Variational Inequalities: an Elementary Approach

Abstract

We will deal with the following problem: Let M be an n × n matrix with real entries. Under which conditions the family of inequalities:{\bf x}\in {\open {R}}^n; {\bf x} \ge {\bf 0};{\bf M}\cdot{\bf x}\ge {\bf 0}has non–trivial solutions? We will prove that a sufficient condition is given bym_{i,j}+m_{j,i}\ge0\quad(1\le i,j\le n);from this result we will derive an elementary proof of the existence theorem for Variational Inequalities in the framework of Monotone Operators.