Convergence analysis of implicit iterative algorithms for asymptotically nonexpansive mappings Xiaolong Qin a , Yeol Je Cho b, * , Meijuan Shang c a Department of Mathematics, Gyeongsang National University, Chinju 660-701, Republic of Korea b Department of Mathematics Education and the RINS, Gyeongsang National University, Chinju 660-701, Republic of Korea c Department of Mathematics, Shijiazhuang University, Shijiazhuang 050035, China article info Keywords: Asymptotically nonexpansive mapping Implicit iterative algorithm Banach space Common fixed point abstract In this paper, we consider the weak and strong convergence of an implicit iterative process with errors for two finite families of asymptotically nonexpansive mappings in the frame- work of Banach spaces. Our results presented in this paper improve and extend the recent ones announced by many others. Ó 2009 Elsevier Inc. All rights reserved. 1. Introduction and preliminaries Throughout this paper, we always assume that E is a real Banach space and F(T) is the set of fixed points of the mapping T. Let K be a nonempty, closed and convex subset of E. For a given sequence fx n g K, let x w ðx n Þ denote the weak w-limit set. Recall that T : K ! K is said to be nonexpansive, if the following inequality holds: kTx  Tyk 6 kx  yk; 8x; y  K: Recall that T : K ! K is said to be asymptotically nonexpansive [6], if there exists a positive sequence h n ½1; 1Þ with lim n!1 h n ¼ 1 such that kT n x  T n yk 6 h n kx  yk; 8x; y  k; n  1: Recently concerning the convergence problems of an implicit (or non-implicit) iterative process to a common fixed point for a finite family of nonexpansive mappings and its extensions have been considered by a number of authors (see, for example [1–7,9–20]). In 2001, Xu and Ori [18] introduced the following implicit iteration process for a finite family of nonexpansive mappings fT 1 ; T 2 ; ... ; T N g, with fa n g a real sequence in (0, 1), and an initial point x 0 2 K: x 1 ¼ a 1 x 0 þð1  a 1 ÞT 1 x 1 ; x 2 ¼ a 2 x 1 þð1  a 2 ÞT 2 x 2 ; . . . x N ¼ a N x N1 þð1  a N ÞT N x N ; x Nþ1 ¼ a Nþ1 x N þð1  a Nþ1 ÞT 1 x Nþ1 ; . . . 0096-3003/$ - see front matter Ó 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2009.01.018 * Corresponding author. E-mail addresses: qxlxajh@163.com, ljjhqxl@yahoo.com.cn (X. Qin), yjcho@gsnu.ac.kr (Y.J. Cho), meijuanshang@yahoo.com.cn (M. Shang). Applied Mathematics and Computation 210 (2009) 542–550 Contents lists available at ScienceDirect Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc