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Zbl 1238.54021
Best proximity point theorems generalizing the contraction principle.
(English)
[J] Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 74, No. 17, 5844-5850 (2011). ISSN 0362-546X

Let $A$ and $B$ be non-void subsets of a metric space $(X,d)$ and $d(A,B)= \text{inf}\{d(x,y): x\in A$ and $y\in B\}$. An element $x\in A$ is said to be a best proximity point of the mapping $S: A\to B$ if $d(x,Sx)= d(A,B)$.\par Given non-void closed subsets $A$ and $B$ of a complete metric space, a contraction non-self-mapping $S: A\to B$ 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.
[T. D. Narang (Amritsar)]
MSC 2000:
*54H25 Fixed-point theorems in topological spaces
54E50 Complete metric spaces
41A65 Abstract approximation theory

Keywords: optimal approximate solution; fixed point; best proximity point; contraction; proximal contraction; proximal cyclic contraction

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