\documentclass[12pt, reqno]{amsart}
\usepackage{amsmath, amsthm, amscd, amsfonts, amssymb, graphicx, color}
\usepackage[bookmarksnumbered, colorlinks, plainpages]{hyperref}

\renewcommand{\baselinestretch}{1.1}
\setlength{\textwidth}{16 cm} \setlength{\textheight}{8.75in}
\setlength{\evensidemargin}{0.15in}
\setlength{\oddsidemargin}{0.15in}
\newcommand{\R}{\mbox{$\mathbb{R}$}}
\newcommand{\N}{\mbox{$\mathbb{N}$}}
\newcommand{\Q}{\mbox{$\mathbb{Q}$}}
\newcommand{\C}{\mbox{$\mathbb{C}$}}
\newcommand{\Z}{\mbox{$\mathbb{Z}$}}
\newcommand{\K}{\mbox{$\mathbb{K}$}}
\newcommand{\nwl}{\left (}
\newcommand{\nwp}{\right )}
\newcommand{\wzor}[1]{{\rm (\ref{#1})}}
\newcommand{\ckd}{\hspace{0.5cm} $\square$\medskip\par\noindent}
\newcommand{\dowod}{{\it Proof:}\ \ }
\newcommand{\id}{{\mathcal J}}
\newcommand{\ip}[2]{\left\langle#1|#2\right\rangle}
\newcommand{\kropka}{\hspace{-1.3ex}.\ }
\newcommand{\eps}{\varepsilon}


 \markboth
{$~$ \hfill \footnotesize {\rm H. Azadi Kenary, Kh. Shafaat and H. Keshavarz  } \hfill
 $~$}
 {$~$ \hfill \footnotesize {\rm Approximate  $m$-Lie homomorphisms ...}  \hfill$~$}

\begin{document}
\thispagestyle{empty}
 \setcounter{page}{1}




\begin{center}
{\large\bf Hyers-Ulam-Rassias Approximation   on  $m$-Lie Algebras}


\vskip.35in

{\bf  H. Azadi Kenary , Kh. Shafaat and H. Keshavarz} \\[2mm]

{\footnotesize Department of Mathematics, College of Sciences, Yasouj University, Yasouj 75914-353, Iran
}
\end{center}
\vskip 3mm

\noindent{\footnotesize{\bf Abstract.}
Using the fixed point method, we establish the stability of $m-$Lie homomorphisms and Jordan $m-$Lie homomorphisms on
 $m-$Lie algebras  associated to the  following
additive functional equation $$
2\mu f\left(\sum_{i=1}^m  m x_i\right)=\sum_{i=1}^mf\left(\mu  \left(m x_i + \sum_{j=1~,i\neq j}^m x_j\right)\right)+f\left(\sum_{i=1}^m \mu x_i\right)$$
where $m$ be an integer greater than 2 and all $\mu\in  \Bbb T_{\frac{1}{n_0}} :=\left\{e^{i\theta} \ ;\
0\leq\theta\leq \frac{2\pi}{n_0}\right\}$.
 \vskip.15in
 \footnotetext { \textbf{2000 Mathematics Subject Classification}:17A42, 39B82,
 39B52.}
 \footnotetext { \textbf{Keywords}:  $m-$Lie algebra; homomorphism; Jordan homomorphism; Stability; Fixed point approach;
functional equation
 }
\footnotetext{\textbf{Corresponding author}{\tt :
h.azadikenary@gmail.com}}


\baselineskip=16pt


%\theoremstyle{definition}
  \newtheorem{df}{Definition}[section]
  \newtheorem{rk}[df]{Remark}
%\theoremstyle{plain}
   \newtheorem{lem}[df]{Lemma}
   \newtheorem{thm}[df]{Theorem}
   \newtheorem{pro}[df]{Proposition}
   \newtheorem{cor}[df]{Corollary}
   \newtheorem{ex}[df]{Example}

