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Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: Research Article Modified Hybrid Algorithm for a Family of Quasi-φ-Asymptotically Nonexpansive Mappings | Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2010 Article ID 170701 15 pages doi 10.1155 2010 170701 Research Article Modified Hybrid Algorithm for a Family of Quasi- -Asymptotically Nonexpansive Mappings Yongchun Xu 1 Xin Zhang 2 Jinlong Kang 2 and Yongfu Su2 1 Department of Mathematics Hebei North University Zhangjiakou 075000 China 2 Department of Mathematics Tianjin Polytechnic University Tianjin 300160 China Correspondence should be addressed to Yongfu Su suyongfu@tjpu.edu.cn Received 19 March 2010 Accepted 16 August 2010 Academic Editor L. Gorniewicz Copyright 2010 Yongchun Xu et al. 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. The purpose of this paper is to propose a modified hybrid projection algorithm and prove strong convergence theorems for a family of quasi- -asymptotically nonexpansive mappings. The method of the proof is different from the original one. Our results improve and extend the corresponding results announced by Zhou et al. 2010 Kimura and Takahashi 2009 and some others. 1. Introduction Let E be a real Banach space and C a nonempty closed convex subset of E. A mapping T C C is said to be asymptotically nonexpansive 1 if there exists a sequence kn of positive real numbers with kn 1 such that Tnx - Tny kn x - y 1.1 for all x y e C and all n 1. The class of asymptotically nonexpansive mappings was introduced by Goebel and Kirk 1 in 1972. They proved that if C is a nonempty bounded closed convex subset of a uniformly convex Banach space E then every asymptotically nonexpansive self-mapping T of C has a fixed point. Further the set F T of fixed points of T is closed and convex. Since 1972 a host of authors have studied the weak and strong convergence problems of the iterative algorithms for such a class of mappings see e.g. 1-3 and the references therein
