In this paper we introduce an iterative process for finding a common element
of the set of common fixed points of a countable family of nonexpansive
mappings and the set of solutions of the variational inequality problem for a
monotone, Lipschitz continuous mapping. The iterative process is based on
two known methods - hybrid and extragradient. We obtain a strong
convergence theorem for three sequences generated by this process. Based on
this theorem, we construct an iterative process for solving the generalized
lexicographic variational inequality problem.