A kind of system of multivariate variational inequalities and the existence theorem of solutions

Let K be a nonempty closed convex and bounded subset of a reflexive Banach space X. Let [Formula: see text] be N-variables monotone demi-continuous mappings from [Formula: see text] into X. Then: (1) the system of multivariate variational inequalities [Formula: see text] has a solution [Formula: see text]; (2) the set of solutions of this system of multivariate variational inequalities is closed convex in [Formula: see text]; (3) if [Formula: see text] are also strictly monotone, this system of multivariate variational inequalities has a unique solution.