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 Some Weak Convergence Theorems for a Family of Asymptotically Nonexpansive Nonself Mappings | Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2010 Article ID 218573 11 pages doi 2010 218573 Research Article Some Weak Convergence Theorems for a Family of Asymptotically Nonexpansive Nonself Mappings Yan Hao 1 Sun Young Cho 2 and Xiaolong Qin3 1 School of Mathematics Physics and Information Science Zhejiang Ocean University Zhoushan 316004 China 2 Department of Mathematics Gyeongsang National University Jinju 660-701 South Korea 3 Department of Mathematics Hangzhou Normal University Hangzhou 310036 China Correspondence should be addressed to Yan Hao zjhaoyan@ Received 31 August 2009 Accepted 16 November 2009 Academic Editor Mohamed A. Khamsi Copyright 2010 Yan Hao 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. A one-step iteration with errors is considered for a family of asymptotically nonexpansive nonself mappings. Weak convergence of the purposed iteration is obtained in a Banach space. 1. Introduction and Preliminaries Let E be a real Banach space and E the dual space of E. Let denote the pairing between E and E . The normalized duality mapping J E 2E is defined by J x f e E x f x 2 f II2 Vx e E. Let UE x e E xH 1 where E is said to be smooth or said to have a Gateaux differentiable norm if the limit lim llx 1 tyll 11x11 t 0 t exists for each x y e UE where E is said to have a uniformly Gateaux differentiable norm if for each y e UE the limit is attained uniformly for all x e UE where E is said to be uniformly smooth or said to have a uniformly Frechet differentiable norm if the limit is 2 Fixed Point Theory and Applications attained uniformly for all x y e UE where E is said to be uniformly convex if for any e e 0 2 there exists Ỗ 0 such that for any x y e UE x y 2 llx - y e implies 1 - Ỗ. It is known that a uniformly convex Banach .