\documentclass[12pt,a4paper]{article}
\usepackage{amssymb}
\usepackage{eucal}
\usepackage{amsmath}
\usepackage{amstext}
\begin{document}

\title {The Ces$\grave{a}$ro convergence of triple sequence spaces of $\chi^{3}$ of fuzzy real numbers defined by a sequence of Musielak-Orlicz functions}
\author{N. Subramanian 
\\Department of Mathematics,SASTRA University,
\\Thanjavur-613 401, India.\thanks{nsmaths@yahoo.com}}
\date{}
\maketitle
\begin{abstract}
We have to find the necessary and sufficient Tauberian conditions of convergence follows form $\left[C,1,1,1\right]-$ convergence of triple sequence spaces of $\chi^{3}$ of fuzzy numbers.
\\ \noindent \textbf{Keywords:} analytic sequence, triple sequences, Musielak-Orlicz function, $p-$ metric space, fuzzy number, Tauberian conditons, Ces$\grave{a}$ro convergence. 
\noindent \\\textbf{\textbf{2000 Mathematics subject classification:}} 40A05; 40C05; 46A45; 03E72; 46B20. 
\end{abstract} 
\section{Introduction}
A triple sequence (real or complex) can be defined as a function $x: \mathbb{N}\times \mathbb{N}\times \mathbb{N}\rightarrow \mathbb{R}\left(\mathbb{C}\right),$ where $\mathbb{N},\mathbb{R}$ and $\mathbb{C}$ denote the set of natural numbers, real numbers and complex numbers respectively. The different types of notions of triple sequence was introduced and investigated at the initial by  
$\left(sahiner\hspace{0.1cm}et\hspace{0.1cm}al.,2007,2008;\hspace{0.1cm}Esi\hspace{0.1cm}et\hspace{0.1cm}al.,2014,2015;\hspace{0.1cm}Datta\hspace{0.1cm}et\hspace{0.1cm}al.,2013;\hspace{0.1cm}\right)\\\left(Subramanian\hspace{0.1cm}et\hspace{0.1cm}al., 2015;\hspace{0.1cm}Debnath\hspace{0.1cm} et\hspace{0.1cm}al.,2015\right)$ and many others.  
\\A triple sequence $x=(x_{mnk})$ is said to be triple analytic if
\begin{center}
$sup_{m,n,k}\left|x_{mnk}\right|^{\frac{1}{m+n+k}}<\infty.$ 
\end{center}
The space of all triple analytic sequences are usually denoted by $\Lambda^{3}$.
A triple sequence $x=(x_{mnk})$ is called triple gai sequence if 
\begin{center}
$\left(\left(m+n+k\right)!\left|x_{mnk}\right|\right)^{\frac{1}{m+n+k}}\rightarrow 0$ as $m,n,k\rightarrow \infty.$ 
\end{center}
The space of all triple gai sequences are usually denoted by $\chi^{3}.$ 
\section{Definitions and Preliminaries}
\subsection{Definition}  An Orlicz function $\left(Kamthan\hspace{0.1cm}et\hspace{0.1cm}al.,1981\right)$ is a function $M:\left[0,\infty\right)\rightarrow \left[0,\infty\right)$ which is continuous, non-decreasing and convex with $M\left(0\right)=0,\hspace{0.05cm}M\left(x\right)>0,$ for $x>0$ and $M\left(x\right)\rightarrow \infty$ as $x\rightarrow \infty.$ If convexity of Orlicz function $M$ is replaced by $M\left(x+y\right)\leq M\left(x\right)+M\left(y\right),$ then this function is called modulus function.
\\\indent $\left(Lindenstrauss\hspace{0.1cm}et\hspace{0.1cm}al.,1971\right)$ used the idea of Orlicz function to construct Orlicz sequence space.
\\\indent  A sequence $g=\left(g_{mn}\right)$ defined by 
\begin{center}
$g_{mn}\left(v\right)=sup\left\{\left|v\right|u-\left(f_{mnk}\right)\left(u\right):u\geq 0\right\},m,n,k=1,2,\cdots$  
\end{center}
is called the complementary function of a Musielak-Orlicz function $f$. For a given Musielak-Orlicz  function $f,$ $\left(Musielak, 1983\right)$ the Musielak-Orlicz sequence space $t_{f}$ is defined as follows                             
\begin{center}
$t_{f}=\left\{x\in w^{3}:I_{f}\left(\left|x_{mnk}\right|\right)^{1/m+n+k}\rightarrow 0\hspace{0.05cm}as\hspace{0.05cm}m,n,k\rightarrow \infty\right\},$  
\end{center} 
where $I_{f}$ is a convex modular defined by 
\begin{center}
$I_{f}\left(x\right)=\sum_{m=1}^{\infty}\sum_{n=1}^{\infty}\sum_{k=1}^{\infty}f_{mnk}\left(\left|x_{mnk}\right|\right)^{1/m+n+k}, x=\left(x_{mnk}\right)\in t_{f}.$
\end{center}
We consider $t_{f}$ equipped with the Luxemburg metric 
\begin{center}
$d\left(x,y\right)=\sum_{m=1}^{\infty}\sum_{n=1}^{\infty}\sum_{k=1}^{\infty}f_{mnk}\left(\frac{\left|x_{mnk}\right|^{1/m+n+k}}{mnk}\right)$
\end{center}
is an exteneded real number.
\subsection{Definition} Let $X,Y$ be a real vector space of dimension $w,$ where $n\leq m.$ A real valued function $ d_p(x_1,\dots,x_n) = \|(d_1(x_1,0), \dots, d_n(x_n,0))\|_p$ on $X$ satisfying the following four conditions:
\\(i) $\|(d_1(x_1,0), \dots, d_n(x_n,0))\|_p=0$ if and only if  $d_1(x_1,0), \dots, d_n(x_n,0)$ are linearly dependent,
\\(ii) $\|(d_1(x_1,0), \dots, d_n(x_n,0))\|_p$ is invariant under permutation,
\\(iii) $\|(\alpha d_1(x_1,0), \dots, d_n(x_n,0))\|_p=\left|\alpha \right|\|(d_1(x_1,0), \dots, d_n(x_n,0))\|_p ,\alpha\in \mathbb{R}$  
\\(iv) $d_{p} \left( (x_{1}, y_{1}) , (x_{2}, y_{2})\cdots (x_{n}, y_{n}) \right) = \left( d_{X} (x_{1}, x_{2},\cdots x_{n})^{p} + d_{Y} (y_{1}, y_{2}, \cdots y_{n})^{p} \right)^{1/p}\\for 1 \leq p < \infty;$ (or)
\\(v) $d\left( (x_{1}, y_{1}) , (x_{2}, y_{2}),\cdots (x_{n}, y_{n}) \right) := \sup \left\{ d_{X} (x_{1}, x_{2}, \cdots x_{n}), d_{Y} (y_{1}, y_{2},\cdots y_{n}) \right\},$\\ for $x_{1}, x_{2},\cdots x_{n} \in X, y_{1}, y_{2},\cdots y_{n} \in Y$ is called the $p$ product metric of the Cartesian product of
 $n$ metric spaces $\left(subramanian\hspace{0.1cm}et\hspace{0.1cm}al.,2016\right)$.
