The finite intersection property for equilibrium problems
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The finite intersection property for equilibrium problems John Cotrina1
· Anton Svensson2
Received: 24 May 2020 / Accepted: 18 October 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract The “finite intersection property” for bifunctions is introduced and its relationship with generalized monotonicity properties is studied. Some characterizations are considered involving the Minty equilibrium problem. Also, some results concerning existence of equilibria and quasi-equilibria are established recovering several results in the literature. Furthermore, we give an existence result for generalized Nash equilibrium problems and variational inequality problems. Keywords Quasi-equilibrium problem · Generalized Nash equilibrium problem · Variational inequality · Set-valued map · Generalized monotonicity · Finite intersection property Mathematics Subject Classification 47J20 · 49J35 · 90C37
1 Introduction Given a non-empty subset C of a topological space X and a bifunction f : X × X → R, the equilibrium problem, see Blum and Oettli [11], is the problem of finding: x ∈ C such that f (x, y) ≥ 0 for all y ∈ C.
(EP)
Problem (EP) has been extensively studied in recent years and several existence results have been developed under generalized monotonicity conditions (see [8–12,15,16,22,26,27, 29] and the references therein). A recurrent theme in the analysis of an equilibrium problem is its relation with the so-called Minty equilibrium problem, which corresponds to a sort of dual formulation of the equilibrium problem and consists of finding: x ∈ C such that f (y, x) ≤ 0 for all y ∈ C.
B
(MEP)
John Cotrina [email protected] Anton Svensson [email protected]
1
Universidad del Pacífico, Lima, Peru
2
Universidad de O’Higgins, Rancagua, Chile
123
Journal of Global Optimization
It is well-known that every solution of (EP) is a solution of (MEP), provided that the bifunction is pseudo-monotone. The converse inclusion is also strongly related to pseudomonotonicity of the additive inverse of the associated bifunction, see [17]. A more general setting of (EP) is the so-called quasi-equilibrium problem, in which the constraint set depends on the currently analysed point. More precisely, given a bifunction f : X × X → R and a set-valued map K : C ⇒ C, the quasi-equilibrium problem consists of finding: x ∈ K (x) such that f (x, y) ≥ 0 for all y ∈ K (x).
(QEP)
and the associated Minty quasi-equilibrium problem consists of finding: x ∈ K (x) such that f (y, x) ≤ 0 for all y ∈ K (x).
(MQEP)
These problems have begun to gain more and more attention due to their equivalence with the Stampacchia and Minty quasi-variational inequality problems, respectively, under convexity assumptions, see [4]. Moreover, it is known that they model problems from economics such as Nash equilibrium problems, multi-leader-follower games, abstract economies among others, see for instance [3,13,20,25,30,32] and their references therein. Recent works on the existence of quasi-equilibria involving generalized monoton
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