This is the fourth of a five-volume exposition of the main principles of nonlinear functional analysis and its applications to the natural sciences, economics, and numerical analysis. The presentation is self-contained and accessible to the nonspecialist. Topics covered in this volume include applications to mechanics, elasticity, plasticity, hydrodynamics, thermodynamics, stastical physics, and special and general relativity including cosmology. The book contains a detailed physical motivation of the relevant basic equations and a discussion of particular problems which have played a significant role in the development of physics and through which important mathematical and physical insight may be gained. An attempt is made to combine classical and modern ideas and to build a bridge between the language and thoughts of physicists and mathematicians. Many exercises and a comprehensive bibliography complement the text. This corrected printing contains many revisions as well as a list of new and recent references.
Author(s): Eberhard Zeidler
Publisher: Springer-Verlag Berlin and Heidelberg GmbH & Co. K
Year: 1986
Language: English
Pages: 920
Front cover......Page 1
Juliusz Schauder (1899-1943)......Page 2
Title page......Page 3
Date-line......Page 4
Dedication......Page 5
Preface......Page 7
Contents......Page 15
Introduction......Page 23
FUNDAMENTAL FIXED-POINT PRINCIPLES......Page 31
CHAPTER 1. The Banach Fixed-Point Theorem and Iterative Methods......Page 37
§1.1. The Banach Fixed-Point Theorem......Page 38
§1.2. Continuous Dependence on a Parameter......Page 40
§1.3. The Significance of the Banach Fixed-Point Theorem......Page 41
§1.4. Applications to Nonlinear Equations......Page 44
§1.5. Accelerated Convergence and Newton's Method......Page 47
§1.6. The Picard-Lindeloef Theorem......Page 49
§1.7. The Main Theorem for Iterative Methods for Linear Operator Equations......Page 52
§1.8. Applications to Systems of Linear Equations......Page 57
§1.9. Applications to Linear Integral Equations......Page 58
CHAPTER 2. The Schauder Fixed-Point Theorem and Compactness......Page 70
§2.1. Extension Theorem......Page 71
§2.2. Retracts......Page 72
§2.3. The Brouwer Fixed-Point Theorem......Page 73
§2.4. Existence Principle for Systems of Equations......Page 74
§2.5. Compact Operators......Page 75
§2.6. The Schauder Fixed-Point Theorem......Page 78
§2.7. Peano's Theorem......Page 79
§2.8. Integral Equations with Small Parameters......Page 80
§2.9. Systems of Integral Equations and Semilinear Differential Equations......Page 82
§2.11. Existence Principle for Systems of Inequalities......Page 83
APPLICATIONS OF THE FUNDAMENTAL FIXED-POINT PRINCIPLES......Page 93
CHAPTER 3. Ordinary Differential Equations in B-spaces......Page 95
§3.1. Integration of Vector Functions of One Real Variable $t$......Page 97
§3.2. Differentiation of Vector Functions of One Real Variable $t$......Page 98
§3.3. Generalized Picard-Lindeloef Theorem......Page 100
§3.4. Generalized Peano Theorem......Page 103
§3.5. GronwalPs Lemma......Page 104
§3.6. Stability of Solutions and Existence of Periodic Solutions......Page 106
§3.7. Stability Theory and Plane Vector Fields, Electrical Circuits, Limit Cycles......Page 113
§3.8. Perspectives......Page 121
CHAPTER 4. Differential Calculus and the Implicit Function Theorem......Page 152
§4.1. Formal Differential Calculus......Page 153
§4.2. The Derivatives of Frechet and Gateaux......Page 157
§4.3. Sum Rule, Chain Rule, and Product Rule......Page 160
§4.4. Partial Derivatives......Page 162
§4.5. Higher Differentials and Higher Derivatives......Page 163
§4.6. Generalized Taylor's Theorem......Page 170
§4.7. The Implicit Function Theorem......Page 171
§4.8. Applications of the Implicit Function Theorem......Page 177
§4.9. Attracting and Repelling Fixed Points and Stability......Page 179
§4.10. Applications to Biological Equilibria......Page 184
§4.11. The Continuously Differentiable Dependence of the Solutions of Ordinary Differential Equations in B-spaces on the Initial Values and on the Parameters......Page 187
§4.12. The Generalized Frobenius Theorem and Total Differential Equations......Page 188
§4.13. Diffeomorphisms and the Local Inverse Mapping Theorem......Page 193
§4.14. Proper Maps and the Global Inverse Mapping Theorem......Page 195
§4.15. The Surjective Implicit Function Theorem......Page 198
§4.16. Nonlinear Systems of Equations, Subimmersions, and the Rank Theorem......Page 199
§4.17. A Look at Manifolds......Page 201
