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\title[Riesz Almost Lacunary multiple triple sequence spaces of $\Gamma^{3}$  ]{Riesz Almost Lacunary multiple triple sequence spaces of $\Gamma^{3}$ defined by a Orlicz function}
\vspace{15mm}

\author[N. Subramanian]{N. Subramanian$^1$}
\address{$^1$ Department of Mathematics,
\newline \indent SASTRA University, 
\newline \indent Thanjavur 613 401, India}
\thanks{nsmaths@yahoo.com} 

\author[N. Rajagopal ]{N. Rajagopal$^2$}
\address{$^2$ Department of Mathematics,
\newline \indent SASTRA University, 
\newline \indent Thanjavur 613 401, India}
\thanks{nrg1968sastra@yahoo.co.in} 

\author[P. Thirunavukkarasu]{P. Thirunavukkarasu $^3$  }
\address{$^3$ P.G. and Research Department of Mathematics,
\newline \indent Periyar E.V.R. College (Autonomous),
\newline \indent Tiruchirappalli--620 023, India. }

\thanks{ptavinash1967@gmail.com}


\keywords{ analytic sequence, Orlicz function, multiple triple sequence spaces, entire sequence, Riesz space    \\
\indent 2010 {\it Mathematics Subject Classification}. 40A05,40C05,40D05.\\
\indent {\it Received}: \\
\indent {\it Revised}: \\}




\begin{abstract}
In this paper we introduce a new concept for Riesz almost lacunary $\Gamma^{3}$ sequence spaces strong $P-$ convergent to zero with respect to an Orlicz function and examine some properties of the resulting sequence spaces. We also introduce and study statistical convergence of Riesz almost lacunary $\Gamma^{3}$ sequence spaces and also some inclusion theorems are discussed.  
\end{abstract}

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\maketitle

\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  
\textit{Sahiner et al. [8,9]}, \textit{Esi et al. [1-3]}, \textit{Datta et al. [4]},\textit{Subramanian et al. [10]}, \textit{Debnath et al. [5]} 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 entire sequence if 
\begin{center}
$\left|x_{mnk}\right|^{\frac{1}{m+n+k}}\rightarrow 0$ as $m,n,k\rightarrow \infty.$ 
\end{center}
The space of all triple entire sequences are usually denoted by $\Gamma^{3}$.
\section{Definitions and Preliminaries}
\subsection{Definition}  An Orlicz function ([see [6]) 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 Lindenstrauss and Tzafriri ([7]) used the idea of Orlicz function to construct Orlicz sequence space.
\subsection{Definition} The four dimensional matrix $A$ is said to be RH-regular if it maps every bounded $P-$ convergent sequence into  a  $P-$ convergent sequence with the same $P-$ limit. The assumption of boundedness was made because a triple sequence spaces which is $P-$ convergent is not necessarily bounded.   
\subsection{Definition} A triple sequence $x=\left(x_{mnk}\right)$ of real numbers is called almost $P-$ convergent to a limit 0 if 
\begin{center}
$P-lim_{p,q,u\rightarrow \infty}sup_{r,s,t\geq 0}\frac{1}{pqu}\sum_{m=r}^{r+p-1}\sum_{n=s}^{s+q-1}\sum_{k=t}^{t+u-1}\left|x_{mnk}\right|^{1/m+n+k} \rightarrow 0.$
\end{center}
that is, the average value of $\left(x_{mnk}\right)$ taken over any rectangle \\$\left\{\left(m,n,k\right):r\leq m\leq r+p-1,s\leq n\leq s+q-1,t\leq k\leq t+u-1\right\}$ tends to 0 as both $p,q$ and $u$ to $\infty,$ and this $P-$ convergence is uniform in $i,\ell$ and $j.$ Let denot the set of sequences with this property as $\left[\widehat{\Gamma^{3}}\right].$
\subsection{Definition} Let $\left(q_{rst}\right),\left(\overline{q_{rst}}\right),\left(\overline{\overline{q_{rst}}}\right)$ be sequences of positive numbers and 
\\$Q_{r}=
\begin{bmatrix}
 q_{11} & q_{12} &  .  . .  & q_{1s} & 0 .  .  . \\
  q_{21} & q_{22} &  .  . .  & q_{2s} & 0 .  .  . \\
    . & \\
  . & \\
  . & \\
  q_{r1} & q_{r2} &  .  . .  & q_{rs} & 0 .  .  . \\
  
  0 & 0 &  .  . . 0 & 0 & 0 .  .  . \\
\end{bmatrix}=q_{11}+q_{12}+\ldots + q_{rs}\neq 0,$
\\$\overline{Q}_{s}=
\begin{bmatrix}
  \overline{q}_{11} & \overline{q}_{12} &  .  . .  & \overline{q}_{1s} & 0 .  .  . \\
  \overline{q}_{21} & \overline{q}_{22} &  .  . .  & \overline{q}_{2s} & 0 .  .  . \\
    . & \\
  . & \\
  . & \\
