Let X be a real reflexive Banach space, let C be a closed convex subset of X , and let A be an m -accretive operator with a zero. Consider the iterative method that generates the sequence { x n } by the algorithm x n + 1 = α n f ( x n ) + ( 1 − α n ) J r n x n , where α n and γ n are two sequences satisfying certain conditions, J r denotes the resolvent ( I + r A ) − 1 for r > 0 , and let f : C → C be a fixed contractive mapping. The strong convergence of the algorithm { x n } is proved assuming that X has a weakly continuous duality map.