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Kohlenbach and Leustean have shown in 2010 that any asymptoticallynonexpansive self-mapping of a bounded nonempty $UCW$-hyperbolic space has afixed point. In this paper, we adapt a construction due to Moloney in order toprovide a sequence that converges strongly to such a fixed point.

Construction of fixed points of asymptotically nonexpansive mappings in uniformly convex hyperbolic spaces