Equilibrium Problems and Applications develops a unified variational approach to deal with single-valued, set-valued and quasi-equilibrium problems. The authors promote original results in relationship with classical contributions to the field of equilibrium problems. The content evolved in the general setting of topological vector spaces and it lies at the interplay between pure and applied nonlinear analysis, mathematical economics, and mathematical physics.
This abstract approach is based on tools from various fields, including set-valued analysis, variational and hemivariational inequalities, fixed point theory, and optimization. Applications include models from mathematical economics, Nash equilibrium of non-cooperative games, and Browder variational inclusions. The content is self-contained and the book is mainly addressed to researchers in mathematics, economics and mathematical physics as well as to graduate students in applied nonlinear analysis.
Author(s): Gábor Kassay, Vicențiu Rădulescu
Series: Mathematics in Science and Engineering
Publisher: Academic Press
Year: 2018
Language: English
Pages: 424
Cover......Page 1
Equilibrium Problems
and Applications
......Page 4
Copyright
......Page 5
Dedication
......Page 6
About the Authors......Page 7
Preface......Page 8
Notes......Page 10
Acknowledgments......Page 12
1.1 Elements of Functional Analysis......Page 14
1.1.2 Semicontinuity of Extended Real-Valued Functions......Page 16
1.1.3 Hemicontinuity of Extended Real-Valued Functions......Page 17
1.2.1 The Sperner's Lemma......Page 18
1.2.2 KKM Lemma......Page 19
1.2.3 Brouwer's Fixed Point Theorem......Page 20
1.3.1 Semicontinuity of Set-Valued Mappings......Page 22
1.3.2 Selections of Set-Valued Mappings......Page 24
1.3.3 Elements of Convex Analysis......Page 26
Notes......Page 29
2.1 The Equilibrium Problem and Its Variants......Page 30
2.2.1 The Convex Minimization Problem......Page 32
2.2.4 Nash Equilibrium of Noncooperative Games......Page 33
2.2.5 The Saddle Point/Minimax Problem of Noncooperative Games......Page 34
2.2.6 Variational Inequalities......Page 39
2.2.7 Vector Minimization Problem......Page 40
2.2.8 The Kirszbraun Problem......Page 41
2.3 Equilibria and Inequality Problems With Variational Structure......Page 42
2.3.1 Quasi-Hemivariational Inequalities......Page 43
2.3.2 Browder Variational Inclusions......Page 44
2.3.3 Quasi-Variational Inequalities......Page 45
Notes......Page 46
3.1 KKM Theory and Related Background......Page 47
3.1.1 KKM and Generalized KKM Mappings......Page 48
3.1.2 Intersection Theorems of KKM and Ky Fan Type......Page 49
3.1.3 The Rudiment for Solving Equilibrium Problems. Fixed Point Theorems......Page 51
3.2.1 Finite Dimensional Separation of Convex Sets......Page 53
3.2.2 Results Based on Separation......Page 58
3.2.3 Equivalent Chain of Minimax Theorems Based on Separation Tools......Page 60
Notes......Page 67
4.1 Existence of Solutions of Vector Equilibrium Problems......Page 68
4.1.1 The Strong Vector Equilibrium Problem......Page 69
4.1.2 The Weak Vector Equilibrium Problems......Page 77
4.2.1 The Strong and Weak Set-Valued Equilibrium Problem......Page 82
4.2.2 Concepts of Continuity......Page 84
4.2.3 Strong and Weak Set-Valued Equilibrium Problems: Existence of Solutions......Page 88
4.2.4 Application to Browder Variational Inclusions......Page 92
4.3.1 The Vector-Valued Case......Page 96
4.3.2 The Set-Valued Case......Page 110
4.4 Existence of Solutions of Quasi-Equilibrium Problems......Page 121
4.4.1 A Fixed Point Theory Approach......Page 125
4.4.2 A Selection Theory Approach......Page 130
4.4.3 Approximate Solutions of Quasi-Equilibrium Problems......Page 133
Conclusions......Page 137
5 Well-Posedness for the Equilibrium Problems......Page 139
5.1 Well-Posedness in Optimization and Variational Inequalities......Page 140
5.2.1 Well-Posedness for the Equilibrium Problems Coming From Optimization......Page 142
5.2.2 Well-Posedness for the Equilibrium Problems Coming From Variational Inequalities......Page 146
5.2.3 Relationship Between the Two Kinds of Well-Posedness......Page 148
5.2.4 Hadamard Well-Posedness......Page 150
5.3.1 M-Well-Posedness of Vector Equilibrium Problems......Page 152
5.3.2 B-Well-Posedness of Vector Equilibrium Problems......Page 154
