Strongly essential set of vector Ky Fan's points problem and its applications

Abstract

In this paper, several existence results of strongly essential set of the solution set for Ky Fan's section problems and vector Ky Fan's point problems are obtained. Firstly, two kinds of strongly essential sets of Ky Fan's section problems are defined, and some further results on existence of the strongly essential component of solutions set of Ky Fan's section problems are proved, which generalize the conclusion in ^{[<span class="xref"><a href="#b22" ref-type="bibr">22</a></span>]}, and further generalize the conclusions in ^{[<span class="xref"><a href="#b21" ref-type="bibr">21</a></span>,<span class="xref"><a href="#b28" ref-type="bibr">28</a></span>]}. Secondly, based on the above results, two classes of stronger perturbations of vector-valued inequality functions are proposed respectively, and several existence results of the strongly essential component of set of vector Ky Fan's points are obtained. By comparing several metrics, we give some strong and weak relationships among the various metrics involved in the text. The main results of this paper actually generalize the relevant conclusions in the current literature. Finally, as an application, we obtain an existence result of the strongly essential component of weakly Pareto-Nash equilibrium for multiobjective games.