Introduction to Metric Spaces and Fixed Point Theory

An Introduction to Metric Spaces and Fixed Point Theory includes an extensive bibliography and an appendix which provides a complete summary of the concepts of set theory, including Zorn's Lemma, Tychonoff's Theorem, Zermelo's Theorem, and transfinite induction. Detailed coverage of the newest developments in metric spaces and fixed point theory makes this the most modern and complete introduction to the subject available. MOHAMED A. KHAMSI, PhD, is Professor in the Department of Mathematical Sciences at the University of Texas at El Paso and visiting Professor in the Department of Mathematics at Kuwait University. He is also co-author of Nonstandard Methods in Fixed Point Theory. WILLIAM A. KIRK, PhD, is Professor in the Department of Mathematics at the University of Iowa, Iowa City, Iowa. He has authored over 100 journal articles and is co-author of Topics in Metric Fixed Point.

• Preface ixI Metric Spaces1 Introduction 31.1 The real numbers R 31.2 Continuous mappings in R 51.3 The triangle inequality in R 71.4 The triangle inequality in R" 81.5 Brouwer's Fixed Point Theorem 10Exercises 112 Metric Spaces 132.1 The metric topology 152.2 Examples of metric spaces 192.3 Completeness 262.4 Separability and connectedness 332.5 Metric convexity and convexity structures 35Exercises 383 Metric Contraction Principles 413.1 Banach's Contraction Principle 413.2 Further extensions of Banach's Principle 463.3 The Caristi-Ekeland Principle 553.4 Equivalents of the Caristi-Ekeland Principle 583.5 Set-valued contractions 613.6 Generalized contractions 64Exercises 674 Hyperconvex Spaces 714.1 Introduction 714.2 Hyperconvexity 774.3 Properties of hyperconvex spaces 804.4 A fixed point theorem 844.5 Intersections of hyperconvex spaces 874.6 Approximate fixed points 894.7 Isbell's hyperconvex hull 91Exercises 985 "Normal" Structures in Metric Spaces 1015.1 A fixed point theorem 1015.2 Structure of the fixed point set 1035.3 Uniform normal structure 1065.4 Uniform relative normal structure 1105.5 Quasi-normal structure 1125.6 Stability and normal structure 1155.7 Ultrametric spaces 1165.8 Fixed point set structure—separable case 120Exercises 123II Banach Spaces6 Banach Spaces: Introduction 1276.1 The definition 1276.2 Convexity 1316.3 £2 revisited 1326.4 The modulus of convexity 1366.5 Uniform convexity of the tp spaces 1386.6 The dual space: Hahn-Banach Theorem 1426.7 The weak and weak* topologies 1446.8 The spaces c, CQ, t\ and ^ 1466.9 Some more general facts 1486.10 The Schur property and £j 1506.11 More on Schauder bases in Banach spaces 1546.12 Uniform convexity and reflexivity 1636.13 Banach lattices 165Exercises 1687 Continuous Mappings in Banach Spaces 1717.1 Introduction 1717.2 Brouwer's Theorem 1737.3 Further comments on Brouwer's Theorem 1767.4 Schauder's Theorem 1797.5 Stability of Schauder's Theorem 1807.6 Banach algebras: Stone Weierstrass Theorem 1827.7 Leray-Schauder degree 1837.8 Condensing mappings 1877.9 Continuous mappings in hyperconvex spaces 191Exercises 1958 Metric Fixed Point Theory 1978.1 Contraction mappings 1978.2 Basic theorems for nonexpansive mappings 1998.3 A closer look at ßë 2058.4 Stability results in arbitrary spaces 2078.5 The Goebel-Karlovitz Lemma 2118.6 Orthogonal convexity 2138.7 Structure of the fixed point set 2158.8 Asymptotically regular mappings 2198.9 Set-valued mappings 2228.10 Fixed point theory in Banach lattices 225Exercises 2389 Banach Space Ultrapowers 2439.1 Finite representability 2439.2 Convergence of ultranets 2489.3 The Banach space ultrapower X 2499.4 Some properties of X 2529.5 Extending mappings to X 2559.6 Some fixed point theorems 2579.7 Asymptotically nonexpansive mappings 2629.8 The demiclosedness principle 2639.9 Uniformly non-creasy spaces 264Exercises 270Appendix: Set Theory 273A.l Mappings 273A.2 Order relations and Zermelo's Theorem 274A.3 Zorn's Lemma and the Axiom Of Choice 275A.4 Nets and subnets 277A.5 Tychonoff's Theorem 278A.6 Cardinal numbers 280A. 7 Ordinal numbers and transfinite induction 281A.8 Zermelo's Fixed Point Theorem 284A.9 A remark about constructive mathematics 286Exercises 287Bibliography 289Index 301

"...deserves to be on the bookshelf of everyone who wants to know about fixed-point theory in metric and Banach spaces and experts who want to read an insightful survey of some basic ideas..." (Mathematical Reviews, 2002b) "Clear, friendly exposition." (American Mathematical Monthly, August/September 2002)