Strongly essential set of Ky Fan's points and the stability of Nash equilibrium

In this paper, a stronger perturbation of inequality functions and set-valued mappings is proposed by means of the Hausdorff upper semi-metric, which includes the perturbations defined by sup-norm of function and the maximum Hausdorff metric of section mapping. Based on this perturbation, a class of strongly essential sets of Ky Fan's points is introduced, and the existence of the strongly essential component of Ky Fan's points is proved. As an application, we use the equivalence of Nash equilibrium with Ky Fan's points to obtain the existence of the strongly essential component of equilibrium, which not only give a class of stronger essential sets, but also provide a method to discuss the stability with respect to the perturbation of strategic sets.