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# Zeitschrift für Analysis und ihre Anwendungen

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**Volume 15, Issue 4, 1996, pp. 999–1013**

**DOI: 10.4171/ZAA/742**

Published online: 1996-12-31

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

E.-Ch. Henkel^{[1]}(1) Universität Halle-Wittenberg, Germany

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 $\partial ^≥_{\epsilon k^0}f(x)$ of $f$ are examined. The non-emptyness of $\partial ^≥_{\epsilon k^0}f(x)$ 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 $\epsilon$-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 $Y$.

*Keywords: *Subdifferentials, $\epsilon-subdifferentials, order complete vector lattices, scalarization, properly efficient elements

Henkel E.-Ch.: $\epsilon k^0$-Subdifferentia1s of Convex Functions. *Z. Anal. Anwend.* 15 (1996), 999-1013. doi: 10.4171/ZAA/742