\section{Main Results}  A fuzzy number is a fuzzy set on the real axis, (i.e) a mapping $X:\mathbb{R}\times \mathbb{R}\times \mathbb{R}\rightarrow \left[0,1\right]$ which satisfies the following four conditions.
\\(i) $X$ is normal (i.e) there exists an $\bar{0}\in \mathbb{R}$ such that $X\left(\bar{0}\right)=1.$
\\(ii) $X$ is fuzzy convex, (i.e) $X\left[\lambda X+\left(1-\lambda\right)Y\right]\geq min\left\{X\left(x\right),X\left(y\right)\right\}$  for all $x,y\in \mathbb{R}$ and for all $\lambda\in \left[0,1\right].$
\\(iii) $X$ is upper semi-continuous.
\\(iv) The set $\left[X\right]=\left\{X\in \mathbb{R}\times \mathbb{R}\times \mathbb{R}:X\left(x\right)>0\right\},$ where $\left\{X\in \mathbb{R}\times \mathbb{R}\times \mathbb{R}:X\left(x\right)>0\right\}$ denotes the closure of the set $\left\{X\in \mathbb{R}\times \mathbb{R}\times \mathbb{R}:X\left(x\right)>0\right\}$ in the usual topology of $\mathbb{R}\times \mathbb{R}\times \mathbb{R}.$ The set of all fuzzy numbers on $\mathbb{R}$ is denoted by $F$ and $\alpha-$ level sets $\left[X\right]_{\alpha}$ of $X\in F$ is defined by
$\left[X\right]_{\alpha}=\left\{\frac{\left\{X\in \mathbb{R}\times \mathbb{R}\times \mathbb{R}:X\left(t\right)\geq \alpha\right\},\left(0<\alpha\leq 1\right)}{\left\{X\in \mathbb{R}\times \mathbb{R}\times \mathbb{R}:X\left(t\right)\geq \alpha\right\},\left(\alpha=0\right)}\right\}.$
\\\indent Let $X$ be a non-empty set, then a family of sets $I\subset 2^{X\times X\times X}$ (the class of all subsets of $X$) is called an ideal if and only if for each $A,B\in I,$ we have $A\bigcup B\in I$ and for each $A\in I$ and each each $B\subset A,$ we have $B\in I.$ A non-empty family of sets $F\subset 2^{X\times X\times X}$ is a filter on $X$ if and only if $\phi\notin F,$ for each $A,B\in F,$ we have $A\bigcap B\in F$ and each $A\in F$ and each $A\subset B,$ we have $B\in F.$ An ideal $I$ is called non-trivial ideal if $I\neq \phi$ and $X\notin I.$ Clearly $I\subset 2^{X\times X\times X}$ is a non-trivial ideal if $F=F\left(I\right)=\left\{X/A: A\in I\right\}$ is a filter on $X.$ A non-trivial ideal $I\subset 2^{X\times X\times X}$ is called admissible if and only if $\left\{\left\{x\right\}:x\in X\right\}\subset I.$ Further details on ideals of $2^{X\times X\times X}$ can be found in Kostyrko. The notion was further investigated by Salat, et. al.  and others.   
Throughout the ideals of $2^{N\times N\times N}$ and $2^{N\times N\times N}$ will be denoted by $I$ and $I_{2}$ respectively. 
\\A fuzzy real number $X$ is a fuzzy set on $R,$ a mapping $X:R\times R\times R\rightarrow L\times L\times L\left(=\left[0,1\right]\right)$ associating each real number $t$ with its grade of membership $X\left(t\right).$ The $\alpha-$ level set of a fuzzy real number $X,0<\alpha<1$ denoted by $\left[X\right]^{\alpha}$ is defined as $\left[X\right]^{\alpha}=\left\{t\in R:X\left(t\right)\geq \alpha\right\}.$ A fuzzy real number $X$ is called convex if $X\left(t\right)\geq X\left(s\right)\wedge X\left(r\right)\wedge X\left(v\right)=min\left(X\left(s\right),X\left(r\right),X\left(v\right)\right),$ where $s<t<r<v.$ If there exists $t_{0}\in R$ such that $X\left(t_{0}\right)=1,$ then the fuzzy real number $X$ is called normal. A fuzzy real $X$ is said to be upper semi-continuous if for each $\epsilon>0,X^{-1}\left(\left[0,a+\epsilon\right)\right),$ for all $a\in L$ is open in the usual topology of $R.$ The set of all upper semi continuous, normal convex fuzzy number is denoted by $L\left(R\right).$
\\Throughout a fuzzy real valued triple sequence is denoted by $\left(X_{mnk}\right)$ i.e a triple infinite array of fuzzy real number $X_{mnk}$ for all $m,n,k\in \mathbb{N}.$ 
\\Every real number $r$ can express as a fuzzy real number $\overline{r}$ as follows:
\begin{center}
$\overline{r}=\left\{\begin{array}{cc}1,&\mbox{ if } t=r; \\0, &\mbox{\hspace{0.5cm}   otherwise } \end{array}\right.$ 
\end{center}
Let $D$ be the set of all closed bounded intervals $X=\left[X^{L},X^{R}\right].$ Then $X\leq Y$ if and only if $X^{L}\leq Y^{L}$ and $X^{R}\leq Y^{R}.$ 
\\Also $d\left(X,Y\right)=max\left(\left|X^{L}-Y^{L}\right|,\left|X^{R}-Y^{R}\right|\right).$ Then $\left(D,d\right)$ is a complete metric space.
\\Let $\overline{d}:L\left(R\right)\times L\left(R\right)\times L\left(R\right)\rightarrow R\times R\times R$ be defined by 
\begin{center}
$\overline{d}\left(X,Y\right)=sup_{0\leq \alpha\leq 1}d\left(\left[X\right]^{\alpha},\left[Y\right]^{\alpha}\right)$ for $X,Y\in L\left(R\right).$
\end{center}
Then $\overline{d}$ defined a metric on $L\left(R\right).$      
\subsection{Definition} A triple sequence spaces of $X=\left(X_{mnk}\right)$ of fuzzy numbers is a function $X: \mathbb{N}\times \mathbb{N}\times \mathbb{N}\rightarrow F.$ The fuzzy numbers $X_{mnk}$ denotes the value of the function at $m,n,k\in \mathbb{N}$ and is called the $\left[m,n,k\right]^{th}$ section of the triple sequence spaces. By $w^{3}\left(F\right),$ we denote the set of all triple sequence spaces of fuzzy real numers.                         
