Best proximity point theorems

Let us assume that AA and BB are non-empty subsets of a metric space. In view of the fact that a non-self mapping T:A⟶BT:A⟶B does not necessarily have a fixed point, it is of considerable significance to explore the existence of an element xx that is as close to TxTx as possible. In other words, when the fixed point equation Tx=xTx=x has no solution, then it is attempted to determine an approximate solution xx such that the error d(x,Tx)d(x,Tx) 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 Tx=xTx=x when there is no solution. Because d(x,Tx)d(x,Tx) is at least d(A,B)d(A,B), a best proximity point theorem ascertains an absolute minimum of the error d(x,Tx)d(x,Tx) by stipulating an approximate solution xx of the fixed point equation Tx=xTx=x to satisfy the condition that d(x,Tx)=d(A,B)d(x,Tx)=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.