\section{Introduction
The stability theory of functional equations deals with the following question: When it is true that the solution of an equation differing slightly from a given one, must of necessity be close to the solution of the given equation. In case of a positive answer to the previous problem, we say that the equation in question is \textit{stable}.
This problem was raised by Ulam (see \cite{Ula40}) and answered by Hyers who proved that the Cauchy equation is stable (\cite{Hye41}). Since then, this result has been extended and generalized in several ways (see e.g. \cite{For95}, \cite{Ger94} and \cite{HIR98}).
The investigation of the stability of the exponential Cauchy equation highlighted a new phenomenon which is now usually called \textit{superstability} (see e.g. \cite{HIR98}). The question of superstability is also dealt with in this paper.
Solving a stability problem, raised in \cite{Mak07} we give an affirmative answer to the case of higher dimensional information functions. \\
Throughout this paper let $k$ and $n$ be an arbitrary but fixed positive integers and denote
	\[
	\Gamma_{n}:=\left\{\left(p_{1}, \ldots, p_{n}\right)\in\R^{kn}\vert p_{i}\geq \mathbf{0}, \sum^{n}_{i=1}p_{i}=\mathbf{1}\right\}
\]
and
	\[
	D_{k}:=\left\{\left(x, y\right)\in\R^{2k}\vert x, y\in[0, 1[^{k}, x+y\leq \mathbf{1}\right\}.
\]
Furthermore, let
\[
\R^{k+}=\left\{x\in \R^{k}\vert x>\mathbf{0}\right\}.
\]
Here $\mathbf{1}$ represents the $k$-vector $\left(1, \ldots, 1\right)\in\R^{k}$ and all operations on vectors are to be done componentwise, i.e., $p_{i}\geq\mathbf{0}$ denotes that all coordinates of the vector $p_{i}\in\R^{k}$ are non-negative and we write $x+y\leq\mathbf{1}$ if $x_{i}+y_{i}\leq 1$ holds for all $i=1, \ldots, k$, where $x_{i}$ and $y_{i}$ denote the $i^{th}$ coordinates of the vector $x$ and $y$, respectively. \\
In what follows, we present some basic results from the theory of functional equations which we shall use throughout the paper, these results can be found for instance in \cite{Kuc85}. \\
\noindent
A function $M:[0, 1]^{k}\rightarrow\R$ is called \emph{multiplicative}, if
\begin{equation}
	M\left(x\cdot y\right)=M\left(x\right)\cdot M\left(y\right)
\end{equation}
holds for all $x, y\in\left[0, 1\right]^{k}$.\\
\noindent
We say that $A:\left[0, 1\right]^{k}\rightarrow\R$ is \emph{additive on $D_{k}$} if
\begin{equation}
A\left(x+y\right)=A\left(x\right)+A\left(y\right)
\end{equation}
holds for all pairs $\left(x, y\right)\in D_{k}$.


\begin{Lem}
If $M:[0,1]^{k}\rightarrow\R$ is both multiplicative on $[0,1]^{k}$ and additive on $D_{k}$, then $M$ is either identically zero or a projection, i.e.,
\begin{equation}
	M\left(x\right)=M\left(x_{1}, \ldots, x_{k}\right)=x_{j}, \qquad x\in [0,1]^{k},
\end{equation}
for some $j\in \left\{0, \ldots, k\right\}$.
\end{Lem}
In the proof of our theorem we shall use the following lemma.
\begin{Lem}\label{lemma}
Let $M:[0,1]^{k}\rightarrow\R$ be a multiplicative function. Then the following statements are equivalent.
\begin{enumerate}[(i)]
	\item $M$ is additive on $D_{k}$;
	\item $M\left(x\right)+M\left(\mathbf{1}-x\right)=1$ holds for all $x\in [0,1]^{k}$.
\end{enumerate}
\end{Lem}

\begin{Lem}
Let $M:[0,1]^{k}\rightarrow\R$ be a multiplicative function then
\[
M\left(x\right)\geq 0
\]
holds for all $x\in [0,1]^{k}$.
\end{Lem}

\begin{Lem}
Let $M:[0,1]^{k}\rightarrow\R$ be multiplicative. Then
\begin{equation}
M\left(x\right)=M\left(x_{1}, \ldots, x_{k}\right)=\prod^{k}_{i=1}m_{i}\left(x_{i}\right)
\end{equation}
for all $x\in [0,1]^{k}$, where each $m_{i}:[0,1]\rightarrow\R$ is multiplicative $(i=1, \ldots, k)$.
\end{Lem}
Now we fall to dealing with information measures (see \cite{AD75}, \cite{ESS98}).
\begin{Def}
A sequence of functions $I_{n}:\Gamma_{n}\rightarrow\R$ ($n=2, 3, \ldots$) is called \emph{information measure}.
\end{Def}
The usual information-theoretical interpretation is that $I_n\left(p_{1},\ldots, p_{n}\right)$
is a measure of uncertainty as to the outcome of an experiment having $n$
possible outcomes with probabilities $p_{1},\ldots, p_{n}$. \\
Some desiderata for measures of information can be found in \cite{AD75} as well as in \cite{ESS98}. Although in this paper we will use only the following properties.
\begin{Def}
The sequence of functions $I_{n}:\Gamma_{n}\rightarrow\R$ ($n=2, 3, \ldots$) is
\begin{enumerate}[(i)]
	\item \emph{M-recursive}, if
	\[
	I_{n}\left(p_{1}, \ldots, p_{n}\right)=I_{n-1}\left(p_{1}+p_{2}, p_{3}, \ldots, p_{n}\right)+
	M\left(p_{1}+p_{2}\right)I_{2}\left(\frac{p_{1}}{p_{1}+p_{2}}, \frac{p_{2}}{p_{1}+p_{2}}\right)
\]
holds for all $n=3, 4, \ldots$ and $\left(p_{1}, \ldots, p_{n}\right)\in\Gamma_{n}$, with some multiplicative function
$M:\left[0, 1\right]^{k}\rightarrow\R$ and with the convention $\frac{0}{0+0}=0$.
\item \emph{3-semisymmetric}, if
	\[
	I_{3}\left(p_{1}, p_{2}, p_{3}\right)=I_{3}\left(p_{1}, p_{3}, p_{2}\right)
\]
holds for all $\left(p_{1}, p_{2}, p_{3}\right)\in\Gamma_{3}$.
\end{enumerate}
\end{Def}
\noindent
Using the following theorem the characterization of information measures can be transformed into solving functional equations, see e.g., \cite{ESS98}.
\begin{Thm}
If the sequence of functions $I_{n}:\Gamma_{n}\rightarrow\R$ $\left(n=2, 3, \ldots\right)$ is M-recursive and 3-semisymmetric then the function
$f:[0, 1]^{k}\rightarrow\R$ defined by
	\[
	f(x):=I_{2}\left(\mathbf{1}-x, x\right)
\]
satisfies the so-called fundamental equation of information of multiplicative type M, i.e.,
\begin{equation}\label{FEIM}
f(x)+M\left(\mathbf{1}-x\right)f\left(\frac{y}{\mathbf{1}-x}\right)=
f(y)+M\left(\mathbf{1}-y\right)f\left(\frac{x}{\mathbf{1}-y}\right)
\end{equation}
for all $\left(x, y\right)\in D_{k}$.
\end{Thm}

\section{Known results}

In \cite{Mak07} it is proved that (\ref{FEIM}) is stable, moreover superstable assuming that $k=1$ and the function $M:[0, 1]^{k}\rightarrow\R$ is the power function, i.e.,  the stability of the following equation was investigated
	\[
	f(x)+\left(1-x\right)^{\alpha}f\left(\frac{y}{1-x}\right)=
	f\left(y\right)+\left(1-y\right)^{\alpha}f\left(\frac{x}{1-y}\right),
\]
where $0<\alpha \neq 1$. \\
In \cite{Mor01} a stability type result is proved for $k=1$ and $\alpha=1$, i.e., for the Shannon entropy.
However, Morando's theorem claims stability only on the rationals.




