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El. knyga: Fixed Point Theory in Modular Function Spaces

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  • Formatas: PDF+DRM
  • Išleidimo metai: 24-Mar-2015
  • Leidėjas: Birkhauser Verlag AG
  • Kalba: eng
  • ISBN-13: 9783319140513
Kitos knygos pagal šią temą:
Fixed Point Theory in Modular Function SpacesDidesnis paveikslėlis
  • Formatas: PDF+DRM
  • Išleidimo metai: 24-Mar-2015
  • Leidėjas: Birkhauser Verlag AG
  • Kalba: eng
  • ISBN-13: 9783319140513
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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.?

Recenzijos

The book is essentially self-contained and it provides an outstanding resource for those interested in this line of research. (Marķa Angeles Japón Pineda, Mathematical Reviews, March, 2016)

The book is devoted to a comprehensive treatment of what is currently known about the fixed point theory in modular function spaces. the book will be useful for all mathematicians whose interests lie in nonlinear analysis, in particular, in the theory of function spaces and fixed point theory. (Peter P. Zabreko, zbMATH 1318.47002, 2015)

1 Introduction
1(4)
2 Fixed Point Theory in Metric Spaces: An Introduction
5(42)
2.1 Banach Contraction Principle
5(2)
2.2 Pointwise Lipschitzian Mappings
7(5)
2.3 Caristi-Ekeland Extension
12(1)
2.4 Some Applications
13(2)
2.4.1 ODE and Integral Equations
13(1)
2.4.2 Cantor and Fractal sets
14(1)
2.5 Metric Fixed Point Theory in Banach Spaces
15(16)
2.5.1 Classical Existence Results
16(3)
2.5.2 The Normal Structure Property
19(4)
2.5.3 The Demiclosedness Principle
23(2)
2.5.4 Opial and Kadec-Klee Properties
25(6)
2.6 Ishikawa and Mann Iterations in Banach spaces
31(3)
2.7 Metric Convexity and Convexity Structures
34(3)
2.8 Uniformly Convex Metric Spaces
37(6)
2.9 More on Convexity Structures
43(1)
2.10 Common Fixed Points
44(3)
3 Modular Function Spaces
47(32)
3.1 Foundations
47(11)
3.2 Space Ep, Convergence Theorems, and Vitali Property
58(7)
3.3 An Equivalent Topology
65(2)
3.4 Compactness and Separability
67(2)
3.5 Examples
69(2)
3.6 Generalizations and Special Cases
71(8)
3.6.1 General Definition of Function Modular
71(1)
3.6.2 Nonlinear Operator Valued Measures
72(3)
3.6.3 Nonlinear Fourier Transform
75(4)
4 Geometry of Modular Function Spaces
79(32)
4.1 Uniform Convexity in Modular Function Spaces
79(9)
4.2 Parallelogram Inequality and Minimizing Sequence Property
88(3)
4.3 Uniform Noncompact Convexity in Modular Function Spaces
91(7)
4.4 Opial and Kadec--Klee Properties
98(13)
5 Fixed Point Existence Theorems in Modular Function Spaces
111(60)
5.1 Definitions
111(3)
5.2 Contractions in Modular Function Spaces
114(7)
5.2.1 Banach Contraction Principle in Modular Function Spaces
114(2)
5.2.2 Case of Uniformly Continuous Function Modulars
116(2)
5.2.3 Case of Modular Function Spaces with Strong Opial Property
118(1)
5.2.4 Quasi-Contraction Mappings in Modular Function Spaces
118(3)
5.3 Nonexpansive and Pointwise Asymptotic Nonexpansive Mappings
121(42)
5.3.1 Case of Uniformly Convex Function Modulars
121(6)
5.3.2 Normal Structure Property in Modular Function Spaces
127(7)
5.3.3 Case of Uniformly Lipschitzian Mappings
134(12)
5.3.4 Common Fixed Point Theorems in Modular Function Spaces
146(7)
5.3.5 Asymptotic Nonexpansive Mappings in Modular Function Spaces Satisfying Δ2-type Condition
153(6)
5.3.6 KKM and Ky Fan Theorems in Modular Function Spaces
159(4)
5.4 Applications to Differential Equations
163(8)
6 Fixed Point Construction Processes
171(14)
6.1 Preliminaries
171(1)
6.2 Demiclosedness Principle
172(3)
6.3 Generalized Mann Iteration Process
175(5)
6.4 Generalized Ishikawa Iteration Process
180(4)
6.5 Strong Convergence
184(1)
7 Semigroups of Nonlinear Mappings in Modular Function Spaces
185(34)
7.1 Definitions
185(1)
7.2 Fixed Point Existence for Semigroups of Nonexpansive Mappings
186(4)
7.3 Characterization of the Set of Common Fixed Points
190(8)
7.4 Convergence of Mann Iteration Processes
198(4)
7.5 Convergence of Ishikawa Iteration Processes
202(3)
7.6 Applications to Differential Equations
205(6)
7.7 Asymptotic Pointwise Nonexpansive Semigroups
211(8)
8 Modular Metric Spaces
219(24)
8.1 Definitions
219(4)
8.2 Banach Contraction Principle in Modular Metric Spaces
223(7)
8.3 Nonexpansive Mappings in Modular Metric Spaces
230(13)
References
235(8)
Index 243
Mohamed Amine Khamsi, Ph.D. is a Professor in the Department of Mathematical Sciences at the University of Texas at El Paso, Texas, USA.  His research interests include functional analysis, fixed point theory, discrete dynamical systems, and logic programming. Dr. Khamsi received his Ph.D. at the University Paris VI in 1987.

Wojciech M. (Walter) Kozlowski, Ph.D. is a professor in the School of Mathematics and Statistics at the University of New South Wales in Sydney, Australia. His research interests include functional analysis, function spaces, fixed point theory, approximation theory and applications. He received his doctorate at the Jagiellonian University in Krakow in 1981. Dr. Kozlowski, a Fulbright Scholar at the California University of Technology in Pasadena in years 1986 - 1988, works also in a capacity of the business consultant for the telecommunications industry.
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