  In this paper we introduce two new generalized variational inequalities, and
we give some existence results of the solutions for these variational
inequalities involving operators belonging to a recently introduced class of
operators. We show by examples, that our results fail outside of this class.
Further, we establish a result that may be viewed as a generalization of
Minty's theorem, that is, we show that under some circumstances the set of
solutions of these variational inequalities coincide. We also show that the
operators, involved in these variational inequalities, to belong to the above
mentioned class, is essential in obtaining this result.
  As application, we show that Brouwer's fixed point theorem is an easy
consequence of our results.
