Convergence of Viscosity Iteration Process for a Finite Family of Generalized Asymptotically Quasi-Nonexpansive Mappings

We introduce a general iteration method for a finite family of generalized asymptotically quasi-nonexpansive mappings. The results presented in the paper extend and improve some recent results in the works by Shahzad and Udomene (2006); L. Qihou (2001); Khan et al. (2008). 1. Introduction and Preliminaries Let be a nonempty subset of a real Banach space and a self-mapping of . The set of fixed points of is denoted by and we assume that . The mapping is said to be(i)contractive mapping if there exists a constant in such that , for all ;(ii)asymptotically nonexpansive mapping if there exists a sequence in with such that , for all and ;(iii)asymptotically quasi-nonexpansive if there exists a sequence in with such that , for all , and ;(iv)generalized asymptotically quasi-nonexpansive [1] if there exist two sequences , in with and such that where ;(v)uniformly -Lipschitzian if there exists a constant such that , for all and ;(vi) uniform -Lipschitz if there are constants and such that , for all and ;(vii)semicompact if for a sequence in with , there exists a subsequence of such that . In (1), if for all , then becomes an asymptotically quasi-nonexpansive mapping; if and for all , then becomes a quasi-nonexpansive mapping. It is known that an asymptotically nonexpansive mapping is an asymptotically quasi-nonexpansive and a uniformly -Lipschitzian mapping is uniform -Lipschitz. The mapping is said to be demiclosed at if for each sequence converging weakly to and converging strongly to , we have . A Banach space is said to satisfy Opial’s property if for each and each sequence weakly convergent to , the following condition holds for all : Let be a nonempty closed convex subset of a real Banach space and a finite family of asymptotically nonexpansive mappings of into itself. Suppose that , , and . Then we consider the following mapping of into itself: where (identity mapping). Such a mapping is called the modified -mapping generated by and (see [2, 3]). In the sequel, we assume that . In 2008, Khan et al. [4] introduced the following iteration process for a family of asymptotically quasi-nonexpansive mappings, for an arbitrary : where , , , and proved that the iterative sequence defined by (4) converges strongly to a common fixed point of the family of mappings if and only if , where . With the help of (3), we write (4) as Recently, Chang et al. [5] introduced the following iteration process of asymptotically nonexpansive mappings in Banach space: where and is a fixed contractive mapping, and necessary and sufficient conditions are given for the iterative