§4.18. Submersions and a Look at the Sard-Smale Theorem......Page 205
§4.19. The Parametrized Sard Theorem and Constructive Fixed-Point Theory......Page 210
CHAPTER 5. Newton's Method......Page 228
§5.1. A Theorem on Local Convergence......Page 230
§5.2. The Kantorovic Semi-Local Convergence Theorem......Page 232
CHAPTER 6. Continuation with Respect to a Parameter......Page 248
§6.1. The Continuation Method for Linear Operators......Page 251
§6.2. B-spaces of Hoelder Continuous Functions......Page 252
§6.3. Applications to Linear Partial Differential Equations......Page 255
§6.4. Functional-Analytic Interpretation of the Existence Theorem and its Generalizations......Page 257
§6.5. Applications to Semi-linear Differential Equations......Page 261
§6.6. The Implicit Function Theorem and the Continuation Method......Page 263
§6.7. Ordinary Differential Equations in B-spaces and the Continuation Method......Page 265
§6.8. The Leray-Schauder Principle......Page 267
§6.9. Applications to Quasi-linear Elliptic Differential Equations......Page 268
CHAPTER 7. Positive Operators......Page 291
§7.1. Ordered B-spaces......Page 297
§7.2. Monotone Increasing Operators......Page 299
§7.3. The Abstract Gronwall Lemma and its Applications to Integral Inequalities......Page 303
§7.4. Supersolutions, Subsolutions, Iterative Methods, and Stability......Page 304
§7.5. Applications......Page 307
§7.6. Minorant Methods and Positive Eigensolutions......Page 308
§7.7. Applications......Page 310
§7.8. The Krein-Rutman Theorem and its Applications......Page 311
§7.9. Asymptotic Linear Operators......Page 318
§7.10. Main Theorem for Operators of Monotone Type......Page 320
§7.11. Application to a Heat Conduction Problem......Page 323
§7.12. Existence of Three Solutions......Page 326
§7.13. Main Theorem for Abstract Hammerstein Equations in Ordered B-spaces......Page 329
§7.14. Eigensolutions of Abstract Hammerstein Equations, Bifurcation, Stability, and the Nonlinear Krein-Rutman Theorem......Page 334
§7.15. Applications to Hammerstein Integral Equations......Page 338
§7.16. Applications to Semi-linear Elliptic Boundary-Value Problems......Page 339
§7.17. Application to Elliptic Equations with Nonlinear Boundary Conditions......Page 348
§7.18. Applications to Boundary Initial-Value Problems for Parabolic Differential Equations and Stability......Page 351
CHAPTER 8. Analytic Bifurcation Theory......Page 372
§8.1. A Necessary Condition for Existence of a Bifurcation Point......Page 380
§8.2. Analytic Operators......Page 382
§8.3. An Analytic Majorant Method......Page 385
§8.4. Fredholm Operators......Page 387
§8.5. The Spectrum of Compact Linear Operators (Riesz-Schauder Theory)......Page 394
§8.6. The Branching Equations of Ljapunov-Schmidt......Page 397
§8.7. The Main Theorem on the Generic Bifurcation from Simple Zeros......Page 403
§8.9. Applications to Integral Equations......Page 409
§8.10. Applications to Differential Equations......Page 411
§8.11. The Main Theorem on Generic Bifurcation for Multiparametric Operator Equations—The Bunch Theorem......Page 413
§8.12. Main Theorem for Regular Semi-linear Equations......Page 420
§8.13. Parameter-Induced Oscillation r......Page 423
§8.14. Self-Induced Oscillations and Limit Cycles......Page 430
§8.15. Hopf Bifurcation......Page 433
§8.16. The Main Theorem on Generic Bifurcation from Multiple Zeros......Page 438
§8.17. Stability of Bifurcation Solutions......Page 445
§8.18. Generic Point Bifurcation......Page 450
CHAPTER 9. Fixed Points of Multivalued Maps......Page 469
§9.1. Generalized Banach Fixed-Point Theorem......Page 471
§9.2. Upper and Lower Semi-continuity of Multivalued Maps......Page 472
§9.3. Generalized Schauder Fixed-Point Theorem......Page 474
§9.4. Variational Inequalities and the Browder Fixed-Point Theorem......Page 475
§9.5. An Extremal Principle......Page 478
§9.6. The Minimax Theorem and Saddle Points......Page 479
§9.7. Applications in Game Theory......Page 483
§9.8. Selections and the Marriage Theorem......Page 485
§9.9. Michael's Selection Theorem......Page 488
§9.10. Application to the Generalized Peano Theorem for Differential Inclusions......Page 490
CHAPTER 10. Nonexpansive Operators and Iterative Methods......Page 495
§10.1. Uniformly Convex B-spaces......Page 496
§10.2. Demiclosed Operators......Page 498
§10.3. The Fixed-Point Theorem of Browder, Goehde, and Kirk......Page 500
§10.4. Demicompact Operators......Page 501
§10.5. Convergence Principles in B-spaces......Page 502
§10.6. Modified Successive Approximations......Page 503
