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Research Article Strong Convergence Theorems for Common Fixed Points of Multistep Iterations with Errors in Banach Spaces | Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2009 Article ID 819036 12 pages doi 10.1155 2009 819036 Research Article Strong Convergence Theorems for Common Fixed Points of Multistep Iterations with Errors in Banach Spaces Feng Gu1 and Qiuping Fu2 1 Department of Mathematics Institute of Applied Mathematics Hangzhou Normal University Hangzhou Zhejiang 310036 China 2 Mathematics group West Lake High Middle School Hangzhou Zhejiang 310012 China Correspondence should be addressed to Feng Gu gufeng99@sohu.com Received 19 November 2008 Revised 11 January 2009 Accepted 9 April 2009 Recommended by Yeol Je Cho We establish strong convergence theorem for multi-step iterative scheme with errors for asymptotically nonexpansive mappings in the intermediate sense in Banach spaces. Our results extend and improve the recent ones announced by Plubtieng and Wangkeeree 2006 and many others. Copyright 2009 F. Gu and Q. Fu. 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. 1. Introduction Let C be a subset of real normal linear space X. A mapping T C C is said to be asymptotically nonexpansive on C if there exists a sequence rn in 0 to with limn xrn 0 such that for each x y e C Tnx- Tnyịị 1 rn x- yịị Vn 1. 1.1 If rn 0 then T is known as a nonexpansive mapping. T is called asymptotically nonexpansive in the intermediate sense 1 provided T is uniformly continuous and limsupsup IITnx - Tny - x - y 0. 1.2 n to x yeC 2 Journal of Inequalities and Applications From the above definitions it follows that asymptotically nonexpansive mapping must be asymptotically nonexpansive in the intermediate sense. Let C be a nonempty subset of normed space X and Let Ti C C be m mappings. For a given X1 e C and a fixed m e N N denotes the set of all positive integers compute the iterative sequences x . xm defined .