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Best proximity point theorems for generalized proximal contractions. (English) Zbl 1282.41012
This paper deals with best proximity points of mappings $T$ with no fixed points, that is, mappings for which the equation $Tx=x$ does not need to have a solution and then the interest is to find such points $x$ minimizing $d(x,Tx)$. The typical situation in this paper is a mapping $T: A\to B$ where $A$ and $B$ are non-empty subsets of a complete metric space such that $\text{dist}(A,B):= \inf_{x\in A,y\in B}d(x,y)>0$. $A$ is supposed to be closed and $B$ approximatively compact with respect to $A$. Results obtained here are for the so-called generalized proximal contraction mappings of first and second kind and, at least formally, they extend some other recent results on best proximity points for proximal contraction type mappings.

41A65Abstract approximation theory
46B20Geometry and structure of normed linear spaces
47H10Fixed-point theorems for nonlinear operators on topological linear spaces
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