Topological methods for set-valued nonlinear analysis

Contents......Page 10
Preface......Page 8
1. Introduction......Page 16
2.1 Contraction Mapping Principle in Uniform Topological Spaces and Applications......Page 24
2.2 Banach Contraction Mapping Principle in Uniform Spaces......Page 25
2.2.1 Successive Approximation......Page 29
2.3 Further Generalization of Banach Contraction Mapping Principle......Page 42
2.3.1 Fixed Point Theorems for Some Extension of Contraction Mappings on Uniform Spaces......Page 43
2.3.2 An Interplay Between the Order and Pseudometric Partial Ordering in Complete Uniform Topological Space......Page 47
2.4 Changing Norm......Page 49
2.4.1 Changing the Norm......Page 53
2.4.2 On the Approximate Iteration......Page 58
2.5 The Contraction Mapping Principle Applied to the Cauchy- Kowalevsky Theorem......Page 59
2.5.1 Geometric Preliminaries......Page 60
2.5.2 The Linear Problem......Page 61
2.5.3 The Quasilinear Problem......Page 65
2.6 An Implicit Function Theorem for a Set of Mappings and Its Application to Nonlinear Hyperbolic Boundary Value Problem as Application of Contraction Mapping Principle......Page 68
2.6.1 An Implicit Function Theorem for a Set of Mappings......Page 70
2.6.2 Notations and Preliminaries......Page 75
2.6.3 Results of Smiley on Linear Problem......Page 76
2.6.4 Alternative Problem and Approximate Equations......Page 81
2.6.5 Application to Nonlinear Wave Equations — A Theorem of Paul Rabinowitz......Page 88
2.7 Set-Valued Contractions......Page 98
2.7.1 End Points......Page 103
2.8 Iterated Function Systems (IFS) and Attractor......Page 106
2.8.1 Applications......Page 109
2.9 Large Contractions......Page 118
2.9.1 Large Contractions......Page 119
2.9.2 The Transformation......Page 120
2.9.3 An Existence Theorem......Page 121
2.10 Random Fixed Point and Set-Valued Random Contraction......Page 122
3.2 Fixed Point Theorem on Partially Ordered Sets......Page 128
3.3 Applications to Games and Economics......Page 131
3.3.1 Game......Page 132
3.3.2 Economy......Page 133
3.3.3 Pareto Optimum......Page 134
3.3.4 The Contraction Mapping Principle in Uniform Space via Kleene's Fixed Point Theorem......Page 135
3.3.5 Applications on Double Ranked Sequence......Page 139
3.4 Lattice Theoretical Fixed Point Theorems of Tarski......Page 140
3.5 Applications of Lattice Fixed Point Theorem of Tarski to Integral Equations......Page 146
3.6 The Tarski-Kantorovitch Principle......Page 149
3.7 The Iterated Function Systems on (2X; )......Page 151
3.8 The Iterated Function Systems on (C(X); )......Page 154
3.9 The Iterated Function System on (K(X); )......Page 156
3.10 Continuity of Maps on Countably Compact and Sequential Spaces......Page 157
3.11 Solutions of Impulsive Differential Equations......Page 161
3.11.1 A Comparison Result .......Page 162
3.11.2 Periodic Solutions......Page 164
4.1 Brouwer Fixed Point Theorem......Page 166
4.1.1 Schauder Projection......Page 175
4.1.2 Fixed Point Theorems of Set Valued Mappings with Applications in Abstract Economy......Page 177
4.1.3 Applications......Page 182
4.1.4 Equilibrium Point of Abstract Economy......Page 184
4.2 Fixed Point Theorems and KKM Theorems......Page 186
4.2.1 Duality in Fixed Point Theory of Set Valued Mappings......Page 189
4.3 Applications on Minimax Principles......Page 192
4.3.1 Applications on Sets with Convex Sections......Page 194
4.4 More on Sets with Convex Sections......Page 197
4.5 More on the Extension of KKM Theorem and Ky Fan's Minimax Principle......Page 205
