The finite intersection property for equilibrium problems
The paper essentially deals with quasi equilibrium problems, that is, given \(f:X\times X\to \mathbb{R}\) and \(K\) from a subset \(C\) of \(X\) into the subsets of \(C\), finding \(x\in K(x)\) such that \(f(x,y)\ge 0\) for all \(y\in K(x)\). Such problems are connected with monotonicity properties of \(f\), that in this setting are of the type \(f(x,y)+f(y,x)\le 0\) for all \(x,y\in C\), with several variants being also considered. The main point of the authors is substituting monotonicity requirements with a new property called finite intersection property, where the name comes from the fact that each family of sets with the classical fip can be associated with a function \(f\) satisfying fip. This property reads as: for all \(x_1, x_2,\ldots , x_n \in C\), there exists \(x \in C\) such that \[ \max_{i\in\{1,2,\ldots,n\}}f (x_i , x) \le 0. \] The authors study some relations between various notions of monotonicities and various versions of fip and show that the solvability of the so-called Minty equilibrium problem involving \(f\) is characterized by \(f\) enjoying fip. The main result establishes existence of solutions of a quasi equilibrium problem, under a number of assumptions that involve fip but not directly monotonicity for \(f\). Applications to existence of solutions of variational inequalities as well as to generalized Nash equilibrium problems are also given, together with some examples.
scientific article; zbMATH DE number 7340881
scientific article