ) → 0 by Lemma 7.8 and because ρ satisfies Δ2 . 

Using Lemma 7.9 instead of Lemma 7.6, and Lemma 7.8 instead of Lemma 7.5, and arguing in a similar way as in the proof of Theorem 7.11, we can obtain the following convergence result for the Ishikawa process.

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7 Semigroups of Nonlinear Mappings in Modular Function Spaces

Theorem 7.13. Let ρ ∈ ℜ. Assume that 1. ρ is (UUC1), 2. ρ has strong Opial property, 3. ρ has Δ2 property and is uniformly continuous. Let C ⊂ Lρ be nonempty, ρ − a.e. compact, convex, strongly ρ -bounded, and ρ closed. Let T : C → C be ρ -nonexpansive and σ ∈ (0, 1), τ ∈ (0, 1). Denote by {xk } a sequence of elements of C generated by an Ishikawa process I(T, σ , τ ). Then there exists x ∈ F(T ) such that xn → x ρ -a.e. Again, let us combine Theorem 7.13 with Theorem 7.10 to demonstrate the convergence of an Ishikawa-type, two-step iterative algorithm to a common fixed point of a semigroup of nonlinear mappings in modular function spaces. Please note that, as said before, typical implementations of the Ishikawa algorithm are convergent at a faster pace than the corresponding Mann schema. Theorem 7.14. Let ρ ∈ ℜ. Assume that 1. ρ is (UUC1), 2. ρ has strong Opial property, 3. ρ has Δ2 property and is uniformly continuous. Let C ⊂ Lρ be nonempty, ρ − a.e. compact, convex, strongly ρ -bounded, and ρ closed. Let F = {Tt : t ≥ 0} be a continuous semigroup of ρ -nonexpansive map/ Q. pings on C. Assume that α > 0 and β > 0 are two real numbers such that α /β ∈ Fix λ , κ ∈ (0, 1) such that κ + λ < 1. Define a sequence {xn } in C by x1 ∈ C and   xn+1 = κ Tα κ Tα (xn ) + λ Tβ (xn ) + (1 − κ − λ )xn   (7.32) + λ Tβ κ Tα (xn ) + λ Tβ (xn ) + (1 − κ − λ )xn + (1 − κ − λ )xn , for natural n ≥ 2. Then {xn } ρ -a.e. converges to a common fixed point of the semigroup F . Proof. Similarly as in the Mann process case, let us define a mapping S by S=

κ λ Tα + T κ +λ κ +λ β

and note that S : C → C is ρ -nonexpansive. Fix any x1 ∈ C and let {xn } be generated by the two-step Ishikawa process I(S, σ , τ ), where σ = κ + λ and τ = κ + λ : xn+1 = σ S(τ S(xn ) + (1 − τ )xn ) + (1 − σ )xn , which is exactly the sequence defined by (7.32). By Theorem 7.13, there exists x ∈ F(S) such that xn → x ρ -a.e. By Theorem 7.10, F(F ) = F(S), hence x ∈ F(F ), which completes the proof. 

7.6 Applications to Differential Equations

205

7.6 Applications to Differential Equations One can ask a legitimate question about existence of natural examples of semigroups of nonlinear mappings in modular function spaces and their applications. Let us address this issue by providing a suitable generic example. In the context of Sect. 5.4 and in particular of Theorem 5.28, we introduce the following notation: let f ∈ C, by u f we will denote a solution of the initial value problem ( u(0) = f (7.33) u (t) + (I − T )u(t) = 0, where f ∈ C and A > 0 are fixed and T : C → C is ρ -nonexpansive. The solution u f is assumed to be obtained by the ρ -limit (5.26). For any t ≥ 0, let us define a mapping St : C → C by St (g) = ug (t). Denote S = {St }t≥0 and F(S ) = { f ∈ C : St ( f ) = f , f or all t ≥ 0}. In this section, we will prove that S is a continuous ρ -nonexpansive semigroup of nonlinear mappings. This result will be used in the following section for proving existence and constructing stationary points of the process defined by the system (7.33). First, let us introduce the following convenient notation: U : C × [0, +∞)

(g,t) → U(g,t) = ug (t) = St (g) ∈ C.

Also, for any n ∈ N define Un : C × [0, +∞) → C by following the recurrent system ⎧ ⎨U0 (g,t) = g ⎩Un+1 (g,t) = e−t g +

 t 0

es−t T (Un (g, s))ds.

First we need to prove the following important technical result. Lemma 7.10. Let ρ ∈ ℜ be separable. Let C ⊂ Eρ be a nonempty, convex, ρ bounded, ρ -closed set with the Vitali property. Let T : C → C be a ρ -nonexpansive mapping. Then for any g ∈ C, t ≥ 0, μ ≥ 0, n ∈ N, m ∈ N   n+m+1 ρ Un (U(g, μ ),t) −Un+m (g,t + μ ) ≤ ∑ K i (μ )δρ (C) + K n+1 (t)δρ (C), (7.34) i=n+1

where K(t) = 1 − e−t =

 t 0

es−t ds.

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7 Semigroups of Nonlinear Mappings in Modular Function Spaces

Proof. The proof is by induction on n ∈ N. Assume first that n = 0 and calculate U0 (U(g, μ ),t) −Um (g,t + μ ) = U(g, μ ) −Um (g,t + μ )

(7.35)

= ug (μ ) −Um (g,t + μ ) = ug (μ ) − e−t−μ g −

 t+μ 0

es−t−μ T (Um−1 (g, s))ds.

Using straightforward calculus we get  t+μ 0

=

es−t−μ T (Um−1 (g, s))ds

 μ 0

es−μ T (Um−1 (g, s))ds +

 t 0

es T (Um−1 (g, s + μ ))ds.

Using the definition of Um and the formula (7.36) we obtain the following   Um (g,t + μ ) = e−t e−μ +

+e−t

 t 0

0

μ

es−μ T (Um−1 (g, s))ds

 (7.36)

es T (Um−1 (g, s + μ ))ds  t

= e−t Um (g, μ ) + e−t

0

es T (Um−1 (g, s + μ ))ds.

Substituting (7.36) into (7.35) we get U0 (U(g, μ ),t) −Um (g,t + μ )

 t

= ug (μ ) − e−t Um (g, μ ) − e−t es T (Um−1 (g, s + μ ))ds 0    t  = e−t ug (μ ) −Um (g, μ ) + es−t ug (μ ) − T (Um−1 (g, s + μ )) ds, 0

where we used the fact that e−t ug (μ ) +

 t 0

es−t ug (μ )ds = ug (μ ).

Let us apply Lemma 5.13 with x(t) = ug (μ ) − T (Um−1 (g,t + μ )), y(t) = ug (μ ) −Um (g, μ ).

(7.37)

7.6 Applications to Differential Equations

207

Hence, using (5.22) from Lemma 5.13 and (7.37) we get  t     ρ U0 (U(g, μ ),t) −Um (g,t + μ ) = ρ e−t y(t) + es−t x(s)ds

(7.38)

0

≤ e−t ρ (y(t)) + K(t) sup ρ (x(s)) 0≤s≤t

  = e ρ (ug (μ ) −Um (g, μ )) + K(t) sup ρ ug (μ ) − T (Um−1 (g, s + μ )) −t

0≤s≤t −t

≤ e ρ (ug (μ ) −Um (g, μ )) + K(t)δρ (C). Applying to (7.38) inequality (5.27) from Theorem 5.28 we finally arrive at   ρ U0 (U(g, μ ),t) −Um (g,t + μ ) ≤ K m+1 (t)δρ (C) + K(t)δρ (C), which gives us desired inequality (7.34) with n = 0. Assume now that (7.34) is true for n ∈ N and let us prove it for n + 1. Using the definition of the recurrent sequence combined with (7.36) we have Un+1 (U(g, μ ),t) −Un+m+1 (g,t + μ ) = e−t U(g, μ ) +

 t 0

es−t T (Un (U(g, μ ), s))ds −Un+m+1 (g,t + μ )

 t

= e−t U(g, μ ) + es−t T (Un (U(g, μ ), s))ds 0  t   −t es T (Un+m (g, s + μ ))ds − e Un+m+1 (g, μ ) + e−t 0    t  −t = e U(g, μ )−Un+m+1 (g, μ ) + es−t T (Un (U(g, μ ), s))−T (Un+m (g, s + μ )) ds. 0

Using Lemma 5.13 with the above formula and with x(t) = T (Un (U(g, μ ),t)) − T (Un+m (g,t + μ )), y(t) = U(g, μ ) −Un+m+1 (g, μ ), we conclude that   ρ Un+1 (U(g, μ ),t) −Un+m+1 (g,t + μ ) ≤ e−t ρ (U(g, μ ) −Un+m+1 (g, μ ))   +K(t) sup ρ T (Un (U(g, μ ), s)) − T (Un+m (g, s + μ )) . 0≤s≤t

(7.39)

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7 Semigroups of Nonlinear Mappings in Modular Function Spaces

Using the ρ -nonexpansiveness of T , the inductive assumption and (5.27) from Theorem 5.28 we conclude from (7.39) that   ρ Un+1 (U(g, μ ),t) −Un+m+1 (g,t + μ ) ≤ e−t ρ (U(g, μ ) −Un+m+1 (g, μ )) + K(t)

 n+m+1

∑

 K i (μ ) + K n+1 (t) δρ (C)

i=m+1

≤ e−t K n+m+2 (μ )δρ (C) + K(t)

 n+m+1

∑

 K i (μ ) + K n+1 (t) δρ (C)

i=m+1

≤

n+m+2

∑

K i (μ )δρ (C) + K n+2 (t)δρ (C),

i=m+1

which completes the proof of Lemma 7.10.