\subsection{Definition} A triple sequence spaces $\left(X_{mnk}\right)\subset w^{3}\left(F\right)$ is called convergent with limit $0\in F,$ if and only if for every $\epsilon>0$ there exists an $m_{0}n_{0}k_{0}= m_{0}n_{0}k_{0}\left(\epsilon\right)\in \mathbb{N}$ such that $D\left(X_{mnk},\bar{0}\right)< \epsilon$ for all $m,n,k\geq m_{0}n_{0}k_{0}.$
\subsection{Definition} A triple sequence spaces $X=\left(X_{mnk}\right)$ of fuzzy real numbers is said to be Cauchy if for every $\epsilon>0$ there exists a positive integer $m_{0}n_{0}k_{0}$ such that $D\left(X_{mnk},\bar{0}\right)< \epsilon$ for all $m,n,k\geq m_{0}n_{0}k_{0}.$
\\The Ces$\grave{a}$ro convergence of a triple sequence spaces of fuzzy numbers defined as follows:
\subsection{Definition} Let $\left(X_{mnk}\right)\left(m,n,k=0,1,2,\cdots\right)$ be a triple sequence spaces of fuzzy numbers. The arithmetic means $\sigma_{rst}\left(X_{mnk}\right)$ is defined by 
\begin{equation}
\sigma_{rst}=\frac{1}{\left(rst\right)+1}\sum_{m=0}^{r}\sum_{n=0}^{s}\sum_{k=0}^{t} X_{mnk} \left(r,s,t=0,1,2,\cdots\right).
\end{equation}
We say that the triple sequence spaces of $\left(X_{mnk}\right)$ is Ces$\grave{a}$ro convergent $\left(\left(C,1,1,1\right)-convergent\right)$ to a fuzzy numer $\bar{0}$ if 
\begin{equation}
lim_{rst\rightarrow \infty} \sigma_{rst}=\bar{0}
\end{equation}
\subsection{Definition} Let $A$ be a particular limitation method. Any additional condition on a triple sequence spaces, which together with the $A-$limitability of that triple sequence spaces implies the convergence of that triple sequence spaces, is called a Tauberian condition for the limitation method. The theorem which establishes the validity of the condition is called a Tauberian theorem.
\\\indent In this paper, we introduce some Tauberian type of theorems for triple sequence spaces of fuzzy numbers and defined as following sets.\\Let $f=\left(f_{mnk}\right)$ be a Musielak-Orlicz function, $\left(X,\left\|\left(d\left(x_{1}\right),d\left(x_{2}\right),\cdots, d\left(x_{n-1}\right)\right)\right\|_{p}\right)$ be a triple sequence spaces of fuzzy luxemburg $p-$ metric spaces respectively. (i) \\\\$\left[\chi^{3\left(F\right)}_{f },\left\|\left(d\left(x_{1}\right),d\left(x_{2}\right),\cdots, d\left(x_{n-1}\right)\right)\right\|_{p}\right]=\left[f_{mnk}\left(\left\|\mu_{mnk}\left(X\right),\left(d\left(x_{1}\right),d\left(x_{2}\right),\cdots, d\left(x_{n-1}\right)\right)\right\|_{p}\right)\right],$ where $\mu_{mnk}\left(X\right)=D\left(\left(\left(m+n+k\right)! \left(\Delta^{m} X_{mnk}\right)^{1/m+n+k},\bar{0}\right)\right)\rightarrow \bar{0}$ as $m,n,k\rightarrow \infty$.  
\section{Main Results} 
\subsection{Theorem} If $\left[\chi^{3\left(F\right)}_{f },\left\|\left(d\left(x_{1}\right),d\left(x_{2}\right),\cdots, d\left(x_{n-1}\right)\right)\right\|_{p}\right]$ is convergent then \\$\left[\chi^{3\left(F\right)}_{f },\left\|\left(d\left(x_{1}\right),d\left(x_{2}\right),\cdots, d\left(x_{n-1}\right)\right)\right\|_{p}\right]$ is $\left[C,1,1,1\right]$ convergent.
\\\textbf{Proof:} Let $X=\left(X_{mnk}\right)\in \left[\chi^{3\left(F\right)}_{f },\left\|\left(d\left(x_{1}\right),d\left(x_{2}\right),\cdots, d\left(x_{n-1}\right)\right)\right\|_{p}\right].$ Then, there exists $\bar{0}\in F$ such that $\left[f_{mnk}\left(\left\|\mu_{mnk}\left(X\right),\left(d\left(x_{1}\right),d\left(x_{2}\right),\cdots, d\left(x_{n-1}\right)\right)\right\|_{p}\right)\right]=0$. Write the following inequality
\\$D\left[\sigma_{rst},\bar{0}\right]=D\left[\frac{1}{\left(rst\right)+1}\sum_{m=0}^{r}\sum_{n=0}^{s}\sum_{k=0}^{t}\left[f_{mnk}\left(\left\|\mu_{mnk}\left(X\right),\left(d\left(x_{1}\right),d\left(x_{2}\right),\cdots, d\left(x_{n-1}\right)\right)\right\|_{p}\right)\right]\right] \\\indent \hspace{1.0cm} \leq \frac{1}{\left(rst\right)+1}\sum_{m=0}^{r}\sum_{n=0}^{s}\sum_{k=0}^{t} D \left[f_{mnk}\left(\left\|\mu_{mnk}\left(X\right),\left(d\left(x_{1}\right),d\left(x_{2}\right),\cdots, d\left(x_{n-1}\right)\right)\right\|_{p}\right)\right].$ Since $lim_{mnk\rightarrow \infty}D \left[f_{mnk}\left(\left\|\mu_{mnk}\left(X\right),\left(d\left(x_{1}\right),d\left(x_{2}\right),\cdots, d\left(x_{n-1}\right)\right)\right\|_{p}\right)\right]=0,\\lim_{rst\rightarrow \infty} \frac{1}{\left(rst\right)+1}\sum_{m=0}^{r}\sum_{n=0}^{s}\sum_{k=0}^{t} D \left[f_{rst}\left(\left\|\mu_{rst}\left(X\right),\left(d\left(x_{1}\right),d\left(x_{2}\right),\cdots, d\left(x_{n-1}\right)\right)\right\|_{p}\right)\right]=0.$ We obtain $lim_{rst\rightarrow \infty}D\left[\sigma_{rst},\bar{0}\right]=0.$\\The fact that the converse does not hold follows from the following example:
\subsection{Example} Consider the fuzzy triple sequence spaces $\left[\chi^{3\left(F\right)}_{f },\left\|\left(d\left(x_{1}\right),d\left(x_{2}\right),\cdots, d\left(x_{n-1}\right)\right)\right\|_{p}\right]$ as follows:
\\$\left[\chi^{3\left(F\right)}_{f },\left\|\left(d\left(x_{1}\right),d\left(x_{2}\right),\cdots, d\left(x_{n-1}\right)\right)\right\|_{p}\right]=\left(\mu_{000}\left(X\right), \mu_{000}\left(Y\right), \cdots  \right)$  
\\$\mu_{000}\left(X\left(t\right)\right)=\left\{\begin{array}{cc}1-t,&\mbox{ if } t\in \left[0,1\right]; \\0, &\mbox{\hspace{0.5cm}   otherwise } \end{array}\right.$  and  $\mu_{000}\left(Y\left(t\right)\right)=\left\{\begin{array}{cc}1+t,&\mbox{ if } t\in \left[0,1\right]; \\0, &\mbox{\hspace{0.5cm}   otherwise } \end{array}\right.$ 
\\Then the $\alpha-$ level set of the arithmetic means $\sigma_{rst}\left[\chi^{3\left(F\right)}_{f },\left\|\left(d\left(x_{1}\right),d\left(x_{2}\right),\cdots, d\left(x_{n-1}\right)\right)\right\|_{p}\right]$ are $\left[\sigma_{2\left(rst\right)}\right]_{\alpha}=\left[\frac{-\left(rst\right)}{2\left(rst\right)+1}\left(1-\alpha\right),\frac{\left(rst\right)+1}{2\left(rst\right)+1}\left(1-\alpha\right)\right]$ and $\left[\sigma_{2\left(rst\right)-1}\right]_{\alpha}=\left[-\frac{1}{2}\left(1-\alpha\right),\frac{1}{2}\left(1-\alpha\right)\right].$ So, $\left[\sigma_{\left(rst\right)}\right]$ is convergent to $\mu_{000}\left(Z\right)=\frac{1}{2}\left[\mu_{000}\left(X\right)+\mu_{000}\left(Y\right)\right]$ but \\$\left[\chi^{3\left(F\right)}_{f },\left\|\left(d\left(x_{1}\right),d\left(x_{2}\right),\cdots, d\left(x_{n-1}\right)\right)\right\|_{p}\right]$ is not convergent. 
