International Journal of Mathematics and Mathematical Sciences
Volume 2007 (2007), Article ID 48648, 10 pages
Research Article

Viscosity Approximation Methods for Nonexpansive Nonself-Mappings in Hilbert Spaces

Rabian Wangkeeree

Department of Mathematics, Faculty of Science, Naresuan University, Phitsanulok 65000, Thailand

Received 26 October 2006; Revised 22 January 2007; Accepted 28 January 2007

Academic Editor: Andrei I. Volodin

Copyright © 2007 Rabian Wangkeeree. 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.


Viscosity approximation methods for nonexpansive nonself-mappings are studied. Let C be a nonempty closed convex subset of Hilbert space H, P a metric projection of H onto C and let T be a nonexpansive nonself-mapping from C into H. For a contraction f on C and {tn}(0,1), let xn be the unique fixed point of the contraction xtnf(x)+(1tn)(1/n)j=1n(PT)jx. Consider also the iterative processes {yn} and {zn} generated by yn+1=αnf(yn)+(1αn)(1/(n+1))j=0n(PT)jyn, n0, and zn+1=(1/(n+1))j=0nP(αnf(zn)+(1αn)(TP)jzn),n0, where y0,z0C,{αn} is a real sequence in an interval [0,1]. Strong convergence of the sequences {xn},{yn}, and {zn} to a fixed point of T which solves some variational inequalities is obtained under certain appropriate conditions on the real sequences {αn} and {tn}.