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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, XM and F:X2 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 AM such that there is xA satisfying d(x,F(x))d(y,F(x)), for all yA;

2) The invariant approximation problem consisting in finding conditions for the mapping F with invariant set A and with Fix(F), implying that Fix(F)P A (p), where P A (p) is the metric projection of pFix(F) onto A;

3) The best proximity pair problem: Find conditions on F:A2 B , A and B implying that there is a point xA such that d(x,F(x))=inf{a,b):aA, bB}·

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 XM is admissible (respectively, bounded externally hyperconvex) and F:X2 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).

41A65Abstract approximation theory
54H25Fixed-point and coincidence theorems in topological spaces
41A50Best approximation, Chebyshev systems
47H10Fixed point theorems for nonlinear operators on topological linear spaces