Weak and strong convergence to fixed points of asymptotically nonexpansive mappings

Abstract

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 = I n Tn (xn) + (1 - In) xn e converges weakly to some fixed point of T. In arbitrary uniformly convex Banach spaces similar results are obtained concerning the strong convergence of (xn) to a fixed point of T, provided T possesses a compact iterate or satisfies a Frum-Ketkov condition of the fourth kind. © 1991, Australian Mathematical Society. All rights reserved.