  \overline{q}_{r1} & \overline{q}_{r2} &  .  . .  & \overline{q}_{rs} & 0 .  .  . \\
  
  0 & 0 &  .  . . 0 & 0 & 0 .  .  . \\
\end{bmatrix}=\overline{q}_{11}+\overline{q}_{12}+\ldots + \overline{q}_{rs}\neq 0,$
\\$\overline{\overline{Q}}_{t}=
\begin{bmatrix}
  \overline{\overline{q}}_{11} & \overline{\overline{q}}_{12} &  .  . .  & \overline{\overline{q}}_{1s} & 0 .  .  . \\
  \overline{\overline{q}}_{21} & \overline{\overline{q}}_{22} &  .  . .  & \overline{\overline{q}}_{2s} & 0 .  .  . \\
    . & \\
  . & \\
  . & \\
  \overline{\overline{q}}_{r1} & \overline{\overline{q}}_{r2} &  .  . .  & \overline{\overline{q}}_{rs} & 0 .  .  . \\
  
  0 & 0 &  .  . . 0 & 0 & 0 .  .  . \\
\end{bmatrix}=\overline{q}_{11}+\overline{q}_{12}+\ldots + \overline{q}_{rs}\neq 0.$ Then the transformation is given by 
\\$T_{rst}=\frac{1}{Q_{r}\overline{Q}_{s}\overline{\overline{Q}}_{t}}\sum_{m=1}^{r}\sum_{n=1}^{s}\sum_{k=1}^{t}q_{m}\overline{q}_{n}\overline{\overline{q}}_{k}\left|x_{mnk}\right|^{1/m+n+k}$ is called the Riesz mean of triple sequence $x=\left(x_{mnk}\right).$ If $P-lim_{rst}T_{rst}\left(x\right)=0,0\in \mathbb{R},$ then the sequence $x=\left(x_{mnk}\right)$ is said to be Riesz convergent to 0. If $x=\left(x_{mnk}\right)$ is Riesz convergent to 0, then we write $P_{R}-limx=0.$
\subsection{Definition} The triple sequence $\theta_{i,\ell,j}=\left\{\left(m_{i},n_{\ell},k_{j}\right)\right\}$ is called triple lacunary if there exist three increasing sequences of integers such that 
\begin{center}
$m_{0}=0,h_{i}=m_{i}-m_{r-1}\rightarrow \infty$ as $i\rightarrow \infty$ and 
\\$n_{0}=0, \overline{h_{\ell}}=n_{\ell}-n_{\ell-1}\rightarrow \infty$ as $\ell\rightarrow \infty.$
\\$k_{0}=0, \overline{h_{j}}=k_{j}-k_{j-1}\rightarrow \infty$ as $j\rightarrow \infty.$
\end{center}
Let $m_{i,\ell,j}=m_{i}n_{\ell}k_{j},h_{i,\ell,j}=h_{i}\overline{h_{\ell}}\overline{h_{j}},$ and $\theta_{i,\ell,j}$ is determine by 
\\$I_{i,\ell,j}=\left\{\left(m,n,k\right):m_{i-1}< m< m_{i}\hspace{0.05cm}and\hspace{0.05cm}n_{\ell-1}< n\leq n_{\ell} \hspace{0.05cm}and\hspace{0.05cm}k_{j-1}< k\leq k_{j} \right\}, q_{k}=\frac{m_{k}}{m_{k-1}},\overline{q_{\ell}}=\frac{n_{\ell}}{n_{\ell-1}},\overline{q_{j}}=\frac{k_{j}}{k_{j-1}}.$
\\Using the notations of lacunary sequence and Riesz mean for triple sequences.
\\$\theta_{i,\ell,j}=\left\{\left(m_{i},n_{\ell},k_{j}\right)\right\}$ be a triple lacunary sequence and $q_{m}\overline{q}_{n}\overline{\overline{q}}_{k}$ be sequences of positive real numbers such that $Q_{m_{i}}=\sum_{m\in \left(0,m_{i}\right]}p_{m_{i}},Q_{n_{\ell}}=\sum_{n\in \left(0,n_{\ell}\right]}p_{n_{\ell}},Q_{n_{j}}=\sum_{k\in \left(0,k_{j}\right]}p_{k_{j}}$ and $H_{i}=\sum_{m\in \left(0,m_{i}\right]}p_{m_{i}},\overline{H}=\sum_{n\in \left(0,n_{\ell}\right]}p_{n_{\ell}},\overline{\overline{H}}=\sum_{k\in \left(0,k_{j}\right]}p_{k_{j}}.$ Clearly, $H_{i}=Q_{m_{i}}-Q_{m_{i-1}},\overline{H}_{\ell}=Q_{n_{\ell}}-Q_{n_{\ell-1}},\overline{\overline{H}}_{j}=Q_{k_{j}}-Q_{k_{j-1}}.$ If the Riesz transformation of triple sequences is RH-regular, and $H_{i}=Q_{m_{i}}-Q_{m_{i-1}}\rightarrow \infty$ as $i\rightarrow \infty,\overline{H}=\sum_{n\in \left(0,n_{\ell}\right]}p_{n_{\ell}}\rightarrow \infty$ as $\ell\rightarrow \infty,\overline{\overline{H}}=\sum_{k\in \left(0,k_{j}\right]}p_{k_{j}}\rightarrow \infty$ as $j\rightarrow \infty,$ then $\theta^{'}_{i,\ell,j}=\left\{\left(m_{i},n_{\ell},k_{j}\right)\right\}=\left\{\left(Q_{m_{i}}Q_{n_{j}}Q_{k_{k}}\right)\right\}$ is a triple lacunary sequence. If the assumptions $Q_{r}\rightarrow \infty$ as $r\rightarrow \infty,$  $\overline{Q}_{s}\rightarrow \infty$ as $s\rightarrow \infty$ and $\overline{\overline{Q}}_{t}\rightarrow \infty$ as $t\rightarrow \infty$ may be not enough to obtain the conditions $H_{i}\rightarrow \infty$ as $i\rightarrow \infty,\overline{H}_{\ell}\rightarrow \infty$ as $\ell\rightarrow \infty$  and $\overline{\overline{H}}_{j}\rightarrow \infty$ as $j\rightarrow \infty$ respectively. For any lacunary sequences $\left(m_{i}\right),\left(n_{\ell}\right)$ and $\left(k_{j}\right)$ are integers.
