• DOI:10.1533/9780857099693.237
• Corpus ID: 116897361

Fixed-point theorems

@inproceedings{Zeidler1986FixedpointT,
  title={Fixed-point theorems},
  author={Eberhard H. Zeidler and P. Wadsack},
  year={1986}
}

• E. Zeidler, P. Wadsack
• Published 1986
• Mathematics

Fundamental Fixed-Point Principles.- 1 The Banach Fixed-Point Theorem and Iterative Methods.- 1.1. The Banach Fixed-Point Theorem.- 1.2. Continuous Dependence on a Parameter.- 1.3. The Significance of the Banach Fixed-Point Theorem.- 1.4. Applications to Nonlinear Equations.- 1.5. Accelerated Convergence and Newton's Method.- 1.6. The Picard-Lindelof Theorem.- 1.7. The Main Theorem for Iterative Methods for Linear Operator Equations.- 1.8. Applications to Systems of Linear Equations.- 1.9. Applications to Linear Integral Equations.- 2 The Schauder Fixed-Point Theorem and Compactness.- 2.1. Extension Theorem.- 2.2. Retracts.- 2.3. The Brouwer Fixed-Point Theorem.- 2.4. Existence Principle for Systems of Equations.- 2.5. Compact Operators.- 2.6. The Schauder Fixed-Point Theorem.- 2.7. Peano's Theorem.- 2.8. Integral Equations with Small Parameters.- 2.9. Systems of Integral Equations and Semilinear Differential Equations.- 2.10. A General Strategy.- 2.11. Existence Principle for Systems of Inequalities.- Applications of the Fundamental Fixed-Point Principles.- 3 Ordinary Differential Equations in B-spaces.- 3.1. Integration of Vector Functions of One Real Variable t.- 3.2. Differentiation of Vector Functions of One Real Variable t.- 3.3. Generalized Picard-Lindelof Theorem.- 3.4. Generalized Peano Theorem.- 3.5. Gronwall's Lemma.- 3.6. Stability of Solutions and Existence of Periodic Solutions.- 3.7. Stability Theory and Plane Vector Fields, Electrical Circuits, Limit Cycles.- 3.8. Perspectives.- 4 Differential Calculus and the Implicit Function Theorem.- 4.1. Formal Differential Calculus.- 4.2. The Derivatives of Frechet and Gateaux.- 4.3. Sum Rule, Chain Rule, and Product Rule.- 4.4. Partial Derivatives.- 4.5. Higher Differentials and Higher Derivatives.- 4.6. Generalized Taylor's Theorem.- 4.7. The Implicit Function Theorem.- 4.8. Applications of the Implicit Function Theorem.- 4.9. Attracting and Repelling Fixed Points and Stability.- 4.10. Applications to Biological Equilibria.- 4.11. The Continuously Differentiable Dependence of the Solutions of Ordinary Differential Equations in B-spaces on the Initial Values and on the Parameters.- 4.12. The Generalized Frobenius Theorem and Total Differential Equations.- 4.13. Diffeomorphisms and the Local Inverse Mapping Theorem.- 4.14. Proper Maps and the Global Inverse Mapping Theorem.- 4.15. The Suijective Implicit Function Theorem.- 4.16. Nonlinear Systems of Equations, Subimmersions, and the Rank Theorem.- 4.17. A Look at Manifolds.- 4.18. Submersions and a Look at the Sard-Smale Theorem.- 4.19. The Parametrized Sard Theorem and Constructive Fixed-Point Theory.- 5 Newton's Method.- 5.1. A Theorem on Local Convergence.- 5.2. The Kantorovi? Semi-Local Convergence Theorem.- 6 Continuation with Respect to a Parameter.- 6.1. The Continuation Method for Linear Operators.- 6.2. B-spaces of Holder Continuous Functions.- 6.3. Applications to Linear Partial Differential Equations.- 6.4. Functional-Analytic Interpretation of the Existence Theorem and its Generalizations.