Existence and Uniqueness of a Solution for Some Nonlinear Programming Problems

Abstract

Let us consider the mappings \({S : A \to A}\) and \({T : A \to B}\), where A and B are two nonempty subsets of a metric space (X, d). The purpose of this paper is to provide sufficient conditions to ensure the existence and uniqueness of an element \({p \in A}\) such that \({d(Sp, Tp) = {\rm dist}(A, B) := {\rm inf}\{d(x, y) : (x, y) \in A \times B\}}\). In particular, the existence and uniqueness of a best proximity point for non-self-mappings are obtained. Examples are given to show usability of our results.