Nonlinear Functional Analysis and Its Applications

Fundamental Fixed-Point Principles.- 1 The Banach Fixed-Point Theorem and Iterative Methods.- x1.1. The Banach Fixed-Point Theorem.- x1.2. Continuous Dependence on a Parameter.- x1.3. The Significance of the Banach Fixed-Point Theorem.- x1.4. Applications to Nonlinear Equations.- x1.5. Accelerated Convergence and Newton's Method.- x 1.6. The Picard-Lindelof Theorem.- x1.7. The Main Theorem for Iterative Methods for Linear Operator Equations.- x1.8. Applications to Systems of Linear Equations.- x1.9. Applications to Linear Integral Equations.- 2 The Schauder Fixed-Point Theorem and Compactness.- x2.1. Extension Theorem.- x2.2. Retracts.- x2.3. The Brouwer Fixed-Point Theorem.- x2.4. Existence Principle for Systems of Equations.- x2.5. Compact Operators.- x2.6. The Schauder Fixed-Point Theorem.- x2.7. Peano's Theorem.- x2.8. Integral Equations with Small Parameters.- x2.9. Systems of Integral Equations and Semilinear Differential Equations.- x2.10. A General Strategy.- x2.11. Existence Principle for Systems of Inequalities.- Applications of the Fundamental Fixed-Point Principles.- 3 Ordinary Differential Equations in B-spaces.- x3.1. Integration of Vector Functions of One Real Variable t.- x3.2. Differentiation of Vector Functions of One Real Variable t.- x3.3. Generalized Picard-Lindelof Theorem.- x3.4. Generalized Peano Theorem.- x3.5. Gronwall's Lemma.- x3.6. Stability of Solutions and Existence of Periodic Solutions.- x3.7. Stability Theory and Plane Vector Fields, Electrical Circuits, Limit Cycles.- x3.8. Perspectives.- 4 Differential Calculus and the Implicit Function Theorem.- x4.1. Formal Differential Calculus.- x4.2. The Derivatives of Frechet and Gateaux.- x4.3. Sum Rule, Chain Rule, and Product Rule.- x4.4. Partial Derivatives.- x4.5. Higher Differentials and Higher Derivatives.- x4.6. Generalized Taylor's Theorem.- x4.7. The Implicit Function Theorem.- x4.8. Applications of the Implicit Function Theorem.- x4.9. Attracting and Repelling Fixed Points and Stability.- x4.10. Applications to Biological Equilibria.- x4.11. The Continuously Differentiable Dependence of the Solutions of Ordinary Differential Equations in B-spaces on the Initial Values and on the Parameters.- x4.12. The Generalized Frobenius Theorem and Total Differential Equations.- x4.13. Diffeomorphisms and the Local Inverse Mapping Theorem.- x4.14. Proper Maps and the Global Inverse Mapping Theorem.- x4.15. The Suijective Implicit Function Theorem.- x4.16. Nonlinear Systems of Equations, Subimmersions, and the Rank Theorem.- x4.17. A Look at Manifolds.- x4.18. Submersions and a Look at the Sard-Smale Theorem.- x4.19. The Parametrized Sard Theorem and Constructive Fixed-Point Theory.- 5 Newton's Method.- x5.1. A Theorem on Local Convergence.- x5.2. The Kantorovi? Semi-Local Convergence Theorem.- 6 Continuation with Respect to a Parameter.- x6.1. The Continuation Method for Linear Operators.- x6.2. B-spaces of Holder Continuous Functions.- x6.3. Applications to Linear Partial Differential Equations.- x6.4. Functional-Analytic Interpretation of the Existence Theorem and its Generalizations.- x6.5. Applications to Semi-linear Differential Equations.- x6.6. The Implicit Function Theorem and the Continuation Method.- x6.7. Ordinary Differential Equations in B-spaces and the Continuation Method.- x6.8. The Leray-Schauder Principle.