Existence of an upper hemi-continuous and convex-valued demand sub-correspondence

In this paper we show that a strictly open, non-saturated and acyclically convex preference relation admits an extension which is ordered by inclusion (a weaker property than regularity), strictly open, locally non saturated and convex; in turn, this result permits to prove the existence of an upper hemi-continuous and convex-valued demand sub-correspondence. By directly applying standard fixed-point techniques to these sub-correspondences, it is therefore possible to demonstrate the existence of general economic equilibrium even if consumers' preference relations are not regular.