Sufficient conditions for the convergence of a new multistep iteration to a common fixed point of a finite family of asymptotically quasi-nonexpansive mappings in the framework of convex metric spaces are obtained. As an application, related results for a new three step iteration are derived. Our convergence results generalize and refine many known results.

In this work we prove that $M$-iteration process converges strongly faster than $S$-iteration and Picard-$S$ iteration processes. Moreover $M-$ iteration process is faster than $S_n$ iteration process with a sufficient condition for weak contractive mapping defined on a normed linear space. We also give two numerical reckoning examples to support our main theorem. For approximating fixed points, all codes were written in MAPLE \textcircled{c}2018 All rights reserved.

In this paper, we study strong convergence of an iteration process for a finite family of asymptotically quasi-nonexpansive mappings in convex metric spaces. We prove some strong convergence theorems for this iterative algorithm. Our results improve and extend the corresponding results of Kettapun et al. [2].

In this paper, we first give the modified version of the iteration process of Thakur et al. [15] which is faster than Picard, Mann, Ishikawa, Noor, Agarwal et al. [2] and Abbas et al. [1] processes. Secondly, we prove weak and strong convergence theorems of this iteration process for multivalued quasi nonexpansive mappings in uniformly convex Banach spaces. Thirdly, we support our theorems with analytical examples. Finally, we compare rates of convergence for multivalued version of iteration processes mentioned above via a numerical example.

Abstract. In this paper, we propose a new multistep iteration for a finite family of asymptotically quasi-nonexpansive mappings in convex cone metric spaces. Then we show that our iteration converges to a common fixed point of this class of mappings under suitable conditions. Our result generalizes the corresponding result of Lee [5] from the closed convex subset of a convex cone metric space to whole space.