 \setcounter{section}{0}
 \numberwithin{equation}{section}
\vskip .2in
%--------------------------------------------------------------------------------------------------------------
\section{\bf Introduction}

Let $n$ be a natural number greater or equal to 3.
The notion of an $n-$Lie algebra was introduced by V.T. Filippov in 1985 \cite{1}.
The Lie product is taken between $n$ elements of the algebra
instead of two. This new bracket is $n-$linear, anti--symmetric and satisfies a
generalization of the Jacobi identity. \\
An $n-$Lie algebra is a natural
generalization of a Lie algebra. Namely:

 A vector space $V$ together with a multi--linear, antisymmetric
$n-$ary operation $[~~~~ ] : \Lambda^nV \rightarrow V $ is called an $n-$Lie algebra, $n \geq 3$, if the $n-$ary
bracket is a derivation with respect to itself, i.e,
\begin{equation}\label{00}
[[x_1,...,x_n],x_{n+1},...,x_{2n-1}]=\sum_{i=1}^n
[x_1,...x_{i-1}[x_i, x_{n+1},...,x_{2n-1}],...,x_n]
\end{equation}
where $x_1,x_2,...,x_{2n-1} \in V$.
Equation $(\ref{00})$ is called the generalized Jacobi identity.
The meaning of this identity is similar to that of the usual Jacobi identity for a Lie algebra (which is a
$2-$Lie algebra).\\
From now on, we  only consider
$n-$Lie algebras over the field of complex numbers.
An $n-$Lie algebra $A$ is a normed $n-$Lie algebra if there exists a norm $||\ ||$ on $A$ such that
 $||[x_1,x_2,...,x_n]||\leq ||x_1|| ||x_2||... ||x_n||$ for all $x_1,x_2,...,x_n \in A$.
 A normed  $n-$Lie  algebra $A$ is called a Banach $n-$Lie algebra, if
$(A,||\ ||)$ is a Banach space.

Let  $(A,[~~]_A)$ and $(B,[~~]_B)$ be two Banach $n-$Lie  algebras.
 A $\Bbb C-$linear mapping $H:(A,[~]_A)\rightarrow(B,[~]_B)$ is
called an $n-$Lie homomorphism  if
$H([x_1x_2...x_n]_A)=[H(x_1)H(x_2)...H(x_n)]_B$
 for  all $x_1,x_2,...,x_n \in A.$
A $\Bbb C-$linear mapping $H:(A,[~]_A)\rightarrow(B,[~]_B)$ is
called a Jordan  $n-$Lie homomorphism  if
$H([xx...x]_A)=[H(x)H(x)...H(x)]_B$
 for  all $x \in A.$




 The
study of stability problems had been formulated by Ulam \cite{10}
during a talk in 1940: Under what condition does there exist a
homomorphism near an approximate homomorphism? In the following
year, Hyers \cite{11} was answered affirmatively the question of
Ulam for Banach spaces, which states that if $\varepsilon
> 0$ and $f :X\rightarrow¨ Y$ is a map with $X$ a normed space, $Y$ a Banach
spaces such that
\begin{equation}\label{E0}
\|f(x + y)-f(x)-f(y)\|\leq \varepsilon
\end{equation}
for all $x, y \in X,$ then there exists a unique additive map $T :
X \rightarrow Y$ such that $\|f(x)-T(x)\|\leq \varepsilon $ for
all $x \in X.$ A generalized version of the theorem of Hyers  for approximately linear mappings
 was presented by Rassias \cite{12} in 1978 by considering the
case when inequality $(\ref{E0})$ is unbounded. Due to that fact,
the additive functional equation $f(x + y)=f(x)+f(y)$ is said to
have the generalized Hyers--Ulam--Rassias stability property.  A large list of
references concerning the stability of functional equations can be
found in \cite{az}--\cite{azc}.\\
In this paper, by using the fixed point method,
we establish the stability of $m-$Lie homomorphisms and Jordan $m-$Lie homomorphisms on
 $m-$Lie Banach algebras  associated to the
following generalized Jensen type functional equation
\begin{equation}\label{e11}
2\mu f\left(\sum_{i=1}^m  m x_i\right)-\sum_{i=1}^mf\left(\mu  \left(m x_i + \sum_{j=1~,i\neq j}^m x_j\right)\right)-f\left(\sum_{i=1}^m \mu x_i\right)=0
\end{equation}
for all $\mu\in  \Bbb T_{\frac{1}{n_o}} :=\left\{e^{i\theta} \ ;\
0\leq\theta\leq \frac{2\pi}{n_o}\right\},$
where $m\geq2.$ Throughout this paper, assume that
$(A,[~]_A),(B,[~]_B)$ are two $m$-Lie Banach algebras.
 \vskip .2in
%--------------------------------------------------------------------------------------------------------------
\section{\bf Main Results}



