Let
Y
Y
be a real vector metric space and
K\subset Y
K
⊂
Y
be a closed convex cone with
K\cap (-K)=\{0\}
K
∩
(
−
K
)
=
{
0
}
. We study properties of set-valued maps
F\colon\mathbb{R}\to 2^Y\setminus\{\emptyset\}
F
:
R
→
2
Y
∖
{
∅
}
which are additive modulo
K
K
, i.e.
F(x+y)+K=F(x)+F(y)+K
F
(
x
+
y
)
+
K
=
F
(
x
)
+
F
(
y
)
+
K
for
x,y\in \mathbb{R}
x
,
y
∈
R
, and satisfy condition
F(xy)+K=xF(y)+yF(x)+K
F
(
x
y
)
+
K
=
x
F
(
y
)
+
y
F
(
x
)
+
K
for
x,y\in [0,\infty)
x
,
y
∈
[
0
,
∞
)
(or
x,y\in \mathbb{R}
x
,
y
∈
R
). Such maps are called set-valued derivations modulo
K
K
and generalize the well-known single-valued derivations of
\mathbb{R}
R
.