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 a nonexpansive mapping satisfying the weakly inward condition, i.e., there exists p∈E such that ∀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.