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

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 x^∗ in a, 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.