Journal of Applied Mathematics and Stochastic Analysis
Volume 2009 (2009), Article ID 320820, 11 pages
doi:10.1155/2009/320820
Research Article

Modified Iterative Algorithms for Nonexpansive Mappings

Yonghong Yao,1 Muhammad Aslam Noor,2 and Syed Tauseef Mohyud-Din2

1Department of Mathematics, Tianjin Polytechnic University, Tianjin 300160, China
2Mathematics Department, Faculty of Sciences, COMSATS Institute of Information Technology, Islamabad, Pakistan

Received 22 November 2008; Accepted 2 March 2009

Academic Editor: Hong Kun Xu

Copyright © 2009 Yonghong Yao 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 H be a real Hilbert space, let S, T be two nonexpansive mappings such that F(S)∩F(T)≠∅, let f be a contractive mapping, and let A be a strongly positive linear bounded operator on H. In this paper, we suggest and consider the strong converegence analysis of a new two-step iterative algorithms for finding the approximate solution of two nonexpansive mappings as xn+1=βnxn+(1−βn)Syn, yn=αnγf(xn)+(I−αnA)Txn, n≥0 is a real number and {αn}, {βn} are two sequences in (0,1) satisfying the following control conditions: (C1) lim⁡n→∞ αn=0, (C3) 0<lim⁡inf⁡n→∞ βn≤lim⁡sup⁡n→∞ βn<1, then ‖xn+1−xn‖→0. We also discuss several special cases of this iterative algorithm.