- 6.5. Applications to Semi-linear Differential Equations.- 6.6. The Implicit Function Theorem and the Continuation Method.- 6.7. Ordinary Differential Equations in B-spaces and the Continuation Method.- 6.8. The Leray-Schauder Principle.- 6.9. Applications to Quasi-linear Elliptic Differential Equations.- 7 Positive Operators.- 7.1. Ordered B-spaces.- 7.2. Monotone Increasing Operators.- 7.3. The Abstract Gronwall Lemma and its Applications to Integral Inequalities.- 7.4. Supersolutions, Subsolutions, Iterative Methods, and Stability.- 7.5. Applications.- 7.6. Minorant Methods and Positive Eigensolutions.- 7.7. Applications.- 7.8. The Krein-Rutman Theorem and its Applications.- 7.9. Asymptotic Linear Operators.- 7.10. Main Theorem for Operators of Monotone Type.- 7.11. Application to a Heat Conduction Problem.- 7.12. Existence of Three Solutions.- 7.13. Main Theorem for Abstract Hammerstein Equations in Ordered B-spaces.- 7.14. Eigensolutions of Abstract Hammerstein Equations, Bifurcation, Stability, and the Nonlinear Krein-Rutman Theorem.- 7.15. Applications to Hammerstein Integral Equations.- 7.16. Applications to Semi-linear Elliptic Boundary-Value Problems.- 7.17. Application to Elliptic Equations with Nonlinear Boundary Conditions.- 7.18. Applications to Boundary Initial-Value Problems for Parabolic Differential Equations and Stability.- 8 Analytic Bifurcation Theory.- 8.1. A Necessary Condition for Existence of a Bifurcation Point.- 8.2. Analytic Operators.- 8.3. An Analytic Majorant Method.- 8.4. Fredholm Operators.- 8.5. The Spectrum of Compact Linear Operators (Riesz-Schauder Theory).- 8.6. The Branching Equations of Ljapunov-Schmidt.- 8.7. The Main Theorem on the Generic Bifurcation From Simple Zeros.- 8.8. Applications to Eigenvalue Problems.- 8.9. Applications to Integral Equations.- 8.10. Application to Differential Equations.- 8.11. The Main Theorem on Generic Bifurcation for Multiparametric Operator Equations-The Bunch Theorem.- 8.12. Main Theorem for Regular Semi-linear Equations.- 8.13. Parameter-Induced Oscillation.- 8.14. Self-Induced Oscillations and Limit Cycles.- 8.15. Hopf Bifurcation.- 8.16. The Main Theorem on Generic Bifurcation from Multiple Zeros.- 8.17. Stability of Bifurcation Solutions.- 8.18. Generic Point Bifurcation.- 9 Fixed Points of Multivalued Maps.- 9.1. Generalized Banach Fixed-Point Theorem.- 9.2. Upper and Lower Semi-continuity of Multivalued Maps.- 9.3. Generalized Schauder Fixed-Point Theorem.- 9.4. Variational Inequalities and the Browder Fixed-Point Theorem.- 9.5. An Extremal Principle.- 9.6. The Minimax Theorem and Saddle Points.- 9.7. Applications in Game Theory.- 9.8. Selections and the Marriage Theorem.- 9.9. Michael's Selection Theorem.- 9.10. Application to the Generalized Peano Theorem for Differential Inclusions.- 10 Nonexpansive Operators and Iterative Methods.- 10.1. Uniformly Convex B-spaces.- 10.2. Demiclosed Operators.- 10.3. The Fixed-Point Theorem of Browder, Gohde, and Kirk.- 10.4. Demicompact Operators.- 10.5. Convergence Principles in B-spaces.- 10.6. Modified Successive Approximations.- 10.7. Application to Periodic Solutions.- 11 Condensing Maps and the Bourbaki-Kneser Fixed-Point Theorem.- 11.1. A Noncompactness Measure.- 11.2. Applications to Generalized Interval Nesting.- 11.3. Condensing Maps.