Using the concept of the spaces proposed by Khamsi et al. [Khamsi, M. A. and Shukri, S. A., Generalized spaces. Bull. Belg. Math. Soc. Simon Stevin, 24 (2017), No. 3, 417–426], we establish -con\-ver\-gence and strong convergence of a perturbed variant of Agarwal et al. -iteration process
for nonexpansive mapping. Finally, from them we deduce results valid for -nonexpansive mappings.