Viscosity approximation methods for nonexpansive mappings and monotone mappings

Viscosity approximation methods for nonexpansive mappings are studied. Consider the iteration process {xn}, where x0 ∈ C is arbitrary and xn+1 = αnf (xn)+(1−αn)SPC(xn −λnAxn), f is a contraction on C, S is a nonexpansive self-mapping of a closed convex subset C of a Hilbert space H. It is shown that {xn} converges strongly to a common element of the set of fixed points of nonexpansive mapping and the set of solutions of the variational inequality for an inverse strongly-monotone mapping which solves some variational inequality. © 2007 Elsevier Inc. All rights reserved.