Let E be a reflexive Banach space with a uniformly Gâteaux differentiable norm, let K be a nonempty closed convex subset of E , and let T : K → K be a uniformly continuous pseudocontraction. If f : K → K is any contraction map on K and if every nonempty closed convex and bounded subset of K has the fixed point property for nonexpansive self-mappings, then it is shown, under appropriate conditions on the sequences of real numbers { α n } , { μ n } , that the iteration process z 1 ∈ K , z n + 1 = μ n ( α n T z n + ( 1 − α n ) z n ) + ( 1 − μ n ) f ( z n ) , n ∈ ℕ , strongly converges to the fixed point of T , which is the unique solution of some variational inequality, provided that K is bounded.