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

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  • Songnian He
  • Wenlong Zhu

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.

Suggested Citation

  • Songnian He & Wenlong Zhu, 2013. "A Modified Mann Iteration by Boundary Point Method for Finding Minimum-Norm Fixed Point of Nonexpansive Mappings," Abstract and Applied Analysis, Hindawi, vol. 2013, pages 1-6, March.
    • Handle: RePEc:hin:jnlaaa:768595
    DOI: 10.1155/2013/768595
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