Best proximity point theorems Original Research Article

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