An introduction to metric spaces and fixed point theory [[electronic resource] /] / Mohamed A. Khamsi, William A. Kirk

• Record Nr.: UNINA9910139576003321
• Autore: Khamsi Mohamed A
• Pubbl/distr/stampa: New York, : John Wiley, c2001
• ISBN: 1-283-30626-3; 9786613306265; 1-118-03307-8; 1-118-03132-6
• Descrizione fisica: 1 online resource (318 p.)
• Collana: Pure and applied mathematics
• Altri autori (Persone): KirkW. A
• Disciplina: 514; 514.32; 514/.32
• Soggetti: Fixed point theory; Metric spaces
• Lingua di pubblicazione: Inglese
• Formato: Materiale a stampa
• Livello bibliografico: Monografia
• Note generali: Description based upon print version of record.
• Nota di bibliografia: Includes bibliographical references (p. 289-299) and index.
• Nota di contenuto: An Introduction to Metric Spaces and Fixed Point Theory; Contents; Preface; I Metric Spaces; 1 Introduction; 1.1 The real numbers R; 1.2 Continuous mappings in R; 1.3 The triangle inequality in R; 1.4 The triangle inequality in Rn; 1.5 Brouwer's Fixed Point Theorem; Exercises; 2 Metric Spaces; 2.1 The metric topology; 2.2 Examples of metric spaces; 2.3 Completeness; 2.4 Separability and connectedness; 2.5 Metric convexity and convexity structures; Exercises; 3 Metric Contraction Principles; 3.1 Banach's Contraction Principle; 3.2 Further extensions of Banach's Principle; 3.3 The Caristi-Ekeland Principle3.4 Equivalents of the Caristi-Ekeland Principle; 3.5 Set-valued contractions; 3.6 Generalized contractions; Exercises; 4 Hyperconvex Spaces; 4.1 Introduction; 4.2 Hyperconvexity; 4.3 Properties of hyperconvex spaces; 4.4 A fixed point theorem; 4.5 Intersections of hyperconvex spaces; 4.6 Approximate fixed points; 4.7 Isbell's hyperconvex hull; Exercises; 5 ""Normal"" Structures in Metric Spaces; 5.1 A fixed point theorem; 5.2 Structure of the fixed point set; 5.3 Uniform normal structure; 5.4 Uniform relative normal structure; ; 5.5 Quasi-normal structure; 5.6 Stability and normal structure5.7 Ultrametric spaces; 5.8 Fixed point set structure-separable case; Exercises; II Banach Spaces; 6 Banach Spaces: Introduction; 6.1 The definition; 6.2 Convexity; 6.3 l2 revisited; 6.4 The modulus of convexity; 6.5 Uniform convexity of the lp spaces; 6.6 The dual space: Hahn-Banach Theorem; 6.7 The weak and weak* topologies; 6.8 The spaces c, c0, l1 and l(infinity); 6.9 Some more general facts; 6.10 The Schur property and l1; 6.11 More on Schauder bases in Banach spaces; 6.12 Uniform convexity and reflexivity; 6.13 Banach lattices; Exercises; 7 Continuous Mappings in Banach Spaces7.1 Introduction; 7.2 Brouwer's Theorem; 7.3 Further comments on Brouwer's Theorem; 7.4 Schauder's Theorem; 7.5 Stability of Schauder's Theorem; 7.6 Banach algebras: Stone Weierstrass Theorem; 7.7 Leray-Schauder degree; 7.8 Condensing mappings; 7.9 Continuous mappings in hyperconvex spaces; Exercises; 8 Metric Fixed Point Theory; 8.1 Contraction mappings; 8.2 Basic theorems for nonexpansive mappings; 8.3 A closer look at l1; 8.4 Stability results in arbitrary spaces; 8.5 The Goebel-Karlovitz Lemma; 8.6 Orthogonal convexity; 8.7 Structure of the fixed point set8.8 Asymptotically regular mappings; 8.9 Set-valued mappings; 8.10 Fixed point theory in Banach lattices; Exercises; 9 Banach Space Ultrapowers; 9.1 Finite representability; 9.2 Convergence of ultranets; 9.3 The Banach space ultrapower X; 9.4 Some properties of X; 9.5 Extending mappings to X; 9.6 Some fixed point theorems; 9.7 Asymptotically nonexpansive mappings; 9.8 The demiclosedness principle; 9.9 Uniformly non-creasy spaces; Exercises; Appendix: Set Theory; A.1 Mappings; A.2 Order relations and Zermelo's Theorem; A.3 Zorn's Lemma and the Axiom Of Choice
• Sommario/riassunto: Presents up-to-date Banach space results.* Features an extensive bibliography for outside reading.* Provides detailed exercises that elucidate more introductory material.