Nonlinear functional analysis: Applications to mathematical physics

Contents

Preface vii

Introduction 1


FUNDAMENTAL FIXED-POINT PRINCIPLES

CHAPTER 1
The Banach Fixed-Point Theorem and Iterative Methods 15
1.1. The Banach Fixed-Point Theorem 16
1.2. Continuous Dependence on a Parameter 18
1.3. The Significance of the Banach Fixed-Point Theorem 19
1.4. Applications to Nonlinear Equations 22
1.5. Accelerated Convergence and Newton\'s Method 25
1.6. The Picard-Lindelöf Theorem 27
1.7. The Main Theorem for Iterative Methods for Linear Operator Equations 30
1.8. Applications to Systems of Linear Equations 35
1.9. Applications to Linear Integral Equations 36

CHAPTER 2
The Schauder Fixed-Point Theorem and Compactness 48
2.1. Extension Theorem 49
2.2. Retracts 50
2.3. The Brouwer Fixed-Point Theorem 51
2.4. Existence Principle for Systems of Equations 52
2.5. Compact Operators 53
2.6. The Schauder Fixed-Point Theorem 56
2.7. Peano\'s Theorem 57
2.8. Integral Equations with Small Parameters 58
2.9. Systems of Integral Equations and Semilinear Differential Equations 60
2.1 o. A General Strategy 61
2.11. Existence Principle for Systems of Inequalities 61


APPLICATIONS OF THE FUNDAMENTAL FIXED-POINT PRINCIPLES

CHAPTER 3
Ordinary Differential Equations in ${\\bf\\rm B}$-spaces 73
3.1. Integration of Vector Functions of One Real Variable $t$ 75
3.2. Differentiation of Vector Functions of One Real Variable $t$ 76
3.3. Generalized Picard- Lindelof Theorem 78
3.4. Generalized Peano Theorem 81
3.5. Gronwall\'s Lemma 82
3.6. Stability of Solutions and Existence of Periodic Solutions 84
3.7. Stability Theory and Plane Vector Fields, Electrical Circuits, Limit Cycles 91
3.8. Perspectives 99

CHAPTER 4
Differential Calculus and the Implicit Function Theorem 130
4.1. Formal Differential Calculus 131
4.2. The Derivatives of Frechet and Gâteaux 135
4.3. Sum Rule, Chain Rule, and Product Rule 138
4.4. Partial Derivatives 140
4.5. Higher Differentials and Higher Derivatives 141
4.6. Generalized Taylor\'s Theorem 148
4.7. The Implicit Function Theorem 149
4.8. Applications of the Implicit Function Theorem 155
4.9. Attracting and Repelling Fixed Points and Stability 157
4.10. Applications to Biological Equilibria 162
4.11. The Continuously Differentiable Dependence of the Solutions of Ordinary Differential Equations in ${\\bf\\rm B}$-spaces on the Initial Values and on the Parameters 165
4.12. The Generalized Frobenius Theorem and Total Differential Equations 166
4.13. Diffeomorphisms and the Local Inverse Mapping Theorem 171
4.14. Proper Maps and the Global Inverse Mapping Theorem 173
4.15. The Surjective Implicit Function Theorem 176
4.16. Nonlinear Systems of Equations, Subimmersions, and the Rank Theorem 177
4.17. A Look at Manifolds 179
4.18. Submersions and a Look at the Sard-Smale Theorem 183
4.19. The Parametrized Sard Theorem and Constructive Fixed-Point Theory 188

CHAPTER 5
Newton\'s Method 206
5.1. A Theorem on Local Convergence 208
5.2. The Kantorovic Semi-Local Convergence Theorem 210

CHAPTER 6
Continuation with Respect to a Parameter 226
6.1. The Continuation Method for Linear Operators 229
6.2. B-spaces of Holder Continuous Functions 230
6.3. Applications to Linear Partial Differential Equations 233
6.4. Functional-Analytic Interpretation of the Existence Theorem and its Generalizations 235
6.5. Applications to Semi-linear Differential Equations 239
6.6. The Implicit Function Theorem and the Continuation Method 241
6.7. Ordinary Differential Equations in ${\\bf\\rm B}$-spaces and the Continuation Method 243
6.8. The Leray-Schauder Principle 245
6.9. Applications to Quasi-linear Elliptic Differential Equations 246

