Open Access

Iterative Approximation to Convex Feasibility Problems in Banach Space

Fixed Point Theory and Applications20072007:046797

https://doi.org/10.1155/2007/46797

Received: 7 November 2006

Accepted: 6 February 2007

Published: 11 April 2007

Abstract

The convex feasibility problem (CFP) of finding a point in the nonempty intersection is considered, where is an integer and each is assumed to be the fixed point set of a nonexpansive mapping , where is a reflexive Banach space with a weakly sequentially continuous duality mapping. By using viscosity approximation methods for a finite family of nonexpansive mappings, it is shown that for any given contractive mapping , where is a nonempty closed convex subset of and for any given the iterative scheme is strongly convergent to a solution of (CFP), if and only if and satisfy certain conditions, where and is a sunny nonexpansive retraction of onto . The results presented in the paper extend and improve some recent results in Xu (2004), O'Hara et al. (2003), Song and Chen (2006), Bauschke (1996), Browder (1967), Halpern (1967), Jung (2005), Lions (1977), Moudafi (2000), Reich (1980), Wittmann (1992), Reich (1994).

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Authors’ Affiliations

(1)
Department of Mathematics, Yibin University
(2)
Department of Mathematics, Sichuan University
(3)
Department of Applied Mathematics, National Sun Yat-Sen University
(4)
Department of Mathematics Education, Kyungnam University
(5)
Department of Mathematics, Southwest University of Science and Technology

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© Shih-Sen Chang et al. 2007

This article is published under license to BioMed Central Ltd. 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.