A best proximity point theorem for weakly contractive non-self-mappings

摘要

Let us consider a map T:A→B, where A and B are two nonempty subsets of a metric space X. The aim of this article is to provide sufficient conditions for the existence of a unique point * in A, called the best proximity point, which satisfies d(x*,Tx*)= dist(A,B):=infd(a,b):a∈A,b∈B. Our result generalizes a result due to Rhoades [B.E. Rhoades, Some theorems on weakly contractive maps, Nonlinear Analysis TMA, 47(2001), 26832693] and hence it provides an extension of Banach's contraction principle to the case of non-self-mappings.