Theorems of Leray-Schauder Type and Applications (Series in mathematical analysis and applications, 1028-8600 ; v. 3)

Theorems of Leray-Schauder Type and Applications presents a systematic and unified treatment of Leray-Schauder continuation theorems in nonlinear analysis. In particular fixed theory is established for many classes of maps, for example, contractive, non-expansive, accretive and compact. This book also presents concidence and multiplicty results. Many applications of current interest in the theory of nonlinear differential equations are given to complement the theory. The text is essentially self-contained so it may also be used as an introduction to topological methods in nonlinear analysis. Front cover......Page 1 Series......Page 2 Title page......Page 3 Date-line......Page 4 Dedication......Page 5 Contents......Page 7 Preface......Page 9 1. Overview......Page 11 2.1 The Continuation Principle for Contractions on Spaces with Two Metrics......Page 19 2.2 Global Solutions to the Cauchy Problem on a Bounded Set in Banach Spaces......Page 28 2.3 Boundary Value Problems on a Bounded Set in Banach Spaces......Page 35 3.1 Continuation Theorems......Page 53 3.2 Elements of Geometry of Sobolev Spaces......Page 57 3.3 Sturm-Liouville Problems in Uniformly Convex Banach Spaces......Page 60 4.1 Properties of Accretive Maps......Page 63 4.2 Continuation Principles for Accretive Maps......Page 65 4.3 Applications to Boundary Value Problems in Hilbert Spaces......Page 71 5.1 Monch Continuation Principle......Page 75 5.2 Granas'Topological Transversality Theorem......Page 78 5.3 Measures of Noncompactness on C(I;E)......Page 80 5.4 The Cauchy Problem in Banach Spaces......Page 84 5.5 Sturm-Liouville Problems in Banach Spaces......Page 89 6. Applications to Semilinear Elliptic Problems......Page 93 6.1 Basic Results from the Theory of Linear Elliptic Equations......Page 94 6.2 Applications of the Banach, Schauder and Darbo Fixed Point Theorems......Page 98 6.3 Applications of the Leray-Schauder Type Theorems......Page 101 7.1 Continuation Principles for Coincidences......Page 111 7.2 Application to Periodic Solutions of Differential Systems......Page 117 8.1 Selective Continuation Principles......Page 121 8.2 Continua of Solutions......Page 125 8.3 Continuation with Respect to a Functional......Page 128 8.4 Periodic Solutions ofSuperlinear Singular Boundary Value Problems......Page 130 9.1 A General Continuation Principle......Page 147 9.2 A General Fixed Point Continuation Principle......Page 153 9.3 Continuation Theorems for Maps of Monotone Type......Page 157 10.1 Leray-Schauder Theorems of Compression-Expansion Type......Page 165 10.2 Multiple Solutions of Focal Boundary Value Problems......Page 173 11. Local Continuation Theorems......Page 181 11.1 Local Continuation Theorems for Contractions......Page 182 11.2 The Classical Implicit Function Theorem......Page 186 11.3 Two Special Local Continuation Theorems......Page 189 11.4 Continuation and Stability......Page 191 Epilogue......Page 199 Bibliography......Page 201 Index......Page 215 Back cover......Page 217 This volume presents a systematic and unified treatment of Leray-Schauder continuation theorems in nonlinear analysis. In particular, fixed point theory is established for many classes of maps, such as contractive, non-expansive, accretive, and compact maps, to name but a few. This book also presents coincidence and multiplicity results. Many applications of current interest in the theory of nonlinear differential equations are presented to complement the theory. The text is essentially self-contained, so it may also be used as an introduction to topological methods in nonlinear analysis. This volume will appeal to graduate students and researchers in mathematical analysis and its applications. Overview. Theorems of Leray-Schauder Type for Contractions. Continuation Theorems for Nonexpansive Maps. Theorems of Leray-Schauder Type for Accretive Maps. Continuation Theorems Involving Compactness. Applications to Semilinear Elliptic Problems. Theorems of Leray-Schauder Type for Coincidences. Theorems of Selective Continuation. The Unified Theory. Multiplicity. Local Continuation Theorems This volume presents a systematic and unified treatment of Leray-Schauder continuation theorems in nonlinear analysis. In particular, fixed point theory is established for many classes of maps, for example, contractive, non-expansive, accretive, and compact, to name but a few. It also presents coincidence and multiplicity results The theorems of Leray-Schauder type, also called continuation theorems, represent a powerful existence tool in studying operator equations and inclusions (of particular interest is the theory of nonlinear differential equations).