An Introduction to Metric Spaces and Fixed Point Theory [Hardcover]

Preface ix

I Metric Spaces

1 Introduction 3

1.1 The real numbers R 3

1.2 Continuous mappings in R 5

1.3 The triangle inequality in R 7

1.4 The triangle inequality in R 8

1.5 Brouwer's Fixed Point Theorem 10

Exercises 11

2 Metric Spaces 13

2.1 The metric topology 15

2.2 Examples of metric spaces 19

2.3 Completeness 26

2.4 Separability and connectedness 33

2.5 Metric convexity and convexity structures 35

Exercises 38

3 Metric Contraction Principles 41

3.1 Banach's Contraction Principle 41

3.2 Further extensions of Banach's Principle 46

3.3 The Caristi-Ekeland Principle 55

3.4 Equivalents of the Caristi-Ekeland Principle 58

3.5 Set-valued contractions 61

3.6 Generalized contractions 64

Exercises 67

4 Hyperconvex Spaces 71

4.1 Introduction 71

4.2 Hyperconvexity 77

4.3 Properties of hyperconvex spaces 80

4.4 A fixed point theorem 84

4.5 Intersections of hyperconvex spaces 87

4.6 Approximate fixed points 89

4.7 Isbell's hyperconvex hull 91

Exercises 98

5 Normal Structures in Metric Spaces 101

5.1 A fixed point theorem 101&l£ƒ