\\Throughout the paper, we assume that $Q_{r}=q_{11}+q_{12}+\ldots + q_{rs}\rightarrow \infty \left(r\rightarrow \infty\right),\overline{Q}_{s}=\overline{q}_{11}+\overline{q}_{12}+\ldots +\overline{q}_{rs}\rightarrow \infty \left(s\rightarrow \infty\right),\overline{\overline{Q}}_{t}=\overline{\overline{q}}_{11}+\overline{\overline{q}}_{12}+\ldots +\overline{\overline{q}}_{rs}\rightarrow \infty \left(t\rightarrow \infty\right),$ such that $H_{i}=Q_{m_{i}}-Q_{m_{i-1}}\rightarrow \infty$ as $i\rightarrow \infty,\overline{H}_{\ell}=Q_{n_{\ell}}-Q_{n_{\ell-1}}\rightarrow \infty$ as $\ell\rightarrow \infty$ and $\overline{\overline{H}}_{j}=Q_{k_{j}}-Q_{k_{j-1}}\rightarrow \infty$ as $j\rightarrow \infty.$ 
\\Let $Q_{m_{i},n_{\ell},k_{j}}=Q_{m_{i}}\overline{Q}_{n_{\ell}}\overline{\overline{Q}}_{k_{j}},H_{i\ell j}=H_{i}\overline{H}_{\ell}\overline{\overline{H}}_{j}, \\I^{'}_{i\ell j}=\left\{\left(m,n,k\right):Q_{m_{i-1}}< m< Q_{m_{i}},\overline{Q}_{n_{\ell-1}}< n< Q_{n_{\ell}}\hspace{0.1cm}and\hspace{0.1cm}\overline{Q}_{k_{j-1}}< k< \overline{Q}_{k_{j}}\right\},\\ V_{i}=\frac{Q_{m_{i}}}{Q_{m_{i-1}}},\overline{V}_{\ell}=\frac{Q_{n_{\ell}}}{Q_{n_{\ell-1}}}$ and $\overline{\overline{V}}_{j}=\frac{Q_{k_{j}}}{Q_{k_{j-1}}}.$ and $V_{i\ell j}=V_{i}\overline{V}_{\ell}\overline{\overline{V}}_{j}.$
\\If we take $q_{m}=1,\overline{q}_{n}=1\hspace{0.1cm} and \hspace{0.1cm}\overline{\overline{q}}_{k}=1$ for all $m,n$ and $k$ then $H_{i\ell j},Q_{i\ell j},V_{i\ell j}$ and $I^{'}_{i\ell j}$ reduce to $h_{i\ell j},q_{i\ell j},v_{i\ell j}$ and $I_{i\ell j}.$
\\\indent Let $f$ be an Orlicz function and $p=\left(p_{mnk}\right)$ be any factorable triple sequence of strictly positive real numbers, we define the following sequence spaces:
\\$\left[\Gamma^{3}_{R},\theta_{i\ell j},q,f,p\right]=
\\\left\{P-lim_{i,\ell, j\rightarrow \infty}\frac{1}{H_{i,\ell j}}\sum_{i\in I_{i\ell j}}\sum_{\ell\in I_{i\ell j}}\sum_{j\in I_{i\ell j}}q_{m}\overline{q}_{n}\overline{\overline{q}}_{k}\left[f\left(\left|x_{m+i,n+\ell,k+j}\right|\right)^{p_{mnk}}\right]=0\right\},$ uniformly in $i,\ell$ and $j.$
\\$\left[\Lambda^{3}_{R},\theta_{i\ell j},q,f,p\right]=
\\\left\{x=\left(x_{mnk}\right):P-sup_{i,\ell, j}\frac{1}{H_{i,\ell j}}\sum_{i\in I_{i\ell j}}\sum_{\ell\in I_{i\ell j}}\sum_{j\in I_{i\ell j}}q_{m}\overline{q}_{n}\overline{\overline{q}}_{k}\left[f\left|x_{m+i,n+\ell,k+j}\right|^{p_{mnk}}\right]< \infty\right\},$  uniformly in $i,\ell$ and $j.$
\\Let $f$ be an Orlicz function, $p=p_{mnk}$ be any factorable double sequence of strictly positive real numbers and
and $q_{m},\overline{q}_{n}$ and $\overline{\overline{q}}_{k}$ be sequences of positive numbers and $Q_{r}=q_{11}+\cdots q_{rs}$, $\overline{Q}_{s}=\overline{q}_{11}\cdots \overline{q}_{rs}$ and $\overline{\overline{Q}}_{t}=\overline{\overline{q}}_{11}\cdots \overline{\overline{q}}_{rs},$
\\If we choose $q_{m}= 1,\overline{q}_{n}= 1$ and $\overline{\overline{q}}_{k}=1$ for all $m,n$ and $k$, then we obtain the following sequence spaces.
\\$\left[\Gamma^{3}_{R},q,f,p\right]=
\\\left\{P-lim_{r,s,t \rightarrow \infty}\frac{1}{Q_{r}\overline{Q}_{s}\overline{\overline{Q}}_{t}}\sum_{m=1}^{r}\sum_{n=1}^{s}\sum_{k=1}^{t} q_{m}\overline{q}_{n}\overline{\overline{q}}_{k}\left[f\left(\left|x_{m+i,n+\ell,k+j}\right|\right)^{p_{mnk}}\right]=0\right\},$ uniformly in $i,\ell$ and $j.$
\\$\left[\Lambda^{3}_{R},q,f,p\right]=
\\\left\{P-sup_{r,s,t}\frac{1}{Q_{r}\overline{Q}_{s}\overline{\overline{Q}}_{t}}\sum_{m=1}^{r}\sum_{n=1}^{s}\sum_{k=1}^{t} q_{m}\overline{q}_{n}\overline{\overline{q}}_{k}\left[f\left(\left|x_{m+i,n+\ell,k+j}\right|\right)^{p_{mnk}}\right]<\infty\right\},$ uniformly in $i,\ell$ and $j.$
\subsection{Definition} Let $f$ be an Orlicz function and $p=\left(p_{mnk}\right)$ be any factorable triple sequence of strictly positive real numbers, we define the following sequence space:
\\$\theta_{i,\ell,j}=\left\{\left(m_{i},n_{\ell},k_{j}\right)\right\}$ be a triple lacunary sequence 
$\Gamma^{3}_{f}\left[AC_{\theta_{i,\ell,j}},p\right]=\\\left\{P-lim_{i,\ell,j}\frac{1}{h_{i\ell j}}\sum_{m\in I_{i,\ell, j}}\sum_{n\in I_{i,\ell, j}}\sum_{k\in I_{i,\ell, j}}\left[f\left(\left|x_{m+i,n+\ell,k+j}\right|\right)^{1/m+n+k}\right]^{p_{mnk}}=0,\right\},$ uniformly in $i,\ell$ and $j$. 
\\\indent We shall denote $\Gamma^{3}_{f}\left[AC_{\theta_{i,\ell,j}},p\right]$ as $\Gamma^{3}\left[AC_{\theta_{i,\ell, j}},p\right]$ respectively when $p_{mnk}=1$ for all $m,n$ and $k$ If $x$ is in $\Gamma^{3}\left[AC_{\theta_{i,\ell, j}},p\right],$ we shall say that $x$ is almost lacunary $\Gamma^{3}$ strongly $p-$convergent with respect to the Orlicz function $f$. Also note if $f\left(x\right)=x,p_{mnk}=1$ for all $m,n$ and $k$ then $\Gamma^{3}_{f}\left[AC_{\theta_{i,\ell, j}},p\right]=\Gamma^{3}\left[AC_{\theta_{i,\ell, j}}\right]$ which are defined as follows:  
$\Gamma^{3}\left[AC_{\theta_{i,\ell, j}}\right]=\\\left\{P-lim_{i,\ell, j}\frac{1}{h_{i\ell j}}\sum_{m\in I_{i,\ell, j}}\sum_{n\in I_{i,\ell, j}}\sum_{k\in I_{i,\ell, j}}\left[f\left(\left|x_{m+i,n+\ell,k+j}\right|\right)^{1/m+n+k}\right]=0,\right\},$ uniformly in $i,\ell$ and $j$. 
