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Best approximation theorems for nonexpansive and condensing mappings in hyperconvex spaces. (English) Zbl 1163.41005
Let $(M,d)$ be a hyperconvex metric space, $X\subset M$ and $F:X\rightarrow 2^{M}$ a given multivalued mapping. The authors consider the following problems in this setting: 1) The best approximation problem: Find the condition on $F$ and on a set $A\subset M$ such that there is $x\in A$ satisfying $d(x,F(x))\leq d(y,F(x)),$ for all $y\in A;$ 2) The invariant approximation problem consisting in finding conditions for the mapping $F$ with invariant set $A$ and with $Fix(F)\neq\emptyset,$ implying that $Fix(F)\cap P_{A}(p)\neq\emptyset,$ where $P_{A}(p)$ is the metric projection of $p\in Fix(F)$ onto A; 3) The best proximity pair problem: Find conditions on $F:A\rightarrow2^{B},$ $A$ and $B$ implying that there is a point $x\in A$ such that $d(x,F(x))=\inf \{a,b):a\in A,$ $b\in B\}.$ These three problems are studied when the multivalued mapping $F$ is condensing or nonexpansive, and one obtains sufficient conditions for the existence of the solutions of these problems. For example, if the set $X\subset M$ is admissible (respectively, bounded externally hyperconvex) and $F:X\rightarrow2^{M}$ is condensing, continuous in the Hausdorff metric of $(M,d)$ and the values of $F$ are nonempty bounded externally hyperconvex (respectively admissible) subset of $(M,d),$ then $F$ has a point of best approximation (Th. 3.1).

##### MSC:
 41A65 Abstract approximation theory 54H25 Fixed-point and coincidence theorems in topological spaces 41A50 Best approximation, Chebyshev systems 47H10 Fixed-point theorems for nonlinear operators on topological linear spaces
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