\section{Main result


In this section we will show stability type
results concerning the fundamental equation of
information of multiplicative type. Our main result is the following
\begin{Thm}\label{main}
Let $\varepsilon\geq 0$ be arbitrary, $M:[0,1]^{k}\rightarrow\R$ be multiplicative but not additive and $f:[0,1]^{k}\rightarrow\R$ be a function. Assume that
\begin{equation}\label{stab}
\left|f\left(x\right)+M\left(\mathbf{1}-x\right)f\left(\frac{y}{\mathbf{1}-x}\right)-
f\left(y\right)-M\left(\mathbf{1}-y\right)f\left(\frac{x}{\mathbf{1}-y}\right)\right|\leq \varepsilon
\end{equation}
holds for all $\left(x, y\right)\in D_{k}$. Then there exist $a, b\in\R$ such that
\begin{equation}\label{stab.ineq}
\begin{array}{l}
\left|f\left(x\right)-\left(aM\left(x\right)+b\left(M\left(\mathbf{1}-x\right)-1\right)\right)\right|\leq  \\
\left|M\left(q*\right)+M\left(\mathbf{1}-q*\right)-1\right|^{-1} \cdot
\left(4\varepsilon +3\varepsilon M\left(\mathbf{1}-xq*\right)\right)
\end{array}
\end{equation}
holds for all $x\in [0, 1]^{k}$, where $q*\in ]0, 1[^{k}$ is such that
$M\left(q*\right)+M\left(\mathbf{1}-q*\right)\neq 1$.
\end{Thm}
\begin{proof}
Define the function $F$ on $]0,1[^{k}\times [0,1]^{k}$ by
\begin{equation}
F\left(p, q\right)= f\left(\mathbf{1}-p\right)+M\left(p\right)f\left(q\right)-
f\left(pq\right)-M\left(\mathbf{1}-pq\right)f\left(\frac{\mathbf{1}-p}{\mathbf{1}-pq}\right).
\end{equation}
Then equation (\ref{stab}) with the substitution
$x=\mathbf{1}-p$ and $y=pq$ implies that
\begin{equation}\label{eps}
\left|F\left(p, q\right)\right|\leq \varepsilon
\end{equation}
holds for all $p, q\in ]0, 1[^{k}$.
On the other hand we have that
\begin{equation}
\begin{array}{l}
\left|\left[M\left(q\right)+M\left(\mathbf{1}-q\right)-1\right]\cdot
\left[f\left(p\right)-f\left(\mathbf{1}\right)M\left(p\right)\right]\right. \\
-\left[M\left(p\right)+M\left(\mathbf{1}-p\right)-1\right]\cdot
\left[f\left(q\right)-f\left(\mathbf{1}\right)M\left(q\right)\right]\left.\right| \\
=F\left(q, p\right)+F\left(p, q\right)-F\left(q, \mathbf{1}\right)+F\left(p, \mathbf{1}\right) \\
+M\left(\mathbf{1}-pq\right)\left[F\left(\frac{\mathbf{1}-p}{\mathbf{1}-pq}, \mathbf{1}\right)
+F\left(\frac{\mathbf{1}-p}{\mathbf{1}-pq}, \mathbf{1}\right)
-F\left(\frac{\mathbf{1}-p}{\mathbf{1}-pq}, q\right)\right]
\end{array}
\end{equation}
holds for all $p, q \in ]0,1[^{k}$. \\
Now using equation (\ref{eps}) we get that
\begin{equation}\label{becsl}
\begin{array}{l}
\left|\left[M\left(q\right)+M\left(\mathbf{1}-q\right)-1\right]\cdot
\left[f\left(p\right)-f\left(\mathbf{1}\right)M\left(p\right)\right]\right. \\
-\left[M\left(p\right)+M\left(\mathbf{1}-p\right)-1\right]\cdot
