Let E be a reflexive Banach space with a uniformly Gâteaux differentiable norm. Suppose that every weakly compact convex subset of E has the fixed point property for nonexpansive mappings. Let C be a nonempty closed convex subset of E , f : C     →     C a contractive mapping (or a weakly contractive mapping), and T : C     →     C nonexpansive mapping with the fixed point set F ( T )     ≠     ∅ . Let { x n } be generated by a new composite iterative scheme: y n = λ n f ( x n ) + ( 1 − λ n ) T x n , x n + 1 = ( 1 − β n ) y n + β n T y n , ( n ≥ 0 ) . It is proved that { x n } converges strongly to a point in F ( T ) , which is a solution of certain variational inequality provided that the sequence { λ n } ⊂ ( 0 , 1 ) satisfies lim n → ∞ λ n = 0 and ∑ n = 1 ∞ λ n = ∞ , { β n } ⊂ [ 0 , a ) for some 0 < a < 1 and the sequence { x n } is asymptotically regular.