Journal of Applied Mathematics
Volume 2012 (2012), Article ID 327878, 9 pages
http://dx.doi.org/10.1155/2012/327878
Research Article

Necessary and Sufficient Condition for Mann Iteration Converges to a Fixed Point of Lipschitzian Mappings

Chang-He Xiang,1 Jiang-Hua Zhang,2 and Zhe Chen1

1College of Mathematics, Chongqing Normal University, Chongqing 400047, China
2School of Management, Shandong University, Shandong Jinan 250100, China

Received 18 July 2012; Revised 2 September 2012; Accepted 2 September 2012

Academic Editor: Jian-Wen Peng

Copyright © 2012 Chang-He Xiang 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 is a real normed linear space, is a nonempty convex subset of , is a Lipschitzian mapping, and is a fixed point of . For given , suppose that the sequence is the Mann iterative sequence defined by , where is a sequence in [0, 1], , . We prove that the sequence strongly converges to if and only if there exists a strictly increasing function with such that .