 We are now ready to prove the main result of this section: Theorem 7.15. Let ρ ∈ ℜ be separable, C ⊂ Eρ be a nonempty, convex, ρ -bounded, ρ -closed set with the Vitali property. Assume T : C → C to be a ρ -nonexpansive mapping. Denote St ( f ) = u f (t), where t ≥ 0, f ∈ C, and u f (t) is a solution of the initial value problem (5.25). Then {St }t≥0 is a continuous ρ -nonexpansive semigroup of nonlinear mappings on C. Proof. First let us fix t ≥ 0 and demonstrate that St is ρ -nonexpansive. Fix f , g ∈ C and recall that by Theorem 5.28,

ρ (Un ( f ,t) −U( f ,t)) → 0 ρ (Un (g,t) −U(g,t)) → 0

(7.40)

as n → ∞. By Proposition 3.14, it follows from (7.40) that

ρ (U( f ,t) −U(g,t)) ≤ lim inf ρ (Un ( f ,t) −Un (g,t)). n→∞

Applying Lemma 5.13 with x(t) = Un ( f ,t) −Un (g,t), y(t) = f − g, we have

ρ (Un+1 ( f ,t) −Un+1 (g,t))  t  t   = ρ e−t f + es−t Un ( f , s)ds − e−t g − es−t Un (g, s)ds 0

0

≤ e−t ρ ( f − g) + K(t) sup ρ (Un ( f , s) −Un (g, s)). 0≤s≤t

(7.41)

7.6 Applications to Differential Equations

209

By the obvious induction, keeping in mind that K(t) = 1 − e−t we can easily deduce from (7.41) that ρ (Un ( f ,t) −Un (g,t)) ≤ ρ ( f − g), which together with (7.6) gives us finally

ρ (St ( f ) − St (g)) = ρ (U( f ,t) −U(g,t)) ≤ ρ ( f − g), hence St is ρ -nonexpansive. Now let us prove that {St }t ≥ 0 forms a semigroup of nonlinear mappings. Since it is obvious that S0 ( f ) = u f (0) = f , it remains to prove that Sμ +t = Sμ ◦ St for any μ ,t ≥ 0. Let us fix temporarily n ∈ N and note that by Theorem 5.28,

ρ (Un+m ( f ,t + μ ) −U( f ,t + μ )) → 0 as m → ∞. Hence, by Proposition 3.14 and by inequality (7.34) from Lemma 7.10, using also the fact that the series ∑i K i (μ ) is convergent, we obtain the following   ρ Un (U( f , μ ),t) −U( f ,t + μ ) (7.42)   ≤ lim inf ρ Un (U( f , μ ),t) −Un+m ( f ,t + μ ) m→∞

≤ lim inf m→∞

n+m+1

∑

K i (μ )δρ (C) + K n+1 (t)δρ (C)

i=n+1

≤ K n+1 (t)δρ (C). From (7.42) we see that     ρ Un (U( f , μ ),t) − St+μ ( f ) = ρ Un (U( f , μ ),t) −U( f ,t + μ ) → 0 as n → ∞. On the other hand, it follows from Theorem 5.28 that     ρ Un (U( f , μ ),t) − St (Sμ ( f )) = ρ Un (Sμ ( f ),t) − St (Sμ ( f )) → 0

(7.43)

as well. The uniqueness of the ρ -limit yields to St+μ ( f ) = St (Sμ ( f )) for every t, μ ≥ 0 and every f ∈ C. To prove ρ -continuity of the semigroup {St }t≥0 , let us observe that if tn → t then

ρ (Stn ( f ) − St ( f )) = ρ (u f (tn ) − u f (t)) → 0 because u f is continuous as a solution of the differential equation (7.33). This completes the proof of Theorem 7.15. 

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7 Semigroups of Nonlinear Mappings in Modular Function Spaces

We proved above that S = {St }t≥0 , where St (g) = ug (t) for any g ∈ C, is a continuous ρ -nonexpansive semigroup of nonlinear mappings. Such a situation is quite typical in mathematics and its applications. For instance, in the theory of dynamical systems, the modular function space Lρ would define the state space and the mapping (t, x) → St (x) would represent the evolution function of a dynamical system. The question about the existence of common fixed points, and about the structure of the set of common fixed points, can be interpreted as a question whether there exist stationary points for this process, that is, elements of C that are fixed during the state space transformation St at any given point of time t, and if yes—what the structure of a set of such points may look like and how such points can be constructed algorithmically. In the setting of this chapter, the state space may be an infinite dimensional. Therefore, it is natural to apply these result to not only to deterministic dynamical systems but also to stochastic dynamical systems. It is immediate that if f ∈ C is a common fixed point of S then St ( f ) = f for all t ≥ 0 which means that u f (t) = f for every such t. Therefore, the process defined by the Eq. (7.33) has a stationary point at such f . Let us now interpret our previous results in this context. Combining Theorem 7.2, Theorem 5.28, and Theorem 7.15, we arrive at the following result. Theorem 7.16. Let ρ ∈ ℜ be separable and (UUC1), C ⊂ Eρ be a nonempty, convex, ρ -bounded, ρ -closed set with the Vitali property. Assume T : C → C to be a ρ -nonexpansive mapping. Denote St ( f ) = u f (t), where t ≥ 0, f ∈ C, and u f (t) is a solution of the initial value problem (7.33). Then S = {St }t≥0 forms a continuous ρ -nonexpansive semigroup of nonlinear mappings on C. Moreover, the set of the stationary points for the process defined by the differential equation (7.33) with the evolution function (t, x) → St (x), is equal to F(S ), the set of all common fixed points of S , which is nonempty, ρ -closed, and convex. Let us finish this section by providing an example how the results of the preceding sections can be utilized for constructing a stationary point of a process defined by the Urysohn operator:  1

T ( f )(x) = 0

k(x, y, | f (y)|)dy + f0 (x),

with the assumptions as per Example 3.12. Using Theorem 5.28 we see that, given f ∈ C, the initial value problem ( u(0) = f u (t) + (I − T )u(t) = 0, has a solution u f : [0, +∞] → C. As proved in our Theorem 7.15 the formula St ( f ) = u f (t)

7.7 Asymptotic Pointwise Nonexpansive Semigroups

211

defines the semigroup of ρ -nonexpansive mappings. Note that ρ in this example is orthogonally additive and hence it has the strong Opial property. Therefore, assuming ρ is (UUC1) and uniformly continuous, we can use the algorithmic methods previously established for the Mann process in Theorem 7.11, and for the Ishikawa process in Theorem 7.13 to construct a common fixed point of the semigroup {St } which will be a stationary point of the Urysohn process defined by the evolution function (t, f ) → u f (t) ∈ C.

7.7 Asymptotic Pointwise Nonexpansive Semigroups In this section, we will extend some of the results obtained above for the ρ nonexpansive semigroups to the case of asymptotic pointwise ρ -nonexpansive semigroups. This theory, initiated in [55, 56], is still in its infancy and the authors believe that there are a lot of interesting aspects of the theory that are waiting to be discovered. Definition 7.8. A one-parameter family F = {Tt : t ≥ 0} of mappings from C into itself is said to be an asymptotic pointwise ρ -nonexpansive semigroup on C if F satisfies the following conditions: 1. T0 ( f ) = f for f ∈ C; 2. Tt+s ( f ) = Tt (Ts ( f )) for f ∈ C and t, s ∈ [0, ∞); 3. for each t ≥ 0, Tt is an asymptotic pointwise nonexpansive mapping, i.e., there exists a function αt : C → [0, ∞) such that

ρ (Tt ( f ) − Tt (g)) ≤ αt ( f )ρ ( f − g) f or all f , g ∈ C. such that lim sup αt ( f ) ≤ 1 for every f ∈ C. t→∞

4. for each f ∈ C, the mapping t → Tt ( f ) is ρ -continuous. Note that without loss of generality we may assume αt ( f ) ≥ 1 for any t ≥ 0 and f ∈ C and lim sup αt ( f ) = lim αt ( f ) = 1. t→∞

t→∞

The definition of a ρ -type used in this book was based on a given sequence. In this section, we generalize this definition to be adapted to one-parameter family of mappings. Definition 7.9. Let K ⊂ Lρ be convex and ρ -bounded. 1. A function τ : K → [0, ∞] is called a ρ -type (or shortly a type) if there exists a one-parameter family {ht }t≥0 of elements of K such that for any f ∈ K there holds   τ ( f ) = inf sup ρ (ht − f ) . M>0

t≥M

212

7 Semigroups of Nonlinear Mappings in Modular Function Spaces

2. Let τ be a type. A sequence {gn } is called a minimizing sequence of τ if lim τ (gn ) = inf{τ ( f ) : f ∈ K}.

n→∞

Note that τ is convex provided ρ is convex. The next result extends the minimizing sequence property for types defined by sequences to the case of one-parameter semigroup of nonlinear mappings, compare with Lemma 4.6. Lemma 7.11. [55] Assume ρ ∈ ℜ is (UUC1). Let C be a nonempty, ρ -bounded, ρ closed, and convex subset of Lρ . Let τ be a type defined by a one-parameter family {ht }t≥0 in C. (i) If τ ( f1 ) = τ ( f2 ) = inf τ ( f ), then f1 = f2 . f ∈C

(ii) Any minimizing sequence { fn } of τ is ρ -convergent. Moreover the ρ -limit of { fn } is independent of the minimizing sequence. Proof. First let us prove (i). Let f1 , f2 ∈ C be such that τ ( f1 ) = τ ( f2 ) = inf τ ( f ). f ∈C

