Let be a reflexive Banach space with a uniformly Gateaux differentiable norm. Suppose that every weakly compact convex subset of has the fixed point property for nonexpansive mappings. Let be a nonempty closed convex subset of , be a contractive mapping (or a weakly contractive mapping), and be a nonexpansive mapping with nonempty fixed point set . Let the sequence be generated by the following composite iterative scheme:
It is proved that converges strongly to a point in , which is a solution of a certain variational inequality, provided that the sequence satisfies and , for some , and the sequence is asymptotically regular.