Some properties of quasi-convex functions on abstract convex spaces

Abstract

In this paper, we introduce F-convex sets which are modified Econvex sets and show that F-convex sets are abstract convex spaces. Using generalized KKM maps on abstract convex spaces, we obtain a fixed point theorem and an equilibrium result for generalized quasiconcave functions on F-convex sets. We discuss properties of quasi-convex functions on abstract convex spaces and apply them to F-convex sets. We also give new results of semi-convex programming.