\\Again note if $p_{mnk}=1$ for all $m,n$ and $k$ then $\Gamma^{3}_{f}\left[AC_{\theta_{i,\ell, j}},p\right]=\Gamma^{3}_{f}\left[AC_{\theta_{i,\ell, j}}\right].$ we define 
$\Gamma^{3}_{f}\left[AC_{\theta_{i,\ell, j}},p\right]=\\\left\{P-lim_{i,\ell, j}\frac{1}{h_{i\ell j}}\sum_{m\in I_{k,\ell ,j}}\sum_{n\in I_{i,\ell, j}}\sum_{k\in I_{i,\ell, j}}\left[f\left(\left|x_{m+i,n+\ell,k+j}\right|\right)^{1/m+n+k}\right]^{p_{mnk}}=0,\right\},$ uniformly in $i,\ell$ and $j$.  
\subsection{Definition} Let $f$ be an Orlicz function $p=\left(p_{mnk}\right)$ be any factorable triple sequence of strictly positive real numbers, we define the following sequence space: 
$\Gamma^{3}_{f}\left[p\right]=\\\left\{P-lim_{r,s,t\rightarrow \infty}\frac{1}{rst}\sum_{m=1}^{r}\sum_{n=1}^{s}\sum_{k=1}^{t}\left[f\left(\left|x_{m+i,n+\ell,k+j}\right|\right)^{1/m+n+k}\right]^{p_{mnk}}= 0\right\},$ uniformly in $i,\ell$ and $j$.
\\\indent If we take $f\left(x\right)=x,p_{mnk}=1$ for all $m,n$ and $k$ then $\Gamma^{3}_{f}\left[p\right]=\Gamma^{3}.$
\subsection{Definition} Let $\theta_{i,\ell, j}$ be a triple lacunary sequence; the triple number sequence $x$ is $\widehat{S_{\theta{i,\ell, j}}}-p-$ convergent to 0 then
\\$P-lim_{i,\ell, j}\frac{1}{h_{i,\ell, j}}max_{i,\ell,j}\left|\left\{\left(m,n,k\right)\in I_{i,\ell, j}:f\left(\left|x_{m+i,n+\ell,k+j}-0\right|\right)^{1/m+n+k}\right\}\right|=0.$
\\In this case we write $\widehat{S_{\theta{i,\ell, j}}}-lim\left(f\left|x_{m+i,n+\ell,k+j}-0\right|\right)^{1/m+n+k}=0.$
\section{Main Results}
\subsection{Theorem}If $f$ be any Orlicz function and a bounded factorable positive triple number sequence $p_{mnk}$ then $\Gamma^{3}_{f}\left[AC_{\theta_{i,\ell,j}},P\right]$ is linear space
\\Proof: The proof is easy. Theorefore omit the proof. 
\subsection{Theorem} For any Orlicz function $f,$ we have $\Gamma^{3}\left[AC_{\theta_{i,\ell,j}}\right]\subset \Gamma^{3}_{f}\left[AC_{\theta_{i,\ell,j}}\right]$
\\Proof: Let $x\in \Gamma^{3}\left[AC_{\theta_{i,\ell,j}}\right]$ so that for each $i,\ell$ and $j$
\\$\Gamma^{3}\left[AC_{\theta_{i,\ell, j}}\right]=\\\left\{lim_{i,\ell, j}\frac{1}{h_{i \ell j}}\sum_{m\in I_{i,\ell, j}}\sum_{n\in I_{i,\ell, j}}\sum_{k\in I_{i,\ell, j}}\left[\left(\left|x_{m+i,n+\ell,k+j}\right|\right)^{1/m+n+k}\right]=0\right\}.$  
\\Since $f$ is continuous at zero, for $\varepsilon>0$ and choose $\delta$ with $0<\delta <1$ such that $f\left(t\right)< \epsilon$ for every $t$ with $0\leq t \leq \delta.$ We obtain the following, 
$\\\frac{1}{h_{i\ell j}}\left(h_{i\ell j}\epsilon\right)+\frac{1}{h_{i\ell j}}\sum_{m\in I_{i,\ell,j}}\sum_{n\in I_{i,\ell,j}}\sum_{k\in I_{i,\ell,j}\hspace{0.05cm}and\hspace{0.05cm}\left|x_{m+i,n+\ell,k+j}-0\right|> \delta} \\f\left[\left(\left|x_{m+i,n+\ell,k+j}\right|\right)^{1/m+n+k}\right]\\\frac{1}{h_{i\ell j}}\left(h_{i\ell j}\epsilon\right)+ \frac{1}{h_{i\ell j}}K\delta^{-1}f\left(2\right)h_{i\ell j}\hspace{0.05cm}\Gamma^{3}\left[AC_{\theta_{i,\ell, j}}\right].$
\\Hence $i,\ell$ and $j$ goes to infinity, we are granted $x\in \Gamma^{3}_{f}\left[AC_{\theta_{i,\ell, j}}\right].$
\subsection{Theorem} Let $\theta_{i,\ell, j}=\left\{m_{i},n_{\ell},k_{j}\right\}$ be a triple lacunary sequence with $liminf_{i}q_{i}>1,\hspace{0.2cm}liminf_{\ell}\overline{q_{\ell}}>1\hspace{0.2cm}and\hspace{0.2cm}liminf_{j}q_{j}>1$ then for any Orlicz function $f,$ $\Gamma^{3}_{f}\left(P\right)\subset \Gamma^{3}_{f}\left(AC_{\theta_{i,\ell, j}},P\right)$
