An alternative theorem for generalized variational inequalities and solvability of nonlinear quasi-PM*-complementarity problems

In this paper, an alternative theorem, and hence a sucient 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 quasimonotone maps and P -maps as the special cases. An existence theorem for this class of complementarity problems is established, which signi®cantly generalizes several previous existence results in the literature. Ó 2000 Elsevier Science Inc. All rights reserved.