\left[f\left(q\right)-f\left(\mathbf{1}\right)M\left(q\right)\right]\left.\right| \\
\leq 4\varepsilon +3\varepsilon M\left(\mathbf{1}-pq\right)
\end{array}
\end{equation}
Since $M$ is not additive there exists a $q*\in ]0,1[^{k}$ such that
\begin{equation}\label{neq}
M\left(q*\right)+M\left(\mathbf{1}-q*\right)\neq 1.
\end{equation}
\noindent
Then with the substitution $q=q*$ in (\ref{becsl}) we have that
\begin{equation}
\begin{array}{l}
\left|\left[M\left(q*\right)+M\left(\mathbf{1}-q*\right)-1\right]\cdot
\left[f\left(p\right)-f\left(\mathbf{1}\right)M\left(p\right)\right]\right. \\
-\left[M\left(p\right)+M\left(\mathbf{1}-p\right)-1\right]\cdot
\left[f\left(q*\right)-f\left(\mathbf{1}\right)M\left(q*\right)\right]\left.\right| \\
\leq 4\varepsilon +3\varepsilon M\left(\mathbf{1}-pq*\right)
\end{array}
\end{equation}
Due to (\ref{neq}) we obtain that
\begin{equation}
\begin{array}{r}
\left|
\left[f\left(p\right)-f\left(\mathbf{1}\right)M\left(p\right)\right]-
\frac{f\left(q*\right)-f\left(\mathbf{1}\right)M\left(q*\right)}
{M\left(q*\right)+M\left(\mathbf{1}-q*\right)-1} \cdot
\left[M\left(p\right)+M\left(\mathbf{1}-p\right)-1\right]\right| \\
\leq \left|M\left(q*\right)+M\left(\mathbf{1}-q*\right)-1\right|^{-1} \cdot
\left(4\varepsilon +3\varepsilon M\left(\mathbf{1}-pq*\right)\right)
\end{array}
\end{equation}
Let
\[
a=f\left(\mathbf{1}\right)M\left(p\right)+
\frac{f\left(q*\right)-f\left(\mathbf{1}\right)M\left(q*\right)}
{M\left(q*\right)+M\left(\mathbf{1}-q*\right)-1}
\]
and
\[
b=\frac{f\left(q*\right)-f\left(\mathbf{1}\right)M\left(q*\right)}
{M\left(q*\right)+M\left(\mathbf{1}-q*\right)-1},
\]
\noindent
Therefore
\begin{equation}
\begin{array}{l}
\left|f\left(p\right)-\left[aM\left(p\right)+b\left(M\left(\mathbf{1}-p\right)-1\right)\right]\right| \leq \\
\left|M\left(q*\right)+M\left(\mathbf{1}-q*\right)-1\right|^{-1} \cdot
\left(4\varepsilon +3\varepsilon M\left(\mathbf{1}-pq*\right)\right)
\end{array}
\end{equation}
holds for all $p\in ]0,1[^{k}$. \\
\noindent
A direct calculation shows that (\ref{stab.ineq}) holds in case $p\in [0,1]^{k}\setminus ]0,1[^{k}$.
\end{proof}
\noindent
In what follows we define a function $K:[0,1]^{k}\rightarrow\R$ by
\begin{equation}\label{def.K}
K(x)=\frac{4\varepsilon+3\varepsilon M\left(\mathbf{1}-xq*\right)}
{\left|M\left(q*\right)+M\left(\mathbf{1}-q*\right)-1\right|}
\end{equation}
where $M:[0,1]^{k}\rightarrow\R$ is multiplicative but not additive,
$\varepsilon\geq 0$ arbitrary but fixed and $q*\in [0,1]^{k}$ is such that
$M\left(q*\right)+M\left(\mathbf{1}-q*\right)\neq 1$. \\
\noindent
Using the previous theorem we shall conclude the following.

