Assume that $C$ is a nonempty closed convex subset of a Hilbert space $H$ and $B: C\to H$ is a relaxed $(\gamma , r)$-cocoercive, $\mu$-Lipschitzian mapping with $r>\gamma\mu^2$. Assume also that $\mathcal{F}$ 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 $x_0\in H$. The sequence $(x_n)$ is shown to converge in norm to the unique solution of the variational inequality $VI(B, \mathcal{F})$ under suitable conditions.
