Abstract
In this paper, some further results on the stability of Ky Fan’s points are proposed by introducing a type of stronger perturbation of section mappings defined by a semi-metric called the maximum Hausdorff semi-metric, and the existence of the essential components of the set of Ky Fan’s points to this perturbation is proved. As an application, the existence of the essential component of the Nash equilibrium is presented using the proposed method, and strong robustness to payoff function perturbation is achieved.
Highlights
Let us consider a function f : X × X → R, where X is a nonempty compact convex subset of a Hausdorff topological linear space.The Ky Fan minimax inequality (Ky Fan’s inequality, for short, [ ]) problem, denoted by (KF), consists in finding an element y∗ ∈ X such that f x, y∗ ≤ (x ∈ X).The function f is called an inequality function, and the element y∗ is called a Ky Fan point
Because of the wide application of Ky Fan’s inequality in optimization, convex analysis, variational inequality, control theory, fixed point theory and mathematical economics, it has been generalized in various ways, such as the implicit variational inequality, equilibrium problem, vector variational inequality and mixed implicit variational inequality ([ – ])
The perturbation of (f ) is defined by the inequality functions corresponding to the payoff functions of (f ), by using result ( ) from Proposition, we see that the essential component with respect to ρu is an essential component with respect to ρm( (f ), (g)) and ρu( (f ), (g)), and it has stronger stability
Summary
Let us consider a function f : X × X → R, where X is a nonempty compact convex subset of a Hausdorff topological linear space. A nonempty closed subset e(E) of Fs(E) is said to be an essential set of Fs(E) with respect to ρsu if, given any number > , there exists a corresponding number δ > such that Fs(E ) ∩ [e(E) + B ( )] = ∅ for all E ∈ E such that ρsu(E , E) < δ. A nonempty closed subset e(φ) of Fk(φ) is said to be an essential set of Fk(φ) with respect to ρku (or ρm) if, given any number > , there exists a corresponding number δ > such that Fk(φ ) ∩ [e(φ) + B ( )] = ∅ for all φ ∈ M such that ρku(φ , φ) < δ. Corollary For every φ ∈ M, Fk(φ) has at least one essential connected component with respect to ρ
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