Fixed Point Theory for Lipschitzian-type Mappings with Applications (eBook)
X, 368 Seiten
Springer New York (Verlag)
978-0-387-75818-3 (ISBN)
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.
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.
Contents 9
Preface 7
Fundamentals 11
Topological spaces 11
Normed spaces 18
Dense set and separable space 30
Linear operators 32
Space of bounded linear operators 35
Hahn-Banach theorem and applications 38
Compactness 42
Reflexivity 44
Weak topologies 46
Continuity of mappings 53
Convexity, Smoothness, and Duality Mappings 58
Strict convexity 58
Uniform convexity 62
Modulus of convexity 67
Duality mappings 76
Convex functions 88
Smoothness 100
Modulus of smoothness 103
Uniform smoothness 107
Banach limit 115
Metric projection and retraction mappings 124
Geometric Coefficients of Banach Spaces 135
Asymptotic centers and asymptotic radius 135
The Opial and uniform Opial conditions 144
Normal structure 154
Normal structure coefficient 161
Weak normal structure coefficient 170
Maluta constant 173
GGLD property 180
Existence Theorems in Metric Spaces 183
Contraction mappings and their generalizations 183
Multivalued mappings 196
Convexity structure and fixed points 205
Normal structure coefficient and fixed points 209
Lifschitz's coefficient and fixed points 214
Existence Theorems in Banach Spaces 218
Non-self contraction mappings 218
Nonexpansive mappings 229
Multivalued nonexpansive mappings 244
Asymptotically nonexpansive mappings 250
Uniformly L-Lipschitzian mappings 257
Non-Lipschitzian mappings 266
Pseudocontractive mappings 271
Approximation of Fixed Points 286
Basic properties and lemmas 286
Convergence of successive iterates 293
Mann iteration process 295
Nonexpansive and quasi-nonexpansive mappings 299
The modified Mann iteration process 307
The Ishikawa iteration process 310
The S-iteration process 314
Strong Convergence Theorems 321
Convergence of approximants of self-mappings 321
Convergence of approximants of non-self mappings 330
Convergence of Halpern iteration process 333
Applications of Fixed Point Theorems 338
Attractors of the IFS 338
Best approximation theory 340
Solutions of operator equations 341
Differential and integral equations 344
Variational inequality 346
Variational inclusion problem 348
Appendix A 354
Basic inequalities 354
Partially ordered set 355
Ultrapowers of Banach spaces 355
Bibliography 358
Index 370
| Erscheint lt. Verlag | 12.6.2009 |
|---|---|
| Reihe/Serie | Topological Fixed Point Theory and Its Applications | Topological Fixed Point Theory and Its Applications |
| Zusatzinfo | X, 368 p. |
| Verlagsort | New York |
| Sprache | englisch |
| Themenwelt | Mathematik / Informatik ► Mathematik ► Analysis |
| Technik | |
| Schlagworte | Banach spaces • convergence theory • Convexity • Fixed Point Theory • Fixed-Point Theory • Functional Analysis • iterative processes • operator theory • problem of existence • Smooth function |
| ISBN-10 | 0-387-75818-6 / 0387758186 |
| ISBN-13 | 978-0-387-75818-3 / 9780387758183 |
| Haben Sie eine Frage zum Produkt? |
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