Open Access

Convergece Theorems for Finite Families of Asymptotically Quasi-Nonexpansive Mappings

Journal of Inequalities and Applications20072007:068616

DOI: 10.1155/2007/68616

Received: 20 October 2006

Accepted: 31 January 2007

Published: 19 March 2007

Abstract

Let https://static-content.springer.com/image/art%3A10.1155%2F2007%2F68616/MediaObjects/13660_2006_Article_1725_IEq1_HTML.gif be a real Banach space, https://static-content.springer.com/image/art%3A10.1155%2F2007%2F68616/MediaObjects/13660_2006_Article_1725_IEq2_HTML.gif a closed convex nonempty subset of https://static-content.springer.com/image/art%3A10.1155%2F2007%2F68616/MediaObjects/13660_2006_Article_1725_IEq3_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2007%2F68616/MediaObjects/13660_2006_Article_1725_IEq4_HTML.gif asymptotically quasi-nonexpansive mappings with sequences (resp.) https://static-content.springer.com/image/art%3A10.1155%2F2007%2F68616/MediaObjects/13660_2006_Article_1725_IEq5_HTML.gif satisfying https://static-content.springer.com/image/art%3A10.1155%2F2007%2F68616/MediaObjects/13660_2006_Article_1725_IEq6_HTML.gif as https://static-content.springer.com/image/art%3A10.1155%2F2007%2F68616/MediaObjects/13660_2006_Article_1725_IEq7_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2007%2F68616/MediaObjects/13660_2006_Article_1725_IEq8_HTML.gif . Let https://static-content.springer.com/image/art%3A10.1155%2F2007%2F68616/MediaObjects/13660_2006_Article_1725_IEq9_HTML.gif be a sequence in https://static-content.springer.com/image/art%3A10.1155%2F2007%2F68616/MediaObjects/13660_2006_Article_1725_IEq10_HTML.gif . Define a sequence https://static-content.springer.com/image/art%3A10.1155%2F2007%2F68616/MediaObjects/13660_2006_Article_1725_IEq11_HTML.gif by https://static-content.springer.com/image/art%3A10.1155%2F2007%2F68616/MediaObjects/13660_2006_Article_1725_IEq12_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2007%2F68616/MediaObjects/13660_2006_Article_1725_IEq13_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2007%2F68616/MediaObjects/13660_2006_Article_1725_IEq14_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2007%2F68616/MediaObjects/13660_2006_Article_1725_IEq15_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2007%2F68616/MediaObjects/13660_2006_Article_1725_IEq16_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2007%2F68616/MediaObjects/13660_2006_Article_1725_IEq17_HTML.gif . Let https://static-content.springer.com/image/art%3A10.1155%2F2007%2F68616/MediaObjects/13660_2006_Article_1725_IEq18_HTML.gif . Necessary and sufficient conditions for a strong convergence of the sequence https://static-content.springer.com/image/art%3A10.1155%2F2007%2F68616/MediaObjects/13660_2006_Article_1725_IEq19_HTML.gif to a common fixed point of the family https://static-content.springer.com/image/art%3A10.1155%2F2007%2F68616/MediaObjects/13660_2006_Article_1725_IEq20_HTML.gif are proved. Under some appropriate conditions, strong and weak convergence theorems are also proved.

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Authors’ Affiliations

(1)
Mathematics Section, The Abdus Salam International Centre for Theoretical Physics
(2)
Department of Mathematical Sciences, Bayero University

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Copyright

© C. E. Chidume and B. Ali. 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.