4.6 A Fixed Point Theorem Equivalent to the Fan–Knaster– Kuratowski–Mazurkiewicz Theorem......Page 210
4.7 More on Fixed Point Theorems......Page 215
4.8 Applications of Fixed Point Theorems to Equilibrium Analysis in Mathematical Economics and Game Theory......Page 221
4.8.1 Fixed Point and Equilibrium Point......Page 222
4.8.2 Existence of Maximal Elements......Page 226
4.8.3 Equilibrium Existence Theorems......Page 228
4.9 Fixed Point of  -Condensing Mapping, Maximal Elements and Equilibria......Page 239
4.9.1 Equilibrium on Paracompact Spaces......Page 252
4.9.2 Equilibria of Generalized Games......Page 255
4.9.3 Applications......Page 258
4.10 Coincidence Points and Related Results, an Analysis on H-Spaces......Page 259
4.11 Applications to Mathematical Economics: An Analogue of Debreu's Social Equilibrium Existence Theorem......Page 276
5.1.1 Variational Inequalities for Single Valued Functions......Page 280
5.1.2 Solutions of Simultaneous Nonlinear Variational Inequalities......Page 283
5.1.3 Application to Nonlinear Boundary Value Problem for Quasilinear Operator of Order 2m in Generalized Divergence Form......Page 291
5.1.4 Minimization Problems and Related Results......Page 295
5.1.5 Extension of a Karamardian Theorem......Page 297
5.2 Variational Inequalities for Setvalued Mappings......Page 299
5.2.1 Simultaneous Variational Inequalities......Page 302
5.2.2 Implicit Variational Inequalities — The Monotone Case......Page 307
5.2.3 Implicit Variational Inequalities — The USC Case......Page 311
5.3 Variational Inequalities and Applications......Page 316
5.3.1 Application to Minimization Problems......Page 319
5.4 Duality in Variational Inequalities......Page 321
5.4.1 Some Auxiliary Results......Page 324
5.5 A Variational Inequality in Non-Compact Sets with Some Applications......Page 327
5.6.1 A Minimax Inequality......Page 336
5.6.2 An Existence Theorem of Variational Inequalities......Page 337
5.7.1 Some Generalized Variational Inequalities......Page 340
5.7.2 Applications to Minimization Problems......Page 348
5.8 Some Results of Tarafdar and Yuan on Generalized Variational Inequalities in Locally Convex Topological Vector Spaces......Page 350
5.8.1 Some Generalized Variational Inequalities......Page 352
5.9 Generalized Variational Inequalities for Quasi-Monotone and Quasi- Semi-Monotone Operators......Page 355
5.9.1 Generalization of Ky Fan's Minimax Inequality......Page 361
5.9.2 Generalized Variational Inequalities......Page 363
5.9.3 Fixed Point Theorems......Page 373
5.10 Generalization of Ky Fan's Minimax Inequality with Applications to Generalized Variational Inequalities for Pseudo-Monotone Type I Operators and Fixed Point Theorems......Page 378
5.10.1 Generalization of Ky Fan's Minimax Inequality......Page 380
5.10.2 Generalized Variational Inequalities......Page 387
5.10.3 Applications to Fixed Point Theorems......Page 392
5.11 Generalized Variational-Like Inequalities for Pseudo-Monotone Type I Operators......Page 394
5.11.1 Existence Theorems for GV LI(T;  ; h; X; F)......Page 398
5.12.1 Generalized Quasi-Variational Inequalities for Monotone and Lower Semi-Continuous Mappings......Page 403
5.12.2 Generalized Quasi-Variational Inequalities for Upper Semi- Continuous Mappings Without Monotonicity......Page 408
5.13 Generalized Quasi-Variational Inequalities for Lower and Upper Hemi-Continuous Operators on Non-Compact Sets......Page 412
5.13.1 Generalized Quasi-Variational Inequalities for Lower Hemi- Continuous Operators......Page 413
5.13.2 Generalized Quasi-Variational Inequalities for Upper Hemi- Continuous Operators......Page 419
