We present fixed point theorems for a nonexpansive mapping from a closed convex subset of a uniformly convex Banach space into itself under some asymptotic contraction assumptions. They generalize results valid for bounded convex sets or asymptotically compact sets. In this note we generalize a famous result by Browder [3], Göhde [6] and Kirk [8], recently extended by Luc in [14], by using the ...

We introduce an iterative algorithm for finding a common element of the set of common fixed points of a finite family of closed quasi-φ-asymptotically nonexpansive mappings, the set of solutions of an equilibrium problem, and the set of solutions of the variational inequality problem for a γ-inverse strongly monotone mapping in Banach spaces. Then we study the strong convergence of the algorith...

The purpose of this paper is to propose an algorithm for solving the split feasibility problems for total quasi-asymptotically nonexpansive mappings in infinite-dimensional Hilbert spaces. The results presented in the paper not only improve and extend some recent results of Moudafi [Nonlinear Anal. 74:4083-4087, 2011; Inverse Problem 26:055007, 2010], but also improve and extend some recent res...

In this paper we established strong and weak convergence theorems for a multi-step iterative scheme with errors for nonself asymptotically nonexpansive mappings in the real uniformly convex Banach space. Our results extend and improve the ones announced by Lin Wang [Lin Wang, Strong and weak convergence theorems for common fixed points of nonself asymptotically nonexpansive mappings, J. Math. A...

The asymptotically nonexpansive mappings have been introduced by Goebel and Kirk in 1972. Since then, a large number of authors have studied the weak and strong convergence problems of the iterative algorithms for such a class of mappings. It is well known that the asymptotically nonexpansive mappings is a proper subclass of the class of asymptotically pseudocontractive mappings. In the present...

In this paper, we introduce the modified general iterative methods for finding a common fixed point of asymptotically nonexpansive semigroups, which is a unique solution of some variational inequality. We prove the strong convergence theorems of such iterative scheme in a real Banach space which has a uniformly Gâteaux differentiable norm and admits the duality mapping jφ and uniform normal str...

In this paper, by combining a modified extragradient scheme with the viscosity approximation technique, an iterative scheme is developed for computing the common element of the set of fixed points of a sequence of asymptotically nonexpansive mappings and the set of solutions of the variational inequality problem for an α-inverse strongly monotone mapping. We prove a strong convergence theorem f...

In this paper, we propose a three-step iteration scheme for nonself asymptotically nonexpansive type mappings in a complete CAT(0) metric space and establish necessary and sufficient conditions for convergence of this process to a fixed point of nonself asymptotically nonexpansive type mappings. We also establish a strong convergence result. These results generalize and unify many important res...

Let C be a nonempty closed convex subset of a real Banach space E. Let S : C → C be a quasi-nonexpansive mapping, let T : C → C be an asymptotically demicontractive and uniformly Lipschitzian mapping, and let F := {x ∈ C : Sx = x and Tx = x} 6 = ∅. Let {xn}n≥0 be the sequence generated from an arbitrary x0 ∈ C by xn+1 = (1− cn)Sxn + cnT xn, n ≥ 0. We prove necessary and sufficient conditions fo...