Nonconvex proximal normal structure in convex metric spaces

Abstract

Given that $A$ and $B$ are two nonempty subsets of the convex metric space $(X,d,\mathcal{W})$, a mapping $T:A\cup B\to A\cup B$ is noncyclic relatively nonexpansive, provided that $T\left(A\right)\subseteq A$, $T\left(B\right)\subseteq B$, and $d(Tx,Ty)\le d(x,y)$ for all $(x,y)\in A\times B$. A point $(p,q)\in A\times B$ is called a best proximity pair for the mapping $T$ if $p=Tp$, $q=Tq$, and $d(p,q)=dist(A,B)$. In this work, we study the existence of best proximity pairs for noncyclic relatively nonexpansive mappings by using the notion of nonconvex proximal normal structure. In this way, we generalize a main result of Eldred, Kirk, and Veeramani. We also establish a common best proximity pair theorem for a commuting family of noncyclic relatively nonexpansive mappings in the setting of convex metric spaces, and as an application we conclude a common fixed-point theorem.