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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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