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,W), a mapping T:A∪B→A∪B is noncyclic relatively nonexpansive, provided that T(A)⊆A, T(B)⊆B, and d(Tx,Ty)≤d(x,y) for all (x,y)∈A×B. A point (p,q)∈A×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.