## 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

### Keywords:

Fan’s best approximation

### Citations:

Zbl 0412.47033; Zbl 0765.47016