Let and be non-void subsets of a metric space and and . An element is said to be a best proximity point of the mapping if .
Given non-void closed subsets and of a complete metric space, a contraction non-self-mapping is improbable to have a fixed point. So, it is quite natural to seek an element such that is minimal, which implies that and are in close proximity to each other. The fact that is at least , best proximity point theorems guarantee the existence of an element such that . The famous Banach contraction principle asserts that every contraction self-mapping on a complete metric space has a unique point. This article explores some interesting generalizations of the contraction principle to the case of non-self-mappings. The proposed extensions are presented as best proximity point theorems for non-self-proximal contractions.