Viscosity approximation methods for nonexpansive mappings

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 xt ∈ C be the unique fixed point of the contraction x →tf (x)+(1−t)T x. Consider also the iteration process {xn}, where x0 ∈ C is arbitrary and xn+1 = αnf (xn) + (1 − αn)T xn for n 1, where {αn} ⊂ (0, 1). If X is either Hilbert or uniformly smooth, then it is shown that {xt } and, under certain appropriate conditions on {αn}, {xn} converge strongly to a fixed point of T which solves some variational inequality.  2004 Published by Elsevier Inc