Open Access

An approximation method for continuous pseudocontractive mappings

Journal of Inequalities and Applications20062006:28950

DOI: 10.1155/JIA/2006/28950

Received: 20 March 2006

Accepted: 28 May 2006

Published: 26 July 2006

Abstract

Let https://static-content.springer.com/image/art%3A10.1155%2FJIA%2F2006%2F28950/MediaObjects/13660_2006_Article_1582_IEq1_HTML.gif be a closed convex subset of a real Banach space https://static-content.springer.com/image/art%3A10.1155%2FJIA%2F2006%2F28950/MediaObjects/13660_2006_Article_1582_IEq2_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2FJIA%2F2006%2F28950/MediaObjects/13660_2006_Article_1582_IEq3_HTML.gif is continuous pseudocontractive mapping, and https://static-content.springer.com/image/art%3A10.1155%2FJIA%2F2006%2F28950/MediaObjects/13660_2006_Article_1582_IEq4_HTML.gif is a fixed https://static-content.springer.com/image/art%3A10.1155%2FJIA%2F2006%2F28950/MediaObjects/13660_2006_Article_1582_IEq5_HTML.gif -Lipschitzian strongly pseudocontractive mapping. For any https://static-content.springer.com/image/art%3A10.1155%2FJIA%2F2006%2F28950/MediaObjects/13660_2006_Article_1582_IEq6_HTML.gif , let https://static-content.springer.com/image/art%3A10.1155%2FJIA%2F2006%2F28950/MediaObjects/13660_2006_Article_1582_IEq7_HTML.gif be the unique fixed point of https://static-content.springer.com/image/art%3A10.1155%2FJIA%2F2006%2F28950/MediaObjects/13660_2006_Article_1582_IEq8_HTML.gif . We prove that if https://static-content.springer.com/image/art%3A10.1155%2FJIA%2F2006%2F28950/MediaObjects/13660_2006_Article_1582_IEq9_HTML.gif has a fixed point and https://static-content.springer.com/image/art%3A10.1155%2FJIA%2F2006%2F28950/MediaObjects/13660_2006_Article_1582_IEq10_HTML.gif has uniformly Gâteaux differentiable norm, such that every nonempty closed bounded convex subset of https://static-content.springer.com/image/art%3A10.1155%2FJIA%2F2006%2F28950/MediaObjects/13660_2006_Article_1582_IEq11_HTML.gif has the fixed point property for nonexpansive self-mappings, then https://static-content.springer.com/image/art%3A10.1155%2FJIA%2F2006%2F28950/MediaObjects/13660_2006_Article_1582_IEq12_HTML.gif converges to a fixed point of https://static-content.springer.com/image/art%3A10.1155%2FJIA%2F2006%2F28950/MediaObjects/13660_2006_Article_1582_IEq13_HTML.gif as https://static-content.springer.com/image/art%3A10.1155%2FJIA%2F2006%2F28950/MediaObjects/13660_2006_Article_1582_IEq14_HTML.gif approaches to 0. The results presented extend and improve the corresponding results of Morales and Jung (2000) and Hong-Kun Xu (2004).

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Authors’ Affiliations

(1)
College of Mathematics and Information Science, Henan Normal University
(2)
Department of Mathematics, Tianjin Polytechnic University

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Copyright

© Song and Chen 2006

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.