Topological Degree Theory and Applications / Edition 1

by Yeol Je Cho, Yu-Qing Chen

Overview

Since the 1960s, many researchers have extended topological degree theory to various non-compact type nonlinear mappings, and it has become a valuable tool in nonlinear analysis. Presenting a survey of advances made in generalizations of degree theory during the past decade, this book focuses on topological degree theory in normed spaces and its applications.

The authors begin by introducing the Brouwer degree theory in Rn, then consider the Leray-Schauder degree for compact mappings in normed spaces. Next, they explore the degree theory for condensing mappings, including applications to ODEs in Banach spaces. This is followed by a study of degree theory for A-proper mappings and its applications to semilinear operator equations with Fredholm mappings and periodic boundary value problems. The focus then turns to construction of Mawhin's coincidence degree for L-compact mappings, followed by a presentation of a degree theory for mappings of class (S+) and its perturbations with other monotone-type mappings. The final chapter studies the fixed point index theory in a cone of a Banach space and presents a notable new fixed point index for countably condensing maps.

Examples and exercises complement each chapter. With its blend of old and new techniques, Topological Degree Theory and Applications forms an outstanding text for self-study or special topics courses and a valuable reference for anyone working in differential equations, analysis, or topology.

Product Details

ISBN-13:	9780367390983
Publisher:	Taylor & Francis
Publication date:	09/05/2019
Series:	Mathematical Analysis and Applications , #10
Pages:	232
Product dimensions:	6.12(w) x 9.19(h) x (d)

About the Author

Cho, Yeol Je; Chen, Yu-Qing

Table of Contents

1 Brouwer Degree Theory 1

1.1 Continuous and Differentiable Functions 2

1.2 Construction of Brouwer Degree 4

1.3 Degree Theory for Functions in VMO 15

1.4 Applications to ODEs 19

1.5 Exercises 22

2 Leray Schauder Degree Theory 25

2.1 Compact Mappings 25

2.2 Leray Schauder Degree 30

2.3 Leray Schauder Degree for Multi-Valued Mappings 38

2.4 Applications to Bifurcations 43

2.5 Applications to ODEs and PDEs 46

2.6 Exercises 53

3 Degree Theory for Set Contractive Maps 55

3.1 Measure of Noncompactness and Set Contractions 55

3.2 Degree Theory for Countably Condensing Mappings 65

3.3 Applications to ODEs in Banach Spaces 68

3.4 Exercises 71

4 Generalized Degree theory for A-Proper Maps 75

4.1 A-Proper Mappings 75

4.2 Generalized Degree for A-Proper Mappings 80

4.3 Equations with Fredholm Mappings of Index Zero 82

4.4 Equations with Fredholm Mappings of Index Zero Type 87

4.5 Applications of the Generalized Degree 95

4.6 Exercises 101

5 Coincidence Degree Theory 105

5.1 Fredholm Mappings 105

5.2 Coincidence Degree for L-Compact Mappings 110

5.3 Existence Theorems for Operator Equations 116

5.4 Applications to ODEs 119

5.5 Exercises 124

6 Degree Theory for Monotone-Type Maps 127

6.1 Monotone Type-Mappings in Reflexive Banach Spaces 128

6.2 Degree Theory for Mappings of Class (S₊) 142

6.3 Degree for Perturbations of Monotone-Type Mappings 145

6.4 Degree Theory for Mappings of Class (S₊)L 149

6.5 Coincidence Degree for Mappings of Class L-(S₊) 152

6.6 Computation of Topological Degree 156

6.7 Applications to PDEs and Evolution Equations 159

6.8 Exercises 167

7 Fixed Point Index Theory 169

7.1 Cone in Normed Spaces 169

7.2 Fixed Point Index Theory 176

7.3 Fixed Point Theorems in Cones 179

7.4 Perturbations of Condensing Mappings 186

7.5 Index Theory for Nonself Mappings 189

7.6 Applications to Integral and Differential Equations 191

7.7 Exercises 193

8 References 195

9 Subject Index 217