§10.7. Application to Periodic Solutions......Page 504
CHAPTER 11. Condensing Maps and the Bourbaki-Kneser Fixed-Point Theorem......Page 510
§11.1. A Noncompactness Measure......Page 514
§11.2. Applications to Generalized Interval Nesting......Page 517
§11.3. Condensing Maps......Page 518
§11.4. Operators with Closed Range and an Approximation Technique for Constructing Fixed Points......Page 519
§11.5. Sadovskii's Fixed-Point Theorem for Condensing Maps......Page 522
§11.6. Fixed-Point Theorems for Perturbed Operators......Page 523
§11.7. Application to Differential Equations in B-spaces......Page 524
§11.8. The Bourbaki-Kneser Fixed-Point Theorem......Page 525
§11.9. The Fixed-Point Theorems of Amann and Tarski......Page 528
§11.10. Application to Interval Arithmetic......Page 530
§11.11. Application to Formal Languages......Page 532
THE MAPPING DEGREE AND THE FIXED-POINT INDEX......Page 539
§12.1. Intuitive Background and Basic Concepts......Page 541
§12.2. Homotopy......Page 549
§12.3. The System of Axioms......Page 551
§12.4. An Approximation Theorem......Page 555
§12.5. Existence and Uniqueness of the Fixed-Point Index in $\mathbb{R}^N$......Page 557
§12.6. Proof of Theorem 12.A......Page 559
§12.7. Existence and Uniqueness of the Fixed-Point Index in B-spaces......Page 564
§12.8. Product Theorem and Reduction Theorem......Page 568
CHAPTER 13. Applications of the Fixed-Point Index......Page 576
§13.1. A General Fixed-Point Principle......Page 577
§13.2. A General Eigenvalue Principle......Page 579
§13.3. Existence of Multiple Solutions......Page 582
§13.4. A Continuum of Fixed Points......Page 586
§13.5. Applications to Differential Equations......Page 588
§13.6. Properties of the Mapping Degree......Page 590
§13.7. The Leray Product Theorem and Homeomorphisms......Page 596
§13.8. The Jordan-Brouwer Separation Theorem and Brouwer's Invariance of Dimension Theorem......Page 602
§13.9. A Brief Glance at the History of Mathematics......Page 604
§13.10. Topology and Intuition......Page 614
§13.11. Generalization of the Mapping Degree......Page 622
CHAPTER 14. The Fixed-Point Index of Differentiable and Analytic Maps......Page 635
§14.1. The Fixed-Point Index of Classical Analytic Functions......Page 638
§14.2. The Leray-Schauder Index Theorem......Page 640
§14.3. The Fixed-Point Index of Analytic Mappings on Complex B-spaces......Page 643
§14.4. The Schauder Fixed-Point Theorem with Uniqueness......Page 646
§14.5. Solution of Analytic Operator Equations......Page 647
§14.6. The Global Continuation Principle of Leray-Schauder......Page 650
§14.7. Unbounded Solution Components......Page 652
§14.9. Applications to Integral Equations......Page 655
§14.11. Applications to Integral Power Series......Page 656
CHAPTER 15. Topological Bifurcation Theory......Page 675
§15.2. Applications to Systems of Equations......Page 679
§15.3. Duality Between the Index Jump Principle and the Leray-Schauder Continuation Principle......Page 680
§15.4. The Geometric Heart of the Continuation Method......Page 683
§15.5. Stability Change and Bifurcation......Page 685
§15.6. Local Bifurcation......Page 687
§15.7. Global Bifurcation......Page 689
§15.8. Application to Systems of Equations......Page 691
§15.9. Application to Integral Equations......Page 692
§15.10. Application to Differential Equations......Page 693
§15.11. Application to Bifurcation at Infinity......Page 695
§15.12. Proof of the Main Theorem......Page 697
§15.13. Preventing Secondary Bifurcation......Page 703
§16.1. Intuitive Introduction......Page 714
§16.2. Essential Mappings and their Homotopy Invariance......Page 719
§16.3. The Antipodal Theorem......Page 722
§16.4. The Invariance of Domain Theorem and Global Homeomorphisms......Page 726
§16.5. The Borsuk-Ulam Theorem and its Applications......Page 730
§16.6. The Mapping Degree and Essential Maps......Page 732
§16.7. The Hopf Theorem......Page 733
§16.8. A Glance at Homotopy Theory......Page 736
CHAPTER 17. Asymptotic Fixed-Point Theorems......Page 745
§17.2. The Fixed-Point Index of Iterated Mappings......Page 746
§17.4. Application to Dissipative Dynamical Systems......Page 747
§17.5. Perspectives......Page 748
Appendix......Page 766
References......Page 830
List of Symbols......Page 873
List of Theorems......Page 881
List of the Most Important Definitions......Page 884
Schematic Overviews......Page 886
General References to the Literature......Page 887
List of Important Principles......Page 888
Contents of the Other Parts......Page 893
Index......Page 899
Back cover......Page 920
