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 A(1), A(2),..., A(N) be N-variables monotone demi-continuous mappings from K-N into X. Then: (1) the system of multivariate variational inequalities { < A(1)(x(1), x(2),..., x(N)), y(1) - x(1)> >= 0 for all y(1) is an element of K, < A(2)(x(1), x(2),..., x(N)),y(2) - x(2)> >= 0, for all y(2) is an element of K, ... < A(N)(x(1), x(2),..., x(N)), y(N) - x(N) > >= 0, for all y(N) is an element of K, has a solution (x(1*), x(2*),..., x(N*)) is an element of K-N; (2) the set of solutions of this system of multivariate variational inequalities is closed convex in KN; (3) if A(1), A(2),..., A(N) are also strictly monotone, this system of multivariate variational inequalities has a unique solution.