Copyright © 2009 Jarosław Górnicki. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The purpose of this paper is to prove, by asymptotic center techniques and the methods of Hilbert spaces, the following theorem. Let H be a Hilbert space, let C be a nonempty bounded closed convex subset of H, and let M=[an,k]n,k≥1 be a strongly ergodic matrix. If T:C→C is a lipschitzian mapping such that lim⁡inf⁡n→∞inf⁡m=0,1,...∑k=1∞an,k·‖Tk+m‖2<2, then the set of fixed points Fix T={x∈C:Tx=x} is a retract of C. This result extends and improves the corresponding results of [7, Corollary 9] and [8, Corollary 1].