Fixed Point Theory in Ordered Sets and Applications (eBook)

Preface 8
Contents 10
1 Introduction 16
2 Fundamental Order-Theoretic Principles 37
2.1 Recursions and Iterations in Posets 37
2.2 Fixed Point Results in Posets 40
2.2.1 Fixed Points for Set-Valued Functions 40
2.2.2 Fixed Points for Single-Valued Functions 44
2.2.3 Comparison and Existence Results 46
2.2.4 Algorithmic Methods 48
2.3 Solvability of Operator Equations and Inclusions 50
2.3.1 Inclusion Problems 51
2.3.2 Single-Valued Problems 52
2.4 Special Cases 55
2.4.1 Fixed Point Results in Ordered Topological Spaces 55
2.4.2 Equations and Inclusions in Ordered Normed Spaces 58
2.5 Fixed Point Results for Maximalizing Functions 63
2.5.1 Preliminaries 63
2.5.2 Main Results 65
2.5.3 Examples and Remarks 66
2.6 Notes and Comments 69
3 Multi-Valued Variational Inequalities 70
3.1 Introductory Example 70
3.2 Multi-Valued Elliptic Variational Inequalities 75
3.2.1 The Sub-Supersolution Method 79
3.2.2 Directedness of Solution Set 90
3.2.3 Extremal Solutions 98
3.2.4 Equivalence to Variational-Hemivariational Inequality 101
3.3 Multi-Valued Parabolic Variational Inequalities 105
3.3.1 Notion of Sub-Supersolution 111
3.3.2 Multi-Valued Parabolic Equation 114
3.3.3 Parabolic Variational Inequality 130
3.4 Notes and Comments 141
4 Discontinuous Multi-Valued Elliptic Problems 143
4.1 Nonlocal and Discontinuous Elliptic Inclusions 143
4.1.1 Hypotheses, Main Result, and Preliminaries 144
4.1.2 Proof of Theorem 4.1 153
4.1.3 Extremal Solutions 157
4.1.4 Application: Difference of Clarke’s Gradient and Subdifferential 160
4.2 State-Dependent Clarke’s Gradient Inclusion 164
4.2.1 Statement of the Problem 164
4.2.2 Notions, Hypotheses, and Preliminaries 167
4.2.3 Existence and Comparison Result 171
4.2.4 Application: Multiplicity Results 175
4.3 Discontinuous Elliptic Problems via Fixed Points for Multifunctions 178
4.3.1 Abstract Fixed Point Theorems for Multi-Functions 178
4.3.2 Discontinuous Elliptic Functional Equations 180
4.3.3 Implicit Discontinuous Elliptic Functional Equations 184
4.4 Notes and Comments 190
5 Discontinuous Multi-Valued Evolutionary Problems 191
5.1 Discontinuous Parabolic Inclusions with Clarke’s Gradient 191
5.2 Implicit Functional Evolution Equations 196
5.2.1 Preliminaries 197
5.2.2 Main Result 200
5.2.3 Generalization and Special Cases 202
5.2.4 Application 204
5.3 Notes and Comments 206
6 Banach-Valued Ordinary Differential Equations 208
6.1 Cauchy Problems 209
6.1.1 Preliminaries 209
6.1.2 A Uniqueness Theorem of Nagumo Type 209
6.1.3 Existence Results 211
6.1.4 Existence and Uniqueness Results 216
6.1.5 Dependence on the Initial Value 220
6.1.6 Well-Posedness of a Semilinear Cauchy Problem 221
6.2 Nonlocal Semilinear Differential Equations 223
6.2.1 Existence and Comparison Results 224
6.2.2 Applications to Multipoint Initial Value Problems 229
6.3 Higher Order Differential Equations 230
6.3.1 Well-Posedness Results 230
6.3.2 Semilinear Problem 231
6.3.3 Extremal Solutions 233
6.4 Singular Differential Equations 236
6.4.1 First Order Explicit Initial Value Problems 236
6.4.2 First Order Implicit Initial Value Problems 241
6.4.3 Second Order Initial Value Problems 246
6.4.4 Second Order Boundary Value Problems 253
6.5 Functional Differential Equations Containing Bochner Integrable Functions 260
6.5.1 Hypotheses and Preliminaries 261
6.5.2 Existence and Comparison Results 265
6.6 Notes and Comments 270