Let us consider two cases. Case 1: inf τ ( f ) = 0. Since f ∈C

ρ

f1 − f2 2



=ρ

f1 − ht + ht − f2 2

 ≤ ρ ( f1 − ht ) + ρ (ht − f2 )

for any t ≥ 0, we get 

f1 − f2 ≤ sup ρ ( f1 − ht ) + sup ρ (ht − f2 ) ρ 2 t≥M t≥M for any M > 0. Since

τ ( f ) = inf

M>0



 sup ρ ( f − ht ) = lim sup ρ ( f − ht ), M→∞ t≥M

t≥M

for any f ∈ C, we get

ρ

f1 − f2 2

 ≤ τ ( f1 ) + τ ( f2 ) = 0,

which implies f1 = f2 as claimed. Case 2: inf τ ( f ) > 0. Assume to the contrary that f1 = f2 . Set f ∈C

R = inf τ ( f ) and ε = f ∈C

ρ ( f1 − f2 ) . 2R

7.7 Asymptotic Pointwise Nonexpansive Semigroups

213

Let ν ∈ (0, R). Then ρ ( f1 − f2 ) = 2Rε ≥ (R + ν )ε . Using the definition of τ , we deduce that there exists Mν > 0 such that sup ρ ( f1 − ht ) ≤ τ ( f1 ) + ν = R + ν and sup ρ ( f2 − ht ) ≤ τ ( f2 ) + ν = R + ν .

t≥Mν

t≥Mν

Since ρ is (UUC1), there exists η (R, ε ) > 0 such that

δ (R + ν , ε ) ≥ η (R, ε ) for any ν ∈ (0, R). So for any t ≥ Mν , we have

     f1 + f2 − ht ≤ (R + ν ) 1 − δ (R + ν , ε ) ≤ (R + ν ) 1 − η (R, ε ) . ρ 2 Hence,

τ

f1 + f2 2



≤ sup ρ t≥Mν

f1 + f2 − ht 2



  ≤ (R + ν ) 1 − η (R, ε ) .

Since C is convex, we get

R≤τ

f1 + f2 2



  ≤ (R + ν ) 1 − η (R, ε ) .

If we let ν → 0, we will get   R ≤ R 1 − η (R, ε ) , which is impossible since R > 0 and η (R, ε ) > 0. Therefore, we must have f1 = f2 . Next we prove (ii). Denote R = inf τ (g). For any n ≥ 1, let us set g∈C

 Kn = convρ ht : t ≥ n , where convρ (A) is the intersection of all ρ -closed convex subset of C which contains for any n ≥ 1. Property A ⊂ C. Since C is itself  ρ -closed and convex, we get Kn ⊂ C / Let us fix then arbitrary f ∈ Kn , g ∈ C, and ε > 0. (R) will then imply Kn = 0. By definition of τ (g), there exists Mε > 0 such that sup ρ (g − ht ) ≤ τ (g) + ε . Let t≥Mε

n ≥ Mε . Then for any t ≥ n, we have ρ (g − ht ) ≤ τ (g) + ε , i.e., ht ∈ Bρ (g, τ (g) + ε ). Since Bρ (g, τ (g) + ε ) is ρ -closed and convex, we get Kn ⊂ Bρ (g, τ (g) + ε ). Hence f ∈ Bρ (g, τ (g) + ε ), i.e., ρ (g − f ) ≤ τ (g) + ε . (7.44) Since ε was taken arbitrarily greater than 0, we get ρ (g − f ) ≤ τ (g), for any g ∈ C. Let { fn } be a minimizing sequence for τ . If R = 0 then, since { fn } is a

214

7 Semigroups of Nonlinear Mappings in Modular Function Spaces

minimizing sequence, we get lim τ ( fn ) = R = 0. Using (7.44) we can see that n→∞

ρ ( fn − f ) ≤ τ ( fn ), for any n ≥ 1. Hence { fn } is ρ -convergent to f . Since selection of f was independent of { fn }, it follows that any minimizing sequence is ρ convergent to f if R = 0. We can assume therefore that R > 0. For any n ≥ 1, let us set (7.45) dn = sup ρ ( fi − f j ). i, j≥n

We claim that { fn } is ρ -Cauchy. Assume to the contrary that this is not the case. Since the sequence {dn } is decreasing and { fn } is not ρ -Cauchy, we get d > 0. Let us fix arbitrary ν ∈ (0, R). Since lim τ ( fn ) = R, d := inf dn > 0. Set ε = n→∞ n≥1 4R there exists n0 ≥ 1 such that for any n ≥ n0 , we have

ν τ ( fn ) ≤ R + . 2

(7.46)

Let n ≥ n0 . By (7.45) there exists in , jn ≥ 1 such that

ρ ( fin − f jn ) > dn −

d d ≥ = 2Rε > (R + ν )ε . 2 2

Using the definition of τ and (7.46), we deduce the existence of M > 0 such that sup ρ ( fin − ht ) ≤ τ ( fin ) +

ν ≤ R + ν, 2

sup ρ ( f jn − ht ) ≤ τ ( f jn ) +

ν ≤ R + ν. 2

t≥M

and t≥M

Hence

ρ

f in + f jn − ht 2



  ≤ (R + ν ) 1 − δ (R + ν , ε ) ,

for any t ≥ M. Since ρ is (UUC1), there exists η1 (R, ε ) > 0 such that δ1 (R + ν , ε ) ≥ η1 (R, ε ). Hence

   f in + f jn − ht ≤ (R + ν ) 1 − η (R, ε ) , ρ 2 for any t ≥ M. Hence 

   f in + f jn f jn + f jn ≤ sup ρ − ht ≤ (R + ν ) 1 − η (R, ε ) , R≤τ 2 2 t≥M for any ν ∈ (0, R). If we let ν → 0, we get R ≤ R(1 − η1 (R, ε )). This contradiction implies that { fn } is ρ -Cauchy. Since Lρ is ρ -complete, we deduce that { fn } is ρ -convergent as claimed.

7.7 Asymptotic Pointwise Nonexpansive Semigroups

215

In order to finish the proof of (ii), let us show that the ρ -limit of { fn } is independent of the minimizing sequence. Indeed, let {gn } be another minimizing sequence of τ . The previous proof will show that {gn } is also ρ -convergent. In order to prove that the ρ -limits of { fn } and {gn } are equal, let us show that lim ρ ( fn − gn ) = 0. n→∞

Assume not, i.e., lim ρ ( fn − gn ) = 0. Without loss of generality we may assume n→∞ d > 0. Let that there exists d > 0 such that ρ ( fn − gn ) ≥ d, for any n ≥ 1. Set ε = 2R ν ∈ (0, R). Since lim τ ( fn ) = lim τ (gn ) = R, there exists n0 ≥ 1 such that for any n→∞ n→∞ ν ν n ≥ 1, we have τ ( fn ) ≤ R + , and τ (gn ) ≤ R + . Fix n ≥ n0 . Then 2 2

ρ ( fn − gn ) ≥ d = 2Rε > (R + ν )ε . Using the definition of τ , we deduce the existence of M > 0 such that sup ρ ( fn − ht ) ≤ τ ( fn ) +

ν ≤ R + ν, 2

sup ρ (gn − ht ) ≤ τ (gn ) +

ν ≤ R + ν. 2

t≥M

and t≥M

Hence

ρ

fn + gn − ht 2



  ≤ (R + ν ) 1 − δ (R + ν , ε ) ,

for any t ≥ M. Since ρ is (UUC1), there exists η (R, ε ) > 0 such that δ (R + ν , ε ) ≥ η (R, ε ), for any ν > 0. Hence

   fn + gn − ht ≤ (R + ν ) 1 − η (R, ε ) , ρ 2 for any t ≥ M. Hence 

   fn + gn fn + gn − ht ≤ (R + ν ) 1 − η (R, ε ) . ≤ sup ρ τ 2 2 t≥M Using the definition of R, we get   R ≤ (R + ν ) 1 − η (R, ε ) , for any ν ∈ (0, R). If we let ν → 0, we get R ≤ R(1 − η (R, ε )). This contradiction implies lim ρ ( fn − gn ) = 0. The Fatou property will finally imply n→∞

ρ ( f − g) ≤ lim inf ρ ( fn − gn ), n→∞

216

7 Semigroups of Nonlinear Mappings in Modular Function Spaces

where f is the ρ -limit of { fn } and g is the ρ -limit of {gn }. Hence ρ ( f − g) = 0, i.e., f = g. 

Using Lemma 7.11, we can prove our common fixed point result for asymptotic pointwise nonexpansive semigroups. Theorem 7.17. [55] Assume ρ ∈ ℜ is (UUC1). Let C be a ρ -closed ρ -bounded convex nonempty subset of Lρ . Let F = {Tt : t ≥ 0} be an asymptotic pointwise nonexpansive semigroup on C. Then F has a common fixed point and the set F(F ) of common fixed points is ρ -closed and convex. Proof. Let us fix f ∈ C and define the type function τ by   τ (g) = inf sup ρ (Tt ( f ) − g) . M>0

t≥M

Since C is ρ -bounded, we have τ (g) ≤ diamρ (C) < +∞, for any g ∈ C. Hence τ0 = inf{τ (g) : g ∈ C} exists and is finite. For any n ≥ 1, there exists gn ∈ C, such that 1 τ0 ≤ τ (gn ) < τ0 + . n Therefore, lim τ (zn ) = τ0 , i.e., {gn } is a minimizing sequence for τ . By Lemma n→∞

7.11, there exists g ∈ C such that {gn } ρ -converges to g. Let us now prove that g ∈ F(F ). Note that

ρ (Ts+t ( f ) − Ts (h)) ≤ αs (h)ρ (Tt ( f ) − h), for s,t > 0, and h ∈ C. Using the definition of τ , we get

τ (Ts (h)) ≤ sup ρ (Ts+t ( f ) − Ts (h)) ≤ αs (h) sup ρ (Tt ( f ) − h), t+s≥M

t≥M−s

for any M > s, which implies

τ (Ts (h)) ≤ αs (h)τ (h).

