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.
