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 , ∀ y 1 ∈ K , 〈 A 2 ( x 1 , x 2 , … , x N ) , y 2 − x 2 〉 ≥ 0 , ∀ y 2 ∈ K , ⋯ 〈 A N ( x 1 , x 2 , … , x N ) , y N − x N 〉 ≥ 0 , ∀ y N ∈ K , has a solution ( x 1 ∗ , x 2 ∗ , … , x N ∗ ) ∈ K N ; (2) the set of solutions of this system of multivariate variational inequalities is closed convex in K N ; (3) if A 1 , A 2 , … , A N are also strictly monotone, this system of multivariate variational inequalities has a unique solution.