On selections of the metric projection and best proximity pairs in hyperconvex spaces

In this work we present new results on nonexpansive retractions and best proximity pairs in hyperconvex metric spaces. We sharpen the main results of R. Esp´ınola et al. in [3] (Nonexpansive retracts in hyperconvex spaces, J. Math. Anal. Appl. 251 (2000), 557–570) on existence of nonexpansive selections of the metric projection. More precisely we characterize those subsets of a hyperconvex metric space with the property that the metric projection onto them admits a nonexpansive selection as a subclass of sets introduced in [3]. This is a rather exceptional property with a lot of applications in approximation theory, in particular we apply it to answer in the positive the main question posed by Kirk et al. in [5] (Proximinal retracts and best proximity pair theorems, Num. Funct. Anal. Opt. 24 (2003), 851–862).Ministerio de Ciencia y TecnologíaJunta de Andalucí

Diversities, hyperconvexity and fixed points

Diversities have been recently introduced as a generalization of metrics for which a rich tight span theory could be stated. In this work we take up a number of questions about hyperconvexity, diversities and fixed points of nonexpansive mappings. Most of these questions are motivated by the study of the connection between a hyperconvex diversity and its induced metric space for which we provide some answers. Examples are given, for instance, showing that such a metric space need not be hyperconvex but still we prove, as our main result, that they enjoy the fixed point property for nonexpansive mappings provided the diversity is bounded and that this boundedness condition cannot be transferred from the diversity to the induced metric space.Ministerio de Ciencia e InnovaciónJunta de Andalucí

Geodesic Ptolemy spaces and fixed point theory

We prove that geodesic Ptolemy spaces with a continuous midpoint map are strictly convex. Moreover, we show that geodesic Ptolemy spaces with a uniformly continuous midpoint map are reflexive and that in such a setting bounded sequences have unique asymptotic centers. These properties will then be applied to yield a series of fixed point results specific to CAT(0) spaces.Dirección General de Enseñanza SuperiorJunta de AndalucíaSectoral Operational Programme Human Resources Development (Romania

Mutually nearest and farthest points of sets and the Drop Theorem in geodesic spaces

Let A and X be nonempty, bounded and closed subsets of a geodesic metric space (E, d). The minimization (resp. maximization) problem denoted by min(A, X) (resp. max(A, X)) consists in finding (a0,x0)∈A×X(a0,x0)∈A×X such that d(a0,x0)=inf{d(a,x):a∈A,x∈X}d(a0,x0)=inf{d(a,x):a∈A,x∈X} (resp. d(a0,x0)=sup{d(a,x):a∈A,x∈X}d(a0,x0)=sup{d(a,x):a∈A,x∈X}). We give generic results on the well-posedness of these problems in different geodesic spaces and under different conditions considering the set A fixed. Besides, we analyze the situations when one set or both sets are compact and prove some specific results for CAT(0) spaces. We also prove a variant of the Drop Theorem in Busemann convex geodesic spaces and apply it to obtain an optimization result for convex functions.Dirección General de Enseñanza SuperiorJunta de AntalucíaThe Sectoral Operational Programme Human Resources Developmen

On a result of W. A. Kirk

W. A. Kirk has recently proved a constructive fixed point theorem for continuous mappings in compact hyperconvex metric spaces [6]. In the present work we use the concept of hyperconvex hull of a metric space to obtain a noncompact counterpart of Kirk’s result

On best proximity points in metric and Banach spaces

In this paper we study the existence and uniqueness of best proximity points of cyclic contractions as well as the convergence of iterates to such proximity points. We do it from two different approaches, leading each one of them to different results which complete, if not improve, other similar results in the theory. Results in this paper stand for Banach spaces, geodesic metric spaces and metric spaces. We also include an appendix on CAT(0) spaces where we study the particular behavior of these spaces regarding the problems we are concerned with

CAT(k)-spaces, weak convergence and fixed points

In this paper we show that some of the recent results on ¯xed point for CAT(0) spaces still hold true for CAT(1) spaces, and so for any CAT(k) space, under natural boundedness conditions. We also introduce a new notion of convergence in geodesic spaces which is related to the ¢-convergence and applied to study some aspects on the geometry of CAT(0) spaces. At this point, two recently posed questions in [12] (W.A. Kirk and B. Panyanak, A concept of convergence in geodesic spaces, Nonlinear Anal. 68 (12) (2008), 3689-3696) are answered in the negative. The work ¯nishes with the study of the Lif¸sic characteristic and property (P) of Lim-Xu to derive ¯xed point results for uniformly lipschitzian mappings in CAT(k) spaces. A conjecture raised in [4] (S. Dhompongsa, W.A. Kirk and B. Sims, Fixed points of uniformly lipschitzian mappings, Nonlinear Anal., 65 (2006), 762{772) on the Lif¸sic characteristic function of CAT(k) spaces is solved in the positive

Fixed points and approximate fixed points in product spaces

The paper deals with the general theme of what is known about the existence of fixed points and approximate fixed points for mappings which satisfy geometric conditions in product spaces. In particular it is shown that if X and Y are metric spaces each of which has the fixed point property for nonexpansive mappings, then the product space (X ×Y )∞ has the fixed point property for nonexpansive mappings satisfying various contractive conditions. It is also shown that the product space H = (M × K)∞ has the approximate fixed point property for nonexpansive mappings whenever M is a metric space which has the approximate fixed point property for such mappings and K is a bounded convex subset of a Banach space.Dirección General de Investigación Científica y Técnic

Metric fixed point theory on hyperconvex spaces: recent progress

In this survey we present an exposition of the development during the last decade of metric fixed point theory on hyperconvex metric spaces. Therefore we mainly cover results where the conditions on the mappings are metric. We will recall results about proximinal nonexpansive retractions and their impact into the theory of best approximation and best proximity pairs. A central role in this survey will be also played by some recent developments on R-trees. Finally, some considerations and new results on the extension of compact mappings will be shown