Let be a nonempty convex subset of a real Banach space . Let be a nearly weak uniformly -Lipschitzian mapping. A modified Mann-type iteration scheme is proved to converge strongly to the unique fixed point of . Our result is a significant improvement and generalization of several known results in this area of research. We give a specific example to support our result. Furthermore, an interesting equivalence of -stability result between the convergence of modified Mann-type and modified Mann iterations is included.