Presents state-of-the-art advancements in the field of modular function theory
Provides a self-contained overview of the topic
Includes open problems, extensive bibliographic references, and suggestions for further development
This monograph provides a concise introduction to the main results and methods of the fixed point theory in modular function spaces. Modular function spaces are natural generalizations of both function and sequence variants of many important spaces like Lebesgue, Orlicz, Musielak-Orlicz, Lorentz, Orlicz-Lorentz, Calderon-Lozanovskii spaces, and others. In most cases, particularly in applications to integral operators, approximation and fixed point results, modular type conditions are much more natural and 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. The material is presented in a systematic and rigorous manner that allows readers to grasp the key ideas and to gain a working knowledge of the theory. Despite the fact that the work is largely self-contained, extensive bibliographic references are included, and open problems and further development directions are suggested when applicable.
The monograph is targeted mainly at the mathematical research community but it is also accessible to graduate students interested in functional analysis and its applications. It could also serve as a text for an advanced course in fixed point theory of mappings acting in modular function spaces.
Content Level » Research
Keywords » Fixed Point - Iterative Processes - Metric Fixed Point Theory - Modular Function Space - Modular Metric Space - Orlicz Space
Author(s): Mohamed A. Khamsi, Wojciech M. Kozlowski
Publisher: Birkhäuser
Year: 2015
Language: English
Pages: C, X, 245
Cover
S Title
Fixed Point Theory in Modular Function Spaces
Copyright
Springer International Publishing Switzerland 2015
ISBN 978-3-319-14050-6
ISBN 978-3-319-14051-3 (eBook)
DOI 10.1007/978-3-319-14051-3
Foreword
Preface
Contents
1 Introduction
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
3 Modular Function Spaces
3.1 Foundations
3.2 Space E., 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
4 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
5 Fixed Point Existence Theorems in Modular Function Spaces
5.1 Definitions
5.2 Contractions in Modular Function Spaces
5.2.1 Banach Contraction Principle in Modular Function Spaces
5.2.2 Case of Uniformly Continuous Function Modulars
5.2.3 Case of Modular Function Spaces with Strong Opial Property
5.2.4 Quasi-Contraction Mappings in Modular Function Spaces
5.3 Nonexpansive and Pointwise Asymptotic Nonexpansive Mappings
5.3.1 Case of Uniformly Convex Function Modulars
5.3.2 Normal Structure Property in Modular Function Spaces
5.3.3 Case of Uniformly Lipschitzian Mappings
5.3.4 Common Fixed Point Theorems in Modular Function Spaces
5.3.5 Asymptotic Nonexpansive Mappings in Modular Function Spaces Satisfying .2-type Condition
5.3.6 KKM and Ky Fan Theorems in Modular Function Spaces
5.4 Applications to Differential Equations
6 Fixed Point Construction Processes
6.1 Preliminaries
6.2 Demiclosedness Principle
6.3 Generalized Mann Iteration Process
6.4 Generalized Ishikawa Iteration Process
6.5 Strong Convergence
7 Semigroups of Nonlinear Mappings in Modular Function Spaces
7.1 Definitions
7.2 Fixed Point Existence for Semigroups of Nonexpansive Mappings
7.3 Characterization of the Set of Common Fixed Points
7.4 Convergence of Mann Iteration Processes
7.5 Convergence of Ishikawa Iteration Processes
7.6 Applications to Differential Equations
7.7 Asymptotic Pointwise Nonexpansive Semigroups
8 Modular Metric Spaces
8.1 Definitions
8.2 Banach Contraction Principle in Modular Metric Spaces
8.3 Nonexpansive Mappings in Modular Metric Spaces
References
Index
