Fixed Point Theory for Lipschitzian-Type Mappings with Applications Book

Preface vii

1 Fundamentals 1

1.1 Topological spaces 1

1.2 Normed spaces 8

1.3 Dense set and separable space 20

1.4 Linear operators 22

1.5 Space of bounded linear operators 25

1.6 Hahn-Banach theorem and applications 28

1.7 Compactness 32

1.8 Reflexivity 34

1.9 Weak topologies 36

1.10 Continuity of mappings 43

2 Convexity, Smoothness, and Duality Mappings 49

2.1 Strict convexity 49

2.2 Uniform convexity 53

2.3 Modulus of convexity 58

2.4 Duality mappings 67

2.5 Convex functions 79

2.6 Smoothness 91

2.7 Modulus of smoothness 94

2.8 Uniform smoothness 98

2.9 Banach limit 106

2.10 Metric projection and retraction mappings 115

3 Geometric Coefficients of Banach Spaces 127

3.1 Asymptotic centers and asymptotic radius 127

3.2 The Opial and uniform Opial conditions 136

3.3 Normal structure 146

3.4 Normal structure coefficient 153

3.5 Weak normal structure coefficient 162

3.6 Maluta constant 165

3.7 GGLD property 172

4 Existence Theorems in Metric Spaces 175

4.1 Contraction mappings and their generalizations 175

4.2 Multivalued mappings 188

4.3 Convexity structure and fixed points 197

4.4 Normal structure coefficient and fixed points 201

4.5 Lifschitz's coefficient and fixed points 206

5 Existence Theorems in Banach Spaces 211

5.1 Non-self contraction mappings 211

5.2 Nonexpansive mappings 222

5.3 Multivalued nonexpansive mappings 237

5.4 Asymptotically nonexpansive mappings 243

5.5 Uniformly L-Lipschitzian mappings 250

5.6 Non-Lipschitzian mappings 259

5.7 Pseudocontractive mappings 264

6 Approximation of Fixed Points 279

6.1 Basic properties and lemmas 279

6.2 Convergence of successive iterates286

6.3 Mann iteration process 288

6.4 Nonexpansive and quasi-nonexpansive mappings 292

6.5 The modified Mann iteration process 300

6.6 The Ishikawa iteration process 303

6.7 The S-iteration process 307

7 Strong Convergence Theorems 315

7.1 Convergence of approximants of self-mappings 315

7.2 Convergence of approximants of non-self mappings 324

7.3 Convergence of Halpern iteration process 327

8 Applications of Fixed Point Theorems 333

8.1 Attractors of the IFS 333

8.2 Best approximation theory 335

8.3 Solutions of operator equations 336

8.4 Differential and integral equations 339

8.5 Variational inequality 341

8.6 Variational inclusion problem 343

Appendix A 349

A.1 Basic inequalities 349

A.2 Partially ordered set 350

A.3 Ultrapowers of Banach spaces 350

Bibliography 353

Index 365