\subsection{Theorem} If a triple sequence spaces $\left[\chi^{3\left(F\right)}_{f },\left\|\left(d\left(x_{1}\right),d\left(x_{2}\right),\cdots, d\left(x_{n-1}\right)\right)\right\|_{p}\right]$ is $\left[C,1,1,1\right]-$ convergent to a fuzzy number $\bar{0}$, then for each $\lambda>1$,
\begin{equation}
lim_{rst\rightarrow \infty}\frac{1}{\lambda_{rst}-\left(rst\right)}\sum_{m=r+1}^{\lambda_{r}}\sum_{n=s+1}^{\lambda_{s}}\sum_{k=t+1}^{\lambda_{t}}\left[f_{mnk}\left(\left\|\mu_{mnk}\left(X\right),\left(d\left(x_{1}\right),d\left(x_{2}\right),\cdots, d\left(x_{n-1}\right)\right)\right\|_{p}\right)\right]=\bar{0}
\end{equation}
and for each $0<\lambda <1$\\$lim_{rst\rightarrow \infty}\frac{1}{\left(rst\right)-\lambda_{rst}}\sum_{m=\lambda_{r}+1}^{r}\sum_{n=\lambda_{s}+1}^{s}\sum_{k=\lambda_{t}+1}^{t}\\\left[f_{mnk}\left(\left\|\mu_{mnk}\left(X\right),\left(d\left(x_{1}\right),d\left(x_{2}\right),\cdots, d\left(x_{n-1}\right)\right)\right\|_{p}\right)\right]=\bar{0},$ where $\lambda_{rst}$ we denote the integral part of the product $\lambda\left(rst\right),$ in symbol $\lambda_{rst}:= \left[\lambda\hspace{0.1cm}rst\right].$ 
\\\textbf{Proof:} Case $\lambda>1.$ If $\lambda>1$ and $\left(rst\right)$ is large in the sense that $\lambda_{rst}>rst$, then 
\\$D\left[\frac{1}{\lambda_{rst}-\left(rst\right)}\sum_{m=r+1}^{\lambda_{r}}\sum_{n=s+1}^{\lambda_{s}}\sum_{k=t+1}^{\lambda_{t}}\left[f_{mnk}\left(\left\|\mu_{mnk}\left(X\right),\left(d\left(x_{1}\right),d\left(x_{2}\right),\cdots, d\left(x_{n-1}\right)\right)\right\|_{p}\right)\right]\right]=\\D\left[\frac{1}{\lambda_{rst}-\left(rst\right)}\sum_{m=r+1}^{\lambda_{r}}\sum_{n=s+1}^{\lambda_{s}}\sum_{k=t+1}^{\lambda_{t}}\left[f_{mnk}\left(\left\|\mu_{mnk}\left(X\hspace{0.1cm}\sigma_{rst}\right),\left(d\left(x_{1}\right),d\left(x_{2}\right),\cdots, d\left(x_{n-1}\right)\right)\right\|_{p}\right)\right]\right]= \\D\left[\frac{1}{\lambda_{rst}-\left(rst\right)}\sum_{m=r+1}^{\lambda_{r}}\sum_{n=s+1}^{\lambda_{s}}\sum_{k=t+1}^{\lambda_{t}}\left[f_{mnk}\left(\left\|\mu_{mnk}\left(X\right)\hspace{0.1cm}\sigma_{rst},\left(d\left(x_{1}\right),d\left(x_{2}\right),\cdots, d\left(x_{n-1}\right)\right)\right\|_{p}\right)\right]\right]+\\D\left[\sigma_{rst},\bar{0}\right]$ and so \\\\$D\left[\frac{1}{\lambda_{rst}-\left(rst\right)}\sum_{m=r+1}^{\lambda_{r}}\sum_{n=s+1}^{\lambda_{s}}\sum_{k=t+1}^{\lambda_{t}}\left[f_{mnk}\left(\left\|\mu_{mnk}\left(X\right)\hspace{0.1cm}\sigma_{rst},\left(d\left(x_{1}\right),d\left(x_{2}\right),\cdots, d\left(x_{n-1}\right)\right)\right\|_{p}\right)\right]\right]=\\D\left[\frac{1}{\lambda_{rst}-\left(rst\right)}\sum_{m=r+1}^{\lambda_{r}}\sum_{n=s+1}^{\lambda_{s}}\sum_{k=t+1}^{\lambda_{t}}\left[f_{mnk}\left(\left\|\mu_{mnk}\left(X\right)\hspace{0.1cm}\sigma_{rst},\left(d\left(x_{1}\right),d\left(x_{2}\right),\cdots, d\left(x_{n-1}\right)\right)\right\|_{p}\right)\right]\right]=\\\left[D\right]\\\frac{1}{\lambda_{rst}-\left(rst\right)}\sum_{m=r+1}^{\lambda_{r}}\sum_{n=s+1}^{\lambda_{s}}\sum_{k=t+1}^{\lambda_{t}}\left[f_{mnk}\left(\left\|\mu_{mnk}\left(X\right)\hspace{0.1cm}\sigma_{rst},\left(d\left(x_{1}\right),d\left(x_{2}\right),\cdots, d\left(x_{n-1}\right)\right)\right\|_{p}\right)\right],\\\frac{1}{\left(rst\right)+1}\sum_{m=0}^{r}\sum_{n=0}^{s}\sum_{k=0}^{t}\left[f_{mnk}\left(\left\|\mu_{mnk}\left(X\right),\left(d\left(x_{1}\right),d\left(x_{2}\right),\cdots, d\left(x_{n-1}\right)\right)\right\|_{p}\right)\right]=\\\left[D\right]\\\frac{1}{\lambda_{rst}-\left(rst\right)}\sum_{m=r+1}^{\lambda_{r}}\sum_{n=s+1}^{\lambda_{s}}\sum_{k=t+1}^{\lambda_{t}}\left[f_{mnk}\left(\left\|\mu_{mnk}\left(X\right),\left(d\left(x_{1}\right),d\left(x_{2}\right),\cdots, d\left(x_{n-1}\right)\right)\right\|_{p}\right)\right]+\\\frac{1}{\lambda_{rst}-\left(rst\right)}\sum_{m=0}^{r}\sum_{n=0}^{s}\sum_{k=0}^{t}\left[f_{mnk}\left(\left\|\mu_{mnk}\left(X\right),\left(d\left(x_{1}\right),d\left(x_{2}\right),\cdots, d\left(x_{n-1}\right)\right)\right\|_{p}\right)\right],\\\frac{1}{\left(rst\right)+1}\sum_{m=0}^{r}\sum_{n=0}^{s}\sum_{k=0}^{t}\left[f_{mnk}\left(\left\|\mu_{mnk}\left(X\right)\hspace{0.1cm}\sigma_{rst},\left(d\left(x_{1}\right),d\left(x_{2}\right),\cdots, d\left(x_{n-1}\right)\right)\right\|_{p}\right)\right]+\\\frac{1}{\lambda_{rst}-\left(rst\right)}\sum_{m=0}^{r}\sum_{n=0}^{s}\sum_