Journal of Convex Analysis 32 (2025), No. 4, 1135--1144
Copyright Heldermann Verlag 2025
On Set-Valued Derivations Modulo K
Eliza Jablonska
Faculty of Applied Mathematics, AGH University of Krakow, Kraków, Poland
elizajab@agh.edu.pl
[Abstract-pdf]
Let $Y$ be a real vector metric space and $K\subset Y$ be a closed convex cone with $K\cap (-K)=\{0\}$. We study properties of set-valued maps $F\colon\mathbb{R}\to 2^Y\setminus\{\emptyset\}$ which are additive modulo $K$, i.e. $F(x+y)+K=F(x)+F(y)+K$ for $x,y\in \mathbb{R}$, and satisfy condition $F(xy)+K=xF(y)+yF(x)+K$ for $x,y\in [0,\infty)$ (or $x,y\in \mathbb{R}$). Such maps are called set-valued derivations modulo $K$ and generalize the well-known single-valued derivations of $\mathbb{R}$.
Keywords: K-additive set-valued map, (strong) set-valued K-derivation, K-lower boundedness, weak K-upper boundedness, K-continuity, null-finite set.
MSC: 39B62; 54C60, 26B25.
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