We prove the demiclosed principle for asymptotically nonexpansive mappings in CAT 0 spaces. As a consequence, we obtain a Δ-convergence theorem of the Krasnosel'skii-Mann iteration for asymptotically nonexpansive mappings in this setting. Our results extend and improve many results in the literature.
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 ⊂ 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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