Fixed Point Theory. With 14 Illustrations. %1 Springer

Transcription

1 Andrzej Granas James Dugundji Fixed Point Theory With 14 Illustrations %1 Springer

2 Contents Preface vii 0. Introduction 1 1. Fixed Point Spaces 1 2. Forming New Fixed Point Spaces from Old 3 3. Topological Transversality 4 4. Factorization Technique 6 I. Elementary Fixed Point Theorems 1. Results Based on Completeness 9 1. Banach Contraction Principle 9 2. Elementary Domain Invariance Continuation Method for Contractive Maps Nonlinear Alternative for Contractive Maps Extensions of the Banach Theorem Miscellaneous Results and Examples Notes and Comments Order-Theoretic Results The Knaster-Tarski Theorem Order and Completeness. Theorem of Bishop-Phelps Fixed Points for Set-Valued Contractive Maps Applications to Geometry of Banach Spaces Applications to the Theory of Critical Points Miscellaneous Results and Examples Notes and Comments 34

3 X Contents 3. Results Based on Convexity KKM-Maps and the Geometric KKM-Principle Theorem of von Neumann and Systems of Inequalities Fixed Points of Affine Maps. Markoff-Kakutani Theorem Fixed Points for Families of Maps. Theorem of Kakutani Miscellaneous Results and Examples Notes and Comments Further Results and Applications Nonexpansive Maps in Hilbert Space Applications of the Banach Principle to Integral and Differential Equations Applications of the Elementary Domain Invariance Elementary KKM-Principle and its Applications Theorems of Mazur-Orlicz and Hahn-Banach Miscellaneous Results and Examples Notes and Comments 81 II. Theorem of Borsuk and Topological Transversality 5. Theorems of Brouwer and Borsuk Preliminary Remarks Basic Triangulation of S n A Combinatorial Lemma The Lusternik-Schnirelmann-Borsuk Theorem Equivalent Formulations. The Borsuk-Ulam Theorem Some Simple Consequences Brouwer's Theorem Topological KKM-Principle Miscellaneous Results and Examples Notes and Comments Fixed Points for Compact Maps in Normed Linear Spaces Compact and Completely Continuous Operators Schauder Projection and Approximation Theorem Extension of the Brouwer and Borsuk Theorems Topological Transversality. Existence of Essential Maps Equation x = F{x). The Leray-Schauder Principle Equation x = XF(x). Birkhoff-Kellogg Theorem 125

4 Contents XI 7. Compact Fields Equation y = x F(x). Invariance of Domain Miscellaneous Results and Examples Notes and Comments Further Results and Applications Applications of the Topological KKM-Principle Some Applications of the Antipodal Theorem The Schauder Theorem and Differential Equations Topological Transversality and Differential Equations Application to the Galerkin Approximation Theory The Invariant Subspace Problem Absolute Retracts and Generalized Schauder Theorem Fixed Points for Set-Valued Kakutani Maps Theorem of Ryll-Nardzewski Miscellaneous Results and Examples Notes and Comments 190 III. Homology and Fixed Points 8. Simplicial Homology Simplicial Complexes and Polyhedra Subdivisions Simplicial Maps and Simplicial Approximations Vertex Schemes, Realizations, and Nerves of Coverings Simplicial Homology Chain Transformations and Chain Homotopies Induced Homomorphism Triangulated Spaces and Polytopes Relative Homology Miscellaneous Results and Examples Notes and Comments The Lefschetz Hopf Theorem and Brouwer Degree Algebraic Preliminaries The Lefschetz-Hopf Fixed Point Theorem The Euler Number of a Map. Periodic Points Applications The Brouwer Degree of Maps S n -> S n 234