CHAPTER 7
Positive Operators 269
7.1. Ordered B-spaces 275
7.2. Monotone Increasing Operators 277
7.3. The Abstract Gronwall Lemma and its Applications to Integral Inequalities 281
7.4. Supersolutions, Subsolutions, Iterative Methods, and Stability 282
7.5. Applications 285
7.6. Minorant Methods and Positive Eigensolutions 286
7.7. Applications 288
7.8. The Krein-Rutman Theorem and its Applications 289
7.9. Asymptotic Linear Operators 296
7.10. Main Theorem for Operators of Monotone Type 298
7.11. Application to a Heat Conduction Problem 301
7.12. Existence of Three Solutions 304
7.13. Main Theorem for Abstract Hammerstein Equations in Ordered B-spaces 307
7.14. Eigensolutions of Abstract Hammerstein Equations, Bifurcation, Stability, and the Nonlinear Krein-Rutman Theorem 312
7.15. Applications to Hammerstein Integral Equations 316
7.16. Applications to Semi-linear Elliptic Boundary-Value Problems 317
7.17. Application to Elliptic Equations with Nonlinear Boundary Conditions 326
7.18. Applications to Boundary Initial-Value Problems for Parabolic Differential Equations and Stability 329

CHAPTER 8
Analytic Bifurcation Theory 350
8.1. A Necessary Condition for Existence of a Bifurcation Point 358
8.2. Analytic Operators 360
8.3. An Analytic Majorant Method 363
8.4. Fredholm Operators 365
8.5. The Spectrum of Compact Linear Operators (Riesz-Schauder Theorem) 372
8.6. The Branching Equations of Ljapunov-Schmidt 375
8.7. The Main Theorem on the Generic Bifurcation from Simple Zeros 381
8.8. Applications to Eigenvalue Problems 387
8.9. Applications to Integral Equations 387
8.10. Applications to Differential Equations 389
8.11. The Main Theorem on Generic Bifurcation for Multiparametric Operator Equations-The Bunch Theorem 391
8.12. Main Theorem for Regular Semi-linear Equations 398
8.13. Parameter-Induced Oscillation 401
8.14. Self-Induced Oscillations and Limit Cycles 408
8.15. Hopf Bifurcation 411
8.16. The Main Theorem on Generic Bifurcation from Multiple Zeros 416
8.17. Stability of Bifurcation Solutions 423
8.18. Generic Point Bifurcation 428

CHAPTER 9
Fixed Points of Multi-valued Maps 447
9.1. Generalized Banach Fixed-Point Theorem 449
9.2. Upper and Lower Semi-continuity of Multivalued Maps 450
9.3. Generalized Schauder Fixed-Point Theorem 452
9.4. Variational Inequalities and the Browder Fixed-Point Theorem 453
9.5. An Extremal Principle 456
9.6. The Minimax Theorem and Saddle Points 457
9.7. Applications in Game Theory 461
9.8. Selections and the Marriage Theorem 463
9.9. Michael\'s Selection Theorem 466
9.10. Application to the Generalized Peano Theorem for Differential Inclusions 468

CHAPTER 10
Nonexpansive Operators and Iterative Methods 473
10.1. Uniformly Convex B-spaces 474
10.2. Demiclosed Operators 476
10.3. The Fixed-Point Theorem of Browder, Gohde, and Kirk 478
10.4. Demicompact Operators 479
10.5. Convergence Principles in ${\\bf\\rm B}$-spaces 480
10.6. Modified Successive Approximations 481
10.7. Application to Periodic Solutions 482

CHAPTER 11
Condensing Maps and the Bourbaki-Kneser Fixed-Point Theorem 488
11.1. A N oncompactness Measure 492
11.2. Applications to Generalized Interval Nesting 495
11.3. Condensing Maps 496
11.4. Operators with Closed Range and an Approximation Technique for Constructing Fixed Points 497
11.5. Sadovskii\'s Fixed-Point Theorem for Condensing Maps 500
11.6. Fixed-Point Theorems for Perturbed Operators 501
11.7. Application to Differential Equations in ${\\bf\\rm B}$-spaces 502
11.8. The Bourbaki-Kneser Fixed-Point Theorem 503
11.9. The Fixed-Point Theorems of Amann and Tarski 506
11.10. Application to Interval Arithmetic 508
11.11. Application to Formal Languages 510


