Best proximity pair results for relatively nonexpansive mappings in geodesic spaces

TL;DR

This paper extends the theory of best proximity points for relatively nonexpansive mappings from Banach spaces to more general metric spaces, including Busemann convex, CAT(0), and uniformly convex geodesic spaces.

Contribution

It generalizes existing results by establishing the existence of best proximity points in broader metric space settings and explores conditions like proximal normal structure.

Findings

Existence of best proximity points in Busemann convex reflexive metric spaces.

Results specialized to CAT(0) and uniformly convex geodesic spaces.

Proximal normal structure is sufficient but not necessary for existence.

Abstract

Given $A$ and $B$ two nonempty subsets in a metric space, a mapping $T : A \cup B \rightarrow A \cup B$ is relatively nonexpansive if $d(Tx,Ty) \leq d(x,y) \text{for every} x\in A, y\in B.$ A best proximity point for such a mapping is a point $x \in A \cup B$ such that $d(x,Tx)=\text{dist}(A,B)$. In this work, we extend the results given in [A.A. Eldred, W.A. Kirk, P. Veeramani, Proximal normal structure and relatively nonexpansive mappings, Studia Math., 171 (2005), 283-293] for relatively nonexpansive mappings in Banach spaces to more general metric spaces. Namely, we give existence results of best proximity points for cyclic and noncyclic relatively nonexpansive mappings in the context of Busemann convex reflexive metric spaces. Moreover, particular results are proved in the setting of CAT(0) and uniformly convex geodesic spaces. Finally, we show that proximal normal structure is a sufficient but not necessary condition for the existence in $A \times B$ of a pair of best proximity points.