$\epsilon k^0$-Subdifferentia1s of Convex Functions

Abstract

The paper as a contribution to convex analysis in ordered linear topological spaces. For any convex function $f$ from a Banach space $X$ into a partially ordered one $Y$ endowed with a convex cone $K$ some properties of the $\epsilon k^0$-subdifferential ${\partial }_{\epsilon k^0}^{\le }f(x)$ of $f$ are examined. The non-emptyness of ${\partial }_{\epsilon k^0}^{\le }f(x)$ is proved, whenever $Y$ is a normal order complete vector lattice and $f$ belongs to the class of functions which are continuous and convex with respect to the cone $K$. For the real-valued case Bronsted and Rockafellar have proved that the set of subgradients of a lower semicontinuous function f on a Banach space $X$ is dense in the set of $\epsilon$-subgradients [21]. We deduce a similar result for a class of $\epsilon k^0$-subdifferentials of functions which takes values in an ordered linear topological space $Y$.