On Set-Valued Derivations Modulo K

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 $\mathbb{R}$.