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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 $\left(X,d\right)$ and $d\left(A,B\right)=\text{inf}\left\{d\left(x,y\right):x\in A$ and $y\in B\right\}$. An element $x\in A$ is said to be a best proximity point of the mapping $S:A\to B$ if $d\left(x,Sx\right)=d\left(A,B\right)$.

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 $d\left(x,Sx\right)$ is minimal, which implies that $x$ and $Sx$ are in close proximity to each other. The fact that $d\left(x,Sx\right)$ is at least $d\left(A,B\right)$, best proximity point theorems guarantee the existence of an element $x$ such that $d\left(x,Sx\right)=d\left(A,B\right)$. 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:
 54H25 Fixed-point and coincidence theorems in topological spaces 54E50 Complete metric spaces 41A65 Abstract approximation theory