Before proceeding to the main results, we recall a fundamental
result in fixed point theory.
\begin{thm}\label{t1.2}\cite{12} Let $(\Omega,d)$ be a  complete
generalized metric space  and $T:\Omega\rightarrow\Omega$ be  a strictly contractive
function  with Lipschitz constant $L$.
Then for each given $x\in\Omega$, either
$d(T^m x, T^{m+1} x)=\infty~$ for all $m\geq0,$

 or other exists a natural number $m_{0}$ such that: $(i)$ $d(T^m x, T^{m+1} x)<\infty ~$for all $m \geq m_{0};$

$\bullet$ the sequence $\{T^m x\}$ is convergent to a fixed point
$y^*$ of $~T$;

$\bullet$ $y^*$ is the unique fixed point of $~T$ in
$~\Lambda=\{y\in\Omega:d(T^{m_{0}} x, y)<\infty\};$

$\bullet$ $d(y,y^*)\leq\frac{1}{1-L}d(y, Ty)$ for all
$~y\in\Lambda.$
\end{thm}


\begin{thm}\cite{azc}
Let $V$ and $W$ be real vector spaces. A mapping $f:V\rightarrow W$ satisfies  the following functional equation
$$2 f\left(\sum_{i=1}^m  m x_i\right)=\sum_{i=1}^mf\left(m x_i + \sum_{j=1~,i\neq j}^m x_j\right)+f\left(\sum_{i=1}^m  x_i\right)$$
if and only if $f$ is additive.
\end{thm}


