EXISTENCE OF MAXIMAL ELEMENTS IN A BINARY RELATION RELAXING THE CONVEXITY CONDITION

In this paper we relax the convexity condition in some classical results on the existence of maximal elements in binary relations in order to generalize them. To do this, we replace the linear segments in the usual convexity with a family of previously fixed paths joining up each two points. From these paths, we introduce a family of sets which generalizes the usual convex sets, and in this context we extend Sonnenschein's theorem on the existence of maximal elements and Browder's theorem on the existence of continuous selection and f ixed point to correspondences.