\begin{Cor}
In case $\varepsilon=0$ in Theorem \ref{main}., then we get the general solution of equation
(\ref{FEIM}).
\end{Cor}

\begin{Cor}\label{rem1}
If the function $M:[0, 1]^{k}\rightarrow\R$ is bounded above by a constant $B\in\R$ then we get that inequality
(\ref{stab}) on $D_{k}$ implies
\begin{equation}
\begin{array}{l}
\left|f(x)-\left(aM\left(x\right)+b\left(M\left(\mathbf{1}-x\right)-1\right)\right)\right|\leq  \\
\left|M\left(q*\right)+M\left(\mathbf{1}-q*\right)-1\right|^{-1} \cdot
\left(4\varepsilon +3B\varepsilon\right)
\end{array}
\end{equation}
on $[0, 1]^{k}$, where $q*\in ]0, 1[^{k}$ is such that $M\left(q*\right)+M\left(\mathbf{1}-q*\right)\neq 1$.
\end{Cor}

\begin{Cor}
Due to Corollary \ref{rem1}. we obtain that the equation
\begin{equation}
f\left(x\right)+M\left(\mathbf{1}-x\right)f\left(\frac{y}{\mathbf{1}-x}\right)=
f\left(y\right)+M\left(\mathbf{1}-y\right)f\left(\frac{x}{\mathbf{1}-y}\right)
\end{equation}
is superstable on $D_{k}$ in case $M$ is bounded above.
\end{Cor}


\begin{Rem}
If $M\left(x\right)=x^{\alpha}$, $\left(x\in [0, 1]\right)$, where $0<\alpha\neq 1$
 then we get the result of Maksa (see \cite{Mak07}).
\end{Rem}
\noindent
Finally, the following theorem concerns the stability of a system of equations.

