Variational Inequalities: an Elementary Approach

We will deal with the following problem: Let M be an n×n matrix with real entries. Under which conditions the family of inequalities: x∈? ⁿ ;x?0;M·x?0has non–trivial solutions? We will prove that a sufficient condition is given by m_i,j+m_j,i?0 (1?i,j?n); from this result we will derive an elementary proof of the existence theorem for Variational Inequalities in the framework of Monotone Operators.