Fixed Point Theory and Applications
Volume 2008 (2008), Article ID 484050, 13 pages
doi:10.1155/2008/484050
Research Article
Composite Implicit General Iterative Process for a Nonexpansive Semigroup in Hilbert Space
1Department of Mathematic and Physics, Hebei Normal University of Science and Technology Qinhuangdao, Hebei 066004, China
2Department of Mathematics, Tianjin Polytechnic University, Tianjin 300160, China
Received 19 March 2008; Accepted 14 August 2008
Academic Editor: Hélène Frankowska
Copyright © 2008 Lihua Li et al. 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
Let C be nonempty closed convex subset of real
Hilbert space H.
Consider C a nonexpansive semigroup ℑ={T(s):s≥0} with a common fixed point, a contraction f with coefficient 0<α<1,
and a strongly positive linear bounded operator A with coefficient γ¯>0. Let 0<γ<γ¯/α.
It is proved that the sequence {xn} generated iteratively by xn=(I−αnA)(1/tn)∫0tnT(s)ynds+αnγf(xn),yn=(I−βnA)xn+βnγf(xn) converges strongly to a common fixed point x∗∈F(ℑ) which solves the variational inequality 〈(γf−A)x∗,z−x∗〉≤0 for all z∈F(ℑ).