\begin{Thm}
Let $I_{n}$ be the sequence of functions
$I_{n}:\Gamma_{n}\rightarrow\R$ $(n\geq 2)$,
$M:[0,1]^{k}\rightarrow\R$ be a multiplicative function.
Suppose that there exist a sequence $\left(\varepsilon_{n}\right)$
of non-negative real numbers such that
\begin{equation}\label{ineq.syst}
\begin{array}{l}
\left| I_{n}\left(p_{1}, \ldots, p_{n} \right)-
I_{n-1}\left( p_{1}+p_{2}, p_{3}, \ldots, p_{n}\right) \right.\\
-M\left(p_{1}+p_{2}\right)
\left. I_{2}\left(\frac{p_{1}}{p_{1}+p_{2}}, \frac{p_{2}}{p_{1}+p_{2}}\right)\right|\leq \varepsilon_{n-1}
\end{array}
\end{equation}
holds for all $n\geq 3$ and $\left(p_{1}, \ldots, p_{n}\right)\in\Gamma_{n}$, and
\[
\left|I_{3}\left(p_{1}, p_{2}, p_{3}\right)-
I_{3}\left(p_{1}, p_{3, p_{2}}\right)\right|\leq \varepsilon_{1}
\]
holds on $\Gamma_{3}$.
Then there exist $c, d\in\R$ and a
$q*\in [0,1]^{k}$ such that
\begin{equation}\label{stab.syst}
\begin{array}{l}
\left|
I_{n}\left(p_{1}, \ldots, p_{n}\right)-
\left[c\left(\sum^{n}_{i=1}M(p_{i})-1\right)-
d\left(M\left(p_{1}\right)-1\right)\right]\right| \\
\leq
\sum^{n-1}_{k=2}\varepsilon_{k}+
\left(1+\left(n-2\right)M\left(p_{1}+p_{2}\right)\right)K\left(p_{2}\right)
\end{array}
\end{equation}
holds for all $n\geq 2$ and $\left(p_{1}, \ldots, p_{n}\right)\in\Gamma_{n}$, where
the convention $\sum^{1}_{k=2}\varepsilon_{k}=0$ is adapted and the function $K$
is defined by formula (\ref{def.K}).
\end{Thm}
\begin{proof}
The proof runs by induction on $n$.
Let $\left(x, y\right)\in D_{k}$, $n=3$ and substitute
\[
p_{1}=1-x-y
\quad
p_{2}=y
\quad
p_{3}=x
\]
into (\ref{ineq.syst}). Then
\[
\left|
I_{3}\left(\mathbf{1}-x-y, y, x\right)-I_{2}\left(\mathbf{1}-x, x\right)-
M\left(\mathbf{1}-x\right)
I_{2}\left(\mathbf{1}-\frac{y}{\mathbf{1}-x}, \frac{y}{\mathbf{1}-x}\right)
\right|
\leq \varepsilon_{2}
\]
holds.
Hence we get that the function $f:[0,1]^{k}\rightarrow\R$ defined by
\[
f(x)=I_{2}\left(\mathbf{1}-x, x\right) \qquad \left(x\in [0,1]^{k}\right)
\]
satisfies
\begin{equation}
\begin{array}{l}
\left|
f\left(x\right)+M\left(\mathbf{1}-x\right)f\left(\frac{y}{\mathbf{1}-x}\right)-
f(x)-M\left(\mathbf{1}-x\right)f\left(\frac{x}{\mathbf{1}-y}\right)
\right| \\
\leq
\left|
f(x)+M\left(\mathbf{1}-x\right)f\left(\frac{y}{\mathbf{1}-x}\right)-
I_{3}\left(\mathbf{1}-x-y, y, x\right)
\right| \\
+ \left|
I_{3}\left(\mathbf{1}-x-y, y, x\right)-I_{3}\left(\mathbf{1}-y-x, x, y\right)
\right| \\
+
\left |
I_{3}\left(\mathbf{1}-y-x, y, x\right)-f(y)-M\left(\mathbf{1}-y\right)f\left(\frac{y}{\mathbf{1}-y}\right)
\right|
\leq \varepsilon_{1}+2\varepsilon_{2}
\end{array}
\end{equation}
for all $(x, y)\in D_{k}$.
Thus, by Theorem \ref{main}. we get that there exist $a, b\in\R$ and
a $q*\in [0,1]$ such that
\begin{equation}
\begin{array}{l}
\left|f\left(x\right)-\left[aM\left(x\right)+b\left(M\left(\mathbf{1}-x\right)-1\right)\right]\right|\\
 \leq \left|M\left(q*\right)+M\left(\mathbf{1}-q*\right)-1\right|^{-1} \cdot \\
\left(4\left(\varepsilon_{1}+2\varepsilon_{2}\right) +
3\left(\varepsilon_{1}+2\varepsilon_{2}\right) M\left(\mathbf{1}-xq*\right)\right)
