Convergence of iterates of asymptotically nonexpansive mappings in Banach spaces with the uniform Opial property

for every x ∈ C, and that T be continuous for some N ≥ 1. The stronger definition (cf. Goebel and Kirk [8]) requires that each iterate T be Lipschitzian with Lipschitz constants Ln → 1 as n → ∞. For our iteration method we find it convenient to introduce a definition somewhere between these two: T is asymptotically nonexpansive in the intermediate sense provided T is uniformly continuous and lim sup n→∞ sup x,y∈C (‖Tx− Ty‖ − ‖x− y‖) ≤ 0 .

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