W. A. Kirk

Topics in metric fixed point theory

Introduction 1. Preliminaries 2. Banachs contraction principle 3. Nonexpansive mappings: introduction 4. The basic fixed point theorems for nonexpansive mappings 5. Scaling the convexity of the unit ball 6. The modulus of convexity and normal structure 7. Normal structure and smoothness 8. Conditions involving compactness 9. Sequential approximation techniques 10. Weak sequential approximations 11. Properties of fixed point sets and minimal sets 12. Special properties of Hilbert space 13. Applications to accretivity 14. Nonstandard methods 15. Set-valued mappings 16. Uniformly Lipschitzian mappings 17. Rotative mappings 18. The theorems of Brouwer and Schauder 19. Lipschitzian mappings 20. Minimal displacement 21. The retraction problem References.

Handbook of metric fixed point theory

Preface. 1. Contraction Mappings and Extensions W.A. Kirk. 2. Examples of Fixed Point Free Mappings B. Sims. 3. Classical Theory of Nonexpansive Mappings K. Goebel, W.A. Kirk. 4. Geometrical Background of Metric Fixed Point Theory S. Prus. 5. Some Moduli and Constants Related to Metric Fixed Point Theory E.L. Fuster. 6. Ultra-Methods in Metric Fixed Point Theory M.A. Khamsi, B. Sims. 7. Stability of the Fixed Point Property for Nonexpansive Mappings J. Garcia-Falset, A. Jimenez-Melado, E. Llorens-Fuster. 8. Metric Fixed Point Results Concerning Measures of Noncompactness T. Dominguez, M.A. Japon, G. Lopez. 9. Renormings of l1 and c0 and Fixed Point Properties P.N. Dowling, C.J. Lennard, B. Turett. 10. Nonexpansive Mappings: Boundary/Inwardness Conditions and Local Theory W.A. Kirk, C.H. Morales. 11. Rotative Mappings and Mappings with Constant Displacement W. Kaczor, M. Koter-Morgowska. 12. Geometric Properties Related to Fixed Point Theory in Some Banach Function Lattices S. Chen, Y. Cui, H. Hudzik, B. Sims. 13. Introduction to Hyperconvex Spaces R. Espinola, M.A. Khamsi. 14. Fixed Points of Holomorphic Mappings: A Metric Approach T. Kuczumow, S. Reich, D. Shoikhet. 15. Fixed Point and Non-Linear Ergodic Theorems for Semigroups of Non-Linear Mappings A. To-Ming Lau, W. Takahashi. 16. Generic Aspects of Metric Fixed Point Theory S. Reich, A.J. Zaslavski. 17. Metric Environment of the TopologicalFixed Point Theorms K. Goebel. 18. Order-Theoretic Aspects of Metric Fixed Point Theory J. Jachymski. 19. Fixed Point and Related Theorems for Set-Valued Mappings G. X.-Z. Yuan. Index.

Fixed point theorems for non-Lipschitzian mappings of asymptotically nonexpansive type

LetX be a Banach space,K a nonempty, bounded, closed and convex subset ofX, and supposeT:K→K satisfies: for eachx∈K, lim supi→∞{supy∈K‖tix−Tiy∼−‖x−y‖}≦0. IfTN is continuous for some positive integerN, and if either (a)X is uniformly convex, or (b)K is compact, thenT has a fixed point inK. The former generalizes a theorem of Goebel and Kirk for asymptotically nonexpansive mappings. These are mappingsT:K→K satisfying, fori sufficiently large, ‖Tix−Tiy‖≦ki‖x−y∼,x,y∈K, whereki→1 asi→∞. The precise assumption in (a) is somewhat weaker than uniform convexity, requiring only that Goebel’s characteristic of convexity, ɛ0 (X), be less than one.

Proximinal Retracts and Best Proximity Pair Theorems

Abstract This note is concerned with proximinality and best proximity pair theorems in hyperconvex metric spaces and in Hilbert spaces. Given two subsets A and B of a metric space and a mapping best proximity pair theorems provide sufficient conditions that ensure the existence of an such that Thus such theorems provide optimal approximate solutions in the case that the mapping T does not have fixed points.

Krasnoselskii's iteration process in hyperbolic space

A well-known iteration scheme due to Krasnoselskii for approximation of fixed points of nonexpansive mappings in Banach spaces is extended to a wider class of spaces. This class includes convex metric spaces of ‘hyperbolic’ type, and the results apply to the study of holomorphic self-mappings of the unit ball in complex Hilbert space.

Fixed Point Theory in Distance Spaces

Preface.- Part 1. Metric Spaces.- Introduction.- Caristis Theorem and Extensions.- Nonexpansive Mappings and Zermelos Theorem.- Hyperconvex metric spaces.- Ultrametric spaces.- Part 2. Length Spaces and Geodesic Spaces.- Busemann spaces and hyperbolic spaces.- Length spaces and local contractions.- The G-spaces of Busemann.- CAT(0) Spaces.- Ptolemaic Spaces.- R-trees (metric trees).- Part 3. Beyond Metric Spaces.- b-Metric Spaces.- Generalized Metric Spaces.- Partial Metric Spaces.- Diversities.- Bibliography.- Index.

Contraction Mappings and Extensions

A complete survey of all that has been written about contraction mappings would appear to be nearly impossible, and perhaps not really useful. In particular the wealth of applications of Banach’s contraction mapping principle is astonishingly diverse. We only attempt to touch on some of the high points of this profound and seminal development in metric fixed point theory.

Fixed point theorems in spaces and -trees

We show that if is a bounded open set in a complete space , and if is nonexpansive, then always has a fixed point if there exists such that for all . It is also shown that if is a geodesically bounded closed convex subset of a complete -tree with , and if is a continuous mapping for which for some and all , then has a fixed point. It is also noted that a geodesically bounded complete -tree has the fixed point property for continuous mappings. These latter results are used to obtain variants of the classical fixed edge theorem in graph theory.

The Knaster-Kuratowski and Mazurkiewicz theroy in hyperconvex metric spaces and some of its applications

The Knaster-Kuratowski and Mazurkiewicz principle is characterized in hyperconvex metric spaces, leading to a characterization theorem for a family of subsets with the finite intersection property in such setting. The theorem is illustrated by giving hyperconvex versions of Fans celebrated minimax principle and Fans best approximation theorem for set-valued mappings. These are applied to obtain formulations of the Browder-Fan fixed point theorem and the Schauder-Tychonoff fixed point theorem in hyperconvex metric spaces for set-valued mappings. In addition, existence theorems for saddle points, intersection theorems and Nash equilibria are obtained.

On successive approximations for nonexpansive mappings in Banach spaces

Let X be a Banach space and K a convex subset of X . A mapping T of K into K is called a nonexpansive mapping if | T (x) – T (y) | ≦ | x – y | for all x, ye K .