Abstract
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 α p ⊕ ( 1 − α ) T x ⊂ I E ( 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.
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