Strong convergence theorems for the implicit iteration process for a finite family of hemicontractive mappings in Banach space

Abstract

The purpose of this work is to study the following implicit iteration scheme x n = α n x n − 1 + ( 1 − α n ) T n x n , n ≥ 1 , where T n = T n m o d N , and to prove several strongly convergent theorems of the iteration for a finite family of hemicontractive mappings in Banach space. Our results extend a recent result of Haiyun Zhou [Haiyun Zhou, Convergence theorems of common fixed points for a finite family of Lipschitz pseudocontractions in Banach spaces, Nonlinear Anal. 68 (2008) 2977–2983] and Xu and Ori [H.K. Xu, R.G. Ori, An implicit iteration process for nonexpansive mappings, Numer. Funct. Anal. Optim. 22 (2001) 767–773], and we have proved that the sequence { x n } converges strongly to a common fixed point of a finite family of hemicontractive mappings { T i } i = 1 N .