 We start
our work with the main theorem of  our paper.
\begin{thm}\label{t2.1}
Let  $n_0 \in \Bbb N$ be a fixed positive integer. Let $f:A\rightarrow B$ be a
mapping  for which there exists a function
$\varphi:A^{m}\rightarrow [0,\infty)$ such that
\begin{eqnarray}\label{e21}
\begin{split}
\left\|2\mu f\left(\sum_{i=1}^m  m x_i\right)-\sum_{i=1}^mf\left(\mu  \left(m x_i + \sum_{j=1~,i\neq j}^m x_j\right)\right)-f\left(\sum_{i=1}^m \mu x_i\right)\right\|\leq\varphi(x_1,x_2,\cdots,x_m)
\end{split}
\end{eqnarray}
for all $\mu\in  \Bbb T_{\frac{1}{n_o}} :=\left\{e^{i\theta} \ ;\
0\leq\theta\leq \frac{2\pi}{n_0}\right\}$
and all $x_1,\cdots,x_m \in A,$ and that
\begin{eqnarray}\label{e211}
\begin{split}
\|f([x_1x_2 \cdots
x_n]_A)-[f(x_1)f(x_2)\cdots f(x_m)]_B\|_B\leq\varphi(x_1,x_2,\cdots,x_m)
\end{split}
\end{eqnarray}
for all $x_1,\cdots,x_m
\in A.$ If there exists an $L< 1$ such that
\begin{equation}\label{e22}
\varphi(x_1,x_2,\cdots,x_m)\leq m
L\varphi\left(\frac{x_1}{m},\frac{x_2}{m},\cdots,\frac{x_m}{m}\right)
\end{equation}
for all $x_1,\cdots,x_m \in A,$ then there exists a
unique $m-$Lie homomorphism  $H:A\rightarrow B$ such that
\begin{equation}\label{e23}
\|f(x)-H(x)\|\leq
\frac{\varphi(x,0,0,\cdots,0)}{m-mL}
\end{equation}
for all $x \in A.$
\end{thm}
\begin{proof}
Let $\Omega$ be the set of all functions from $A$ into $B$ and let
$d(g,h):=\inf\{C \in \Bbb R^+ :\|g(x)-h(x)\|_B\leq C\varphi(x,0,\cdots,0),  \forall x\in A\}.$
It is easy to show that $(\Omega,d)$ is a generalized complete metric
space \cite{41}.  Now we define the  mapping
$J:\Omega\rightarrow \Omega$ by $J(h)(x)=\frac{1}{m}h(mx)$
for all $x \in A.$ Note that for all $g, h \in \Omega,$
\begin{align*}
d (g,h)<C &~\Rightarrow~~\|g(x)-h(x)\| \leq
C\phi(x,0,\cdots,0),
~\Rightarrow~~\left\|\frac{1}{m}g(m x)-\frac{1}{m}
h(m x)\right\|\leq \frac{C\varphi(m
x,0,\cdots,0)}{m},
\\&~\Rightarrow~~\left\|\frac{1}{m}g(m x)-\frac{1}{m}
h(m x)\right\|\leq L~C~\varphi(x,0,\cdots,0),
~\Rightarrow~~d(J(g),J(h))\leq L~C.
\end{align*}
for all $x\in A$. Hence we see that
$d(J(g),J(h))\leq Ld(g,h)$
for all $g, h \in \Omega.$
 It follows from $(\ref{e22})$ that
\begin{equation}\label{e24}
\lim_{k \to \infty}\frac{\varphi(m^{k}x_1,m^{k}x_2,\cdots,m^{k}x_m)}{m^{k}}\leq \lim_{k \to \infty} L^{k}\varphi(x_1,\cdots,x_m)=0
\end{equation}
for all $x_1,\cdots,x_m \in A.$ Putting $\mu=1,x_1=x$
and $x_j=0$ ($j=2,\cdots,m$) in $(\ref{e21}),$
we obtain
\begin{equation}\label{e25}
\left\|  \frac{f(mx)}{m}-f(x)\right\|\leq\frac{\varphi(x,0,\cdots,0)}{m}
\end{equation}
for all $x \in A.$ Therefore,
\begin{equation}\label{e27}
d(f,J(f))\leq \frac{1}{m} < \infty.
\end{equation}
 By Theorem \ref{t1.2},  $J$ has a unique  fixed point in the set
$X_1:=\{h \in  \Omega:d(f,h)<\infty\}.$ Let $H$ be the fixed point of
$J.$ $H$ is the unique  mapping with
$H(mx)=mH(x)$
for all $x \in A,$ such that there exists $C\in (0,\infty)$
satisfying
