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 ∈ H . The sequence ( x n ) is shown to converge in norm to the unique solution of the variational inequality V I ( B , F ) under suitable conditions.