(7.47)

Since lim αs (g1 ) = 1, there exists s1 > 0 such that for any s ≥ s1 , we have s→∞

αs (g1 ) < 1 + 1. Repeating this argument, one will find s2 > s1 + 1 such that for any 1 s ≥ s2 , we have αs (g2 ) < 1 + . By induction, we will construct a sequence {sn } of 2 1 1 positive numbers such that sn+1 < sn + and for any s ≥ sn we have αs (gn ) < 1 + . n n Let us fix t ≥ 0. Then the inequality (7.47) will imply 

1 τ (Tsn +t (gn )) ≤ αsn +t (gn )τ (gn ) ≤ 1 + τ (gn ), n

7.7 Asymptotic Pointwise Nonexpansive Semigroups

217

for any n ≥ 1. In particular we get {Tsn +t (gn )} is a minimizing sequence of τ . Therefore, Lemma 7.11 implies that {Tsn +t (gn )} ρ -converges to g, for any t ≥ 0. In particular, we have {Tsn (gn )} ρ -converges to g. Since   ρ Tsn +t (gn ) − Tt (g) ≤ αt (g)ρ (Tsn (gn ) − g), we get {Tsn +t (gn )} ρ -converges to Tt (g). Finally using 

    Tt (g) − g ≤ ρ Tt (g) − Tsn +t (gn ) + ρ Tsn +t (gn ) − g , ρ 2 we get Tt (g) = g. Since t > 0 was arbitrarily chosen, we get g ∈ F(F ), i.e., F(F ) is not empty. Next let us prove that F(F ) is ρ -closed. Let { fn } in F(F ) ρ -convergent to f . Since

ρ (Ts ( fn ) − Ts ( f )) ≤ αs ( f )ρ ( fn − f ), for any n ≥ 1 and s > 0, we get {Ts ( fn )} ρ -converges to Ts ( f ). Since fn ∈ F(F ), we get {Ts ( fn )} = { fn }. In other words, { fn } ρ -converges to Ts ( f ) and f . The uniqueness of the ρ -limit, implies then that Ts ( f ) = f , for any s ≥ 0, i.e., f ∈ F(F ). Therefore F(F ) is ρ -closed. Let us finish the proof of Theorem 7.17 by showing that F(F ) is convex. It is sufficient to show that f +g h= ∈ F(F ), 2 for any f , g ∈ F(F ). Without loss of generality, we will assume that f = g. Let s > 0. We have

ρ ( f − Ts (h)) = ρ (Ts ( f ) − Ts (h)) ≤ αs ( f )ρ ( f − h) and

ρ (g − Ts (h)) = ρ (Ts (g) − Ts (h)) ≤ αs (g)ρ (g − h) . 

f −g , and Since ρ ( f − h) = ρ (g − h) = ρ 2 

f −g 1 1 ≤ ρ ( f − Ts (h)) + ρ (g − Ts (h)), ρ 2 2 2 we conclude that

lim ρ ( f − Ts (h)) = lim ρ (g − Ts (h)) = ρ

s→∞

s→∞

 f −g . 2

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Similarly we have 

1 h + Ts (h) 1 ≤ ρ ( f − h) + ρ ( f − Ts (h)), ρ f− 2 2 2 

1 h + Ts (h) 1 ≤ ρ (g − h) + ρ (g − Ts (h)). ρ g− 2 2 2

and

Since

ρ

f −g 2



1 ≤ ρ 2

f−

 

1 h + Ts (h) h + Ts (h) + ρ g− , 2 2 2

we conclude that   

f −g h + Ts (h) h + Ts (h) = lim ρ g − =ρ . lim ρ f − s→∞ s→∞ 2 2 2 Therefore, we have

lim ρ ( f − Ts (h)) = lim ρ

s→∞

s→∞

h + Ts (h) f− 2

 = ρ ( f − h).

Lemma 4.2 applied to At = f −Ts (h) and Bt = Ts (h)−g implies that ρ (At −Bt ) → 0. Hence A −B  t t ≤ lim ρ (At − Bt ) = 0. lim ρ (h − Ts (h)) = lim ρ s→∞ s→∞ s→∞ 2 Clearly we will get lim ρ (h − Ts+t (h)) = 0, for any t ≥ 0. Since s→∞

ρ (Tt (h) − Ts+t (h)) ≤ αt (h)ρ (h − Ts (h)), we have lim ρ (Tt (h) − Ts+t (h)) = 0. Finally using the inequality s→∞

ρ

h − Tt (h) 2



1 1 ≤ ρ (h − Ts+t (h)) + ρ (Tt (h) − Ts+t (h)), 2 2

by letting s → ∞, we get Tt (h) = h, for any t ≥ 0, i.e., h ∈ F(F ).



Chapter 8

Modular Metric Spaces

The concept of a metric space is closely related to our intuitive understanding of space in the 3-dimensional Euclidean space. In fact, the notion of metric is a generalization of the Euclidean metric arising from the basic long known properties of the Euclidean distance. Maurice Fr´echet1 is credited as the mathematician who introduced the abstract definition of a metric space [74]. Metric spaces are seen as a nonlinear version of vector spaces endowed with a norm. Following the same direction, one may think of a nonlinear version of modular function spaces. Indeed, throughout this book, we have seen that a modular function space is a vector space endowed with a modular function. Therefore, it is natural to consider a nonlinear version of function modular spaces. The first to consider such generalization was V. Chistyakov [46, 47]. Note that Chistyakov’s generalization of modular function spaces had a physical interpretation. Informally speaking, whereas a metric on a set represents nonnegative finite distances between any two points of the set, a modular on a set attributes a nonnegative (possibly, infinite valued) “field of (generalized) velocities” to each “time” λ > 0 (the absolute value of) an average velocity wλ (x, y) which is associated in such a way that in order to cover the “distance” between points x, y ∈ X it takes time λ to move from x to y with velocity wλ (x, y). The nonlinear approach to modular function spaces was initiated in [1, 2, 3].

8.1 Definitions Let X be a nonempty set. Throughout this chapter, for a given function

ω : (0, +∞) × X × X → [0, +∞], 1

The term metric space was in fact introduced by F. Hausdorff in Grundz¨uge der Mengenlehre, Leipzig, 1914.

c Springer International Publishing Switzerland 2015  M. A. Khamsi, W. M. Kozlowski, Fixed Point Theory in Modular Function Spaces, DOI 10.1007/978-3-319-14051-3 8

219

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8 Modular Metric Spaces

we will write

ωλ (x, y) = ω (λ , x, y), for all λ > 0 and x, y ∈ X. Definition 8.1. A function ω : (0, +∞) × X × X → [0, +∞] is said to be a modular metric on X if it satisfies the following axioms: (i) x = y if and only if ωλ (x, y) = 0, for all λ > 0; (ii) ωλ (x, y) = ωλ (y, x), for all λ > 0, and x, y ∈ X; (iii) ωλ +μ (x, y) ≤ ωλ (x, z) + ωμ (z, y), for all λ , μ > 0 and x, y, z ∈ X. If instead of (i), we only have the condition (i’) ωλ (x, x) = 0, for all λ > 0, x ∈ X, then ω is said to be a pseudomodular (metric) on X. A modular metric ω on X is said to be regular if the following weaker version of (i) is satisfied: x = y if and only if ωλ (x, y) = 0, for some λ > 0. Finally, ω is said to be convex if for λ , μ > 0 and x, y, z ∈ X, it satisfies the inequality

ωλ +μ (x, y) ≤

λ μ ωλ (x, z) + ωμ (z, y). λ +μ λ +μ

Remark 8.1. Note that for a metric pseudomodular ω on a set X, and any x, y ∈ X, the function λ → ωλ (x, y) is nonincreasing on (0, +∞). Indeed if 0 < μ < λ , then

ωλ (x, y) ≤ ωλ −μ (x, x) + ωμ (x, y) = wμ (x, y). Definition 8.2. Let ω be a pseudomodular on X. Fix x0 ∈ X. The two sets Xω = Xω (x0 ) = {x ∈ X : ωλ (x, x0 ) → 0 as λ → +∞} and Xω∗ = Xω∗ (x0 ) = {x ∈ X : ∃λ = λ (x) > 0 such that ωλ (x, x0 ) < +∞} are said to be modular spaces (around x0 ) . It is clear that Xω ⊂ Xω∗ , but this inclusion may be proper in general. If ω is a modular on X, then the modular space Xω can be equipped with a (nontrivial) metric, generated by ω and given by dω (x, y) = inf{λ > 0 : ωλ (x, y) ≤ λ },

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221

for any x, y ∈ Xω . If ω is a convex modular on X, then the two modular spaces coincide, i.e., Xω∗ = Xω , and this common set can be endowed with the metric dω∗ given by dω∗ (x, y) = inf{λ > 0 : ωλ (x, y) ≤ 1}, for any x, y ∈ Xω . These distances will be called Luxemburg distances similarly as we did in the linear case. In the next example, we show how the linear modular function spaces are connected to modular metric spaces as defined above. Example 8.1. A modular function space furnishes a wonderful example of a modular metric space. Indeed, define the function ω by

 f −g , ωλ ( f , g) = ρ λ for all λ > 0, and f , g ∈ Lρ . Then ω is a modular metric on Lρ . Note that ω is convex if and only if ρ is convex. Moreover, we have  f − gρ = dω∗ ( f , g), for any f , g ∈ Lρ . Remark 8.2. Note that regular metric spaces are also modular metric spaces. Indeed, let (M, d) be a metric space. Then the functional ω : (0, +∞) × M × M → [0, +∞) defined by d(x, y) ωλ (x, y) = λ is a regular modular metric on M. Note that ω is convex. Note that in this case we have dω∗ = d. For the regular metric spaces, a natural topology is introduced as follows. Definition 8.3. Let (X, ω ) be a modular metric space. (1) The sequence {xn }n∈N in Xω is said to be ω -convergent to x ∈ Xω if and only if ωλ (xn , x) → 0 as n → +∞, for some λ > 0. (2) The sequence {xn }n∈N in Xω is said to be ω -Cauchy if ωλ (xm , xn ) → 0 as m, n → +∞, for some λ > 0. (3) A subset C of Xω is said to be ω -closed if the limit of a ω -convergent sequence of C always belong to C. (4) A subset C of Xω is said to be ω -complete if any ω -Cauchy sequence in C is a convergent sequence and its limit is in C. (5) A subset C of Xω is said to be ω -bounded if for some λ > 0, we have

δω (C) = sup{ωλ (x, y) : x, y ∈ C} < +∞.

