Let X be a real uniformly smooth Banach space, K a nonempty subset of X such that K+K?K. Let T:K→K be a bounded φ-hemicontractive map. Let {un｝n=0∞ and {vn｝n=0∞ be two sequences in K and {αn｝n=0∞ and {βn｝n=0∞ be two real sequences in[0,1)satisfying (?) If {Tyn}is bounded, then{xn}converges strongly to the unique fixed point of T. Some related results are deduced.
