We introduce an iterative scheme for finding a common element of the set of solutions of generalizedmixed equilibrium problems and the set of fixed points for countable families of total quasi-φasymptotically nonexpansive mappings in Banach spaces. We prove a strong convergence theorem of the iterative sequence generated by the proposed iterative algorithm in an uniformly smooth and strictly co...

for all x, y ∈ C and each n ≥ 1. The class of asymptotically nonexpansive mappings was introduced by Goebel and Kirk [1] as an important generalization of nonexpansive mappings. It was proved in [1] that if C is a nonempty bounded closed convex subset of a real uniformly convex Banach space and T is an asymptotically nonexpansive self mapping on C, then F (T ) is nonempty closed convex subset o...

An iterative algorithm is introduced to solve the split common fixed point problems for asymptotically quasi-nonexpansive mappings in Hilbert spaces. The strong and weak convergence of the presented algorithm to some split common fixed point are obtained. The results presented in this paper improve and extend some recent results of Moudafi [Nonlinear Anal. 74 (2011), 4083-4087], Xu [Inverse Pro...

The purpose of this paper is to study sufficient and necessary conditions for finite-step iterative sequences withmean errors for a finite family of asymptotically quasi-nonexpansive and type mappings in Banach spaces to converge to a common fixed point. The results presented in this paper improve and extend the recent ones announced by GhostDebnath, Liu, Xu and Noor, Chang, Shahzad et al., Sha...

The purpose of this paper is to propose a modified hybrid projection algorithm and prove a strong convergence theorem for a family of quasi-φ-nonexpansive mappings. The strong convergence theorem is proven in the more general reflexive, strictly convex, and smooth Banach spaces with the property K . The results of this paper improve and extend the results of S. Matsushita and W. Takahashi 2005 ...

The purpose of this article is to discuss a modified Halpern-type iteration algorithm for a countable family of uniformly totally quasi- -asymptotically nonexpansive multi-valued mappings and establish some strong convergence theorems under certain conditions. We utilize the theorems to study a modified Halpern-type iterative algorithm for a system of equilibrium problems. The results improve ...

In this paper, we proposed a new two-step iteration scheme of hybrid mixed type for two asymptotically nonexpansive self mappings and two total asymptotically nonexpansive non-self mappings and establish some weak convergence theorems for mentioned scheme and mappings in the setting of uniformly convex Banach spaces. Our results extend and generalize several results from the current existing li...

We study nearly equal and nearly convex sets, ranges of maximally monotone operators, and ranges and fixed points of convex combinations of firmly nonexpansive mappings. The main result states that the range of an average of firmly nonexpansive mappings is nearly equal to the average of the ranges of the mappings. A striking application of this result yields that the average of asymptotically r...

In this paper, based on a generalized projection, we introduce a new modified Halpern-type iteration algorithm for finding a common element of the set of solutions of an equilibrium problem and the set of a common fixed point of an infinitely countable family of relatively quasi-nonexpansive mappings in the framework of Banach spaces. We establish the strong convergence theorem and obtain some ...

In this paper, we introduce two modified Mann-type iterative algorithms for finding a common element of the set of common fixed points of a family of quasi-φ-asymptotically nonexpansive mappings and the set of solutions of an equilibrium problem in Banach spaces. Then we study the strong convergence of the algorithms. Our results improve and extend the corresponding results announced by many ot...