{k=0}^{t}\left[f_{mnk}\left(\left\|\mu_{mnk}\left(X\right),\left(d\left(x_{1}\right),d\left(x_{2}\right),\cdots, d\left(x_{n-1}\right)\right)\right\|_{p}\right)\right]=\\\left[D\right]\\\frac{1}{\lambda_{rst}-\left(rst\right)}\sum_{m=r+1}^{\lambda_{r}}\sum_{n=s+1}^{\lambda_{s}}\sum_{k=t+1}^{\lambda_{t}}\left[f_{mnk}\left(\left\|\mu_{mnk}\left(X\right),\left(d\left(x_{1}\right),d\left(x_{2}\right),\cdots, d\left(x_{n-1}\right)\right)\right\|_{p}\right)\right],\\\frac{\lambda_{rst}+1}{\lambda_{rst}-\left(rst\right)}\frac{1}{\left(rst\right)+1}\sum_{m=0}^{r}\sum_{n=0}^{s}\sum_{k=0}^{t}\left[f_{mnk}\left(\left\|\mu_{mnk}\left(X\right),\left(d\left(x_{1}\right),d\left(x_{2}\right),\cdots, d\left(x_{n-1}\right)\right)\right\|_{p}\right)\right]=\\\left[D\right]\\\frac{1}{\lambda_{rst}+1}\frac{\lambda_{rst}+1}{\lambda_{rst}-\left(rst\right)}\sum_{m=r+1}^{\lambda_{r}}\sum_{n=s+1}^{\lambda_{s}}\sum_{k=t+1}^{\lambda_{t}}\left[f_{mnk}\left(\left\|\mu_{mnk}\left(X\right),\left(d\left(x_{1}\right),d\left(x_{2}\right),\cdots, d\left(x_{n-1}\right)\right)\right\|_{p}\right)\right],\\\frac{\lambda_{rst}+1}{\lambda_{rst}-\left(rst\right)}\frac{1}{\left(rst\right)+1}\sum_{m=0}^{r}\sum_{n=0}^{s}\sum_{k=0}^{t}\left[f_{mnk}\left(\left\|\mu_{mnk}\left(X\right),\left(d\left(x_{1}\right),d\left(x_{2}\right),\cdots, d\left(x_{n-1}\right)\right)\right\|_{p}\right)\right]=\\\left[\frac{\lambda_{rst}+1}{\lambda_{rst}-\left(rst\right)}\right]D\\\frac{1}{\lambda_{rst}+1}\sum_{m=0}^{\lambda_{r}}\sum_{n=0}^{\lambda_{s}}\sum_{k=0}^{\lambda_{t}}\left[f_{mnk}\left(\left\|\mu_{mnk}\left(X\right),\left(d\left(x_{1}\right),d\left(x_{2}\right),\cdots, d\left(x_{n-1}\right)\right)\right\|_{p}\right)\right],\\\frac{1}{\left(rst\right)+1}\sum_{m=0}^{r}\sum_{n=0}^{s}\sum_{k=0}^{t}\left[f_{mnk}\left(\left\|\mu_{mnk}\left(X\right),\left(d\left(x_{1}\right),d\left(x_{2}\right),\cdots, d\left(x_{n-1}\right)\right)\right\|_{p}\right)\right]=\\\frac{\lambda_{rst}+1}{\lambda_{rst}-\left(rst\right)}D\left[\sigma_{\lambda_{rst}},\sigma_{rst}\right].$ \\Now, (3) follows from (2) and the fact that for large enough$\left(rst\right),$ \\$\frac{\lambda}{\lambda-1}=\frac{\lambda_{rst}}{\lambda \left(rst\right)-\left(rst\right)}<\frac{\lambda_{rst}+1}{\lambda_{rst}-\left(rst\right)}<\frac{\lambda \left(rst\right)+1}{\lambda\left(rst\right)-\left(rst\right)-1}\leq \frac{2\lambda}{\lambda-1}.$ \\In case $0<\lambda <1$ then the following inequality:
\\$D\left[\frac{1}{\left(rst\right)-\lambda_{rst}}\sum_{m=\lambda_{r}+1}^{r}\sum_{n=\lambda_{s}+1}^{s}\sum_{k=\lambda_{t}+1}^{t}\left[f_{mnk}\left(\left\|\mu_{mnk}\left(X\right),\left(d\left(x_{1}\right),d\left(x_{2}\right),\cdots, d\left(x_{n-1}\right)\right)\right\|_{p}\right)\right]\right]\leq \frac{\lambda_{rst}+1}{\left(rst\right)-\lambda_{rst}}D\left[\sigma_{\lambda_{rst}},\sigma_{rst}\right]+D\left[\sigma_{rst},\bar{0}\right].$
\\Suppose $\left(rst\right)$ is large, in the sense that $\lambda_{rst}< rst$; then the inequality for large $ \left(rst\right), \frac{\lambda_{rst}+1}{\left(rst\right)-\lambda_{rst}}\leq \frac{2\lambda}{\lambda-1}.$ 
\subsection{Theorem} If a triple sequence spaces $\left[\chi^{3\left(F\right)}_{f },\left\|\left(d\left(x_{1}\right),d\left(x_{2}\right),\cdots, d\left(x_{n-1}\right)\right)\right\|_{p}\right]$ is $\left[C,1,1,1\right]-$ convergent to a fuzzy number $\bar{0}$, then $lim_{rst\rightarrow \infty}\left[\chi^{3\left(F\right)}_{f },\left\|\left(d\left(x_{1}\right),d\left(x_{2}\right),\cdots, d\left(x_{n-1}\right)\right)\right\|_{p}\right]=\bar{0}$ if and only if one of the following two conditions are satisfied $lim_{rst\rightarrow \infty} \\D\left[\frac{1}{\lambda_{rst}-\left(rst\right)}\sum_{m=r+1}^{\lambda_{r}}\sum_{n=s+1}^{\lambda_{s}}\sum_{k=t+1}^{\lambda_{t}}\left[f_{mnk}\left(\left\|\mu_{mnk}\left(X\right),\left(d\left(x_{1}\right),d\left(x_{2}\right),\cdots, d\left(x_{n-1}\right)\right)\right\|_{p}\right)\right]\right]=0$ is extended real number (or) \\$lim_{rst\rightarrow \infty} \\D\left[\frac{1}{\left(rst\right)-\lambda_{rst}}\sum_{m=\lambda_{r}+1}^{r}\sum_{n=\lambda_{s}+1}^{{s}}\sum_{k=\lambda_{t}+1}^{t}\left[f_{mnk}\left(\left\|\mu_{mnk}\left(X\right),\left(d\left(x_{1}\right),d\left(x_{2}\right),\cdots, d\left(x_{n-1}\right)\right)\right\|_{p}\right)\right]\right]=0$ is extended real number.
