Exceptional families and existence theorems for variational inequality problems

This paper introduces the concept of exceptional family for nonlinear varia tional inequality problems. Among other things, we show that the nonexisten ce of an exceptional family is a sufficient condition for the existence of a solution to variational inequalities. This sufficient condition is weaker than many known solution conditions and it is also necessary for pseudomon otone variational inequalities. From the results in this paper, we believe that the concept of exceptional families of variational inequalities provid es a new powerful tool for the study of the existence theory for variationa l inequalities.