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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:XCB(X)

is called multivalued nonexpansive if

H(Tx,Ty)d(x,y)

for all x,yX. 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:XK(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:
47H10Fixed point theorems for nonlinear operators on topological linear spaces
47H09Mappings defined by “shrinking” properties
54H25Fixed-point and coincidence theorems in topological spaces
51K10Synthetic differential geometry
05C05Trees
53C70Direct methods (G-spaces of Busemann, etc.)
58C30Fixed point theorems on manifolds