Abstract and Applied Analysis

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

Songnian He and Wenlong Zhu

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Abstract

Let H be a real Hilbert space and C H  a closed convex subset. Let T : C C be a nonexpansive mapping with the nonempty set of fixed points F i x ( T ) . Kim and Xu (2005) introduced a modified Mann iteration x 0 = x C , y n = α n x n + ( 1 α n ) T x n , x n + 1 = β n u + ( 1 β n ) y n , where u C is an arbitrary (but fixed) element, and { α n } and { β n } are two sequences in ( 0 , 1 ) . In the case where 0 C , the minimum-norm fixed point of T can be obtained by taking u = 0 . But in the case where 0 C , this iteration process becomes invalid because x n may not belong to C . 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  T and prove its strong convergence under some assumptions. Since our algorithm does not involve the computation of the metric projection P C , 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.

Article information

Source
Abstr. Appl. Anal., Volume 2013 (2013), Article ID 768595, 6 pages.

Dates
First available in Project Euclid: 27 February 2014

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1393511808

Digital Object Identifier
doi:10.1155/2013/768595

Mathematical Reviews number (MathSciNet)
MR3039125

Zentralblatt MATH identifier
06209437

Citation

He, Songnian; Zhu, Wenlong. A Modified Mann Iteration by Boundary Point Method for Finding Minimum-Norm Fixed Point of Nonexpansive Mappings. Abstr. Appl. Anal. 2013 (2013), Article ID 768595, 6 pages. doi:10.1155/2013/768595. https://projecteuclid.org/euclid.aaa/1393511808


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