Proximal pairs and relatively nonexpansive mappings in hyperconvex spaces

In the current paper, we present new and more general conditions to ensure the nonemptyness of proximal pairs in hyperconvex spaces and use them to investigate the existence of a best proximity point for multivalued non-self mappings in such spaces. In this way, we extend and improve the main conclusions of Kirk et al. (Numer Funct Anal Opt 24:851-862, 2003). We also discuss the existence of best proximity points (pairs) for cyclic (noncyclic) relatively nonexpansive mappings and obtain counterpart results of Eladred et al. (Stud Math 17:283-293, 2005) in the framework of hyperconvex metric spaces and in a special case, in the non-reflexive Banach space & ell;infinity\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell _\infty $$\end{document}, where it is bounded, closed and convex subsets may not be weakly compact.