Lim’s theorems for multivalued mappings in CAT(0) spaces

Let X be a complete CAT(0) space. We prove that, if E is a nonempty bounded closed convex subset of X and T :E→K(X) a nonexpansive mapping satisfying the weakly inward condition, i.e., there exists p ∈ E such that αp ⊕(1 − α)T x ⊂ IE(x) ∀x ∈ E, ∀α ∈ [0, 1], then T has a fixed point. In Banach spaces, this is a result of Lim [On asymptotic centers and fixed points of nonexpansive mappings, Canad. J. Math. 32 (1980) 421–430]. The related result for unbounded R-trees is given.  2005 Elsevier Inc. All rights reserved