\\\textbf{Proof: } Suppose $liminf_{i}q_{i}>1,\hspace{0.2cm}liminf_{\ell}\overline{q_{\ell}}>1$ and $liminf_{j}q_{j}>1$ then there exists $\delta>0$ such that $q_{i}>1+\delta,\hspace{0.2cm}\overline{q_{\ell}}>1+\delta$ and $q_{j}>1+\delta$ This implies $\frac{h_{i}}{m_{i}}\geq \frac{\delta}{1+\delta},\hspace{0.2cm}\frac{h_{\ell}}{n_{\ell}}\geq \frac{\delta}{1+\delta}$ and $\frac{h_{j}}{k_{j}}\geq \frac{\delta}{1+\delta}$ Then for $x\in \Gamma^{3}_{f}\left(P\right),$ we can write for each $r,s$ and $u.$  
\\$B_{i,\ell, j}=\frac{1}{h_{i\ell j}}\sum_{m\in I_{i,\ell, j}}\sum_{n\in I_{i,\ell, j}}\sum_{k\in I_{i,\ell, j}}f\left[\left(\left|x_{m+i,n+\ell,k+j}\right|\right)^{1/m+n+k}\right]^{p_{mnk}}=\\\indent\hspace{0.8cm}\frac{1}{h_{i\ell j}}\sum_{m=1}^{m_{i}}\sum_{n=1}^{n_{\ell}}\sum_{k=1}^{k_{j}}f\left[\left(\left|x_{m+i,n+\ell,k+j}\right|\right)^{1/m+n+k}\right]^{p_{mnk}}-\\\indent\hspace{0.8cm}\frac{1}{h_{i\ell j}}\sum_{m=1}^{m_{i-1}}\sum_{n=1}^{n_{\ell-1}}\sum_{k=1}^{k_{i-1}}f\left[\left(\left|x_{m+i,n+\ell,k+j}\right|\right)^{1/m+n+k}\right]^{p_{mnk}}-\\\indent\hspace{0.8cm}\frac{1}{h_{i\ell j}}\sum_{m=m_{i-1}+1}^{m_{i}}\sum_{n=1}^{n_{\ell-1}}\sum_{k=1}^{k_{j-1}}f\left[\left(\left|x_{m+i,n+\ell,k+j}\right|\right)^{1/m+n+k}\right]^{p_{mnk}}-\\\indent\hspace{0.8cm}\frac{1}{h_{i\ell j}}\sum_{k=k_{j}+1}^{k_{j}}\sum_{n=n_{\ell-1}+1}^{n_{\ell}}\sum_{m=1}^{m_{k-1}}f\left[\left(\left|x_{m+i,n+\ell,k+j}\right|\right)^{1/m+n+k}\right]^{p_{mnk}}
\\\indent\hspace{0.1cm}=\frac{m_{i}n_{\ell}k_{j}}{h_{i\ell j}}\left(\frac{1}{m_{i}n_{\ell}k_{j}}\sum_{m=1}^{m_{i}}\sum_{n=1}^{n_{\ell}}\sum_{k=1}^{k_{j}}f\left[\left(\left|x_{m+i,n+\ell,k+j}\right|\right)^{1/m+n+k}\right]^{p_{mnk}}\right)-\\\indent\hspace{0.7cm}\frac{m_{k-1}n_{\ell-1}k_{j-1}}{h_{i\ell j}}\left(\frac{1}{m_{i-1}n_{\ell-1}k{j-1}}\sum_{m=1}^{m_{i-1}}\sum_{n=1}^{n_{\ell-1}}\sum_{k=1}^{k_{j-1}}f\left[\left(\left|x_{m+i,n+\ell,k+j}\right|\right)^{1/m+n+k}\right]^{p_{mnk}}\right)\\-\indent\hspace{0.7cm}\frac{k_{j-1}}{h_{i\ell j}}\left(\frac{1}{k_{j-1}}\sum_{m=m_{i-1}+1}^{m_{i}}\sum_{n=1}^{n_{\ell-1}}\sum_{k=1}^{k_{j}}f\left[\left(\left|x_{m+i,n+\ell,k+j}\right|\right)^{1/m+n+k}\right]^{p_{mnk}}\right)\\-\indent\hspace{0.7cm}\frac{n_{\ell-1}}{h_{i\ell j}}\left(\frac{1}{n_{\ell-1}}\sum_{m=m_{k-1}+1}^{m_{k}}\sum_{n=1}^{n_{\ell-1}}\sum_{k=1}^{k_{j}}f\left[\left(\left|x_{m+i,n+\ell,k+j}\right|\right)^{1/m+n+k}\right]^{p_{mnk}}\right)-\\\indent\hspace{0.7cm}\frac{m_{k-1}}{h_{i\ell j}}\left(\frac{1}{m_{k-1}}\sum_{k=1}^{k_{j}}\sum_{n=n_{\ell-1}+1}^{n_{\ell}}\sum_{m=1}^{m_{k-1}}f\left[\left(\left|x_{m+i,n+\ell,k+j}\right|\right)^{1/m+n+k}\right]^{p_{mnk}}\right).$
\\Since $x\in \Gamma^{3}_{f}\left(P\right)$ the last three terms tend to zero uniformly in $m,n,k$  in the sense, thus, for each $r,s$ and $u$ 
\\$B_{i,\ell ,j}=\frac{m_{i}n_{\ell}k_{j}}{h_{i\ell j}}\left(\frac{1}{m_{i}n_{\ell}k_{j}}\sum_{m=1}^{m_{i}}\sum_{n=1}^{n_{\ell}}\sum_{k=1}^{k_{j}}f\left[\left(\left|x_{m+i,n+\ell,k+j}\right|\right)^{1/m+n+k}\right]^{p_{mnk}}\right)-\\\indent\hspace{0.7cm}\frac{m_{i-1}n_{\ell-1}k_{j-1}}{h_{i\ell j}}\left(\frac{1}{m_{i-1}n_{\ell-1}k_{j-1}}\sum_{m=1}^{m_{i-1}}\sum_{n=1}^{n_{\ell-1}}\sum_{k=1}^{k_{j-1}}f\left[\left(\left|x_{m+i,n+\ell,k+j}\right|\right)^{1/m+n+k}\right]^{p_{mnk}}\right)+O\left(1\right).$
\\Since $h_{i\ell j}=m_{i}n_{\ell}k_{j}-m_{i-1}n_{\ell-1}k_{j-1}$ we are granted for each $i,\ell$ and $j$ the following 
\begin{center}
$\frac{m_{i}n_{\ell}k{j}}{h_{i\ell j}}\leq \frac{1+\delta}{\delta}$ and $\frac{m_{i-1}n_{\ell-1}k_{j-1}}{h_{i\ell j}}\leq \frac{1}{\delta}.$ 
\end{center}
The terms 
\\$\left(\frac{1}{m_{i}n_{\ell}k_{j}}\sum_{m=1}^{m_{i}}\sum_{n=1}^{n_{\ell}}\sum_{k=1}^{k_{j}}f\left[\left(\left|x_{m+i,n+\ell,k+j}\right|\right)^{1/m+n+k}\right]^{p_{mnk}}\right)$ and 