\end{array}
\end{equation}
holds for all $x\in [0,1]^{k}$. Let now
$\left(p_{1}, p_{2}\right)\in \Gamma_{2}$, then we obtain that
\begin{equation}
\begin{array}{l}
\left|
I_{2}\left(p_{1}, p_{2}\right)-
\left[aM\left(p_{2}\right)+b\left(M\left(p_{1}\right)-1\right)\right]\right. \\
\left. \leq
\left|M\left(q*\right)+M\left(\mathbf{1}-q*\right)-1\right|^{-1} \cdot
\left(4\varepsilon +3\varepsilon M\left(\mathbf{1}-p_{2}q*\right)\right)
\right|
\end{array}
\end{equation}
holds. Define $c=a$ and $d=b-a$, then we get that
\[
\begin{array}{l}
\left|
I_{2}\left(p_{1}, p_{2}\right)-
c\left[\sum^{2}_{k=1}M\left(p_{k}\right)-d\left(M(p_{1})-1\right)\right]\right. \\
\left. \leq
\left|M\left(q*\right)+M\left(\mathbf{1}-q*\right)-1\right|^{-1} \cdot
\left(4\varepsilon +3\varepsilon M\left(\mathbf{1}-p_{2}q*\right)\right)
\right|\\
= \sum^{2-1}_{k=2}\varepsilon_{k}+
\left(1-\left(1-1\right)M\left(p_{1}+p_{2}\right)\right)\cdot K\left(p_{2}\right),
\end{array}
\]
hence the statement holds for $n=2$.  Assume now that (\ref{ineq.syst}) holds and
introduce the following notation
\[
J_{n}\left(p_{1}, \ldots, p_{n}\right)=c\left(\sum^{n}_{k=1}M\left(p_{k}\right)-1\right)
+d\left(M\left(p_{1}\right)-1\right)
\]
for all $n\geq 2$, $\left(p_{1}, \ldots, p_{n}\right)\in\Gamma_{n}$.
It can be easily seen that $J_{n}:\Gamma_{n}\rightarrow\R$ is an
$M$--recursive and $3$-semisymmetric information measure $\left(n\in\N\right)$.
Therefore we get that
\begin{equation}
\begin{array}{l}
\left|I_{n+1}\left(p_{1}, \ldots, p_{n+1}\right)-J_{n+1}\left(p_{1}, \ldots, p_{n+1}\right) \right| \\
=
\left|I_{n+1}\left(p_{1}, \ldots, p_{n+1}\right)-J_{n}\left(p_{1}+p_{2}, \ldots, p_{n+1}\right)-
M\left(p_{1}+p_{2}\right)I_{2}\left(\frac{p_{1}}{p_{1}+p_{2}}, \frac{p_{2}}{p_{1}+p_{2}}\right) \right|\\
\leq
\left|I_{n+1}\left(p_{1}, \ldots, p_{n+1}\right)-
I_{n}\left(p_{1}+p_{2}, \ldots, p_{n+1}\right)-
M\left(p_{1}+p_{2}\right)I_{2}\left(\frac{p_{1}}{p_{1}+p_{2}}, \frac{p_{2}}{p_{1}+p_{2}}\right)\right| \\
+
\left|
I_{n}\left(p_{1}+p_{2}, \ldots, p_{n+1}\right)- J_{n}\left(p_{1}+p_{2}, \ldots, p_{n}\right)
\right| \\
+
\left|
M\left(p_{1}+p_{2}\right)I_{2}\left(\frac{p_{1}}{p_{1}+p_{2}}, \frac{p_{2}}{p_{1}+p_{2}}\right)-
M\left(p_{1}+p_{2}\right)J_{2}\left(\frac{p_{1}}{p_{1}+p_{2}}, \frac{p_{2}}{p_{1}+p_{2}}\right)
\right| \\
\leq
\varepsilon_{n}+ \sum^{n-1}_{k=2}\varepsilon_{n}+
\left(1+\left(n-2\right)M\left(p_{1}+p_{2}\right)\right)K\left(p_{2}\right)+
M\left(p_{1}+p_{2}\right)K\left(p_{2}\right)\\
=
\sum^{n}_{k=2}\varepsilon_{k}+\left(1+\left(n-1\right)M\left(p_{1}+p_{2}\right)\right)K\left(p_{2}\right),
\end{array}
\end{equation}
 for all $\left(p_{1}, \ldots, p_{n}\right)\in \Gamma_{n+1}$, that is, (\ref{stab.syst}) holds for
 $n+1$ instead of $n$, which ends the proof.
\end{proof}

\begin{Rem}
Our argument does not work in case $M$ is a projection, i.e., we cannot prove stability concerning the
fundamental equation of information in this case neither on the closed nor on the open domain.
\end{Rem}




