Weak and strong convergence of the Ishikawa iterative process for a finite family of asymptotically nonexpansive mappings

In this paper, we introduce an Ishikawa implicit iterative process with errors for a finite family of N asymptotically nonexpansive mappings as follows: {x(n) = (1 - alpha(n) - gamma(n))x(n-1) + alpha(n)T(i(n >)(k(n >)y(n) + gamma(n)u(n), y(n) = (1 - beta(n) - delta(n))x(n) + beta(n)T(i(n >)(k(n >)x(n) + delta(n)v(n), n >= 1, where, for any n is an element of N fixed, k(n) - 1 denotes the quotient of the division of n by N and i(n) the rest, i.e. n = (k(n) - 1)N + i(n), i(n) is an element of {1, ..., N}. The sequences {alpha(n)}, {beta(n)}, {gamma(n)}, {delta(n)} are four real sequences in [0,1] satisfying alpha(n) + gamma(n) <= 1 and beta(n) + delta(n) <= 1 for all n >= 1, {u(n)}, {v(n)} are two bounded sequences and x(0) is a given point. In the setting of uniformly convex Banach spaces we give some results of weak and strong convergence of the above iterative process. The results presented here are situated on the line of research of the corresponding results of Sun [J. Math. Anal. Appl. 286 (1) (2003) 351-358], Osilike [J. Math. Anal. Appl. 294 (1) (2004) 73-81], Chang et al. [J. Math. Anal. Appl. 313 (1) (2006) 273-283], Gu [J. Math. Anal. Appl. 329 (2) (2007) 766-776], Huang and Noor [Appl. Math. Comput. 190 (1) (2007) 356-361], Su and Qin [Appl. Math. Comput. 186 (1) (2007) 271-278), Zhou et al. [Appl. Math. Comput. 173 (1) (2006) 196-212] and some others. (C) 2010 Elsevier Inc. All rights reserved.