THE MAPPING DEGREE AND THE FIXED-POINT INDEX

CHAPTER 12
The Leray-Schauder Fixed-Point Index 519
12.1. Intuitive Background and Basic Concepts 519
12.2. Homotopy 527
12.3. The System of Axioms 529
12.4. An Approximation Theorem 533
12.5. Existence and Uniqueness of the Fixed-Point Index in ${\\mathbb R}^N$ 535
12.6. Proof of Theorem 12.A. 537
12.7. Existence and Uniqueness of the Fixed-Point Index in ${\\bf\\rm B}$-spaces 542
12.8. Product Theorem and Reduction Theorem 546

CHAPTER 13
Applications of the Fixed-Point Index 554
13.1. A General Fixed-Point Principle 555
13.2. A General Eigenvalue Principle 557
13.3. Existence of Multiple Solutions 560
13.4. A Continuum of Fixed Points 564
13.5. Applications to Differential Equations 566
13.6. Properties of the Mapping Degree 568
13.7. The Leray Product Theorem and Homeomorphisms 574
13.8. The Jordan-Brouwer Separation Theorem and Brouwer\'s Invariance of Dimension Theorem 580
13.9. A Brief Glance at the History of Mathenlatics 582
13.10. Topology and Intuition 592
13.11. Generalization of the Mapping Degree 600

CHAPTER 14
The Fixed-Point Index of Differentiable and Analytic Maps 613
14.1. The Fixed-Point Index of Classical Analytic Functions 616
14.2. The Leray-Schauder Index Theorem 618
14.3. The Fixed-Point Index of Analytic Mappings on Complex ${\\bf\\rm B}$-spaces 621
14.4. The Schauder Fixed-Point Theorem with Uniqueness 624
14.5. Solution of Analytic Operator Equations 625
14.6. The Global Continuation Principle of Leray-Schauder 628
14.7. Unbounded Solution Components 630
14.8. Applications to Systems of Equations 633
14.9. Applications to Integral Equations 633
14.10. Applications to Boundary-Value Problems 634
14.11. Applications to Integral Power Series 634

CHAPTER 15
Topological Bifurcation Theory 653
15.1. The Index Jump Principle 657
15.2. Applications to Systems of Equations 657
15.3. Duality Between the Index Jump Principle and the Leray-Schauder Continuation Principle 658
15.4. The Geometric Heart of the Continuation Method 661
15.5. Stability Change and Bifurcation 663
15.6. Local Bifurcation 665
15.7. Global Bifurcation 667
15.8. Application to Systems of Equations 669
15.9. Application to Integral Equations 670
15.10. Application to Differential Equations 671
15.11. Application to Bifurcation at Infinity 673
15.12. Proof of the Main Theorem 675
15.13. Preventing Secondary Bifurcation 681

CHAPTER 16
Essential Mappings and the Borsuk Antipodal Theorem 692
16.1. Intuitive Introduction 692
16.2. Essential Mappings and their Homotopy Invariance 697
16.3. The Antipodal Theorem 700
16.4. The Invariance of Domain Theorem and Global Homeomorphisms 704
16.5. The Borsuk-Ulam Theorem and its Applications 708
16.6. The Mapping Degree and Essential Maps 710
16.7. The Hopf Theorem 711
16.8. A Glance at Homotopy Theory 714

CHAPTER 17
Asymptotic Fixed-Point Theorems 723
17.1. The Generalized Banach Fixed-Point Theorem 724
17.2. The Fixed-Point Index of Iterated Mappings 724
17.3. The Generalized Schauder Fixed-Point Theorem 725
17.4. Application to Dissipative Dynamical Systems 725
17.5. Perspectives 726

Appendix 744

References 808

List of Symbols 851

List of Theorems 859

List of the Most Important Definitions 862

Schematic Overviews 864

General References to the Literature 865

List of Important Principles 866

Contents of the Other Parts 871

Index 877