\\$\left(\frac{1}{m_{i-1}n_{\ell-1}k_{j-1}}\sum_{m=1}^{m_{i-1}}\sum_{n=1}^{n_{\ell-1}}\sum_{k=1}^{k_{j-1}}f\left[\left(\left|x_{m+i,n+\ell,k+j}\right|\right)^{1/m+n+k}\right]^{p_{mnk}}\right)$ are both gai sequences for all $i,\ell$ and $j.$ Thus $B_{i\ell j}$ is a gai sequence for each $i,\ell$ and $j.$ Hence $x\in \Gamma^{3}_{f}\left(AC_{\theta_{i,\ell, j}},P\right).$    
\subsection{Theorem} Let $\theta_{i,\ell, j}=\left\{m,n,k\right\}$ be a triple lacunary sequence with $limsup_{\eta}q_{\eta}<\infty$ and $limsup_{i}\overline{q_{i}}< \infty$ then for any Orlicz function $f,\hspace{0.5cm}\Gamma^{3}_{f}\left(AC_{\theta_{i,\ell, j}},P\right)\subset \Gamma^{3}_{f}\left(p\right).$
\\\textbf{Proof: } Since $limsup_{i}q_{i}<\infty$ and $limsup_{i}\overline{q_{i}}< \infty$ there exists $H>0$ such that $q_{i}<H,\hspace{0.2cm}\overline{q_{\ell}}<H$ and $q_{j}<H$  for all $i,\ell$ and $j.$ Let $x\in \Gamma^{3}_{f}\left(AC_{\theta_{i,\ell, j}},P\right).$ Also there exist $i_{0}>0,\ell_{0}>0$ and $j_{0}>0$ such that for every $a\geq i_{0}\hspace{0.2cm}b\geq \ell_{0}$ and $c\geq j_{0}$ and $i,\ell$ and $j.$
\begin{center}
$A^{'}_{abc}=\frac{1}{h_{abc}}\sum_{m\in I_{a,b,c}}\sum_{n\in I_{a,b,c}} \sum_{k\in I_{a,b,c}}f\left[\left(\left|x_{m+i,n+\ell,k+j}\right|\right)^{1/m+n+k}\right]^{p_{mnk}}\rightarrow 0\hspace{0.05cm}as\hspace{0.05cm}m,n,k\rightarrow \infty.$
\end{center}
Let $G^{'}=max\left\{A^{'}_{a,b,c}:1\leq a\leq i_{0},\hspace{0.2cm}1\leq b\leq \ell_{0}\hspace{0.2cm} and \hspace{0.2cm}1\leq c\leq j_{0}\right\}$ and $p,q$ and $t$ be such that $m_{i-1}<p\leq m_{i},\hspace{0.2cm}n_{\ell-1}<q\leq n_{\ell}$ and $m_{j-1}<t\leq m_{j}.$ Thus we obtain the following: 
\\$\frac{1}{pqt}\sum_{m=1}^{p}\sum_{n=1}^{q}\sum_{k=1}^{t}\left[\left(\left|x_{m+i,n+\ell,k+j}\right|\right)^{1/m+n+k}\right]^{p_{mnk}}
\\\leq \frac{1}{m_{i-1}n_{\ell-1}k_{j-1}}\sum_{m=1}^{m_{i}}\sum_{n=1}^{n_{\ell}}\sum_{k=1}^{k_{j}}\left[\left(\left|x_{m+i,n+\ell,k+j}\right|\right)^{1/m+n+k}\right]^{p_{mnk}}\\\leq \frac{1}{m_{i-1}n_{\ell-1}k_{j-1}}\sum_{a=1}^{i}\sum_{b=1}^{\ell}\sum_{c=1}^{j}\\\left(\sum_{m\in I_{a,b,c}}\sum_{n\in I_{a,b,c}}\sum_{k\in I_{a,b,c}}\left[\left(\left|x_{m+i,n+\ell,k+j}\right|\right)^{1/m+n+k}\right]^{p_{mnk}}\right)\\= \frac{1}{m_{i-1}n_{\ell-1}k_{j-1}}\sum_{a=1}^{i_{0}}\sum_{b=1}^{\ell_{0}}\sum_{c=1}^{j_{0}}h_{a,b,c}A^{'}_{a,b,c}+\frac{1}{m_{k-1}n_{\ell-1}k_{j-1}}\sum_{\left(i_{0}<a\leq i\right)\bigcup \left(\ell_{0}<b\leq \ell\right)\bigcup \left(j_{0}<c\leq j\right)}h_{a,b,c}A^{'}_{a,b,c}\\\leq \frac{G^{'}}{m_{i-1}n_{\ell-1}k_{j-1}}\sum_{a=1}^{i_{0}}\sum_{b=1}^{\ell_{0}}\sum_{c=1}^{j_{0}}h_{a,b,c}+\frac{1}{m_{i-1}n_{\ell-1}k_{j-1}}\sum_{\left(i_{0}<a\leq i\right)\bigcup \left(\ell_{0}<b\leq \ell\right)\bigcup  \left(j_{0}<c\leq \j\right)}h_{a,b,c}A^{'}_{a,b,c}\\\leq \frac{G^{'}m_{i_{0}}n_{\ell_{0}}k_{j_{0}}i_{0}\ell_{0}j_{0}}{m_{i-1}n_{\ell-1}k_{j-1}}+\frac{1}{m_{i-1}n_{\ell-1}j_{j-1}}\sum_{\left(i_{0}<a\leq i\right)\bigcup \left(\ell_{0}<b\leq \ell\right)\bigcup \left(j_{0}<c\leq j\right)}h_{a,b,c}A^{'}_{a,b,c}\\\leq \frac{G^{'}m_{i_{0}}n_{\ell_{0}k_{j_{0}}}i_{0}\ell_{0}j_{0}}{m_{i-1}n_{\ell-1}k_{j-1}}+\left(sup_{a\geq i_{0}\bigcup b\geq \ell_{0}\bigcup c\geq j_{0}}A^{'}_{a,b,c}\right)\frac{1}{m_{i-1}n_{\ell-1}k_{j-1}}\sum_{\left(i_{0}<a\leq i\right)\bigcup \left(\ell_{0}<b\leq \ell\right)\bigcup \left(j_{0}<c\leq j\right)}h_{a,b,c}\\\leq \frac{G^{'}m_{i_{0}}n_{\ell_{0}k_{j_{0}}}i_{0}\ell_{0}j_{0}}{m_{i-1}n_{\ell-1}k_{j-1}}+\frac{\epsilon}{m_{i-1}n_{\ell-1}k_{j-1}} \sum_{\left(i_{0}<a\leq i\right)\bigcup \left(\ell_{0}<b\leq \ell\right)\bigcup \left(j_{0}<c\leq j\right)}h_{a,b,c}\\\leq \frac{G^{'}m_{i_{0}}n_{\ell_{0}k_{j_{0}}}i_{0}\ell_{0}j_{0}}{m_{i-1}n_{\ell-1}k_{j-1}}+\epsilon H^{3}.$
\\Since $m_{i},\hspace{0.2cm}n_{\ell}$ and $k_{j}$ both approaches infinity as both $p,q$ and $t$ approaches infinity, it follows that 