\\\textbf{Proof:} (Necessity) The necessity of condition (3) follows from theorem (4.1).
\\(Sufficiency): Suppose that condition (3) holds. Then, for any given $\epsilon>0,$ there exists $\lambda>0$ such that,   \\ $lim_{rst\rightarrow \infty} \\D\left[\frac{1}{\lambda_{rst}-\left(rst\right)}\sum_{m=r+1}^{\lambda_{r}}\sum_{n=s+1}^{\lambda_{s}}\sum_{k=t+1}^{\lambda_{t}}\left[f_{mnk}\left(\left\|\mu_{mnk}\left(X\right),\left(d\left(x_{1}\right),d\left(x_{2}\right),\cdots, d\left(x_{n-1}\right)\right)\right\|_{p}\right)\right]\right]<\epsilon$.  
\\On the other hand, since $D\left[\left[f_{mnk}\left(\left\|\mu_{mnk}\left(X\right),\left(d\left(x_{1}\right),d\left(x_{2}\right),\cdots, d\left(x_{n-1}\right)\right)\right\|_{p}\right)\right]\right]=\\\left[D\right]\left[f_{mnk}\left(\left\|\mu_{mnk}\left(X\right),\left(d\left(x_{1}\right),d\left(x_{2}\right),\cdots, d\left(x_{n-1}\right)\right)\right\|_{p}\right)\right]+\\\frac{1}{\lambda_{rst}-\left(rst\right)}\sum_{m=r+1}^{\lambda_{r}}\sum_{n=s+1}^{\lambda_{s}}\sum_{k=t+1}^{\lambda_{t}}\left[f_{mnk}\left(\left\|\mu_{mnk}\left(X\right),\left(d\left(x_{1}\right),d\left(x_{2}\right),\cdots, d\left(x_{n-1}\right)\right)\right\|_{p}\right)\right]\leq \frac{1}{\lambda_{rst}-\left(rst\right)}\sum_{m=r+1}^{\lambda_{r}}\sum_{n=s+1}^{\lambda_{s}}\sum_{k=t+1}^{\lambda_{t}}\left[f_{mnk}\left(\left\|\mu_{mnk}\left(X\right),\left(d\left(x_{1}\right),d\left(x_{2}\right),\cdots, d\left(x_{n-1}\right)\right)\right\|_{p}\right)\right]+\\D\left[ \frac{1}{\lambda_{rst}-\left(rst\right)}\sum_{m=r+1}^{\lambda_{r}}\sum_{n=s+1}^{\lambda_{s}}\sum_{k=t+1}^{\lambda_{t}}\left[f_{mnk}\left(\left\|\mu_{mnk}\left(X\right),\left(d\left(x_{1}\right),d\left(x_{2}\right),\cdots, d\left(x_{n-1}\right)\right)\right\|_{p}\right)\right]\right].$\\ We conclude that $\lim_{rst\rightarrow \infty}D\left[\left[f_{mnk}\left(\left\|\mu_{mnk}\left(X\right),\left(d\left(x_{1}\right),d\left(x_{2}\right),\cdots, d\left(x_{n-1}\right)\right)\right\|_{p}\right)\right]\leq \epsilon\right]$ is an extended real number, since $\epsilon$ is arbitrary.
\subsection{Definition} A triple sequence spaces $\left(X_{mnk}\right)$ convergent of fuzzy numbers is said to be slowly oscillating if 
\\$lim_{rst\rightarrow \infty}D\left[\left[f_{mnk}\left(\left\|\mu_{mnk}\left(X\right),\left(d\left(x_{1}\right),d\left(x_{2}\right),\cdots, d\left(x_{n-1}\right)\right)\right\|_{p}\right)\right]\right]=0$ is extended real number.  
