Let E be a reflexive Banach space with a weakly sequentially continuous duality mapping, C be a nonempty closed convex subset of E, f : C $ ightarrow$C a contractive mapping (or a weakly contractive mapping), and T : C $ ightarrow$ C a nonexpansive mapping with the fixed point set F(T) ${ eq}{emptyset}$. Let {$x_n$} be generated by a new composite iterative scheme: $y_n={lambda}_nf(x_n)+(1-{lambda}_n)Tx_n$, $x_{n+1}=(1-{eta}_n)y_n+{eta}_nTy_n$, ($n{geq}0$). It is proved that {$x_n$} converges strongly to a point in F(T), which is a solution of certain variational inequality provided the sequence {$lambda_n$} $subset$ (0, 1) satisfies $lim_{n{ ightarrow}{infty}}{lambda}_n$ = 0 and $sum_{n=0}^{infty}{lambda}_n={infty}$, {$eta_n$} $subset$ [0, a) for some 0 < a < 1 and the sequence {$x_n$} is asymptotically regular.