Viscosity approximation methods for nonexpansive mappings are studied. Consider a nonexpansive self-mapping T of a closed convex subset C of a Banach space X. Suppose that the set Fix(T) of fixed points of T is nonempty. For a contraction f on C and t (0,1), let x t C be the unique fixed point of the contraction x tf(x)+(1-t)Tx. Consider also the iteration process {x n }, where x 0 C is arbitrary and x n + 1 =α n f(x n )+(1-α n )Tx n for n>=1, where {α n } (0,1). If X is either Hilbert or uniformly smooth, then it is shown that {x t } and, under certain appropriate conditions on {α n }, {x n } converge strongly to a fixed point of T which solves some variational inequality.