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Best proximity point theorems generalizing the contraction principle. (English) Zbl 1238.54021

Let A and B be non-void subsets of a metric space (X,d) and d(A,B)=inf{d(x,y):xA and yB}. An element xA is said to be a best proximity point of the mapping S:AB if d(x,Sx)=d(A,B).

Given non-void closed subsets A and B of a complete metric space, a contraction non-self-mapping S:AB is improbable to have a fixed point. So, it is quite natural to seek an element x such that d(x,Sx) is minimal, which implies that x and Sx are in close proximity to each other. The fact that d(x,Sx) is at least d(A,B), best proximity point theorems guarantee the existence of an element x such that d(x,Sx)=d(A,B). 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.

MSC:
54H25Fixed-point and coincidence theorems in topological spaces
54E50Complete metric spaces
41A65Abstract approximation theory
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