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Fan’s best approximation and fixed point theorems for multivalued nonexpansive mappings in Hilbert spaces. (English) Zbl 0797.41022

Let \(K\) be a closed convex unbounded subset of a Hilbert space \(H\) and \({\mathcal {CB}}(K)\) the family of nonempty closed bounded subsets of \(K\). A multivalued map \(T: K\to {\mathcal {CB}}(K)\) is said to be nonexpansive if for every \(x,y\in K\), \({\mathcal D}(Tx,Ty)\leq \| x-y\|\), where \({\mathcal D}(A,B)\) is the Hausdorff distance between the sets \(A\) and \(B\). In this paper some Fan’s best approximation and fixed point theorems for multivalued nonexpansive maps \(T:K\to {\mathcal {CB}}(K)\) are proved. Similar problems was considered by W. A. Kirk and W. O. Ray [Studia Math. 64, 127-138 (1972; Zbl 0412.47033] and by the authors [Nonlinear Anal., Theory Methods Appl. 17, No. 1, 11-20 (1991; Zbl 0765.47016)].

MSC:

41A50 Best approximation, Chebyshev systems
47H10 Fixed-point theorems
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