  In this paper, we introduce and study a new extragradient iterative process
for finding a common element of the set of fixed points of an infinite family
of nonexpansive mappings and the set of solutions of a variational inequality
for an inverse strongly monotone mapping in a real Hilbert space. Also, we
prove that under quite mild conditions the iterative sequence defined by our
new extragradient method converges strongly to a solution of the fixed point
problem for an infinite family of nonexpansive mappings and the classical
variational inequality problem. In addition, utilizing this result, we provide
some applications of the considered problem not just giving a pure extension of
existing mathematical problems.
