A Note on the Nonemptiness and Compactness of Solution Sets of Weakly Homogeneous Variational Inequalities

Recently, Gowda and Sossa studied weakly homogeneous variational inequalities (VIs), which contain the polynomial complementarity problem (PCP) as a special case. A lot of good theoretical results were obtained, and one of the important results is about the nonemptiness and compactness of the solution set of the concerned problem under the copositivity of the involved mapping and some additional conditions. In this note, we aim to generalize such a result. We obtain that the solution set of the weakly homogeneous VI is nonempty and compact when the involved mapping is a generalized copositive mapping and some additional conditions are satisfied. Such a result is a genuine generalization of the corresponding one achieved by Gowda and Sossa in the sense that one of their conditions is removed and every other condition is improved. We give some discussions on the conditions we used and obtain several related results which generalize the corresponding ones for the PCP. Moreover, we also investigate the relationships between the well-known coercivity condition and the conditions used in our main result.