Existence of Solutions and Star-shapedness in Minty Variational Inequalities

Minty Variational Inequalities (for short, MVI) have proved to characterize a kind of equilibrium more qualified than Stampacchia Variational Inequalities (for short, SVI). This conclusion leads to argue that, when a MVI admits a solution and the operator F admits a primitive minimization problem (that is the function f to minimize is such that F = f ′), then f has some regularity property, e.g. convexity or generalized convexity. In this paper we put in terms of the lower Dini directional derivative a problem, referred to as GMV I(f ′ −,K), which can be considered a nonlinear extension of the MVI with F = f ′ (K denotes a subset of R). We investigate, in the caseK star-shaped, the existence of a solution of GMV I(f ′ −,K) and the property of f to increase along rays ∗Université de la Vallée d’Aoste, Facoltà di Scienze Economiche, Strada Cappuccini 2A, 11100 Aosta, Italia. e–mail: [email protected] †Technical University of Varna, Department of Mathematics, 9010 Varna, Bulgaria. e–mail: [email protected] ‡Università dell’Insubria, Dipartimento di Economia, via Ravasi 2, 21100 Varese, Italia. e–mail: [email protected]