7 Banach-Valued Integral Equations 271
7.1 Integral Equations in HL-Spaces 272
7.1.1 Fredholm Integral Equations 272
7.1.2 Volterra Integral Equations 278
7.1.3 Application to Impulsive IVP 282
7.1.4 A Volterra Equation Containing HL Integrable Functions 288
7.2 Integral Equations in Lp-Spaces 290
7.2.1 Preliminaries 290
7.2.2 Urysohn Equations 291
7.2.3 Fredholm Integral Equations 294
7.2.4 Volterra Integral Equations 302
7.2.5 Application to Impulsive IVP 306
7.3 Evolution Equations 308
7.3.1 Well-Posedness Results 308
7.3.2 Existence and Uniqueness Result 310
7.3.3 Continuous Dependence on x0 312
7.3.4 Special Cases 314
7.3.5 Application to a Cauchy Problem 315
7.3.6 Extremal Solutions of Evolution Equations 315
7.3.7 Evolution Equations Containing Bochner Integrable Functions 319
7.3.8 Application 323
7.4 Notes and Comments 325
8 Game Theory 326
8.1 Pure Nash Equilibria for Finite Simple Normal-Form Games 328
8.1.1 Preliminaries 328
8.1.2 Existence and Comparison Results 329
8.1.3 An Application to a Pricing Game 332
8.2 Pure and Mixed Nash Equilibria for Finite Normal-Form Games 334
8.2.1 Preliminaries 335
8.2.2 Existence Result for the Greatest Nash Equilibrium 335
8.2.3 Comparison Result for Utilities 338
8.2.4 Dual Results 339
8.2.5 Applications to Finite Supermodular Games 340
8.2.6 Application to a Multiproduct Pricing Game 345
8.3 Pure Nash Equilibria for Normal-Form Games 347
8.3.1 Extreme Value Results 348
8.3.2 Smallest and Greatest Pure Nash Equilibria 352
8.3.3 Special Cases 358
8.3.4 Applications to a Multiproduct Pricing Game 364
8.3.5 Minimal and Maximal Pure Nash Equilibria 368
8.4 Pure and Mixed Nash Equilibria of Normal-Form Games 373
8.4.1 Definitions and Auxiliary Results 374
8.4.2 Existence and Comparison Results 378
8.4.3 Applications to Supermodular Games 382
8.5 Undominated and Weakly Dominating Strategies and Weakly Dominating Pure Nash Equilibria for Normal-Form Games 388
8.5.1 Existence of Undominated Strategies 388
8.5.2 Existence of Weakly Dominating Strategies and Pure Nash Equilibria 391
8.5.3 Examples 393
8.6 Pursuit and Evasion Game 395
8.6.1 Preliminaries 395
8.6.2 Winning Strategy 396
8.6.3 Applications and Special Cases 402
8.7 Notes and Comments 408
9 Appendix 409
9.1 Analysis of Vector-Valued Functions 409
9.1.1 µ-Measurability and µ-Integrability of Banach-ValuedFunctions 409
9.1.2 HL Integrability 413
9.1.3 Integrals of Derivatives of Vector-Valued Functions 420
9.1.4 Convergence Theorems for HL Integrable Functions 424
9.1.5 Ordered Normed Spaces of HL Integrable Functions 427
9.2 Chains in Ordered Function Spaces 429
9.2.1 Chains in Lp-Spaces 429
9.2.2 Chains of Locally Bochner Integrable Functions 432
9.2.3 Chains of HL Integrable and Locally HL Integrable Functions 434
9.2.4 Chains of Continuous Functions 437
9.2.5 Chains of Random Variables 440
9.2.6 Properties of Order Intervals and Balls in Ordered Function Spaces 441
9.3 Sobolev Spaces 444
9.3.1 Definition of Sobolev Spaces 444
9.3.2 Chain Rule and Lattice Structure 446
9.4 Operators of Monotone Type 448
9.4.1 Main Theorem on Pseudomonotone Operators 448
9.4.2 Leray–Lions Operators 450
9.4.3 Multi-Valued Pseudomonotone Operators 451
9.5 First Order Evolution Equations 455
9.5.1 Evolution Triple and Generalized Derivative 455
9.5.2 Existence Results for Evolution Equations 459
9.6 Calculus of Clarke’s Generalized Gradient 460
List of Symbols 466
References 469
Index 480