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(6) ω is said to satisfy the Fatou property, if for any {xn } ω -convergent to x and {yn } ω -convergent to y, then

ω1 (x, y) ≤ lim inf ω1 (xn , yn ). n→∞

In general if lim ωλ (xn , x) = 0, for some λ > 0, then we may not have n→∞

lim wλ (xn , x) = 0, for all λ > 0. This remark suggests the following definition.

n→∞

Definition 8.4. Let (X, ω ) be a modular metric space. ω is said to satisfy Δ2 condition if and only if lim ωλ (xn , x) = 0, for some λ > 0 implies lim ωλ (xn , x) = 0, n→∞

for all λ > 0.

n→∞

As we did in the linear case, it is easy to show that lim dω (xn , x) = 0 if and only if lim ωλ (xn , x) = 0, for all λ > 0,

n→∞

n→∞

for any {xn } ∈ Xw and x ∈ Xw . In particular, we see that ω -convergence and dω convergence are equivalent if and only if the modular ω satisfies the Δ2 -condition. Moreover, if the modular ω is convex, then we know that dω∗ and dω are equivalent which implies lim d ∗ (xn , x) n→∞ ω

= 0 if and only if lim ωλ (xn , x) = 0, for all λ > 0, n→∞

for any {xn } ∈ Xω and x ∈ Xω . Remark 8.3. Let (X, ω ) be a modular metric space. Assume that ω is regular. Then we have the uniqueness of the ω -limit. Indeed, let {xn } ∈ Xω be a sequence such that {xn } ω -converges to a ∈ Xω and b ∈ Xω . Then there exist λ , μ > 0 such that lim ωλ (xn , a) = 0 and lim ωμ (xn , b) = 0.

n→+∞

n→+∞

Since

ωλ +μ (a, b) ≤ ωλ (xn , a) + ωμ (xn , b), for any n ≥ 1, we get ωλ +μ (a, b) = 0. Since ω is regular, we must have a = b as claimed. Let us finish this section with the modular metric definitions of Lipschitzian mappings. The definitions are straightforward generalizations of their norm and metric equivalents. Definition 8.5. Let (X, ω ) be a modular metric space. Let C be a nonempty subset of Xω . A mapping T : C → C is called an ω -Lipschitzian mapping if there exists a constant k ≥ 0 such that

ω1 (T (x), T (y)) ≤ k ω1 (x, y), for any x, y ∈ C.

8.2 Banach Contraction Principle in Modular Metric Spaces

223

The smallest such constant k will be known as Lipω (T ). If Lipω (T ) < 1, then T is called ω -contraction mapping. And if Lipω (T ) ≤ 1, then T is called ω nonexpansive mapping. In the next definition, we introduce the concept of pointwise Lipschitzian mappings in modular metric spaces. Definition 8.6. Let (X, ω ) be a modular metric space. Let C be a nonempty subset of Xω . A mapping T : C → C is called a pointwise ω -Lipschitzian mapping if there exists a function α : C → [0, +∞) such that

ω1 (T (x), T (y)) ≤ α (x) ω1 (x, y), for any x, y ∈ C. If α (C) ⊂ [0, 1), then T is called a pointwise ω -contraction mapping. And if α (C) ⊂ [0, 1], then T is called a pointwise ω -nonexpansive mapping. In [46, 47], the author defined Lipschitzian mappings in modular metric spaces and proved some fixed point theorems. The above definition is more general. Indeed if we assume ω is convex, then we have

ωλ (T (x), T (y)) ≤ ωλ (x, y), for any λ > 0 if and only if

dω∗ (T (x), T (y)) ≤ dω∗ (x, y).

Moreover, we have seen that in the modular function spaces, we may have

ω1 (T (x), T (y)) ≤ ω1 (x, y), but T is not Lipschitzian with respect to the Luxemburg distance dω∗ with constant 1.

8.2 Banach Contraction Principle in Modular Metric Spaces The statement of the Banach Contraction Principle in modular metric spaces goes as follow: Theorem 8.1. Let (X, ω ) be a modular metric space. Assume ω is regular. Let C be a nonempty subset of Xω . Assume that C is ω -complete and ω -bounded, i.e., δω (C) = sup{ω1 (x, y) : x, y ∈ C} < ∞. Let T : C → C be an ω -contraction. Then T has a unique fixed point x0 . Moreover, the orbit {T n (x)} ω -converges to x0 , for each x ∈ C. Proof. Since T is a contraction, then there exists k ∈ [0, 1) such that

ω1 (T (x), T (y)) ≤ k ω1 (x, y), for any x, y ∈ C.

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First note that T has at most one fixed point. Indeed, let a, b ∈ C be two fixed points of T . Then we have

ω1 (a, b) = ω1 (T (a), T (b)) ≤ k ω1 (a, b). Our assumption on C implies ω1 (a, b) < ∞. This will force ω1 (a, b) = 0, which implies a = b because ω is regular. Next we fix x ∈ C. Then we have

ω1 (T n+h (x), T n (x)) ≤ kn ω1 (T h (x), x) ≤ kn δω (C), for any n ≥ 1 and h ≥ 1. This clearly implies that {T n (x)} is ω -Cauchy. Since C is ω -complete, therefore {T n (x)} ω -converges to some point x0 ∈ C. Next, let us show that x0 is a fixed point of T . Indeed, we have

ω2 (x0 , T (x0 )) ≤ ω1 (x0 , T n (x)) + ω1 (T (x0 ), T n (x)), which implies

ω2 (x0 , T (x0 )) ≤ ω1 (x0 , T n (x)) + k ω1 (x0 , T n−1 (x)), for any n ≥ 1. Since {T n (x)} ω -converges to x0 , we get ω2 (x0 , T (x0 )) = 0 which implies T (x0 ) = x0 since ω is regular. Since T has at most one fixed point, we conclude that any orbit of T , ω -converges to the only fixed point x0 of T . 

Remark 8.4. In the classical Banach contraction principle, the metric space is not supposed to be bounded. In fact, the contractive condition of the mapping implies that any orbit is bounded. In the case of a modular metric space, due to the failure of of the triangle inequality, it is not true that the contractive condition of the mapping will imply the boundedness of the orbit. Note that if T has a fixed point, then it is obvious that an orbit of T is bounded. Conversely, if T has a bounded orbit {T n (x)}, for some x ∈ C, then we have

ω1 (T n+h (x), T n (x)) ≤ kn ω1 (T h (x), x) ≤ kn sup ω1 (T n (x), T m (x)), n,m≥1

for any n ≥ 1 and h ≥ 1. This clearly implies that {T n (x)} is ω -Cauchy. Since C is ω -complete, then {T n (x)} is ω -convergent to some x0 ∈ C. Since

ω1 (T n (x), T (x0 )) ≤ k ω1 (T n−1 (x), x0 ), n = 1, 2, · · · , we conclude that {T n (x)} ω -converges to T (x0 ). Since ω is regular, we must have T (x0 ) = x0 , i.e., x0 is a fixed point of T . Next for any z ∈ C such that ω1 (z, x0 ) < ∞, we have ω1 (T n (z), x0 ) = ω1 (T n (z), T n (x0 )) ≤ kn ω1 (z, x0 ), for any n ≥ 1. Hence the orbit {T n (z)} ω -converges to x0 as well. In this remark, we showed how one has to be careful when dealing with modulars. Indeed, a modular

8.2 Banach Contraction Principle in Modular Metric Spaces

225

may take infinite value. This is the problem that the authors of [165] did not pay attention to. Next we discuss the extension of Theorem 8.1 to the case of pointwise contraction mappings in modular metric spaces. Let (X, ω ) be a modular metric space. Let C be a nonempty subset of Xω . Assume that C is ω -complete. Let T : C → C be a pointwise ω -contraction. Let a be a fixed point of T . Then the orbit {T n (x)} ω converges to a, for each x ∈ C such that ω1 (x, a) < +∞. Indeed, we have

ω1 (a, T n (x)) = ω1 (T n (a), T n (x)) ≤ α (a)n ω1 (a, x), for any n ≥ 1. Since α (a) < 1, we conclude that {T n (x)} ω -converges to a. Moreover, if b is another fixed point of T such that ω1 (a, b) < +∞, then we must have a = b. But it is not clear how to prove the existence of any fixed point. In this case, we have to use a different technique than the one used in the proof of Theorem 8.1. Let (X, ω ) be a modular metric space. Assume that ω satisfies the Fatou property. For any x ∈ Xω and r ≥ 0, define the modular ball Bω (x, r) = {y ∈ Xω : ω1 (x, y) ≤ r}. Note that since ω satisfies the Fatou property, then modular balls are ω -closed. An admissible subset of Xω is defined as an intersection of modular balls. Denote by Aω (Xω ) the family of admissible subsets of Xω . Observe that Aω (Xω ) is stable under intersection. For a subset A of a modular metric space Xω , set:  

B : B is a modular ball and A ⊂ B . covω (A) = Note that if A is ω -bounded, i.e., δω (A) = sup{ω1 (x, y) : x, y ∈ A} < +∞, then covω (A) is well defined and covω (A) ∈ Aω (Xω ). Definition 8.7. Let (X, ω ) be a modular metric space. We will say that Aω (Xω ) is compact if any family {Aα }α ∈Γ of elements of Aω (Xω ) has a nonempty intersection Aα = 0/ for any finite subset F ⊂ Γ . provided α ∈F

Remark 8.5. Very often we only need a weaker version of compactness. Indeed, let (X, ω ) be a modular metric space. We will say that Aω (Xω ) satisfies the property (R) if any decreasing sequence of nonempty elements {An }n≥1 in Aω (Xω ) has a nonempty intersection. As for the classical metric spaces, we have the following technical lemma. Lemma 8.1. Let (X, ω ) be a modular metric space. Assume that Aω (Xω ) is compact. Then Xω is ω -complete.