- 11.4. Operators with Closed Range and an Approximation Technique for Constructing Fixed Points.- 11.5. Sadovskii's Fixed-Point Theorem for Condensing Maps.- 11.6. Fixed-Point Theorems for Perturbed Operators.- 11.7. Application to Differential Equations in B-spaces.- 11.8. The Bourbaki-Kneser Fixed-Point Theorem.- 11.9. The Fixed-Point Theorems of Amann and Tarski.- 11.10. Application to Interval Arithmetic.- 11.11. Application to Formal Languages.- The Mapping Degree and the Fixed-Point Index.- 12 The Leray-Schauder Fixed-Point Index.- 12.1. Intuitive Background and Basic Concepts.- 12.2. Homotopy.- 12.3. The System of Axioms.- 12.4. An Approximation Theorem.- 12.5. Existence and Uniqueness of the Fixed-Point Index in ?N.- 12.6. Proof of Theorem 12.A..- 12.7. Existence and Uniqueness of the Fixed-Point Index in B-spaces.- 12.8. Product Theorem and Reduction Theorem.- 13 Applications of the Fixed-Point Index.- 13.1. A General Fixed-Point Principle.- 13.2. A General Eigenvalue Principle.- 13.3. Existence of Multiple Solutions.- 13.4. A Continuum of Fixed Points.- 13.5. Applications to Differential Equations.- 13.6. Properties of the Mapping Degree.- 13.7. The Leray Product Theorem and Homeomorphisms.- 13.8. The Jordan-Brouwer Separation Theorem and Brouwer's Invariance of Dimension Theorem.- 13.9. A Brief Glance at the History of Mathematics.- 13.10. Topology and Intuition.- 13.11. Generalization of the Mapping Degree.- 14 The Fixed-Point Index of Differentiable and Analytic Maps.- 14.1. The Fixed-Point Index of Classical Analytic Functions.- 14.2. The Leray-Schauder Index Theorem.- 14.3. The Fixed-Point Index of Analytic Mappings on Complex B-spaces.- 14.4. The Schauder Fixed-Point Theorem with Uniqueness.- 14.5. Solution of Analytic Operator Equations.- 14.6. The Global Continuation Principle of Leray-Schauder.- 14.7. Unbounded Solution Components.- 14.8. Applications to Systems of Equations.- 14.9. Applications to Integral Equations.- 14.10. Applications to Boundary-Value Problems.- 14.11. Applications to Integral Power Series.- 15 Topological Bifurcation Theory.- 15.1. The Index Jump Principle.- 15.2. Applications to Systems of Equations.- 15.3. Duality Between the Index Jump Principle and the Leray-Schauder Continuation Principle.- 15.4. The Geometric Heart of the Continuation Method.- 15.5. Stability Change and Bifurcation.- 15.6. Local Bifurcation.- 15.7. Global Bifurcation.- 15.8. Application to Systems of Equations.- 15.9. Application to Integral Equations.- 15.10. Application to Differential Equations.- 15.11. Application to Bifurcation at Infinity.- 15.12. Proof of the Main Theorem.- 15.13. Preventing Secondary Bifurcation.- 16 Essential Mappings and the Borsuk Antipodal Theorem.- 16.1. Intuitive Introduction.- 16.2. Essential Mappings and their Homotopy Invariance.- 16.3. The Antipodal Theorem.- 16.4. The Invariance of Domain Theorem and Global Homeomorphisms.- 16.5. The Borsuk-Ulam Theorem and its Applications.- 16.6. The Mapping Degree and Essential Maps.- 16.7. The Hopf Theorem.- 16.8. A Glance at Homotopy Theory.- 17 Asymptotic Fixed-Point Theorems.- 17.1. The Generalized Banach Fixed-Point Theorem.- 17.2. The Fixed-Point Index of Iterated Mappings.- 17.3. The Generalized Schauder Fixed-Point Theorem.- 17.4. Application to Dissipati ve Dynamical Systems.- 17.5. 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.