Modified hybrid steepest-descent methods for variational inequalities and fixed points

Assume that C is a nonempty closed convex subset of a Hilbert space H and B : C -> H is a strongly monotone mapping. Assume also that F is the intersection of the common fixed points of an infinite family of nonexpansive mappings on C and the set of solutions of a system of equilibrium problems. We devise a modified hybrid steepest-descent method which generates a sequence (x(n)) from an arbitrary initial point x(0) is an element of H. The sequence (x(n)) is shown to converge in norm to the unique solution of the variational inequality VI(B, F) under suitable conditions. (C) 2010 Elsevier Ltd. All rights reserved.