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Proof. Let {xn } be an ω -Cauchy sequence in Xω . Set rn = sup ω1 (xm , xs ), m,s≥n

for any n ≥ 1. Since {xn } is an ω -Cauchy sequence, then lim rn = 0. By defin→∞

nition of {rn }, we get xm ∈ Bω (xn , rn ), for any n ≥ 1 and m ≥ n. Hence for any n1 , n2 , · · · , n p ≥ 1, we have xm ∈



Bω (xni , rni ),

1≤i≤p

for any m ≥ max{n1 , n2 , · · · , n p }. Since Aω (Xω ) is compact, then

Ω=



Bω (xn , rn ) = 0. /

n≥1

If z ∈ Ω , then we have ω1 (xn , z) ≤ rn , for any n ≥ 1. Hence {xn } ω -converges to z, which completes the proof of Lemma 8.1. 

The following result is the modular metric analogue to the main result on pointwise contraction mappings of [125, 127]. Theorem 8.2. Let (X, ω ) be a modular metric space. Let C be a nonempty ω -closed ω -bounded subset of Xω . Assume that the family Aω (C) is compact. Let T : C → C be a pointwise ω -contraction. Then T has a unique fixed point x0 ∈ C. Moreover, the orbit {T n (x)} ω -converges to x0 , for each x ∈ C. Proof. Since Aω (C) is compact, therefore there exists a minimal nonempty K ∈ Aω (C) such that T (K) ⊂ K. It is easy to check that covω (T (K)) = K. Let us prove that δω (K) = sup{ω (x, y) : x, y ∈ K} = 0, i.e., K is reduced to one point. Indeed, since C is ω -bounded, then δω (K) < +∞, i.e. K is also ω -bounded. As T is a pointwise ω -contraction, so there exists a mapping α : C → [0, 1) such that

ω1 (T (x), T (y)) ≤ α (x) ω1 (x, y), for any x, y ∈ C. For any x ∈ K, set rx (K) = sup{ω1 (x, y) : y ∈ K}. We have rx (K) ≤ δω (K), for any x ∈ K. Moreover, it is easy to check that T (K) ⊂ Bω (T (x), α (x)rx (K)), for any x ∈ K. Hence covω (T (K)) ⊂ Bω (T (x), α (x)rx (K)). So rT (x) (K) ≤ α (x) rx (K), for any x ∈ K. Next we fix a ∈ K and define Ka = {x ∈ K : rx (K) ≤ ra (K)}. Clearly, Ka is not empty since a ∈ Ka . Moreover, we have Ka =

 x∈K

Bω (x, ra (K)) ∩ K ∈ Aω (C).

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227

And, since rT (x) (K) ≤ α (x) rx (K), for any x ∈ K, we get T (Ka ) ⊂ Ka . The minimality behavior of K implies Ka = K. In particular, we have rx (K) = ra (K), for any x ∈ K. Hence δω (K)= sup rx (K)=ra (K), for any a ∈ K. Hence δω (K) ≤ α (a) δω (K). x∈K

And since α (a) < 1, we get δω (K) = 0, i.e., K is reduced to one point which is fixed by T . Hence the fixed point set of T is not empty. The remaining conclusion of Theorem 8.2 follows from the general properties of pointwise ω -contractions. 

Next, we extend the previous theorems to the case of asymptotically pointwise contractions in modular metric spaces. Definition 8.8. Let (X, ω ) be a modular metric space. Let C be a nonempty subset of Xω . A mapping T : C → C is called an asymptotic pointwise ω -Lipschitzian mapping if there exists a sequence of mappings αn : C → [0, +∞) such that

ω1 (T n (x), T n (y)) ≤ αn (x) ω1 (x, y), for any x, y ∈ C. (i) if {αn } converges pointwise to α : C → [0, 1), then T is called an asymptotic pointwise ω -contraction ; (ii)if there exists k ∈ [0, 1) such that lim sup αn (x) ≤ k, for any x ∈ C, then T is called n→+∞

a strongly asymptotic pointwise ω -contraction . An extension of Theorem 8.2 to the case of asymptotic pointwise ω -Lipschitzian mappings may be stated as: Theorem 8.3. Let (X, ω ) be a modular metric space. Assume ω is regular. Let C be a nonempty ω -closed ω -bounded subset of Xω . Assume that the family Aω (C) is compact and T : C → C is a strongly asymptotic pointwise ω -contraction. Then T has a unique fixed point z. Moreover any orbit {T n (x)} converges to z, for each x ∈ C. Proof. Let x ∈ C and define the function

Φ (a) = lim sup ω1 (T n (x), a), n→∞

for any a ∈ C. Since Aω (C) is compact, then    Ω (x) = covω {T k (x) : k ≥ n} = 0. / n≥1

For any n, m, h ≥ 1, we have     ω1 T m+n+h (x), T m+h (x) ≤ αh (T m (x)) ω1 T n (x), T m (x) .

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If we let n go to infinity, we get

Φ (T m+h (x)) ≤ αh (T m (x)) Φ (T m (x)). Next, we let h go to infinity to get lim sup Φ (T n (x)) ≤ k Φ (T m (x)), n→∞

for some k ∈ [0, 1), which easily implies that lim sup Φ (T n (x)) = 0. Fix z ∈ Ω (x) n→∞

and notice that

Φ (z) ≤ lim sup Φ (T n (x)). n→∞

Indeed let a ∈ C, then for any ε > 0, there exists n0 ≥ 1 such that for any n ≥ n0 , we have ω1 (T n (x), a) ≤ Φ (a) + ε . In particular, we have T n (x) ∈ Bω (a, Φ (a) + ε ), for any n ≥ n0 . So   Ω (x) ⊂ covω {T n (x); n ≥ n0 } ⊂ Bω (a, Φ (a) + ε ), which implies z ∈ Bω (a, Φ (a) + ε ). This is true for any ε > 0. Hence for any a ∈ C, we have ω1 (z, a) ≤ Φ (a). Therefore,

Φ (z) = lim sup ω1 (T n (x), z) ≤ lim sup Φ (T n (x)). n→∞

n→∞

Therefore, we have Φ (z) = 0, i.e., a fixed point of T . Indeed, we have

{T n (x)}

ω -converges to z. This will force z to be

ω2 (z, T (z)) ≤ ω1 (z, T n (x)) + ω1 (T (z), T n (x)), which implies

ω2 (z, T (z)) ≤ ω1 (z, T n (x)) + α1 (z) ω1 (z, T n−1 (x)), for any n ≥ 1. Since {T n (x)} ω -converges to z, we get ω2 (z, T (z)) = 0, i.e., T (z) = z, since ω is regular. Let c ∈ C be another fixed point of T . Since C is ω -bounded, then we have ω1 (c, z) < +∞. Since

ω1 (c, z) = ω1 (T n (c), T n (z)) ≤ αn (z) ω1 (c, z), for any n ≥ 1. Hence

ω1 (c, z) ≤ lim sup αn (z) ω1 (c, z) ≤ k ω1 (c, z). n→+∞

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229

Since k < 1, we get ω1 (c, z) = 0. Because ω is regular, we get z = c. Therefore T has one fixed point z in C and any orbit of T ω -converges to z. 

Now we will relax the strong behavior of T . Before we state the next result, we need the definition of convex functionals defined on modular metric spaces. Definition 8.9. Let (X, ω ) be a modular metric space. Let C ⊂ Xω be nonempty. A function Φ : C → [0, +∞] is said to be ω -convex if {x ∈ C : Φ (x) ≤ r} ∈ Aω (C), f or any r ∈ [0, +∞). We will say that C is type-stable if for any {xn } ω -bounded sequence in C, the function Φ : C → [0, +∞] defined by

Φ (u) = lim sup ω1 (xn , u), n→∞

is ω -convex. Theorem 8.4. Let (X, ω ) be a modular metric space. Let C be a nonempty ω -closed ω -bounded subset of Xω . Assume that the family Aω (C) is compact. Assume that Aω (C) is type-stable. Let T : C → C be an asymptotic pointwise ω -contraction. Then T has a unique fixed point z ∈ C. Moreover the orbit {T n (x)} converges to z, for each x ∈ C. Proof. As we did in the proof of Theorem 8.3, let x ∈ C and define the function

Φ (a) = lim sup ω1 (T n (x), a), n→∞

for each a ∈ C. Since Aω (C) is compact and type stable, then there exists z ∈ C such that Φ (z) = inf{Φ (a) : a ∈ C}. Let us show that Φ (z) = 0. Indeed, we have

ω1 (T n+m (x), T m (z)) ≤ αm (z) ω1 (T n (x), z), for any n, m ≥ 1. If we let n goes to infinity, we get

Φ (T m (z)) ≤ αm (z) Φ (z), which implies

Φ (z) = inf{Φ (a) : a ∈ M} ≤ Φ (T m (z)) ≤ αm (z) Φ (z). If we let m go to infinity, we get Φ (z) ≤ α (z) Φ (z). Since α (z) < 1, we get Φ (z) = 0, which implies that {T n (x)} ω -converges to z. This will force z to be a fixed point

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of T . The end of the proof of Theorem 8.4 follows the same steps as the end of the proof of Theorem 8.3. 

