Weak and strong convergence to fixed points of asymptotically nonexpansive mappings

Let T be an asymptotically nonexpansive self-mapping of a closed bounded and convex subset of a uniformly convex Banach space which satisfies Opial's condition. It is shown that, under certain assumptions, the sequence given by x n +1 = α n T n ( x n ) + (1 - α n ) x n converges weakly to some fixed point of T . In arbitrary uniformly convex Banach spaces similar results are obtained concerning the strong convergence of ( x n ) to a fixed point of T , provided T possesses a compact iterate or satisfies a Frum-Ketkov condition of the fourth kind.