Fixed Point Theory and Applications
Volume 2010 (2010), Article ID 513956, 12 pages
doi:10.1155/2010/513956
Research Article

Some Krasnonsel'skiĭ-Mann Algorithms and the Multiple-Set Split Feasibility Problem

Huimin He,1 Sanyang Liu,1 and Muhammad Aslam Noor2,3

1Department of Mathematics, Xidian University, Xi'an 710071, China
2Mathematics Department, COMSATS Institute of Information Technology, Islamabad, Pakistan
3Mathematics Department, College of Science, King Saud University, Riyadh 11451, Saudi Arabia

Received 3 April 2010; Revised 7 July 2010; Accepted 13 July 2010

Academic Editor: S. Reich

Copyright © 2010 Huimin He 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

Some variable Krasnonsel'skiĭ-Mann iteration algorithms generate some sequences {xn}, {yn}, and  {zn}, respectively, via the formula xn+1=(1-αn)xn+αnTN⋯T2T1xn, yn+1=(1-βn)yn+βn∑i=1NλiTiyn, zn+1=(1-γn+1)zn+γn+1T[n+1]zn, where T[n]=Tn mod N and the mod function takes values in {1,2,…,N}, {αn}, {βn}, and  {γn} are sequences in (0,1), and {T1,T2,…,TN} are sequences of nonexpansive mappings. We will show, in a fairly general Banach space, that the sequence {xn}, {yn}, {zn} generated by the above formulas converge weakly to the common fixed point of {T1,T2,…,TN}, respectively. These results are used to solve the multiple-set split feasibility problem recently introduced by Censor et al. (2005). The purpose of this paper is to introduce convergence theorems of some variable Krasnonsel'skiĭ-Mann iteration algorithms in Banach space and their applications which solve the multiple-set split feasibility problem.