Bulletin of the Australian Mathematical Society

Research Article

Weak and strong convergence to fixed points of asymptotically nonexpansive mappings

J. Schua1

a1 RWTH Aachen, Lehrstuhl C für Mathematik, Templergraben 55, D-5100 Aachen, Federal Republic of Germany

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 xn+1 = αnTn(xn) + (1 - αn)xn 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.

(Received March 19 1990)