Existence of a solution to nonlinear variational inequality under generalized positive homogeneity

We establish several new existence theorems for nonlinear variational inequalities with generalized positively homogeneous functions. The results presented here are general enough to include two Moré existence theorems of complementarity problems as special cases. We also establish an existence result for nonlinear complementarity problems with an exceptional regularity map. The concept of an exceptional family for a variational inequality plays a key role in our analysis.