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Minimax Theory and its Applications 05 (2020), No. 1, 047--064
Copyright Heldermann Verlag 2020



Generalized Nash Equilibrium Problems and Variational Inequalities in Lebesgue Spaces

Giandomenico Mastroeni
Dip. di Informatica, UniversitÓ di Pisa, 56127 Pisa, Italy
giandomenico.mastroeni@unipi.it

Massimo Pappalardo
Dip. di Informatica, UniversitÓ di Pisa, 56127 Pisa, Italy
massimo.pappalardo@unipi.it

Fabio Raciti
Dip. di Matematica e Informatica, UniversitÓ 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 Karush-Kuhn-Tucker (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) 520--534] in finite dimensional spaces.

Keywords: Generalized Nash equilibrium, variational inequalities, Karush-Kuhn-Tucker conditions.

MSC: 90C33, 58E35, 90C30.

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