Existence of solutions of Minty type scalar and vector variational inequalities

Abstract

Scalar and vector variational inequalities involving bifunctions are considered in this article. Necessary and sufficient conditions for existence of solutions are proposed, based on the finite intersection property of a suitable set-valued function. In the case of a properly quasi-monotone bifunction the relation to the KKM property is established. The result of John (A note on Minty variational inequality and generalized monotonicity. Proc. Generalized Convexity and Monotonicity, 6 (2001), pp. 240–246) concerning the characterization of properly quasi-monotone functions in terms of existence of solutions of variational inequalities (scalar case) is extended to bifunctions both in the scalar and vector cases. Existence of solutions for scalar equilibrium problems with bifunctions are considered Bianchi and Pini (A note on equilibrium problems with properly quasimonotone bifunctions. J. Global Optim. 20 (2001), pp. 67–76) and their results admit a reformulation for variational inequalities as a special class of equilibrium problems. The present article also generalizes this result, even in the scalar case (here the bifunctions are not assumed quasi-convex). As for the vector case, it should be stressed that two types of variational inequalities are studied, and respectively, the quasi-monotonicity is understood in two different ways. Finally, as a particular case variational inequalities of differentiable type are discussed.