# $ϵk_{0}$-Subdifferentia1s of Convex Functions

### E.-Ch. Henkel

Universität Halle-Wittenberg, Germany

## 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 $ϵk_{0}$-subdifferential $∂_{ϵk_{0}}f(x)$ of $f$ are examined. The non-emptyness of $∂_{ϵk_{0}}f(x)$ 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 $ϵ$-subgradients [21]. We deduce a similar result for a class of $ϵk_{0}$-subdifferentials of functions which takes values in an ordered linear topological space $Y$.

## Cite this article

E.-Ch. Henkel, $ϵk_{0}$-Subdifferentia1s of Convex Functions. Z. Anal. Anwend. 15 (1996), no. 4, pp. 999–1013

DOI 10.4171/ZAA/742