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Journal of Function Spaces and Applications
Volume 2012, Article ID 765903, 18 pages
http://dx.doi.org/10.1155/2012/765903
Research Article

q-Hyperconvexity in Quasipseudometric Spaces and Fixed Point Theorems

Department of Mathematics and Applied Mathematics, University of Cape Town, Rondebosch 7701, South Africa

Received 15 May 2012; Accepted 4 July 2012

Academic Editor: Salvador Romaguera

Copyright © 2012 Hans-Peter A. Künzi and Olivier Olela Otafudu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

In a previous work, we started investigating the concept of hyperconvexity in quasipseudometric spaces which we called q-hyperconvexity or Isbell-convexity. In this paper, we continue our studies of this concept, generalizing further known results about hyperconvexity from the metric setting to our theory. In particular, in the present paper, we consider subspaces of q-hyperconvex spaces and also present some fixed point theorems for nonexpansive self-maps on a bounded q-hyperconvex quasipseudometric space. In analogy with a metric result, we show among other things that a set-valued mapping T on a q-hyperconvex T0-quasimetric space (X, d) which takes values in the space of nonempty externally q-hyperconvex subsets of (X, d) always has a single-valued selection T which satisfies d(T(x),T(y))dH(T(x),T(y)) whenever x,yX. (Here, dH denotes the usual (extended) Hausdorff quasipseudometric determined by d on the set 𝒫0(X) of nonempty subsets of X.)