Abstract :
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
