An alternative theorem for generalized variational inequalities and solvability of nonlinear quasi- [formula omitted]-complementarity problems

Abstract

In this paper, an alternative theorem, and hence a sufficient solution condition, is established for generalized variational inequality problems. The concept of exceptional family for generalized variational inequality is introduced. This concept is general enough to include as special cases the notions of exceptional family of elements and the D-orientation sequence for continuous functions. Particularly, we apply the alternative theorem for investigating the solvability of the nonlinear complementarity problems with so-called quasi- P M ∗ -maps, which are broad enough to encompass the quasi-monotone maps and P ∗ -maps as the special cases. An existence theorem for this class of complementarity problems is established, which significantly generalizes several previous existence results in the literature.