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.