Agarwal / O'Regan / Sahu Fixed Point Theory for Lipschitzian-type Mappings with Applications

1. Auflage 2009
ISBN: 978-0-387-75818-3
Verlag: Springer
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, Band 6, 368 Seiten

Reihe: Topological Fixed Point Theory and Its Applications

In recent years, the fixed point theory of Lipschitzian-type mappings has rapidly grown into an important field of study in both pure and applied mathematics. It has become one of the most essential tools in nonlinear functional analysis. This self-contained book provides the first systematic presentation of Lipschitzian-type mappings in metric and Banach spaces. The first chapter covers some basic properties of metric and Banach spaces. Geometric considerations of underlying spaces play a prominent role in developing and understanding the theory. The next two chapters provide background in terms of convexity, smoothness and geometric coefficients of Banach spaces including duality mappings and metric projection mappings. This is followed by results on existence of fixed points, approximation of fixed points by iterative methods and strong convergence theorems. The final chapter explores several applicable problems arising in related fields. This book can be used as a textbook and as a reference for graduate students, researchers and applied mathematicians working in nonlinear functional analysis, operator theory, approximations by iteration theory, convexity and related geometric topics, and best approximation theory.

Autoren/Hrsg.

Agarwal, Ravi P.

O'Regan, Donal

Sahu, D. R.

Inhaltsverzeichnis

1;Contents;9
2;Preface;7
3;Fundamentals;11
3.1;Topological spaces;11
3.2;Normed spaces;18
3.3;Dense set and separable space;30
3.4;Linear operators;32
3.5;Space of bounded linear operators;35
3.6;Hahn-Banach theorem and applications;38
3.7;Compactness;42
3.8;Reflexivity;44
3.9;Weak topologies;46
3.10;Continuity of mappings;53
4;Convexity, Smoothness, and Duality Mappings;58
4.1;Strict convexity;58
4.2;Uniform convexity;62
4.3;Modulus of convexity;67
4.4;Duality mappings;76
4.5;Convex functions;88
4.6;Smoothness;100
4.7;Modulus of smoothness;103
4.8;Uniform smoothness;107
4.9;Banach limit;115
4.10;Metric projection and retraction mappings;124
5;Geometric Coefficients of Banach Spaces;135
5.1;Asymptotic centers and asymptotic radius;135
5.2;The Opial and uniform Opial conditions;144
5.3;Normal structure;154
5.4;Normal structure coefficient;161
5.5;Weak normal structure coefficient;170
5.6;Maluta constant;173
5.7;GGLD property;180
6;Existence Theorems in Metric Spaces;183
6.1;Contraction mappings and their generalizations;183
6.2;Multivalued mappings;196
6.3;Convexity structure and fixed points;205
6.4;Normal structure coefficient and fixed points;209
6.5;Lifschitz's coefficient and fixed points;214
7;Existence Theorems in Banach Spaces;218
7.1;Non-self contraction mappings;218
7.2;Nonexpansive mappings;229
7.3;Multivalued nonexpansive mappings;244
7.4;Asymptotically nonexpansive mappings;250
7.5;Uniformly L-Lipschitzian mappings;257
7.6;Non-Lipschitzian mappings;266
7.7;Pseudocontractive mappings;271
8;Approximation of Fixed Points;286
8.1;Basic properties and lemmas;286
8.2;Convergence of successive iterates;293
8.3;Mann iteration process;295
8.4;Nonexpansive and quasi-nonexpansive mappings;299
8.5;The modified Mann iteration process;307
8.6;The Ishikawa iteration process;310
8.7;The S-iteration process;314
9;Strong Convergence Theorems;321
9.1;Convergence of approximants of self-mappings;321
9.2;Convergence of approximants of non-self mappings;330
9.3;Convergence of Halpern iteration process;333
10;Applications of Fixed Point Theorems;338
10.1;Attractors of the IFS;338
10.2;Best approximation theory;340
10.3;Solutions of operator equations;341
10.4;Differential and integral equations;344
10.5;Variational inequality;346
10.6;Variational inclusion problem;348
11;Appendix A;354
11.1;Basic inequalities;354
11.2;Partially ordered set;355
11.3;Ultrapowers of Banach spaces;355
12;Bibliography;358
13;Index;370