- x6.9. Applications to Quasi-linear Elliptic Differential Equations.- 7 Positive Operators.- x7.1. Ordered B-spaces.- x7.2. Monotone Increasing Operators.- x7.3. The Abstract Gronwall Lemma and its Applications to Integral Inequalities.- x7.4. Supersolutions, Subsolutions, Iterative Methods, and Stability.- x7.5. Applications.- x7.6. Minorant Methods and Positive Eigensolutions.- x7.7. Applications.- x7.8. The Krein-Rutman Theorem and its Applications.- x7.9. Asymptotic Linear Operators.- x7.10. Main Theorem for Operators of Monotone Type.- x7.11. Application to a Heat Conduction Problem.- x7.12. Existence of Three Solutions.- x7.13. Main Theorem for Abstract Hammerstein Equations in Ordered B-spaces.- x7.14. Eigensolutions of Abstract Hammerstein Equations, Bifurcation, Stability, and the Nonlinear Krein-Rutman Theorem.- x7.15. Applications to Hammerstein Integral Equations.- x7.16. Applications to Semi-linear Elliptic Boundary-Value Problems.- x7.17. Application to Elliptic Equations with Nonlinear Boundary Conditions.- x7.18. Applications to Boundary Initial-Value Problems for Parabolic Differential Equations and Stability.- 8 Analytic Bifurcation Theory.- x8.1. A Necessary Condition for Existence of a Bifurcation Point.- x8.2. Analytic Operators.- x8.3. An Analytic Majorant Method.- x8.4. Fredholm Operators.- x8.5. The Spectrum of Compact Linear Operators (Riesz-Schauder Theory).- x8.6. The Branching Equations of Ljapunov-Schmidt.- x8.7. The Main Theorem on the Generic Bifurcation From Simple Zeros.- x8.8. Applications to Eigenvalue Problems.- x8.9. Applications to Integral Equations.- x8.10. Application to Differential Equations.- x8.11. The Main Theorem on Generic Bifurcation for Multiparametric Operator Equations-The Bunch Theorem.- x8.12. Main Theorem for Regular Semi-linear Equations.- x8.13. Parameter-Induced Oscillation.- x8.14. Self-Induced Oscillations and Limit Cycles.- x8.15. Hopf Bifurcation.- x8.16. The Main Theorem on Generic Bifurcation from Multiple Zeros.- x8.17. Stability of Bifurcation Solutions.- x8.18. Generic Point Bifurcation.- 9 Fixed Points of Multivalued Maps.- x9.1. Generalized Banach Fixed-Point Theorem.- x9.2. Upper and Lower Semi-continuity of Multivalued Maps.- x9.3. Generalized Schauder Fixed-Point Theorem.- x9.4. Variational Inequalities and the Browder Fixed-Point Theorem.- x9.5. An Extremal Principle.- x9.6. The Minimax Theorem and Saddle Points.- x9.7. Applications in Game Theory.- x9.8. Selections and the Marriage Theorem.- x9.9. Michael's Selection Theorem.- x9.10. Application to the Generalized Peano Theorem for Differential Inclusions.- 10 Nonexpansive Operators and Iterative Methods.- x10.1. Uniformly Convex B-spaces.- x10.2. Demiclosed Operators.- x10.3. The Fixed-Point Theorem of Browder, Gohde, and Kirk.- x10.4. Demicompact Operators.- x10.5. Convergence Principles in B-spaces.- x10.6. Modified Successive Approximations.- x10.7. Application to Periodic Solutions.- 11 Condensing Maps and the Bourbaki-Kneser Fixed-Point Theorem.- x11.1. A Noncompactness Measure.- x11.2. Applications to Generalized Interval Nesting.- x11.3. Condensing Maps.- x11.4. Operators with Closed Range and an Approximation Technique for Constructing Fixed Points.- x11.5. Sadovskii's Fixed-Point Theorem for Condensing Maps.- x11.6. Fixed-Point Theorems for Perturbed Operators.- x11.7. Application to Differential Equations in B-spaces.