
Minimax Theory and its Applications 05 (2020), No. 1, 047064 Copyright Heldermann Verlag 2020 Generalized Nash Equilibrium Problems and Variational Inequalities in Lebesgue Spaces Giandomenico Mastroeni Dip. di Informatica, Università di Pisa, 56127 Pisa, Italy giandomenico.mastroeni@unipi.it Massimo Pappalardo Dip. di Informatica, Università di Pisa, 56127 Pisa, Italy massimo.pappalardo@unipi.it Fabio Raciti Dip. di Matematica e Informatica, Università di Catania, 95125 Catania, Italy fraciti@dmi.unict.it We study generalized Nash equilibrium problems (GNEPs) in Lebesgue spaces by means of a family of variational inequalities (VIs) parametrized by an L^{∞} vector r(t). The solutions of this family of VIs constitute a subset of the solution set of the GNEP. For each choice of r(t), the VI solutions thus obtained are solutions of the GNEP which can be characterized by a certain relationship among the KarushKuhnTucker (KKT) multipliers of the players. This result extends a previous one, where only the case in which the parameter r is a constant vector was investigated, and can be considered as a full generalization, to Lebesgue spaces, of a classical property proven by J. B. Rosen [Existence and uniqueness of equilibrium points for concave n person games, Econometrica 33 (1965) 520534] in finite dimensional spaces. Keywords: Generalized Nash equilibrium, variational inequalities, KarushKuhnTucker conditions. MSC: 90C33, 58E35, 90C30. [ Fulltextpdf (140 KB)] for subscribers only. 