On the strong convergence of the implicit iterative processes with errors for a finite family of asymptotically nonexpansive mappings

The purpose of this work is to study the sufficient and necessary conditions and sufficient conditions on the strong convergence of the implicit iteration process with errors for a finite family of asymptotically nonexpansive mappings in real uniformly convex Banach spaces. The results presented in this work give an affirmative answer to the open question raised by Xu and Ori [H.K. Xu, M.G. Ori, An implicit iterative process for nonexpansive mappings, Numer. Funct. Anal. Optim. 22 (2001) 767–773] and improve and generalize the corresponding results of Chang et al. [S.S. Chang, K.K. Tan, H.W.J. Lee, Chi Kin Chan, On the convergence of implicit iteration process with error for a finite family of asymptotically nonexpansive mappings, J. Math. Anal. Appl. 313 (2006) 273–283], Sun [Z.H. Sun, Strong convergence of an implicit iteration process for a finite family of asymptotically quasi-nonexpansive mappings, J. Math. Anal. Appl. 286 (2003) 351–358], Zhou and Chang [Y.Y. Zhou, S.S. Chang, Convergence of implicit iterative process for a finite family of asymptotically nonexpansive mappings in Banach spaces, Numer. Funct. Anal. Optim. 23 (2002) 911–921].