Fixed Point Theory (Springer Monographs in Mathematics)

✍ Scribed by Andrzej Granas, James Dugundji

• Publisher: Springer
• Year: 2003
• Tongue: English
• Leaves: 707
• Edition: 2003
• Category: Library

✦ Synopsis

The aim of this monograph is to give a unified account of the classical topics in fixed point theory that lie on the border-line of topology and non­ linear functional analysis, emphasizing developments related to the Leray­ Schauder theory. Using for the most part geometric methods, our study cen­ ters around formulating those general principles of the theory that provide the foundation for many of the modern results in diverse areas of mathe­ matics. The main text is self-contained for readers with a modest knowledge of topology and functional analysis; the necessary background material is collected in an appendix, or developed as needed. Only the last chapter pre­ supposes some familiarity with more advanced parts of algebraic topology. The "Miscellaneous Results and Examples", given in the form of exer­ cises, form an integral part of the book and describe further applications and extensions of the theory. Most of these additional results can be established by the methods developedin the book, and no proof in the main text relies on any of them; more demanding problems are marked by an asterisk. The "Notes and Comments" at the end of paragraphs contain references to the literature and give some further information about the results in the text.

✦ Table of Contents

Front cover
Series
Title page
Date-line
Dedication
Preface
Contents
§0. Introduction
1. Fixed Point Spaces
2. Forming New Fixed Point Spaces from Old
3. Topological Transversality
4. Factorization Technique
I. Elementary Fixed Point Theorems
§1. Results Based on Completeness
1. Banach Contraction Principle
2. Elementary Domain Invariance
3. Continuation Method for Contractive Maps
4. Nonlinear Alternative for Contractive Maps
5. Extensions of the Banach Theorem
6. Miscellaneous Results and Examples
7. Notes and Comments
§2. Order-Theoretic Results
1. The Knaster-Tarski Theorem
2. Order and Completeness. Theorem of Bishop-Phelps
3. Fixed Points for Set-Valued Contractive Maps
4. Applications to Geometry of Banach Spaces
5. Applications to the Theory of Critical Points
6. Miscellaneous Results and Examples
7. Notes and Comments
§3. Results Based on Convexity
1. KKM-Maps and the Geometric KKM-Principle
2. Theorem of von Neumann and Systems of Inequalities
3. Fixed Points of Affine Maps. Markoff-Kakutani Theorem
4. Fixed Points for Families of Maps. Theorem of Kakutani
5. Miscellaneous Results and Examples
6. Notes and Comments
§4. Further Results and Applications
1. Nonexpansive Maps in Hilbert Space
2. Applications of the Banach Principle to Integral and Differential Equations
3. Applications of the Elementary Domain Invariance
4. Elementary KKM-Principle and its Applications
5. Theorems of Mazur-Orlicz and Hahn-Banach
6. Miscellaneous Results and Examples
7. Notes and Comments
II. Theorem of Borsuk and Topological Transversality
§5. Theorems of Brouwer and Borsuk
1. Preliminary Remarks
2. Basic Triangulation of $S^n$
3. A Combinatorial Lemma
4. The Lusternik-Schnirelmann-Borsuk Theorem
5. Equivalent Formulations. The Borsuk-Ulam Theorem
6. Some Simple Consequences
7. Brouwer's Theorem
8. Topological KKM-Principle
9. Miscellaneous Results and Examples
10. Notes and Comments
§6. Fixed Points for Compact Maps in Normed Linear Spaces
1. Compact and Completely Continuous Operators
2. Schauder Projection and Approximation Theorem
3. Extension of the Brouwer and Borsuk Theorems
4. Topological Transversality. Existence of Essential Maps
5. Equation $x = F(x)$. The Leray-Schauder Principle
6. Equation $x = \lambda F(x)$. Birkhoff-Kellogg Theorem
7. Compact Fields
8. Equation $y = x — F(x)$. Invariance of Domain
9. Miscellaneous Results and Examples
10. Notes and Comments
§7. Further Results and Applications
1. Applications of the Topological KKM-Principle
2. Some Applications of the Antipodal Theorem
3. The Schauder Theorem and Differential Equations
4. Topological Transversality and Differential Equations
5. Application to the Galerkin Approximation Theory
6. The Invariant Subspace Problem
7. Absolute Retracts and Generalized Schauder Theorem
8. Fixed Points for Set-Valued Kakutani Maps
9. Theorem of Ryll-Nardzewski
10. Miscellaneous Results and Examples
11. Notes and Comments
III. Homology and Fixed Points
§8. Simplicial Homology
1. Simplicial Complexes and Polyhedra
2. Subdivisions
3. Simplicial Maps and Simplicial Approximations
4. Vertex Schemes, Realizations, and Nerves of Coverings
5. Simplicial Homology
6. Chain Transformations and Chain Homotopies
7. Induced Homomorphism
8. Triangulated Spaces and Polytopes
