Scalar and vector variational inequalities involving bifunctions are considered. 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 [19] 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 consider Bianchi and Pini [3], and their results admit a reformulation for variational inequalities as a special class of equilibrium problems. The present paper 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 type of variational inequalities are studied, and respectively the quasimonotonicity is understood in two different ways. Finally, as a particular case variational inequalities of differentiable type are discussed.

Existence of solutions of Minty type scalar and vector variational inequalities

Crespi G. P.;Ginchev I.;Rocca M.
2009-01-01

Abstract

Scalar and vector variational inequalities involving bifunctions are considered. 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 [19] 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 consider Bianchi and Pini [3], and their results admit a reformulation for variational inequalities as a special class of equilibrium problems. The present paper 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 type of variational inequalities are studied, and respectively the quasimonotonicity is understood in two different ways. Finally, as a particular case variational inequalities of differentiable type are discussed.
2009
Minty variational inequalities; Generalized monotonicity; Existence of solutions
Crespi, G. P.; Ginchev, I.; Rocca, M.
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11383/1714698
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