Let K be a nonempty bounded closed convex subset of a real reflexive Banach space E with a uniformly Gateaux differentiable norm. Let T : K → K be a uniformly continuous pseudocontractive mapping. Suppose every closed convex and bounded subset of K has the fixed point property for nonexpansive mappings. Let {λn} C (0, 1/2] be a sequence satisfying the conditions: (i) limn→∞λn = 0; (ii)Σ∞n=0 λn=∞. Let the sequence {xn} be generated from arbitrary x1 ∈ K by xn+1 = (1-λn)xn+ λnTxn -λn(xn- x1), n ≥1. Suppose limn→∞‖xn - Txn‖ = 0. Then {xn} converges strongly to a fixed point of T.