5.14 Generalized Quasi-Variational Inequalities for Upper Semi- Continuous Operators on Non-Compact Sets......Page 424
5.14.1 Non-Compact Generalized Quasi-Variational Inequalities......Page 425
5.15.1 Generalized Quasi-Variational Inequalities for Strong Pseudo- Monotone Operators......Page 430
5.15.2 Generalized Quasi-Variational Inequalities for Pseudo- Monotone Set-Valued Mappings......Page 436
5.16 Non-Linear Variational Inequalities and the Existence of Equilibrium in Economics with a Riesz Space of Commodities......Page 441
5.16.1 Existence of Equilibrium Lemma......Page 443
5.17.1 Existence of Maximal Elements......Page 445
5.17.2 Existence of Equilibrium for Non-Compact Abstract Economies......Page 449
5.18 Equilibria of Non-Compact Generalized Games......Page 453
5.18.1 Equilibria of Generalized Games......Page 457
5.18.2 Tarafdar and Yuan's Application on Existence Theorem of Equilibria for Constrained Games......Page 460
6. Best Approximation and Fixed Point Theorems for Set-Valued Mappings in Topological Vector Spaces......Page 462
6.1 Single-Valued Case......Page 463
6.2 Set-Valued Case......Page 467
6.2.1 Some Lemmas and Relevant Results......Page 469
7.1 Degree Theory for Set-Valued Ultimately Compact Vector Fields......Page 478
7.1.1 Properties of the Degree of Ultimately Compact Vector Fields......Page 480
7.1.2 k- -Contractive Set Valued Mappings......Page 482
7.2 Coincidence Degree for Non-Linear Single-Valued Perturbations of Linear Fredholm Mappings......Page 486
7.2.1 An Equivalence Theorem......Page 488
7.2.2 Definition of Coincidence Degree......Page 489
7.2.3 Properties of the Coincidence Degree......Page 490
7.3 On the Existence of Solutions of the Equation Lx 2 Nx and a Coincidence Degree Theory......Page 493
7.3.1 Coincidence Degree for Set-Valued k......Page 494
7.4.1 Basic Assumptions and Main Results in Akashi (1988)......Page 512
7.4.2 Akashi's Basic Properties of Coincidence Degree......Page 517
7.4.3 Application to Multitivalued Boundary Value Problem for Elliptic Partial Differential Equation......Page 518
7.5 Applications of Equivalence Theorems with Single-Valued Mappings: An Approach to Non-Linear Elliptic Boundary Value Problems......Page 522
7.5.1 Tarafdar's Application to Elliptic Boundary Value Problems......Page 536
7.6 Further Results in Coincidence Degree Theory......Page 540
7.7 Tarafdar and Thompson's Theory of Bifurcation for the Solutions of Equations Involving Set-Valued Mapping......Page 543
7.7.2 Tarafdar and Thompson's Results on the Theory of Bifurcation......Page 547
7.7.3 Tarafdar and Thompson's Application on the Theory of Bifurcation......Page 554
7.8.1 Measure of Noncompactness and Set Contraction......Page 557
7.8.2 Epi Mappings......Page 561
7.8.3 Tarafdar and Thompson's (p; k)-Epi Mappings on the Whole Space......Page 570
7.8.4 Tarafdar and Thompson's Applications of (p; k)-Epi Mappings in Differential Equations......Page 571
8.1.1 Nonexpansive Mappings......Page 578
8.2 Set-Valued Mappings of Nonexpansive Type......Page 586
8.2.1 Normal Structure and Fixed Point Theorems......Page 587
8.2.2 Another Definition of Nonexpansive Set-Valued Mapping and Corresponding Results on Fixed Point Theorems......Page 590
8.3 Fixed Point Theorems for Condensing Set-Valued Mappings on Locally Convex Topological Vector Spaces......Page 591
8.3.1 Measure of Precompactness and Non-Precompactness......Page 592
8.3.2 Condensing Mappings......Page 593
8.3.3 Fixed Point Theorems......Page 595
Bibliography......Page 598
Index......Page 620