Handbook of Applied Analysis (eBook)

Series Preface 8
Preface 10
Contents 14
1 Smooth and Nonsmooth Calculus 17
Introduction 17
1.1 Gˆateaux and Fr´echet Derivatives 18
1.2 Convex Functions 28
1.3 Locally Lipschitz Functions 43
1.4 Variational Geometry 51
1.5 G-Convergence 61
1.6 Remarks 76
2 Extremal Problems and Optimal Control 78
Introduction 78
2.1 Lower Semicontinuity 79
2.2 Constrained Minimization Problems 87
2.3 Saddle Points and Duality 94
2.4 Variational Principles 104
2.5 Calculus of Variations 113
2.6 Optimal Control 126
2.7 Remarks 159
3 Nonlinear Operators and Fixed Points 161
Introduction 161
3.1 Compact and Fredholm Operators 162
3.2 Monotone and Accretive Operators 176
3.3 Degree Theory 210
3.4 Metric Fixed Points 238
3.5 Topological Fixed Points 251
3.6 Order and Fixed Points 259
3.7 Remarks 277
4 Critical Point Theory and Variational Methods 280
Introduction 280
4.1 Critical Point Theory 281
4.2 Ljusternik–Schnirelmann Theory 302
4.3 Spectrum of the Laplacian and of the p-Laplacian 310
4.4 Abstract Eigenvalue Problems 347
4.5 Bifurcation Theory 353
4.6 Remarks 362
5 Boundary Value Problems–Hamiltonian Systems 364
Introduction 364
5.1 Variational Method 365
5.2 Method of Upper–Lower Solutions 393
5.3 Degree-Theoretic Methods 417
5.4 Nonlinear Eigenvalue Problems 434
5.5 Maximum and Comparison Principles 445
5.6 Periodic Solutions for Hamiltonian Systems 453
5.7 Remarks 463
6 Multivalued Analysis 468
Introduction 468
6.1 Continuity of Multifunctions 469
6.2 Measurability of Multifunctions 483
6.3 Continuous and Measurable Selectors 488
6.4 Decomposable Sets and Set-Valued Integration 500
6.5 Fixed Points and Caratheodory Multifunctions 517
6.6 Convergence of Sets 527
6.7 Remarks 537
7 Economic Equilibrium and Optimal Economic Planning 541
Introduction 541
7.1 Perfectly Competitive Economies: Core and Walras Equilibria 542
7.2 Infinite Horizon Multisector Growth Models 554
7.3 Turnpike Theorems 570
7.4 Stochastic Growth Models 579
7.5 Continuous-Time Growth Models 601
7.6 Expected Utility Hypothesis 611
7.7 Remarks 618
8 Game Theory 621
Introduction 621
8.1 Noncooperative Games–Nash Equilibrium 622
8.2 Cooperative Games 629
8.3 Cournot–Nash Equilibria for Random Games 636
8.4 Bayesian Games 640
8.5 Stochastic Games 644
8.6 Approximate Equilibria 656
8.7 Remarks 661
9 Uncertainty, Information, Decision Making 663
Introduction 663
9.1 Mathematical Space of Information 664
9.2 The ex-Post View 674
9.3 The ex-ante View 680
9.4 Convergence of s-Fields and Prediction Sequences 685
9.5 Games with Incomplete Information 692
9.6 Markov Decision Chains with Unbounded Costs 696
9.7 Remarks 701
10 Evolution Equations 703
Introduction 703
10.1 Lebesgue–Bochner Spaces 704
10.2 Semilinear Evolution Equations 713
10.3 Nonlinear Evolution Equations 730
10.4 Second-Order Nonlinear Evolution Equations 752
10.5 Remarks 762
References 764
List of Symbols 792
Index 794