International Journal of Mathematics and Mathematical Sciences
Volume 2008 (2008), Article ID 687815, 10 pages
Research Article

Convergence to Common Fixed Point for Generalized Asymptotically Nonexpansive Semigroup in Banach Spaces

Yali Li, Jianjun Liu, and Lei Deng

School of Mathematics and Statistics, Southwest University, Chongqing 400715, China

Received 13 May 2008; Accepted 14 August 2008

Academic Editor: Nils Ackermann

Copyright © 2008 Yali 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.


Let K be a nonempty closed convex subset of a reflexive and strictly convex Banach space E with a uniformly Gâteaux differentiable norm, ={T(h):h0} a generalized asymptotically nonexpansive self-mapping semigroup of K, and f:KK a fixed contractive mapping with contractive coefficient β(0,1). We prove that the following implicit and modified implicit viscosity iterative schemes {xn} defined by xn=αnf(xn)+(1αn)T(tn)xn and xn=αnyn+(1αn)T(tn)xn,yn=βnf(xn1)+(1βn)xn1 strongly converge to pF as n and p is the unique solution to the following variational inequality: f(p)p,j(yp)0 for all yF.