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Zbl 1229.54049
Best proximity point theorems.
(English)
[J] J. Approx. Theory 163, No. 11, 1772-1781 (2011). ISSN 0021-9045

Summary: Let us assume that $A$ and $B$ are non-empty subsets of a metric space. In view of the fact that a non-self mapping $T:A\to B$ does not necessarily have a fixed point, it is of considerable significance to explore the existence of an element $x$ that is as close to $Tx$ as possible. In other words, when the fixed point equation $Tx=x$ has no solution, then it is attempted to determine an approximate solution $x$ such that the error $d(x,Tx)$ 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 $Tx=x$ when there is no solution. Because $d(x,Tx)$ is at least $d(A,B)$, a best proximity point theorem ascertains an absolute minimum of the error $d(x,Tx)$ by stipulating an approximate solution $x$ of the fixed point equation $Tx=x$ to satisfy the condition that $d(x,Tx)=d(A,B)$. This article establishes best proximity point theorems for proximal contractions, thereby extending Banach's contraction principle to the case of non-self mappings.
MSC 2000:
*54H25 Fixed-point theorems in topological spaces
54E50 Complete metric spaces
41A50 Best approximation

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

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