$\|f(x)-H(x)\|_B\leq C\varphi(x,0,\cdots,0) $
for all $x \in A.$ On the other hand we have $\lim_{k \to
\infty}d(J^k(f),H)=0,$ and so
\begin{equation}\label{e29}
 \lim_{k\to
\infty}\frac{1}{m^{k}}f(m^{k}x)=H(x)
\end{equation}
 for all $x \in A.$
Also by Theorem \ref{t1.2}, we have
 \begin{equation}\label{e210}
 d(f,H)\leq
\frac{d(f,J(f))}{1-L}
\end{equation}
It follows from $(\ref{e27})$ and $(\ref{e210}),$
that
$d(f,H)\leq \frac{1}{m-mL}.$
This implies the inequality $(\ref{e23}).$
By
$(\ref{e211})$, we have
\begin{eqnarray*}
&&\|H([x_1 x_2 \cdots x_m]_A)-[H(x_1)H(x_2)H(x_3) \cdots
H(x_m)]_B\|\\&&=\lim_{k \to
\infty}\left\|\frac{H([m^k x_1 m^k x_2 \cdots
m^k x_m]_A)}{m^{mk}}-\frac{([H(m^k x_1)H(m^k x_2)H(m^k x_3) \cdots
H(m^k x_m)]_B)}{m^{mk}}\right\| \\&&\leq\lim_{k \to
\infty}\frac{\varphi(m^k x_1, m^k  x_2, \cdots,
m^k x_m)}{m^{mk}}=0
\end{eqnarray*}
for all $x_1,\cdots,x_m \in A.$ Hence
$H([x_1 x_2 \cdots x_m]_A)=[H(x_1)H(x_2)H(x_3) \cdots H(x_m)]_B$ for all $x_1,\cdots,x_m \in A.$ On the other hand, it follows from
$(\ref{e21}),$ $(\ref{e24})$
 and $(\ref{e29})$ that
 \begin{align*}
&\left\|2H\left(\sum_{i=1}^m  m x_i\right)-\sum_{i=1}^mH  \left(m x_i + \sum_{j=1~,i\neq j}^m x_j\right)-H\left(\sum_{i=1}^m  x_i\right)\right\|\\&=\lim_{k \to
\infty}\frac{1}{m^{k}}\left\|2f\left(\sum_{i=1}^m  m^{k+1} x_i\right)-\sum_{i=1}^m f  \left(m^{k+1} x_i + \sum_{j=1~,i\neq j}^m m^k x_j\right)-f\left(\sum_{i=1}^m  m^k x_i\right)\right\| \\& \leq \lim_{m \to
\infty}\frac{\varphi(m^{k}x_1,m^{k}x_2,\cdots,m^{k}x_m)}{m^{k}}=0
\end{align*}
for all $x_1,\cdots,x_m \in A.$ Then
 \begin{equation}\label{e211gh}
 2H\left(\sum_{i=1}^m  m x_i\right)=\sum_{i=1}^mH  \left(m x_i + \sum_{j=1~,i\neq j}^m x_j\right)+H\left(\sum_{i=1}^m  x_i\right)
\end{equation}
for all $x_1,\cdots,x_m \in A.$ So by Theorem 2.2, $H$ is  additive. Letting
$x_i=x$ for all $i=1,2,\cdots,m$ in $(\ref{e21}),$ we obtain
$\|\mu f(x)-f(\mu x)\|_B \leq \varphi(x,x,\cdots,x)$
for all $x \in A.$ It follows that
\begin{align*}
\|H(\mu x)-\mu H( x)\|&=\lim_{k \to
\infty}\frac{\|f(\mu m^k x)-\mu
f(m^k x)\|}{m^k}\leq \lim_{k \to
\infty}\frac{\varphi(m^kx,m^kx,\cdots,m^kx)}{m^k}=0
\end{align*}
for all $\mu\in \Bbb T_{\frac{1}{n_o}} :=\left\{e^{i\theta} \ ;\
0\leq\theta\leq \frac{2\pi}{n_0}\right\},$ and all $x \in A.$ One can show that
the mapping $H:A\rightarrow B$ is $\Bbb C$--linear.
Hence, $H:A\rightarrow B$ is an $m-$Lie homomorphism satisfying
$(\ref{e23}),$ as desired.
\end{proof}
\begin{cor}\label{c22}
Let $\theta $ and $p$ be
non--negative real numbers such that $p <1.$ Suppose that
a function $f:A\rightarrow B$ satisfies
\begin{align}\label{xxx}
\left\|2\mu f\left(\sum_{i=1}^m  m x_i\right)-\sum_{i=1}^m f  \left(\mu \left(m x_i + \sum_{j=1~,i\neq j}^m x_j\right)\right)-f\left(\sum_{i=1}^m  \mu x_i\right)\right\| \leq
\theta\left(\sum^{m}_{i=1}\|x_{i}\|_{A}^{p}\right)
\end{align}
for all $\mu\in \Bbb T_{\frac{1}{n_o}} $ and all $x_1,\cdots,x_m
\in A$ and