8.3 Nonexpansive Mappings in Modular Metric Spaces In this section, we study the fixed point theory of mappings which are nonexpansive in the modular sense. The main result one will try to prove in this setting is the Kirk’s fixed point theorem [120]. A key property used in Kirk’s theorem is the normal structure property. The following notations will be used throughout this section. Let (X, ω ) be a modular metric space. Let C be a nonempty subset of Xω . For any A ∈ Aω (C), not reduced to one point, ω -bounded, write Rω (A) = inf rx (A), x ∈A

where rx (A) = sup ω1 (x, y), and δω (A) = sup ω1 (x, y). y ∈A

x,y ∈A

Definition 8.10. Let (X, ω ) be a modular metric space. Let C be a nonempty subset of Xω . We will say that Aω (C) is normal if for any A ∈ Aω (C), not reduced to one point, ω -bounded, we have Rω (A) < δω (A). We will say that Aω (C) is uniformly normal if there exists c ∈ (0, 1) such that for any A ∈ Aω (C), not reduced to one point, ω -bounded, we have Rω (A) ≤ c δω (A). Theorem 8.5. Let (X, ω ) be a modular metric space. Let C be a nonempty ω -closed ω -bounded subset of Xω . Assume that the family Aω (C) is normal and compact. Let T : C → C be ω -nonexpansive. Then T has a fixed point. Proof. Since Aω (C) is compact there exists a minimal nonempty A ∈ Aω (C) such that T (A) ⊂ A. It is easy to check that covω (T (A)) = A. Let us prove that δω (A) = 0, i.e., A is reduced to one point. Suppose that δω (A) = 0. For any x ∈ A, set rx (A) = sup{ω1 (x, y) : y ∈ A} ≤ δω (A) ≤ δω (C) < +∞.   Since T is ω -nonexpansive, we have T (A) ⊂ Bω T (x), rx (A) , for any x ∈ A. Hence   covω (T (A)) ⊂ Bω T (x), rx (A) . So rT (x) (A) ≤ rx (A), for any x ∈ A. Next we fix a ∈ A and define Aa = {x ∈ A : rx (A) ≤ ra (A)}.

8.3 Nonexpansive Mappings in Modular Metric Spaces

231

Clearly Aa is not empty since a ∈ Aa . Moreover, we have Aa =



Bω (x, ra (A)) ∩ A ∈ Aω (C).

x∈A

And since rT (x) (A) ≤ rx (A), for any x ∈ A, we get T (Aa ) ⊂ Aa . The minimality behavior of A implies Aa = A. In particular, we have rx (A) = ra (A) for any x ∈ A. Hence δω (A) = sup rx (A) = ra (A), for any a ∈ A. Since Aω (C) is normal, we get x∈A

δω (A) < δω (A), which is a contradiction. Thus we must have δω (A) = 0, i.e., A is reduced to one point which is fixed by T . 

Next we give a constructive result discovered by Kirk [123] which relaxes the compactness assumption in the above theorem. The main ingredient in Kirk’s constructive proof is a technical lemma due to Gillespie and Williams [77]. Lemma 8.2. Let (X, ω ) be a modular metric space and C be a nonempty ω -bounded subset of Xω . Let T : C → C be a ω -nonexpansive mapping. Assume that Aω (C) is normal. Let A ∈ Aω (C) be nonempty and T -invariant, i.e., T (A) ⊂ A. Then there exists a nonempty A0 ∈ Aω (C), A0 ⊂ A, which is T -invariant such that

δω (A0 ) ≤

δω (A) + Rω (A) . 2

Proof. Set r = 12 (δω (A) + Rω (A)). We assume that δω (A) > 0, otherwise we can take the set A = A0 . Since Aω (C) is normal, we have Rω (A) < δω (A). Hence Rω (A) < r, which implies the existence of a ∈ A such that ra (A) < r. Therefore, the set  Bω (x, r) ∩ A D = {a ∈ A : A ⊂ Bω (a, r)} = x∈A

is a nonempty admissible subset of C. Note that there is no reason for D to be T invariant. Consider the family F = {M ∈ Aω (C) : D ⊂ M and T (M) ⊂ M}. Note that F is nonempty, since A ∈ F . Set L =



M. The set L is an admissible

M∈F

subset of C which contains D. Note that L ⊂ A. Using the definition of F , we deduce that L is T -invariant. Consider B = D ∪ T (L), and observe that covω (B) = L. Indeed, since B ⊂ T (L) ⊂ L and L ∈ Aω (C), we have covω (B) ⊂ L. From this we obtain T (covω (B)) ⊂ T (L) ⊂ B ⊂ covω (B). Hence covω (B) ∈ F , and L ⊂ covω (B). This gives the desired equality. Define A0 = {x ∈ L : L ⊂ Bω (x, r)}.

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We claim that A0 is the desired set. Observe that A0 is nonempty since it contains D (by definition of D). Using the same argument we can prove that A0 is an admissible subset of C. On the other hand, it is clear that δω (A0 ) ≤ r. To complete the proof, we have to show that A0 is T -invariant. Let x ∈ A0 . By definition of A0 , we have L ⊂ Bω (x, r). Since T is ω -nonexpansive, we have T (L) ⊂ Bω (T (x), r). For any y ∈ D, there holds L ⊂ Bω (y, r). But T (x) ∈ L, so T (x) ∈ Bω (y, r), which implies y ∈ Bω (T (x), r). Hence D ⊂ Bω (T (x), r) holds. Since B = D ∪ T (L), we get B ⊂ Bω (T (x), r). Therefore, we must have covω (B) = L ⊂ Bω (T (x), r). By the definition of A0 , it follows that T (x) ∈ A0 . In other words, A0 is T -invariant. Let x, y ∈ A0 , then x, y ∈ L, which implies ω1 (x, y) ≤ rx (L) ≤ r, i.e.,

δω (A0 ) ≤ r =

δω (A) + Rω (D) . 2



Next we give the analogue of the main fixed point result in [123] and of Theorem 5.8 in this book. Theorem 8.6. Let (X, ω ) be an ω -complete modular metric space and C be a nonempty ω -closed ω -bounded subset of Xω . Assume that the family Aω (C) is uniformly normal and T : C → C is ω -nonexpansive. Then T has a fixed point. Proof. Since C is ω -bounded, then we have C ∈ Aω (C) because C = Bω (x, δω (C)) ∩C, f or any x ∈ C. Now, let us take C = A0 and T (C) ⊂ C. Since Aω (C) is uniformly normal, there exists c ∈ (0, 1) such that Rω (A) ≤ c δω (A), f or any A ∈ Aω (C). By Lemma 8.2, there exists A1 ∈ Aω (C) such that A1 ⊂ A0 , T (A1 ) ⊂ A1 and it satisfies R(A0 ) + δω (A0 ) . δω (A1 ) ≤ 2 Using the induction argument, we build a sequence {An } ⊂ Aω (C) such that An+1 ⊂ An , T (An+1 ) ⊂ An+1 and

δω (An+1 ) ≤

R(An ) + δω (An ) . 2

8.3 Nonexpansive Mappings in Modular Metric Spaces

233

Since Rω (An ) ≤ c δω (An ), we get

δω (An+1 ) ≤ which implies that

δω (An ) ≤ Since δω (C) < +∞ , we get

 1+c δω (An ), 2

1+c 2

n

δω (C).

lim δω (An ) = 0.

n→∞

Let xn ∈ An , for any n ≥ 1. Then {xn } is ω -Cauchy sequence. Since Xω is ω complete and C is ω -closed subset of Xω , then C is ω -complete. Thus {xn } ω converges to x ∈ C. Since A n+1 ⊂ An , and An is ω -closed, for any n ≥ 1, then x ∈ An , for any n ≥ 1. Thus A∞ = {An : n ≥ 1} is not empty and clearly is reduced to the single point x. Indeed, let y ∈ A∞ , then y ∈ An , for any n ≥ 1. Hence

ω1 (x, y) ≤ δω (An ). Since lim δω (An ) = 0, we get ω1 (x, y) = 0. Since ω is regular, we get y = x, i.e., n→∞

A∞ = {x}. Since T (An ) ⊂ An , for any n ≥ 1, we get T (x) = x. 

We need the following technical proposition in order to show an analogue to the main result in [105]. Proposition 8.1. Let (X, ω ) be an ω -complete modular metric space and C be a nonempty ω -closed ω -bounded subset of Xω . Assume that the family Aω (C) is uniformly normal. Consider the cartesian product C∞ = ∏ C. Define n≥1

Ω : (0, ∞) ×C ×C → [0, ∞] by

  Ωλ (xn ), (yn ) = sup ωλ (xn , yn ). n≥1

Then (i) (C∞ , Ω ) is an Ω -complete modular metric space. (ii) C∞ is Ω -bounded with δΩ (C∞ ) = δω (C). (iii) For any (xn ) ∈ C∞ and r ∈ [0, ∞], we have   BΩ (xn ), r = ∏ Bω (xn , r), n≥1

where

  

  BΩ (xn ), r = (yn ) ∈ C∞ : Ω1 (xn ), (yn ) ≤ r .

