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

Let us consider a map T:A→BT:A→B, where AA and BB are two nonempty subsets of a metric space XX. The aim of this article is to provide sufficient conditions for the existence of a unique point x∗x∗ in AA, called the best proximity point, which satisfies d(x∗,Tx∗)=dist(A,B):=inf{d(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), 2683–2693] and hence it provides an extension of Banach’s contraction principle to the case of non-self-mappings.