www.astesj.com 69 Multi-Step Iteration Algorithm of Total Asymptotically Quasi-Nonexpansive Maps Salwa Salman Abed * , Zahra Mahmood Mohamed Hasan Department of Mathematics, college of Education for Pure Sciences (Ibn Al-Haitham) / University of Baghdad, Baghdad, Iraq A R T I C L E I N F O A B S T R A C T Article history: Received: 14 February 2019 Accepted: 28 April, 2019 Online: 21 May, 2019 In Banach spaces an iteration algorithm for two finite families of total asymptotically quasi- nonexpansive maps is introduced. Weak and strong convergence theorems of this algorithm to approximation common fixed points are proved by using suitable conditions. As well as, numerical example by using Mat-lab is given. Keywords: Banach space, total asymptotically quasi- nonexpansive map, weak convergence, strong convergence, common fixed points 1. Introduction and Preliminaries This paper was originally published in the Conference: 2018 International Conference on Advanced Science and Engineering (ICOASE), Iraq [1]. It is well known that the concept of asymptotically nonexpansive introduced by Goebel and Kirk [2]. Additionally, every asymptotically nonexpansive map of a Banach space has a fixed point is proved. In [3], Petryshyn and Williamson proved the weak and strong convergence for quasi-nonexpansive map by using a sufficient and necessary condition. Alber [4], a new class of asymptotically nonexpansive is introduced. As well as, approximating methods for finding their fixed points are studied. In 2014, G. S. Saluja [5] established the strong and weak convergence for approximating common fixed point for generalized asymptotically quasi-nonexpansive maps in a Banach space. Very recently, In [6], the authors proposed an implicit iteration for two finite families of generalized asymptotically quasi- nonexpansive maps. As well as, some strong convergence theorems are established. It is useful to point out our findings in this area which appeared in [7]. Let B be a non-empty closed convex subset of a real Banach space M and T be a self-map of B. The set of all fixed points denoted by F(T). A self-map T from B into M is called nonexpansive map [2] if ‖ − ‖ ≤ ‖ − ‖ , ∈ and is called quasi- nonexpansive map [6] if () ≠ ∅ ‖ − ∗ ‖ ≤ ‖ − ∗ ‖ for all and for all ∗ (). A Banach space M is satisfying: "Opial’s condition if for each sequence ( ) in , is weak convergence to implies that lim →∞ ‖ − ‖ < lim →∞ ‖ − ‖ for all ∈ ℎ ≠ ". "Kadec-Klee property if for each sequence ( ) in is weak convergence to () together with ‖ ‖ converges strongly to ‖‖ imply that ( ) is strong convergence to a point ∈ [7]". The aim of this paper, an iterative scheme for two families of total asymptotically quasi-nonexpansive maps is established. The strong and weak convergence theorems of this scheme for approximation of common fixed points in Banach space by using suitable conditions are established. For this purpose, let us recall the following definitions and lemmas. Definition (1.1): "A map is named asymptotically nonexpansive [1] if there is a sequence ( ) in [0, +∞) with →∞ = 0 and ‖ − ‖ ≤ (1 + )‖ − ‖, for all , ∈ , = 1,2, . .. ASTESJ ISSN: 2415-6698 * Salwa Salman Abed, Email: salwaalbundi@yahoo.com Advances in Science, Technology and Engineering Systems Journal Vol. 4, No. 3, 69-74 (2019) www.astesj.com Special Issue on Advancement in Engineering and Computer Science