Vector-Valued Convex Functions

Abstract

In this paper, we will extend some properties of the convex real functions to the valued functions in a Banach lattice: with adequate definitions, we will establish that an order convex function is continuous on a convex C if and only if it is continuous at a point of C (Theorem 1.2). We will show that order convex functions on a compact satisfy Bauer’s maximal principle (Theorem 2.2). A fixed-point theorem is given for the contracting orders functions (Theorem 2.3).