\begin{center}
$\frac{1}{pqt}\sum_{m=1}^{p}\sum_{n=1}^{q}\sum_{k=1}^{t}\left[\left(\left|x_{m+i,n+\ell,k+j}\right|\right)^{1/m+n+k}\right]^{p_{mnk}}= 0,\hspace{0.05cm}uniformly\hspace{0.05cm}in\hspace{0.05cm}i,\ell\hspace{0.05cm}and\hspace{0.05cm}j.$
\end{center}
Hence $x\in \Gamma^{3}_{f}\left(P\right).$
\subsection{Theorem} Let $\theta_{i,\ell, j}$ be a triple lacunary sequence then 
\\(i) $\left(x_{mnk}\right)\stackrel{P}{\rightarrow} \Gamma^{3}\left(\widehat{S_{\theta_{i,\ell, j}}}\right)$
\\(ii)$\left(AC_{\theta_{i,\ell, j}}\right)$ is a proper subset of $\left(\widehat{S_{\theta_{i,\ell, j}}}\right)$ 
\\(iii) If $x\in \Lambda^{3}$ and $\left(x_{mnk}\right)\stackrel{P}{\rightarrow} \Gamma^{3}\left(\widehat{S_{\theta_{i,\ell, j}}}\right)$ then $\left(x_{mnk}\right)\stackrel{P}{\rightarrow} 
\Gamma^{3} \left(AC_{\theta_{i,\ell, j}}\right)$
\\ (iv) $\Gamma^{3}\left(\widehat{S_{\theta_{i,\ell, j}}}\right)\bigcap \Lambda^{3}= \Gamma^{3}\left[AC_{\theta_{i,,\ell, j}}\right]\bigcap \Lambda^{3}.$
\\\textbf{Proof: (i)} Since for all $i,\ell$ and $j$ 
\\$\left|\left\{\left(m,n,k\right)\in I_{i,\ell, j}:\left(\left|x_{m+i,n+\ell,k+j}-0\right|\right)^{1/m+n+k}\right\}=0\right|\leq \\\sum_{m\in I_{i,\ell, j}} \sum_{n\in I_{i,\ell, j}} \sum_{k\in I_{i,\ell, j}\hspace{0.05cm}and\hspace{0.05cm}  \left|x_{m+i,n+\ell,k+j}\right|=0 }\left(\left|x_{m+i,n+\ell,k+j}-0\right|\right)^{1/m+n+k}\leq \\\sum_{m\in I_{i,\ell,j}}\sum_{n\in I_{i,\ell,j}}\sum_{k\in I_{i,\ell,j}}\left(\left|x_{m+i,n+\ell,k+j}-0\right|\right)^{1/m+n+k},$
for all $i,\ell$ and $j$ 
\\$P-lim_{i,\ell, j}\frac{1}{h_{i\ell j}}\sum_{m\in I_{i,\ell, j}}\sum_{n\in I_{i,\ell, j}} \sum_{k\in I_{i,\ell, j}}\left(\left|x_{m+i,n+\ell,k+j}-0\right|\right)^{1/m+n+k}=0$
\\This implies that for all $i,\ell$ and $j$
\begin{center}
$P-lim_{i,\ell, j}\frac{1}{h_{i,\ell, j}}\left|\left\{\left(m,n,k\right)\in I_{i,\ell, j}:\left(\left|x_{m+i,n+\ell,k+j}-0\right|\right)^{1/m+n+k}=0\right\}\right|=0.$
\end{center}
\textbf{(ii)}let $x=\left(x_{mnk}\right)$ be defined as follows:
\begin{center}
$\left(x_{mnk}\right)=\begin{bmatrix}
  1 & 2 & 3 & .  . . \frac{\left[\sqrt[4]{h_{i,\ell, j}}\right]^{m+n+k}}{\left(1\right)!} & 0 & \ldots\\
  1 & 2 & 3 & .  . . \frac{\left[\sqrt[4]{h_{i,\ell, j}}\right]^{m+n+k}}{\left(1\right)!} & 0 & \ldots\\
  . & \\
  . & \\
  . & \\
  1 & 2 & 3 & .  . . \frac{\left[\sqrt[4]{h_{i,\ell, j}}\right]^{m+n+k}}{\left(1\right)!} & 0 &\ldots \\
  . & \\
  . & \\
  . & \\
  0 & 0 & 0 &  .  . .                             0                                    & 0  & \ldots \\
  . & \\
  . & \\
  . & \\
\end{bmatrix};$ 
\end{center}
Here $x$ is an trible sequence and for all $i,\ell$ and $j$
\\$P-lim_{i,\ell, j}\frac{1}{h_{k,\ell, j}}\left|\left\{\left(m,n,k\right)\in I_{i,\ell, j}:\left(\left|x_{m+i,n+\ell,k+j}-0\right|\right)^{1/m+n+k}=0\right\}\right|=\\P-lim_{i,\ell, j}\frac{1}{h_{i,\ell, j}}\left(\frac{\hspace{0.05cm}\left[\sqrt[4]{h_{i,\ell, j}}\right]^{m+n+k}}{\left(1\right)!}\right)^{1/m+n+k}=0.$
\\Therefore $\left(x_{mnk}\right)\stackrel{P}{\rightarrow} \Gamma^{3}\left(\widehat{S_{\theta_{i,\ell, j}}}\right).$ Also 
\\$P-lim_{i,\ell, j}\frac{1}{h_{i\ell j}}\sum_{m\in I_{i,\ell, j}}\sum_{n\in I_{i,\ell, j}}\sum_{k\in I_{i,\ell, j}}\left(\left|x_{m+i,n+\ell,k+j}\right|\right)^{1/m+n+k}=
\\P-\frac{1}{2}\left(lim_{i,\ell, j}\frac{1}{h_{i,\ell, j}}\left(\frac{\hspace{0.05cm}\left[\sqrt[4]{h_{i,\ell, j}}\right]^{m+n+k}\left[\sqrt[4]{h_{i,\ell, j}}\right]^{m+n+k}\left[\sqrt[4]{h_{i,\ell, j}}\right]^{m+n+k}}{\left(1\right)!}\right)^{1/m+n+k}+1\right)=\frac{1}{2}.$
\\Therefore $\left(x_{mnk}\right)\stackrel{P}{\not \rightarrow}\Gamma^{3} \left(AC_{\theta_{i,\ell, j}}\right).
