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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 768595, 6 pages
http://dx.doi.org/10.1155/2013/768595
Research Article

A Modified Mann Iteration by Boundary Point Method for Finding Minimum-Norm Fixed Point of Nonexpansive Mappings

1College of Science, Civil Aviation University of China, Tianjin 300300, China
2Tianjin Key Laboratory for Advanced Signal Processing, Civil Aviation University of China, Tianjin 300300, China

Received 22 November 2012; Accepted 13 February 2013

Academic Editor: Satit Saejung

Copyright © 2013 Songnian He and Wenlong Zhu. 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 be a real Hilbert space and a closed convex subset. Let be a nonexpansive mapping with the nonempty set of fixed points . Kim and Xu (2005) introduced a modified Mann iteration , , , where is an arbitrary (but fixed) element, and and are two sequences in . In the case where , the minimum-norm fixed point of can be obtained by taking . But in the case where , this iteration process becomes invalid because may not belong to . In order to overcome this weakness, we introduce a new modified Mann iteration by boundary point method (see Section 3 for details) for finding the minimum norm fixed point of and prove its strong convergence under some assumptions. Since our algorithm does not involve the computation of the metric projection , which is often used so that the strong convergence is guaranteed, it is easy implementable. Our results improve and extend the results of Kim, Xu, and some others.