Fixed Point Theory and Applications
Volume 2009 (2009), Article ID 586487, 12 pages
Research Article

The Methods of Hilbert Spaces and Structure of the Fixed-Point Set of Lipschitzian Mapping

Department of Mathematics, Rzeszów University of Technology, P.O. Box 85, 35-595 Rzeszów, Poland

Received 16 May 2009; Accepted 25 August 2009

Academic Editor: William A. Kirk

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.


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,k1 be a strongly ergodic matrix. If T:CC is a lipschitzian mapping such that liminfninfm=0,1,...k=1an,k·Tk+m2<2, then the set of fixed points FixT={xC:Tx=x} is a retract of C. This result extends and improves the corresponding results of [7, Corollary 9] and [8, Corollary 1].