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Best approximation theorems for nonexpansive and condensing mappings in hyperconvex spaces. (English) Zbl 1163.41005

Let $\left(M,d\right)$ be a hyperconvex metric space, $X\subset M$ and $F:X\to {2}^{M}$ 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 $d\left(x,F\left(x\right)\right)\le d\left(y,F\left(x\right)\right),$ for all $y\in A;$

2) The invariant approximation problem consisting in finding conditions for the mapping $F$ with invariant set $A$ and with $Fix\left(F\right)\ne \varnothing ,$ implying that $Fix\left(F\right)\cap {P}_{A}\left(p\right)\ne \varnothing ,$ where ${P}_{A}\left(p\right)$ is the metric projection of $p\in Fix\left(F\right)$ onto A;

3) The best proximity pair problem: Find conditions on $F:A\to {2}^{B},$ $A$ and $B$ implying that there is a point $x\in A$ such that $d\left(x,F\left(x\right)\right)=inf\left\{a,b\right):a\in A,$ $b\in B\right\}·$

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 $F:X\to {2}^{M}$ is condensing, continuous in the Hausdorff metric of $\left(M,d\right)$ and the values of $F$ are nonempty bounded externally hyperconvex (respectively admissible) subset of $\left(M,d\right),$ 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