\subsection{Remark} The triple sequence spaces convergent of fuzzy numbers is slowly oscillating, which follows from the Cauchy criterion. On the other hand, the sequence \\$\left[f_{mnk}\left(\left\|\mu_{mnk}\left(X\right),\left(d\left(x_{1}\right),d\left(x_{2}\right),\cdots, d\left(x_{n-1}\right)\right)\right\|_{p}\right)\right]=\\\sum_{m=0}^{r}\sum_{n=0}^{s}\sum_{k=0}^{t} \left[f_{mnk}\left(\left\|\mu_{mnk}\left(Y\right),\left(d\left(x_{1}\right),d\left(x_{2}\right),\cdots, d\left(x_{n-1}\right)\right)\right\|_{p}\right)\right],$  where \\$\mu_{mnk}\left(Y\left(t\right)\right)=\left\{\begin{array}{cc}1-\left(\left(mnk\right)+1\right)t,&\mbox{ if } \left(0\leq t\leq  \frac{1}{\left(mnk\right)+1}\right); \\\bar{0}, &\mbox{\hspace{0.5cm}   otherwise } \end{array}\right\}$\\is not convergent, but $\mu$ is slowly oscillating since for all $\left(r_{0}s_{0}t_{0}\right)\leq \left(rst\right)\leq \left(mnk\right)\leq \lambda \left(rst\right)$ with $1<\lambda \leq 1+\epsilon,$ \\$D \left[\left[f_{mnk}\left(\left\|\mu_{mnk}\left(X\right),\left(d\left(x_{1}\right),d\left(x_{2}\right),\cdots, d\left(x_{n-1}\right)\right)\right\|_{p}\right)\right]\right]=\\D\left[\sum_{u=0}^{m}\sum_{v=0}^{n}\sum_{w=0}^{k} \left[f_{mnk}\left(\left\|\mu_{mnk}\left(Y\right),\left(d\left(x_{1}\right),d\left(x_{2}\right),\cdots, d\left(x_{n-1}\right)\right)\right\|_{p}\right)\right]\right]\leq \\\sum_{u=r+1}^{m}\sum_{v=s+1}^{n}\sum_{w=t+1}^{k} D\left[\left[f_{mnk}\left(\left\|\mu_{mnk}\left(X\right),\left(d\left(x_{1}\right),d\left(x_{2}\right),\cdots, d\left(x_{n-1}\right)\right)\right\|_{p}\right)\right]\right]=\\\sum_{u=r+1}^{m}\sum_{v=s+1}^{n}\sum_{w=t+1}^{k} \frac{1}{\left(uvw\right)+1}\leq \left(\frac{\left(mnk\right)}{\left(rst\right)}-1\right)\leq \left(\lambda -1\right)\leq \epsilon.$ 
\subsection{Proposition} A triple sequence spaces $\left(X_{mnk}\right)$ of fuzzy numbers be slowly oscillating. Then 
\\$lim_{rst\rightarrow \infty}\sigma_{rst}=\bar{0}\Longrightarrow lim_{rst\rightarrow \infty}[\left[f_{mnk}\left(\left\|\mu_{mnk}\left(X\right),\left(d\left(x_{1}\right),d\left(x_{2}\right),\cdots, d\left(x_{n-1}\right)\right)\right\|_{p}\right)\right]=\bar{0}.$ 
\\\textbf{Proof:} If the triple sequence spaces $\left(X_{mnk}\right)$ is slowly oscillating, then the following from the inequality
\\$\left[D\right]\frac{1}{\lambda_{rst}-\left(rst\right)}\sum_{m=r+1}^{\lambda_{r}}\sum_{n=s+1}^{\lambda_{s}}\sum_{k=t+1}^{\lambda_{t}}\left[f_{mnk}\left(\left\|\mu_{mnk}\left(X\right),\left(d\left(x_{1}\right),d\left(x_{2}\right),\cdots, d\left(x_{n-1}\right)\right)\right\|_{p}\right)\right],\\\left[f_{mnk}\left(\left\|\mu_{mnk}\left(X\right),\left(d\left(x_{1}\right),d\left(x_{2}\right),\cdots, d\left(x_{n-1}\right)\right)\right\|_{p}\right)\right]=\\\left[D\right]\frac{1}{\lambda_{rst}-\left(rst\right)}\sum_{m=r+1}^{\lambda_{r}}\sum_{n=s+1}^{\lambda_{s}}\sum_{k=t+1}^{\lambda_{t}}\left[f_{mnk}\left(\left\|\mu_{mnk}\left(X\right),\left(d\left(x_{1}\right),d\left(x_{2}\right),\cdots, d\left(x_{n-1}\right)\right)\right\|_{p}\right)\right],\\\frac{1}{\lambda_{rst}-\left(rst\right)}\sum_{m=r+1}^{\lambda_{r}}\sum_{n=s+1}^{\lambda_{s}}\sum_{k=t+1}^{\lambda_{t}}\left[f_{mnk}\left(\left\|\mu_{mnk}\left(X\right),\left(d\left(x_{1}\right),d\left(x_{2}\right),\cdots, d\left(x_{n-1}\right)\right)\right\|_{p}\right)\right]\leq \\\frac{1}{\lambda_{rst}-\left(rst\right)}\sum_{m=r+1}^{\lambda_{r}}\sum_{n=s+1}^{\lambda_{s}}\sum_{k=t+1}^{\lambda_{t}}\left[D\right]\left[f_{mnk}\left(\left\|\mu_{mnk}\left(X\right),\left(d\left(x_{1}\right),d\left(x_{2}\right),\cdots, d\left(x_{n-1}\right)\right)\right\|_{p}\right)\right],\\\left[f_{rst}\left(\left\|\mu_{rst}\left(X\right),\left(d\left(x_{1}\right),d\left(x_{2}\right),\cdots, d\left(x_{n-1}\right)\right)\right\|_{p}\right)\right]\leq \\\left[D\right]\left[f_{mnk}\left(\left\|\mu_{mnk}\left(X\right),\left(d\left(x_{1}\right),d\left(x_{2}\right),\cdots, d\left(x_{n-1}\right)\right)\right\|_{p}\right)\right],\\\left[f_{rst}\left(\left\|\mu_{rst}\left(X\right),\left(d\left(x_{1}\right),d\left(x_{2}\right),\cdots, d\left(x_{n-1}\right)\right)\right\|_{p}\right)\right].$ Hence the equation (3) holds.
\subsection{Proposition} Let $\left[\chi^{3\left(F\right)}_{f },\left\|\left(d\left(x_{1}\right),d\left(x_{2}\right),\cdots, d\left(x_{n-1}\right)\right)\right\|_{p}\right]$ be a triple sequence spaces of fuzzy numbers. Then  \\$\left[D\right]\left[f_{rst}\left(\left\|\mu_{rst}\left(X\right),\left(d\left(x_{1}\right),d\left(x_{2}\right),\cdots, d\left(x_{n-1}\right)\right)\right\|_{p}\right)\right] ,\\\left[f_{r-1s-1t-1}\left(\left\|\mu_{r-1s-1t-1}\left(X\right),\left(d\left(x_{1}\right),d\left(x_{2}\right),\cdots, d\left(x_{n-1}\right)\right)\right\|_{p}\right)\right]=O\left(\frac{1}{rst}\right)$ implies that the triple sequence spaces $\left(X_{mnk}\right)$ is slowly oscillating.
