Best approximation theorems for nonexpansive and condensing mappings in hyperconvex spaces
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).
scientific article; zbMATH DE number 5528004
scientific article