The aim of this article is to establish some strong convergence theorems of three-step iteration process with errors for approximating common fixed point for generalized asymptotically quasi-nonexpansive mappings and also establish a weak convergence theorem by using Opial's [\textbf{11}] condition for said iteration scheme and mappings in the framework of Banach spaces. The results presented in this paper extend and improve the corresponding results of [\textbf{2}, \textbf{8}, \textbf{9}, \textbf{13}, \textbf{14}, \textbf{15}, \textbf{18}] and many others.