Search

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 $T(A)\subseteq A$ , $T(B)\subseteq B$ , and $d(Tx,Ty)\leq d(x,y)$ for all $(x,y)\in A\times B$ . A point $(p,q)\inA\times B$ is called a best proximity pair for the mapping $T$ if $p=Tp$ , $q=Tq$ , and $d(p,q)=\operatorname {dist}(A,B)$ . 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.