In this work, we establish several results concerning the existence of solutions for set-valued and single-valued equilibrium problems in real Hausdorff topological vector spaces. Firstly we introduce some generalizations of convexity and continuity conditions to set-valued mappings and then apply them to special dense subsets of the domain to obtain the existence of local dense solutions of equilibrium problems. Then the existence of the global solutions follows from a condition that is weaker than semistrict quasiconvexity. Specifically, we give an existence theorem for noncooperative n-person games, under assumptions imposed on a locally segment-dense subset of the strategy set of each player.

在这项工作中，我们建立了几个关于真实 Hausdorff 拓扑向量空间中集合值和单值均衡问题的解的存在性的结果。首先，我们将凸性和连续性条件的一些推广引入到集值映射，然后将它们应用于域的特殊稠密子集，以获得平衡问题的局部稠密解的存在性。那么全局解的存在是从弱于半严格拟凸性的条件得出的。具体来说，我们给出了非合作n人博弈的存在定理，假设强加于每个玩家策略集的局部分段密集子集。