In this paper, a strong convergence theorem is proved for a generalized Mann type iteration scheme in normed linear spaces. We also consider two, two-step iteration schemes and prove the strong convergence of these iterations in normed linear spaces. We use a generalized $Z$-type condition to prove our results. Our results extend and improve upon, among others, the corresponding results proved by Berinde [1], Yildirim et al. [12] and Bosede [4].