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

Assume that CC is a nonempty closed convex subset of a Hilbert space HH and B:C→HB:C→H is a strongly monotone mapping. Assume also that FF is the intersection of the common fixed points of an infinite family of nonexpansive mappings on CC and the set of solutions of a system of equilibrium problems. We devise a modified hybrid steepest-descent method which generates a sequence (xn)(xn) from an arbitrary initial point x0∈Hx0∈H. The sequence (xn)(xn) is shown to converge in norm to the unique solution of the variational inequality VI(B,F)VI(B,F) under suitable conditions.