\begin{align}\label{yyy}
\|f([x_1x_2 \cdots
x_n]_A)-[f(x_1)f(x_2)\cdots f(x_m)]_B\| \leq
\theta\left(\sum^{m}_{i=1}\|x_{i}\|_{A}^{p}\right)
\end{align}
for  all $x_1,\cdots,x_n
\in A.$ Then there exists a unique $m-$Lie homomorphism
$H:A\rightarrow B$ such that
\begin{equation}\label{e2213}
\|f(x)-H(x)\| \leq \frac{\theta\|x\|_{A}^{p}}{m-m^p}
\end{equation}
for all $x \in A.$
\end{cor}
\begin{proof}
Put  $\varphi(x_1,x_2,\cdots,x_m):=\theta
\sum^{m}_{i=1}(\|x_{i}\|_{A}^{p})$  for all
$x_1,\cdots,x_n \in A$ in Theorem \ref{t2.1}.  Then
$(\ref{e24})$ holds for $p ~<1$,  and
$(\ref{e2213})$ holds when $L=m^{p-1}.$
\end{proof}
Similarly, we have the following and we will omit the proof.
\begin{thm}\label{t2.1x}
Let  $n_0 \in \Bbb N$ be a fixed positive integer. Let $f:A\rightarrow B$ be a
mapping  for which there exists a function
$\varphi:A^{m}\rightarrow [0,\infty)$  satisfying  (\ref{e21})
for all $\mu\in  \Bbb T_{\frac{1}{n_0}} :=\left\{e^{i\theta} \ ;\
0\leq\theta\leq \frac{2\pi}{n_0}\right\}$ and (\ref{e211}).
If there exists an $L< 1$ such that
\begin{equation}\label{e22x}
\varphi\left(\frac{x_1}{m},\frac{x_2}{m},\cdots,\frac{x_m}{m}\right)\leq
\frac{L\varphi(x_1,x_2,\cdots,x_m)}{m}
\end{equation}
for all $x_1,\cdots,x_m \in A,$ then there exists a
unique $m-$Lie homomorphism  $H:A\rightarrow B$ such that
\begin{equation}\label{e23x}
\|f(x)-H(x)\|\leq
\frac{L\varphi(x,0,0,\cdots,0)}{m-mL}
\end{equation}
for all $x \in A.$
\end{thm}

