Brailey Sims

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 Contractive Mappings in Complete -Metric Spaces

We prove some fixed point results for mappings satisfying various contractive conditions on Complete -metric Spaces. Also the Uniqueness of such fixed point are proved, as well as we showed these mappings are -continuous on such fixed points.

A class of spaces with weak normal structure

A Banach space X is said to have the weak fixed poznt property if whenever C is a nonempty weak compact convex subset of X and T : C -+ C is a nonexpansive mapping; (that is, JITx T y J < Jlx yl for all x, y E C ) , then T has a fixed point in C . It is well known that if X fails to have the weak fixed point property then it fails to have weak normal structure; that is, X contains a weak compact convex subset C with more than one point which is diametral in the sense that, for all x E C

Characterisation of normed linear spaces with Mazur's intersection property

Normed linear spaces possessing the euclidean space property that every bounded closed convex set is an intersection of closed balls, are characterised as those with dual ball having weak* denting points norm dense in the unit sphere. A characterisation of Banach spaces whose duals have a corresponding intersection property is established. The question of the density of the strongly exposed points of the ball is examined for spaces with such properties.

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.

The Douglas–Rachford Algorithm in the Absence of Convexity

The Douglas–Rachford iteration scheme, introduced half a century ago in connection with nonlinear heat flow problems, aims to find a point common to two or more closed constraint sets. Convergence of the scheme is ensured when the sets are convex subsets of a Hilbert space, however, despite the absence of satisfactory theoretical justification, the scheme has been routinely used to successfully solve a diversity of practical problems in which one or more of the constraints involved is non-convex. As a first step toward addressing this deficiency, we provide convergence results for a prototypical non-convex two-set scenario in which one of the sets is the Euclidean sphere.

LINEAR HAHN-BANACH EXTENSION OPERATORS

Given any subspace N of a Banach space X , there is a subspace M containing N and of the same density character as N , for which there exists a linear Hahn–Banach extension operator from M * to X *. This result was first proved by Heinrich and Mankiewicz [ 4 , Proposition 3.4] using some of the deeper results of Model Theory. More precisely, they used the Banach space version of the Lowenheim–Skolem theorem due to Stern [ 11 ], which in turn relies on the Lowenheim–Skolem and Keisler–Shelah theorems from Model Theory. Previously Lindenstrauss [ 7 ], using a finite dimensional lemma and a compactness argument, obtained a version of this for reflexive spaces. We shall show that the same finite dimensional lemma leads directly to the general result, without any appeal to Model Theory.

On some Banach space properties sufficient for weak normal structure and their permanence properties

We consider Banach space properties that lie between conditions introduced by Bynum and Landes. These properties depend on the metric behavior of weakly convergent sequences. We also investigate the permanence properties of these conditions.

Denseness of operators which attain their numerical radius

We show that a bounded linear operator on a uniformly convex space may be perturbed by a compact operator of arbitrarily small norm to yield an operator which attains its numerical radius.

Property (M) and the Weak Fixed Point Property

It is shown that in Banach spaces with the property (M) of Kalton, nonexpansive self mappings of nonempty weakly compact convex sets necessarily have fixed points. The stability of this conclusion under renormings is examined and conditions for such spaces to have weak normal structure are considered.