9. Relative Homology
10. Miscellaneous Results and Examples
11. Notes and Comments
§9. The Lefschetz-Hopf Theorem and Brouwer Degree
1. Algebraic Preliminaries
2. The Lefschetz-Hopf Fixed Point Theorem
3. The Euler Number of a Map. Periodic Points
4. Applications
5. The Brouwer Degree of Maps $S^n \to S^n$
6. Theorem of Borsuk Hirsch
7. Maps of Even- and of Odd-Dimensional Spheres
8. Degree and Homotopy. Theorem of Hopf
9. Vector Fields on Spheres
10. Miscellaneous Results and Examples
11. Notes and Comments
IV. Leray-Schauder Degree and Fixed Point Index
§10. Topological Degree in $\mathbb{R}^n$
1. PL Maps of Polyhedra
2. Polyhedral Domains in $\mathbb{R}^n$ Degree for Generic Maps
3. Local Constancy and Homotopy Invariance
4. Degree for Continuous Maps
5. Some Properties of Degree
6. Extension to Arbitrary Open Sets
7. Axiomatics
8. The Main Theorem on the Brouwer Degree in $\mathbb{R}^n$
9. Extension of the Antipodal Theorem
10. Miscellaneous Results and Examples
11. Notes and Comments
§11. Absolute Neighborhood Retracts
1. General Properties
2. ARs and ANRs
3. Local Properties
4. Pasting ANRs Together
5. Theorem of Hanner
6. Homotopy Properties
7. Generalized Leray-Schauder Principle in ANRs
8. Miscellaneous Results and Examples
9. Notes and Comments
§12. Fixed Point Index in ANRs
1. Fixed Point Index in $\mathbb{R}^n$
2. Axioms for the Index
3. The Leray-Schauder Index in Nonned Linear Spaces
4. Commutativity of the Index
5. Fixed Point Index for Compact Maps in ANRs
6. The Leray-Schauder Continuation Principle in ANRs
7. Simple Consequences and Index Calculations
8. Local Index of an Isolated Fixed Point
9. Miscellaneous Results and Examples
10. Notes and Comments
§13. Further Results and Applications
1. Bifurcation Results in ANRs
2. Application of the Index to Nonlinear PDEs
3. The Leray-Schauder Degree
4. Extensions of the Borsuk and Borsuk-Ulam Theorems
5. The Leray-Schauder Index in Locally Convex Spaces
6. Miscellaneous Results and Applications
7. Notes and Comments
V. The Lefschetz-Hopf Theory
§14. Singular Homology
1. Singular Chain Complex and Homology Functors
2. Invariance of Homology under Barycentric Subdivision
3. Excision
4. Axiomatization
5. Comparison of Homologies. Runneth Theorem
6. Homology and Topological Degree
7. Miscellaneous Results and Examples
8. Notes and Comments
§15. Lefschetz Theory for Maps of ANRs
1. The Leray Trace
2. Generalized Lefschetz Number
3. Lefschetz Maps and Lefschetz Spaces
4. Lefschetz Theorem for Compact Maps of ANRs
5. Asymptotic Fixed Point Theorems for ANRs
6. Basic Classes of Locally Compact Maps
7. Asymptotic Lefschetz-Type Results in ANRs
8. Periodicity Index of a Map. Periodic Points
9. Miscellaneous Results and Examples
10. Notes and Comments
§16. The Hopf Index Theorem
1. Normal Fixed Points in Polyhedral Domains
2. Homology of Polyhedra with Attached Cones
3. The Hopf Index Theorem in Polyhedral Domains
4. The Hopf Index Theorem in Arbitrary ANRs
5. The Lefschetz-Hopf Fixed Point Index for ANRs
6. Some Consequences of the Index
7. Miscellaneous Results and Examples
8. Notes and Comments
§17. Further Results and Applications
1. Local Index Theory for ANRs
2. Fixed Points for Self-Maps of Arbitrary Compacta
3. Forming New Lefschetz Spaces from Old by Domination
4. Fixed Points in Linear Topological Spaces
5. Fixed Points in NES(compact) Spaces
6. General Asymptotic Fixed Point Results
7*. Domination of ANRs by Polytopes
8. Miscellaneous Results and Examples
9. Notes and Comments
VI. Selected Topics
§18. Finite-Codimensional Cech Cohomology
1. Preliminaries
2. Continuous Functors
3. The Cech Cohomology Groups $H^{\infty-n}(X)$
4. The Functor $H^{\infty-n}: (\mathfrak{L},\sim) o Ab$
5. Cohomology Theory on $\mathfrac{L}$
6. Miscellaneous Results and Examples
7. Notes and Comments
§19. Vietoris Fractions and Coincidence Theory
1. Preliminary Remarks
2. Category of Fractions
3. Vietoris Maps and Fractions
4. Induced Homomorphisms and the Lefschetz Number
5. Coincidence Spaces
6. Some General Coincidence Theorems
7. Fixed Points for Compact and Acyclic Set-Valued Maps
8. Miscellaneous Results and Examples
9. Notes and Comments
§20. Further Results and Supplements
1. Degree for Equivariant Maps in $\mathbb{R}^n$
2. The Infinite-Dimensional $E^+$-Cohomology
3. Lefschetz Theorem for $\mathcal{NB}$-Maps of Compacta
4. Miscellaneous Results and Examples
5. Notes and Comments
Appendix: Preliminaries
A. Generalities
B. Topological Spaces
C. Linear Topological Spaces
D. Algebraic Preliminaries
E. Categories and Functors
Bibliography
I. General Reference Texts
II. Monographs, Lecture Notes, and Surveys
III. Articles
IV. Additional References
List of Standard Symbols
Index of Names
Index of Terms
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