\begin{cor}
Let $\theta $ and $p$ be
non--negative real numbers such that $p >1.$ Suppose that
a function $f:A\rightarrow B$ satisfying  (\ref{xxx}) and (\ref{yyy}). Then there exists a unique $m-$Lie homomorphism
$H:A\rightarrow B$ such that
\begin{equation}\label{e2213x}
\|f(x)-H(x)\| \leq \frac{m\theta\|x\|_{A}^{p}}{m^{p+1}-m^2}
\end{equation}
for all $x \in A.$
\end{cor}
\begin{proof}
Put  $\varphi(x_1,x_2,\cdots,x_m):=\theta
\sum^{m}_{i=1}(\|x_{i}\|_{A}^{p})$  for all
$x_1,\cdots,x_n \in A$ in Theorem \ref{t2.1x}.  Then
$(\ref{e23x})$ holds for $p ~<1$,  and
$(\ref{e2213x})$ holds when $L=m^{(1-p)}.$
\end{proof}

\begin{thm}\label{t2.2}
Let  $n_0 \in \Bbb N$ be a fixed positive integer. Let $f:A\rightarrow B$ be a
mapping  for which there exists a function
$\varphi:A^{m}\rightarrow [0,\infty)$ such that
\begin{eqnarray}\label{e221}
\begin{split}
\left\|2\mu f\left(\sum_{i=1}^m  m x_i\right)-\sum_{i=1}^m f  \left(\mu \left(m x_i + \sum_{j=1~,i\neq j}^m x_j\right)\right)-f\left(\sum_{i=1}^m  \mu x_i\right)\right\| \leq\varphi(x_1,x_2,\cdots,x_m)
\end{split}
\end{eqnarray}
for all $\mu\in  \Bbb T_{\frac{1}{n_o}} :=\left\{e^{i\theta} \ ;\
0\leq\theta\leq \frac{2\pi}{n_0}\right\}$
and all $x_1,\cdots,x_m \in A,$ and that
\begin{eqnarray}\label{e2211}
\begin{split}\label{jk}
\|f([xx \cdots
x]_A)-[f(x)f(x)\cdots f(x)]_B\|_B\leq\varphi(x,x,\cdots,x)
\end{split}
\end{eqnarray}
for all $x \in A.$ If there exists an $L< 1$ such that
\begin{equation}\label{e222}
\varphi(x_1,x_2,\cdots,x_m)\leq m
L\varphi\left(\frac{x_1}{m},\frac{x_2}{m},\cdots,\frac{x_m}{m}\right)
\end{equation}
for all $x_1,\cdots,x_m \in A,$ then there exists a
unique Jordan $m-$Lie homomorphism  $H:A\rightarrow B$ such that
\begin{equation}\label{e223}
\|f(x)-H(x)\|\leq
\frac{\varphi(x,0,\cdots,0)}{m-mL}
\end{equation}
for all $x \in A.$
\end{thm}

\begin{proof}
 By the same reasoning as in the proof of Theorem
\ref{t2.1}, we can define the mapping $H(x)= \lim_{k \to
\infty}\frac{1}{m^{ k}}f(m^{ k}x)$
 for all $x \in A.$ Moreover, we can show that $H$ is $\Bbb C$--linear. It follows from $(\ref{e2211})$  that
\begin{align*}
\|&H([xx \cdots x]_A)-[H(x)H(x)\cdots H(x)]_B\|\\&=\lim_{k \to
\infty}\left\|\frac{H([m^k x \cdots
m^k x]_A)}{m^{mk}}-\frac{[H(m^k x)H(m^k x) \cdots H(m^k x)]_B}{m^{mk}}\right\|\leq \lim_{k \to
\infty}\frac{1}{m^{mk}}\varphi(m^k x,m^k x,...,m^k x)=0
\end{align*}
for all $x \in A.$ So
$H([xx \cdots x]_A)=[H(x)H(x)\cdots H(x)]_B$ for
all $x \in A.$ Hence $H:A\rightarrow B$ is a Jordan $m-Lie$ homomorphism satisfying $(\ref{e223}).$
\end{proof}

\begin{cor}\label{c22}
Let $\theta $ and $p$ be
non--negative real numbers such that $p <1.$ Suppose that
a function $f:A\rightarrow B$ satisfies
\begin{align*}
\left\|2\mu f\left(\sum_{i=1}^m  m x_i\right)-\sum_{i=1}^m f  \left(\mu \left(m x_i + \sum_{j=1~,i\neq j}^m x_j\right)\right)-f\left(\sum_{i=1}^m  \mu x_i\right)\right\|  \leq
\theta\sum^{n}_{i=1}(\|x_{i}\|_{A}^{p})
\end{align*}
for all $\mu\in \Bbb T_{\frac{1}{n_o}} $ and all $x_1,\cdots,x_m
\in A$ and
$
\|f([xx \cdots
x]_A)-[f(x)f(x)\cdots f(x)]_B\| \leq
n \theta(\|x\|_{A}^{p})
$
for  all $x\in A.$ Then there exists a unique Jordan $m-$Lie homomorphism
$H:A\rightarrow B$ such that
\begin{equation}\label{e213}
\|f(x)-H(x)\|_B \leq \frac{\theta\|x\|_{A}^{p}}{m-m^p}
\end{equation}
for all $x \in A.$
\end{cor}
\begin{proof}
It follows from Theorem \ref{t2.2} by putting  $\varphi(x_1,x_2,\cdots,x_m):=\theta
\sum^{m}_{i=1}(\|x_{i}\|_{A}^{p})$  for all
$x_1,\cdots,x_m \in A$  and $L=m^{(p-1)}.$
\end{proof}


