In this paper, we give necessary and sufficient condition for strong convergence of three-step iteration process with errors for approximating common fixed point for asymptotically quasi-nonexpansive type mappings and also prove weak convergence of three-step iteration process with errors for approximating common fixed point for said mappings in Banach spaces. The results presented in this paper extend and improve the corresponding results [1, 5, 6, 10, 11, 14] and many others.