Variational inequalities and Fixed Point theorems in the Euclidean space for non-continuous operators

The existence of solutions of general variational inequalities is obtained for some maps defined on a nonempty closed convex subset of the Euclidean space. The convex subset is possibly unbounded, the operator is possibly neither continuous nor coercive, the convex function is possibly non-lower semi-continuous. Two generalizations of the fixed points theorem of Brouwer are deduced for some operators which are non-continuous and defined on some unbounded closed convex subsets