Fixed Point Theory and Applications
Volume 2007 (2007), Article ID 064874, 10 pages
doi:10.1155/2007/64874
Research Article
A New Iterative Algorithm for Approximating Common Fixed Points for Asymptotically Nonexpansive Mappings
1Department of Applied Mathematics, North China Electric Power University, Baoding 071003, China
2Department of Mathematics Education and RINS, College of Natural Sciences, Gyeongsang National University, Chinju 660-701, South Korea
Received 28 February 2007; Accepted 13 April 2007
Academic Editor: Nan-Jing Huang
Copyright © 2007 H. Y. Zhou 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
Suppose that K is a nonempty closed convex subset of a real uniformly convex and smooth Banach space E with P as a sunny nonexpansive retraction. Let T1,T2:K→E be two weakly inward and asymptotically nonexpansive mappings with respect to P with sequences {Kn},{ln}⊂[1,∞), limn→∞kn=1, limn→∞ln=1, F(T1)∩F(T2)={x∈K:T1x=T2x=x}≠∅, respectively. Suppose that {xn} is a sequence in K generated iteratively by x1∈K, xn+1=αnxn+βn(PT1)nxn+γn(PT2)nxn, for all n≥1, where {αn}, {βn}, and {γn} are three real sequences in [ε,1−ε] for some ε>0 which satisfy condition αn+βn+γn=1. Then, we have the following. (1) If one of T1 and T2 is completely continuous or demicompact and ∑n=1∞(kn−1)<∞,∑n=1∞(ln−1)<∞, then the strong convergence of {xn} to some q∈F(T1)∩F(T2) is established. (2) If E is a real uniformly convex Banach space satisfying Opial's condition or whose norm is Fréchet differentiable, then the weak convergence of {xn} to some q∈F(T1)∩F(T2) is proved.