5 Xli Contents 6. Theorem of Borsuk-Hirsch Maps of Even- and of Odd-Dimensional Spheres Degree and Homotopy. Theorem of Hopf Vector Fields on Spheres Miscellaneous Results and Examples Notes and Comments 245 IV. Leray Schauder Degree and Fixed Point Index 10. Topological Degree in R n PL Maps of Polyhedra Polyhedral Domains in R n. Degree for Generic Maps Local Constancy and Homotopy Invariance Degree for Continuous Maps Some Properties of Degree Extension to Arbitrary Open Sets Axiomatics The Main Theorem on the Brouwer Degree in R n Extension of the Antipodal Theorem Miscellaneous Results and Examples Notes and Comments Absolute Neighborhood Retracts General Properties ARs and ANRs Local Properties Pasting ANRs Together Theorem of Hanner Homotopy Properties Generalized Leray-Schauder Principle in ANRs Miscellaneous Results and Examples Notes and Comments Fixed Point Index in ANRs Fixed Point Index in R n Axioms for the Index The Leray-Schauder Index in Normed Linear Spaces Commutativity of the Index Fixed Point Index for Compact Maps in ANRs 315

6 Contents Xlll 6. The Leray-Schauder Continuation Principle in ANRs Simple Consequences and Index Calculations Local Index of an Isolated Fixed Point Miscellaneous Results and Examples Notes and Comments Further Results and Applications Bifurcation Results in ANRs Application of the Index to Nonlinear PDEs The Leray-Schauder Degree Extensions of the Borsuk and Borsuk-Ulam Theorems The Leray-Schauder Index in Locally Convex Spaces Miscellaneous Results and Applications Notes and Comments 364 V. The Lefschetz-Hopf Theory 14. Singular Homology Singular Chain Complex and Homology Functors Invariance of Homology under Barycentric Subdivision Excision Axiomatization Comparison of Homologies. Kiinneth Theorem Homology and Topological Degree Miscellaneous Results and Examples Notes and Comments Lefschetz Theory for Maps of ANRs The Leray Trace Generalized Lefschetz Number Lefschetz Maps and Lefschetz Spaces Lefschetz Theorem for Compact Maps of ANRs Asymptotic Fixed Point Theorems for ANRs Basic Classes of Locally Compact Maps Asymptotic Lefschetz-Type Results in ANRs Periodicity Index of a Map. Periodic Points Miscellaneous Results and Examples Notes and Comments 437

7 XIV Contents 16. The Hopf Index Theorem Normal Fixed Points in Polyhedral Domains Homology of Polyhedra with Attached Cones The Hopf Index Theorem in Polyhedral Domains The Hopf Index Theorem in Arbitrary ANRs The Lefschetz-Hopf Fixed Point Index for ANRs Some Consequences of the Index Miscellaneous Results and Examples Notes and Comments Further Results and Applications Local Index Theory for ANRs Fixed Points for Self-Maps of Arbitrary Compacta Forming New Lefschetz Spaces from Old by Domination Fixed Points in Linear Topological Spaces Fixed Points in NES(compact) Spaces General Asymptotic Fixed Point Results 474 7*. Domination of ANRs by Polytopes Miscellaneous Results and Examples Notes and Comments 488 VI. Selected Topics 18. Finite-Codimensional Cech Cohomology Preliminaries Continuous Functors The Cech Cohomology Groups if - n (X) The Functor H 00 ' 71 : (, ~) -> Ab Cohomology Theory on Miscellaneous Results and Examples Notes and Comments Vietoris Fractions and Coincidence Theory Preliminary Remarks Category of Fractions Vietoris Maps and Fractions Induced Homomorphisms and the Lefschetz Number Coincidence Spaces Some General Coincidence Theorems 539

8 Contents XV 7. Fixed Points for Compact and Acyclic Set-Valued Maps Miscellaneous Results and Examples Notes and Comments Further Results and Supplements Degree for Equivariant Maps in R n The Infinite-Dimensional E^-Cohomology Lefschetz Theorem for A^-Maps of Compacta Miscellaneous Results and Examples Notes and Comments 573 Appendix: Preliminaries 588 A. Generalities 588 B. Topological Spaces 590 C. Linear Topological Spaces 599 D. Algebraic Preliminaries 608 E. Categories and Functors 616 Bibliography 620 I. General Reference Texts 620 II. Monographs, Lecture Notes, and Surveys 621 III. Articles 625 IV. Additional References 650 List of Standard Symbols 668 Index of Names 672 Index of Terms 678