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Let K be a nonempty closed bounded convex subset of an arbitrary smooth Banach space X and T:K→KOpen image in new window be a continuous strictly hemicontractive mapping. Under some conditions, we obtain that the Mann iteration method with error term converges strongly to a unique fixed point of T and is almost T-stable on K. As an application of our results, we establish strong convergence of a multi-step iteration process.

Almost stability of the Mann type iteration method with error term involving strictly hemicontractive mappings in smooth Banach spaces