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Then, for any A ∈ AΩ (C∞ ), we have A = ∏ An , where An ∈ AΩ (C). n≥1

(iv) AΩ (C∞ ) is uniformly normal.

Proof. The proofs of (i), (ii), (iii) are easy. Let us prove (iv). Let A ∈ AΩ (C∞ ) be nonempty and not reduced to one point. Then (iii) implies that A = ∏ An , where n≥1

An ∈ Aω (C). Let ε > 0. Since Aω (C) is uniformly normal, there exists c ∈ [0, 1) such that for any n ≥ 1 for which An is not reduced to one point, there exists xn ∈ An such that rxn (An ) ≤ (c + ε ) δω (An ). Hence

    r(xn ) (A) = sup Ω (xn ), (yn ) = sup sup ω1 (xn , yn ) , (yn )∈A

(yn )∈A

n≥1

which implies r(xn ) (A) ≤ (c + ε ) sup δω (An ) = (c + ε ) δΩ (A). n≥1

So RΩ (A) ≤ (c + ε ) δΩ (A). Since ε was arbitrary, we get RΩ (A) ≤ c δΩ (A). This completes the proof of (iv). 

The following theorem shows that although we do not need compactness of the family of admissible sets in Theorem 8.6, its assumptions imply a weaker form of compactness, mainly countable compactness or the property (R). Theorem 8.7. Let (X, ω ) be an ω -complete modular metric space and C be a nonempty ω -closed ω -bounded subset of Xω . Assume that the family Aω (C) is uniformly normal. Then Aω (C) has the property (R). Proof. Let {An } be a decreasing sequence of a nonempty subsets of C, with An ∈ Aω (C). Consider the modular metric space (C∞ , Ω ) defined in Proposition 8.1. Set A = ∏ An . Then A ∈ AΩ (C∞ ). Since AΩ (C∞ ) is uniformly normal, then n

AΩ (A) is uniformly normal. Consider the shift T : A → A defined by T ((xn )) = (xn+1 ). Obviously T is ω -nonexpansive. Theorem 8.6 implies that T has a fixed point, i.e., there exists (xn ) ∈ A such that T ((xn )) = (xn ). The definition of T will force {xn } to be a constant sequence, i.e.,  xn = x, for any n ≥ 1. Obviously, we have x ∈ An , for An = 0. / any n ≥ 1, which implies 

n≥1

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Index

I(T, σ , τ ), 202 M(T, σ ), 199 Δ2 -condition in modular metric spaces, 222 Δ2 -type condition, 56 Δα , 94 Δ χ , 94 Δ2 property, 55 ℜ - class of all nonzero regular convex function modulars, 48 α -uniformly ρ -noncompact convex, 94 χ -uniformly ρ -noncompact convex, 94 δ -ring, 47 M - space of all measurable functions, 48 T (C), 171 Tr (C), 172 ω -Lipschitzian mapping, 222 ω -convex function, 229 ρ -Cauchy sequence, 49 ρ -Chebyshev center of C, 128 ρ -Chebyshev radius of C, 128 ρ -Chebyshev radius of C with respect to f , 128 ρ -Lipschitzian mapping, 185 ρ -Lipschitzian semigroup, 185 ρ -a.e. closed set, 49 ρ -a.e. compact set, 49 ρ -a.e. strong Opial property, 98 ρ -a.e.-Kadec-Klee property, 99 ρ -a.e.-Opial property, 98 ρ -a.e.-uniform Kadec-Klee property, 99 ρ -a.e.-uniform Opial property, 98 ρ -admissible set, 128 ρ -almost everywhere, 48 ρ -approximate fixed point sequence, 172 ρ -bounded set, 49 ρ -closed set, 49 ρ -compact set, 49

ρ -continuous map, 161 ρ -contraction, 111 ρ -contractive semigroup, 186 ρ -convergent sequence, 49 ρ -distance, 49 ρ -hull, 48 ρ -lower semicontinuous function, 57 ρ -modulus of noncompact convexity, 94 ρ -nonexpansive mapping, 112 ρ -nonexpansive retract, 146 ρ -nonexpansive semigroup, 185, 186 ρ -semicontinuous function, 57 ρ -type, 89 ˜ N(C), 128 clρ (conv(A)), 58 coC (D), 128 dρ ( f ,C), 49 (CN) inequality, 40 (SC) condition, 80 (UC1) condition, 80 (UC2) condition, 80 (UUC1) condition, 80 (UUC2) condition, 80 admissible set, 8 admissible set in metrically convex spaces, 36 approximate fixed point property, 16 approximate fixed point sequence, 16 asymptotic center, 16 asymptotic pointwise ω -contraction, 227 asymptotic pointwise ω -Lipschitzian mapping, 227 asymptotic pointwise ρ -contraction mapping, 112 asymptotic pointwise ρ -nonexpansive mapping, 112

c Springer International Publishing Switzerland 2015  M. A. Khamsi, W. M. Kozlowski, Fixed Point Theory in Modular Function Spaces, DOI 10.1007/978-3-319-14051-3

243

244 asymptotic pointwise ρ -nonexpansive semigroup, 211 asymptotic pointwise contraction, 9 asymptotic pointwise mapping, 9, 111 asymptotic pointwise nonexpansive, 9 asymptotic radius, 17 asymptotically ρ -nonexpansive mapping, 112 asymptotically regular mapping, 141 Banach Contraction Principle, 5 bounded away sequence, 82 Brouwer fixed point theorem, 15 Bruhat and Tits inequality, 40 Brønsted partial order, 12 Cantor set, 14 Caristi fixed point theorem, 12 CAT(0) comparison axiom, 40 CAT(0) spaces, 39 characteristic of noncompact convexity, 94 characteristic of uniform convexity, 21 Chebyshev center, 21 Chebyshev radius, 21 coefficient of uniform normal structure property, 22 compact Aρ (C) , 128 comparison triangle, 40 continuous semigroup, 186 contraction mapping, 5 contraction mapping in modular space, 111 convergence locally in measure, 109 convex function in metric space, 9 convexity structure, 43 countably compact Aρ (C) , 128 demiclosed mapping, 23 Demiclosedness Principle in modular function spaces, 173 Egoroff Theorem, 52 Ekeland Variational Principle, 12 Fatou property, 48 fixed point, vii, 5 fixed point theory, 5 fractal sets, 14 generalized contractions, 111 generalized Ishikawa iteration process, 180 generalized Mann iteration process, 175 generalized nonexpansive mappings, 111 geodesic segment, 40 geodesic triangle, 40 growth function ωρ , 56

Index Hadamard space, 40 Hausdorff measure of noncompactness, 92 Hausdorff metric, 14 hyperbolic metric space, 38 hyperconvex metric space, 44 Ishikawa iteration process in modular function spaces, 202 Kirk fixed point theorem, 21 KKM-mapping, 159 Knaster–Kuratowski–Mazurkiewicz mapping, 159 Knaster–Kuratowski–Mazurkiewicz Theorem, 159 Kuratowski measure of noncompactness, 91 Ky Fan Fixed Point Theorem, 162 Lebesgue Dominated Convergence Theorem, 61 Lipschitzian mapping, 5 lower semicontinuity in the sense of convergence ρ -a.e., 102 lower semicontinuous function, l.s.c, 12 Luxemburg norm, 48 Mann process in modular function spaces, 199 metric convex hull, 36 metric convexity, 35 metric segment, 35, 40 minimizing sequence property, 89 modular function space, 48 modular metric, 220 modular spaces (around x0 ), 220 monotone function modular, 47 nondiametral point, 20 nonexpansive mapping, 5 normal Aω (C), 230 normal Aρ (C) , 128 normal structure in metric space, 20 one-local retract, 45 one-local retract in modular function space, 146 Opial property, 25 order continuous, 48 orthogonally additive function modular, 99 orthogonally subadditive function modular, 48 P-uniform convexity, 42 Parallelogram inequality, 41 parallelogram inequality in modular function spaces, 88 Picard iterates, 7 Poincar´e hyperbolic metric, 39

Index pointwise ρ -contraction, 111 pointwise contraction, 7 pointwise contraction mapping, 111 pointwise Lipschitzian mapping, 7 property (R ) in modular function space, 58 property (R) in modular function space, 57 quasi ρ -contraction, 119 quasi-period, 177 quasi-periodic, 177 reflexive metric spaces, 41 regular convex function modular, 48 regular convex function pseudomodular, 47 regular convex function semimodular, 48 regular growth condition, 123 separable function modular, 67 separable set function, 67 strictly convex function modular, 80 strongly ρ -bounded set, 49 strongly ρ -continuous map, 161 strongly asymptotic pointwise ω -contraction, 227 strongly asymptotic pointwise contraction, 9 strongly continuous semigroup, 186

245 T-stable admissible set, 9 type, 9 type-stable set in modular metric space, 229 uniform continuity of a modular, 116 uniform normal structure property, 22 uniform Opial property, 25 uniformly convex, 22 uniformly convex Banach space, 79 uniformly convex metric spaces, 38 uniformly convex modular function space, 79 uniformly Lipschitzian mapping, 42 uniformly non-square, 22 uniformly normal Aω (C), 230 uniformly normal Aρ (C) , 128 unique best approximant property, 84 uniqueness of ρ -limit, 50 very convex Orlicz function, 81 Vitali Convergence Theorem, 60 Vitali property, 63 well defined Ishikawa process, 180 well defined Mann process, 175