Fixed Point Theory and Applications
Volume 2009 (2009), Article ID 586487, 12 pages
doi:10.1155/2009/586487

The methods of Hilbert spaces and structure of the fixed-point set of Lipschitzian mapping

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].