\\\textbf{Proof:} Let $\left[D\right]\left[f_{rst}\left(\left\|\mu_{rst}\left(X\right),\left(d\left(x_{1}\right),d\left(x_{2}\right),\cdots, d\left(x_{n-1}\right)\right)\right\|_{p}\right)\right] ,\\\left[f_{r-1s-1t-1}\left(\left\|\mu_{r-1s-1t-1}\left(X\right),\left(d\left(x_{1}\right),d\left(x_{2}\right),\cdots, d\left(x_{n-1}\right)\right)\right\|_{p}\right)\right]=O\left(\frac{1}{rst}\right).$ Then, there exists $B>0$ such that \\$\left[D\right]\left[f_{rst}\left(\left\|\mu_{rst}\left(X\right),\left(d\left(x_{1}\right),d\left(x_{2}\right),\cdots, d\left(x_{n-1}\right)\right)\right\|_{p}\right)\right] ,\\\left[f_{r-1s-1t-1}\left(\left\|\mu_{r-1s-1t-1}\left(X\right),\left(d\left(x_{1}\right),d\left(x_{2}\right),\cdots, d\left(x_{n-1}\right)\right)\right\|_{p}\right)\right]\leq \frac{B}{rst}$ for $r,s,t\in \mathbb{N}.$ So, for all $1<\left(r_{0}s_{0}t_{0}\right)\leq \left(rst\right)<\left(mnk\right)\leq \lambda_{rst},$ we obtain \\$\left[D\right]\left[f_{rst}\left(\left\|\mu_{rst}\left(X\right),\left(d\left(x_{1}\right),d\left(x_{2}\right),\cdots, d\left(x_{n-1}\right)\right)\right\|_{p}\right)\right] ,\\\left[f_{r-1s-1t-1}\left(\left\|\mu_{r-1s-1t-1}\left(X\right),\left(d\left(x_{1}\right),d\left(x_{2}\right),\cdots, d\left(x_{n-1}\right)\right)\right\|_{p}\right)\right]\leq\\\sum_{u=r+1}^{m}\sum_{v=s+1}^{n}\sum_{w=t+1}^{k}\\\left[D\right]\left[f_{uvw}\left(\left\|\mu_{uvw}\left(X\right),\left(d\left(x_{1}\right),d\left(x_{2}\right),\cdots, d\left(x_{n-1}\right)\right)\right\|_{p}\right)\right] ,\\\left[f_{u-1v-1w-1}\left(\left\|\mu_{u-1v-1w-1}\left(X\right),\left(d\left(x_{1}\right),d\left(x_{2}\right),\cdots, d\left(x_{n-1}\right)\right)\right\|_{p}\right)\right]\leq \\\sum_{u=r+1}^{m}\sum_{v=s+1}^{n}\sum_{w=t+1}^{k}\left(\frac{B}{uvw}\right)\leq B\left(\frac{\left(mnk\right)-\left(rst\right)}{rst}\right)=B\left(\frac{mnk}{rst}-1\right)< B\left(\lambda -1\right).$\\ Hence, for each$\epsilon>0$ and $1\leq \lambda\leq 1+\frac{\epsilon}{B}$ we get for all $\left(r_{0}s_{0}t_{0}\right)\leq \left(rst\right)<\left(mnk\right)\leq \lambda_{rst}.$ We have   
\\$\left[D\right]\left[f_{mnk}\left(\left\|\mu_{mnk}\left(X\right),\left(d\left(x_{1}\right),d\left(x_{2}\right),\cdots, d\left(x_{n-1}\right)\right)\right\|_{p}\right)\right],\\\left[f_{rst}\left(\left\|\mu_{rst}\left(X\right),\left(d\left(x_{1}\right),d\left(x_{2}\right),\cdots, d\left(x_{n-1}\right)\right)\right\|_{p}\right)\right]\leq \epsilon.$  Hence  \\$\left[\chi^{3\left(F\right)}_{f },\left\|\left(d\left(x_{1}\right),d\left(x_{2}\right),\cdots, d\left(x_{n-1}\right)\right)\right\|_{p}\right]$ is slowly oscillating.
\subsection{Corollary} A triple sequence spaces $\left(X_{mnk}\right)$ of fuzzy numbers which is $\left[C,1,1,1\right]-$ convergent a fuzzy number $\bar{0}$.Then \\$\left[D\right]\left[f_{rst}\left(\left\|\mu_{rst}\left(X\right),\left(d\left(x_{1}\right),d\left(x_{2}\right),\cdots, d\left(x_{n-1}\right)\right)\right\|_{p}\right)\right] ,\\\left[f_{r-1s-1t-1}\left(\left\|\mu_{r-1s-1t-1}\left(X\right),\left(d\left(x_{1}\right),d\left(x_{2}\right),\cdots, d\left(x_{n-1}\right)\right)\right\|_{p}\right)\right]=O\left(\frac{1}{rst}\right)\Longrightarrow \\ \left[\chi^{3\left(F\right)}_{f },\left\|\left(d\left(x_{1}\right),d\left(x_{2}\right),\cdots, d\left(x_{n-1}\right)\right)\right\|_{p}\right]$ is convergent to a fuzzy number of $\bar{0}.$  
\\\\\textbf{Competing Interests: } The authors declare that there is not any conflict of interests regarding the publication of this manuscript. 
\begin{thebibliography}{1}

\bibitem{Esi}{Datta, A. J. } {Esi, A. }  and { Tripathy, B.C.} 2013. {Statistically convergent triple sequence spaces defined by Orlicz function }. \textit{Journal of Mathematical Analysis}. \textbf{4(2)}, ~16-22. 

\bibitem{Debnath} {Debnath, S.} {Sarma, B.} and { Das, B.C. } 2015. {Some generalized triple sequence spaces of real numbers }. 
\textit{Journal of nonlinear analysis and optimization}. \textbf{Vol. 6, No. 1}, ~71-79.

\bibitem{Esi}{ Esi, A.} 2014. {On some triple almost lacunary sequence spaces defined by Orlicz functions}. 
\textit{Research and Reviews:Discrete Mathematical Structures}. \textbf{1(2)}, ~16-25. 

\bibitem{Esi}{ Esi, A.} and { Necdet Catalbas, M.} 2014. {Almost convergence of triple sequences}. 
\textit{Global Journal of Mathematical Analysis}. \textbf{2(1)}, ~6-10. 

\bibitem{Esi}{ Esi, A.} and { Savas,  E.} 2015. { On lacunary statistically convergent triple sequences in probabilistic normed space}. \textit{Applied Mathematics and Information Science}. \textbf{9 (5) }, ~2529-2534.

\bibitem{Kamthan}{ Kamthan, P.K.} and { Gupta, M.} 1981. {Sequence spaces and series, Lecture notes, Pure and Applied Mathematics}. \textit{65 Marcel Dekker, In c. New York}. 

\bibitem{Lindenstrauss}{Lindenstrauss, J. } and {Tzafriri, L.} 1971. {On Orlicz sequence spaces}. \textit{Israel Journal of Mathematics}.
\textbf{10}, ~379-390. 

\bibitem{Musielak}{ Musielak, J.} 1983. {Orlicz Spaces,Lectures Notes in Math.,1034}. \textit{Springer-Verlag}.


\bibitem{Sahiner}{Sahiner, A. }, { Gurdal,  M.} and { Duden, F.K.} 2007. {Triple sequences and their statistical convergence}. \textit{Selcuk Journal of Applied Mathematics}. \textbf{8 No. (2)}, ~49-55.  

\bibitem{Sahiner}{ Sahiner, A.}, { Tripathy, B.C.} 2008. {Some $I$ related properties of triple sequences }. \textit{Selcuk Journal of Applied Mathematics} .\textbf{9 No. (2)}, ~9-18.  

\bibitem{Subramanian} { Subramanian, N.} and {Esi, A.} 2015. {The generalized tripled difference of $\chi^{3}$ sequence spaces}. \textit{Global Journal of Mathematical Analysis}. \textbf{3 (2)}, ~54-60.

\bibitem{Subramanian} {Subramanian, N. } and { Murugesan, C.} 2016. {The entire sequence over Musielak $p-$ metric space}. \textit{Journal of the Egyptian Mathematical Society}. \textbf{Vol. 24 (2)} , ~233-238.





\end{thebibliography}
\end{document}




