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:xn=(1-αn-γn)xn-1+αnTi(n)k(n)yn+γnun,yn=(1-βn-δn)xn+βnTi(n)k(n)xn+δnvn,n⩾1,where, for any n∈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)∈{1,…,N}.The sequences {αn}, {βn}, {γn}, {δn} are four real sequences in [0, 1] satisfying αn + γn ⩽ 1 and βn + δn ⩽ 1 for all n ⩾ 1, {un}, {vn} are two bounded sequences and x0 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.