ON A NON-COMPACT GENERALIZATION OF FAN'S MINIMAX THEOREM

Abstract

In this paper, we first introduce the weak convexlike condition which generalizes the convexlike concept due to Fan. Next, using the separation theorem for convex sets, we will prove a non-compact generalization of Fan's minimax theorem by relaxing the concavelike assumption to the weak concavelike condition. Also we give some examples which show that the convex and concave assumptions on Kneser's minimax theorem can not be relaxed with the quasi-convex and quasi-concave conditions simultaneously, and the previous minimax theorems can not be available.