5.3.3 Relationship Between M- and B-Well-Posedness......Page 156
5.3.4 Convexity and Well-Posedness......Page 158
6.1 The Ekeland's Variational Principle for the Equilibrium Problems......Page 161
6.1.1 The Scalar Case......Page 162
6.1.2 The Vector Case......Page 170
6.1.3 The Case of Countable Systems......Page 180
6.2 Metric Regularity, Linear Openness and Aubin Property......Page 188
6.2.1 The Diagonal Subdifferential Operator Associated to an Equilibrium Bifunction......Page 189
6.2.2 Metric Regularity of the Diagonal Subdifferential Operator......Page 190
6.2.3 Metric Subregularity of the Diagonal Subdifferential Operator......Page 194
7 Applications to Sensitivity of Parametric Equilibrium Problems......Page 197
7.1 Preliminaries on Generalized Equations......Page 198
7.2 Hölder Continuity of the Solution Map......Page 200
7.3 Calmness and Aubin Property of the Solution Map......Page 204
7.4 Applications to Sensitivity of Parametric Equilibrium Problems and Variational Systems......Page 205
7.4.1 Pompeiu-Hausdorff Metric and Pseudo-Lipschitz Set-Valued Mappings......Page 207
7.4.2 Fixed Points Sets of Set-Valued Pseudo-Contraction Mappings......Page 211
7.4.3 Inverse of the Sum of Two Set-Valued Mappings......Page 215
7.4.4 Sensitivity Analysis of Variational Inclusions......Page 221
Notes......Page 223
8.1 Introduction......Page 225
8.2 Nash Equilibrium for Perov Contractions......Page 230
8.2.1 Application: Oscillations of Two Pendulums......Page 234
8.3 Nash Equilibrium for Systems of Variational Inequalities......Page 239
8.3.1 Application to Periodic Solutions of Second-Order Systems......Page 245
8.4 Nash Equilibrium of Nonvariational Systems......Page 250
8.4.1 A Localization Critical Point Theorem......Page 251
8.4.2 Localization of Nash-Type Equilibria of Nonvariational Systems......Page 255
8.5.1 Case of a Single Equation......Page 260
8.5.2 Case of a Variational System......Page 263
8.5.3 Case of a Nonvariational System......Page 264
9.1 A Debreu-Gale-Nikaïdo Theorem......Page 270
9.2 Application to Walras Equilibrium Model......Page 272
9.3 Nash Equilibria Within n-Pole Economy......Page 276
9.4 Pareto Optimality for n-Person Games......Page 279
Note......Page 284
10.1 Quasi-Hemivariational Inequalities......Page 285
10.1.1 Some Concepts of Continuity......Page 287
10.1.2 Existence Results for Quasi-Hemivariational Inequalities......Page 290
10.2 Equilibrium Problems Versus Quasi-Hemivariational Inequalities......Page 295
10.3 Browder Variational Inclusions......Page 298
10.3.1 Strong and Weak Set-Valued Equilibrium Problems......Page 299
10.3.2 Browder Variational Inclusions: Existence of Solutions......Page 306
10.3.3 Pseudo-Monotone Case......Page 308
10.4 Fixed Point Theory......Page 324
Notes......Page 329
11 Regularization and Numerical Methods for Equilibrium Problems......Page 330
11.1.1 Bregman Functions and Their Properties......Page 331
11.1.2 The Basic Hypotheses for the Equilibrium Problem......Page 333
11.1.3 A Bregman Regularization for the Equilibrium Problem......Page 334
11.1.4 A Bregman Proximal Method for the Equilibrium Problem......Page 338
11.2 The Tikhonov Regularization Method......Page 341
11.2.1 Auxiliary Results......Page 342
11.3 The Tikhonov Regularization Method for Equilibrium Problems......Page 346
11.3.1 Examples of Suitable Bifunctions......Page 349
11.3.2 Applications to Quasi-Hemivariational Inequalities......Page 353
11.4 A Subgradient Extragradient Method for Solving Equilibrium Problems......Page 358
11.4.1 A Projection Algorithm for Equilibrium Problems......Page 360
11.4.2 Applications to Variational Inequalities......Page 373
11.4.3 Numerical Results......Page 378
Notes......Page 385
Appendix
A Ekeland Variational Principle......Page 386
A.1 Bishop-Phelps Theorem......Page 390
A.2 Case of Szulkin-Type Functionals......Page 391
Note......Page 392
Appendix
B Minimization Problems and Fixed Point Theorems......Page 393
B.1 Caristi and Banach Fixed Point Theorems......Page 398
Appendix
C Nonsmooth Clarke Theory and Generalized Derivatives......Page 400
Appendix
D Elements of Szulkin Critical Point Theory......Page 404
Note......Page 407
Bibliography......Page 408
Index......Page 415
Back Cover......Page 424
