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Lim’s theorems for multivalued mappings in CAT(0) spaces. (English) Zbl 1086.47019
Let $$(X, D)$$ be a metric space, $$CB(X)$$ (resp., $$K(X)$$) the family of nonempty closed bounded (resp., the family of nonempty compact) subsets of $$X$$, and $$H$$ the Hausdorff metric on $$CB(X)$$ induced by $$d$$. Then a map $T: X \rightarrow CB(X)$ is called multivalued nonexpansive if $H(Tx, Ty) \leq d(x, y)$ for all $$x, y \in X$$. The authors attempt to investigate condions under which a multivalued nonexpansive map may have a fixed point. Since such a map on a complete metric space need not have a fixed point, the authors work in a complete CAT(0) space [cf. W. A. Kirk, in: Proceedings of the international conference on fixed-point theory and its applications, Valencia, Spain, July 13–19, 2003, 113–142 (2004; Zbl 1083.53061)] and assume some “inwardness” requirement on the map. Their main result goes as follows. Let $$E$$ be a nonempty bounded closed convex subset of a complete CAT(0) space $$X$$ and $$T: X \rightarrow K(X)$$ a nonexpansive map. Assume that $$T$$ is weakly inward on $$E$$. Then $$T$$ has a fixed point.
Another main result is about the existence of a common fixed point of a single-valued nonexpansive map commuting with a multivalued nonexpansive map.

##### MSC:
 47H10 Fixed-point theorems 47H09 Contraction-type mappings, nonexpansive mappings, $$A$$-proper mappings, etc. 54H25 Fixed-point and coincidence theorems (topological aspects) 51K10 Synthetic differential geometry 05C05 Trees 53C70 Direct methods ($$G$$-spaces of Busemann, etc.) 58C30 Fixed-point theorems on manifolds
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##### References:
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