- x11.8. The Bourbaki-Kneser Fixed-Point Theorem.- x 11.9. The Fixed-Point Theorems of Amann and Tarski.- x11.10. Application to Interval Arithmetic.- x11.11. Application to Formal Languages.- The Mapping Degree and the Fixed-Point Index.- 12 The Leray-Schauder Fixed-Point Index.- x12.1. Intuitive Background and Basic Concepts.- x12.2. Homotopy.- x12.3. The System of Axioms.- x12.4. An Approximation Theorem.- x12.5. Existence and Uniqueness of the Fixed-Point Index in ?N.- x12.6. Proof of Theorem 12.A..- x12.7. Existence and Uniqueness of the Fixed-Point Index in B-spaces.- x12.8. Product Theorem and Reduction Theorem.- 13 Applications of the Fixed-Point Index.- x13.1. A General Fixed-Point Principle.- x13.2. A General Eigenvalue Principle.- x13.3. Existence of Multiple Solutions.- x13.4. A Continuum of Fixed Points.- x13.5. Applications to Differential Equations.- x13.6. Properties of the Mapping Degree.- x13.7. The Leray Product Theorem and Homeomorphisms.- x13.8. The Jordan-Brouwer Separation Theorem and Brouwer's Invariance of Dimension Theorem.- x13.9. A Brief Glance at the History of Mathematics.- x13.10. Topology and Intuition.- x13.11. Generalization of the Mapping Degree.- 14 The Fixed-Point Index of Differentiable and Analytic Maps.- x14.1. The Fixed-Point Index of Classical Analytic Functions.- x14.2. The Leray-Schauder Index Theorem.- x14.3. The Fixed-Point Index of Analytic Mappings on Complex B-spaces.- x14.4. The Schauder Fixed-Point Theorem with Uniqueness.- x14.5. Solution of Analytic Operator Equations.- x14.6. The Global Continuation Principle of Leray-Schauder.- x14.7. Unbounded Solution Components.- x14.8. Applications to Systems of Equations.- x14.9. Applications to Integral Equations.- x14.10. Applications to Boundary-Value Problems.- x14.11. Applications to Integral Power Series.- 15 Topological Bifurcation Theory.- x15.1. The Index Jump Principle.- x15.2. Applications to Systems of Equations.- x15.3. Duality Between the Index Jump Principle and the Leray-Schauder Continuation Principle.- x15.4. The Geometric Heart of the Continuation Method.- x15.5. Stability Change and Bifurcation.- x15.6. Local Bifurcation.- x15.7. Global Bifurcation.- x15.8. Application to Systems of Equations.- x15.9. Application to Integral Equations.- x15.10. Application to Differential Equations.- x15.11. Application to Bifurcation at Infinity.- x15.12. Proof of the Main Theorem.- x15.13. Preventing Secondary Bifurcation.- 16 Essential Mappings and the Borsuk Antipodal Theorem.- x16.1. Intuitive Introduction.- x16.2. Essential Mappings and their Homotopy Invariance.- x16.3. The Antipodal Theorem.- x16.4. The Invariance of Domain Theorem and Global Homeomorphisms.- x16.5. The Borsuk-Ulam Theorem and its Applications.- x16.6. The Mapping Degree and Essential Maps.- x16.7. The Hopf Theorem.- x16.8. A Glance at Homotopy Theory.- 17 Asymptotic Fixed-Point Theorems.- x17.1. The Generalized Banach Fixed-Point Theorem.- x17.2. The Fixed-Point Index of Iterated Mappings.- x17.3. The Generalized Schauder Fixed-Point Theorem.- x17.4. Application to Dissipati ve Dynamical Systems.- x17.5. Perspectives.- References.- Additional References to the Second Printing.- List of Symbols.- List of Theorems.- List of the Most Important Definitions.- Schematic Overviews.- General References to the Literature.- List of Important Principles.- of the Other Parts.