Similarly, we have the following and we will omit the proof.
\begin{thm}\label{t2.1xxx}
Let  $n_0 \in \Bbb N$ be a fixed positive integer. Let $f:A\rightarrow B$ be a
mapping  for which there exists a function
$\varphi:A^{m}\rightarrow [0,\infty)$  satisfying  (\ref{e21})
for all $\mu\in  \Bbb T_{\frac{1}{n_o}} :=\left\{e^{i\theta} \ ;\
0\leq\theta\leq \frac{2\pi}{n_o}\right\}$ and (\ref{e211}).
If there exists an $L< 1$ such that
$
\varphi\left(\frac{x_1}{m},\frac{x_2}{m},\cdots,\frac{x_m}{m}\right)\leq
\frac{L}{m}\varphi(x_1,x_2,\cdots,x_m)
$
for all $x_1,\cdots,x_m \in A,$ then there exists a
unique Jordan $m-$Lie homomorphism  $H:A\rightarrow B$ such that
\begin{equation}\label{e23xx}
\|f(x)-H(x)\|\leq
\frac{L\varphi(x,0,0,\cdots,0)}{m-mL}
\end{equation}
for all $x \in A.$
\end{thm}

\begin{cor}
Let $\theta $ and $p$ be
non--negative real numbers such that $p >1.$ Suppose that
a function $f:A\rightarrow B$ satisfying  (\ref{xxx}) and (\ref{jk}). Then there exists a unique Jordan $m-$Lie homomorphism
$H:A\rightarrow B$ such that
\begin{equation}\label{e2213xx}
\|f(x)-H(x)\|_B \leq \frac{\theta\|x\|_{A}^{p}}{m^{p}-m}
\end{equation}
for all $x \in A.$
\end{cor}
\begin{proof}
Put  $\varphi(x_1,x_2,\cdots,x_m):=\theta
\sum^{m}_{i=1}(\|x_{i}\|_{A}^{p})$  for all
$x_1,\cdots,x_n \in A$ in Theorem \ref{t2.1xxx}.  Then
$(\ref{e23xx})$ holds for $p ~>1$,  and
$(\ref{e2213xx})$ holds when $L=m^{(1-p)}.$
\end{proof}











 \vskip .2in
%-------------------------------------------------------------
\begin{thebibliography}{99}

\bibitem{az}H. Azadi Kenary, {\it Random approximation of an additive functional equation of m-Apollonius type}, Acta Mathematica Scientia Volume 32, Issue 5, September 2012, Pages 1813-1825.


\bibitem{azc} 17.	H. Azadi Kenary and Y. J. Cho, {\it Stability of mixed additive-quadratic Jensen type functional equation in various spaces}, Computer and  Mathematics with Applications, Vol. 61, Issue 9, 2704-2724, 2011.


\bibitem{11} D.H. Hyers, {\it On the stability of the linear
functional equation},  Proc. Natl. Acad. Sci. 27,  222--224 (1941).

\bibitem {12} Th.M. Rassias, {\it On the stability of the linear
mapping in Banach spaces},  Proc. Amer. Math. Soc. 72
297--300 (1978).

\bibitem {10} S.M. Ulam,
   Chapter VI, science ed., Wiley, New York, 1940.

\end{thebibliography}
}
\end{document}