Best proximity point theorems

Abstract

Let us assume that A and B are non-empty subsets of a metric space. In view of the fact that a non-self mapping T : A ⟶ B does not necessarily have a fixed point, it is of considerable significance to explore the existence of an element x that is as close to T x as possible. In other words, when the fixed point equation T x = x has no solution, then it is attempted to determine an approximate solution x such that the error d ( x , T x ) is minimum. Indeed, best proximity point theorems investigate the existence of such optimal approximate solutions, known as best proximity points, of the fixed point equation T x = x when there is no solution. Because d ( x , T x ) is at least d ( A , B ) , a best proximity point theorem ascertains an absolute minimum of the error d ( x , T x ) by stipulating an approximate solution x of the fixed point equation T x = x to satisfy the condition that d ( x , T x ) = d ( A , B ) . This article establishes best proximity point theorems for proximal contractions, thereby extending Banach’s contraction principle to the case of non-self mappings.