\\\textbf{(iii)}$ If $x\in \Lambda^{3}$ and $\left(x_{mnk}\right)\stackrel{P}{\rightarrow} \Gamma^{3}\left(\widehat{S_{\theta_{i,\ell, j}}}\right)$ then $\left(x_{mnk}\right)\stackrel{P}{\rightarrow} 
\Gamma^{3} \left(AC_{\theta_{i,\ell, j}}\right).$
\\Suppose $x\in \Lambda^{3}$ then for all $i,\ell$ and $j,\hspace{0.05cm}\left(\left|x_{m+i,n+\ell,k+j}-0\right|\right)^{1/m+n+k}\leq M$ for all $m,n,k.$ Also for given $\epsilon>0$ and $i,\ell$ and $j$ large for all $i,\ell$ and $j$ we obtain the following:
$\\\frac{1}{h_{i\ell j}}\sum_{m\in I_{i,\ell, j}}\sum_{n\in I_{i,\ell, j}}\sum_{k\in I_{i,\ell, j}}\left(\left|x_{m+i,n+\ell,k+j}-0\right|\right)^{1/m+n+k}=\\\frac{1}{h_{i\ell j}}\sum_{m\in I_{k,\ell}}\sum_{n\in I_{i,\ell, j}}\sum_{k\in I_{k,\ell, j}\hspace{0.05cm}and\hspace{0.05cm}  \left|x_{m+i,n+\ell,k+j}\right|\geq0}\left(\left|x_{m+i,n+\ell,k+j}-0\right|\right)^{1/m+n+k}+\\\frac{1}{h_{i\ell j}}\sum_{m\in I_{i,\ell, j}}\sum_{n\in I_{i,\ell, j}}\sum_{k\in I_{i,\ell, j}\hspace{0.05cm}and\hspace{0.05cm}  \left|x_{m+i,n+\ell,k+j}\right|\leq 0}\left(\left|x_{m+i,n+\ell,k+j}-0\right|\right)^{1/m+n+k}\\\leq \frac{M}{h_{i\ell j}}\left|\left\{\left(m,n,k\right)\in I_{i,\ell, j}:\left(\left|x_{m+i,n+\ell,k+j}-0\right|\right)^{1/m+n+k}\right\}=0\right|+\epsilon.$
\\Therefore $x\in \Lambda^{3}$ and $\left(x_{mnk}\right)\stackrel{P}{\rightarrow} \Gamma^{3}\left(\widehat{S_{\theta_{i,\ell, j}}}\right)$ then $\left(x_{mnk}\right)\stackrel{P}{\rightarrow} 
\Gamma^{3} \left(AC_{\theta_{i,\ell, j}}\right).$  
\\\textbf{(iv)}$\Gamma^{3}\left(\widehat{S_{\theta_{i,\ell, j}}}\right)\bigcap \Lambda^{3}= \Gamma^{3}\left[AC_{\theta_{i,\ell, j}}\right]\bigcap \Lambda^{3}.$ follows from (i),(ii) and (iii).
\subsection{Theorem} If  $f$ be any Orlicz function then  $\Gamma^{3}_{f}\left[AC_{\theta_{i,\ell, j}}\right]\notin \Gamma^{3}\left(\widehat{S_{\theta_{i,\ell, j}}}\right)$ 
\\\textbf{Proof:} Let $x\in \Gamma^{3}_{f}\left[AC_{\theta_{i,\ell, j}}\right],$ for all $i,\ell$ and $j.$ 
\\Therefore we have 
\\$\frac{1}{h_{i\ell j}}\sum_{m\in I_{i,\ell, j}}\sum_{n\in I_{i,\ell, j}}\sum_{k\in I_{i,\ell, j}}f\left[\left(\left|x_{m+i,n+\ell,k+j}-0\right|\right)^{1/m+n+k}\right]\geq \\\frac{1}{h_{i\ell
 j}}\sum_{m\in I_{i,\ell, j}}\sum_{n\in I_{i,\ell, j}} \sum_{k\in I_{i,\ell, j}\hspace{0.05cm}and\hspace{0.05cm}  \left|x_{m+r,n+s,k+u}\right|=0}\\f\left[\left(\left|x_{m+i,n+\ell,k+j}-0\right|\right)^{1/m+n+k}\right]>\\\frac{1}{h_{i\ell j}}f\left(0\right)\left|\left\{\left(m,n,k\right)\in I_{i,\ell, j}:\left(\left|x_{m+i,n+\ell, k+j}-0\right|\right)^{1/m+n+k}\right\}=0\right|.$
\\Hence $x\notin \Gamma^{3}\left(\widehat{S_{\theta_{i,\ell, j}}}\right).$ 
\\\\\textbf{Competing Interests: } The authors declare that there is not any conflict of interests regarding the publication of this manuscript. 

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