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

Songnian He and Wenlong Zhu

Abstract and Applied Analysis, 2013, vol. 2013, 1-6

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.

Date: 2013
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Persistent link: https://EconPapers.repec.org/RePEc:hin:jnlaaa:768595

DOI: 10.1155/2013/768595

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