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\title[The Backward Operator of Double Almost $\left(\lambda_{m}\mu_{n}\right)$ convergence in $\chi^{2}-$Riesz ]{The Backward Operator of Double Almost $\left(\lambda_{m}\mu_{n}\right)$ convergence in $\chi^{2}-$Riesz space defined by a Musielak-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[A.Esi]{A.Esi $^2$  }
\address{$^2$ Department of Mathematics,
\newline \indent Adiyaman University, 
\newline \indent 02040,Adiyaman, Turkey.}
\thanks{aesi23@hotmail.com}




\keywords{ analytic sequence, Museialk-Orlicz function, double sequences, chi sequence,Lambda, Riesz space, strongly, statistical convergent      \\
\indent 2010 {\it Mathematics Subject Classification}. 40A05,40C05,40D05.\\
\indent {\it Received}: \\
\indent {\it Revised}: \\}




\begin{abstract}

In this paper we introduce the backward operator is $\nabla$ and study the notion of $\nabla-$ statistical convergence and $\nabla-$ statistical Cauchy sequence using by almost $\left(\lambda_{m}\mu_{n}\right)$ convergence in $\chi^{2}-$Riesz space and also some inclusion theorems are discussed.  
\end{abstract}

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\maketitle

\section{Introduction}

\hspace{0.5cm} Throughout $w,\chi$ and $\Lambda$ denote the classes of all, gai and analytic scalar valued single sequences, respectively. We write $w^{2}$ for the set of all complex double sequences $(x_{mn}),$ where $m,n\in \mathbb{N},$ the set of positive integers. Then, $w^{2}$ is a linear space under the coordinate wise addition and scalar multiplication.
\\\indent Some initial works on double sequence spaces is found in Tripathy [1] and Mursaleen [2] and Mursaleen and Edely [3,4], Subramanian and Misra [5], Pringsheim [6], Moricz and Rhoades [7], Robison [8], Savas et al. [9], Raj et al. [10], Francesco Tulone [11] and many others.
\\\indent Let $\left(x_{mn}\right)$ be a double sequence of real or complex numbers. Then the series $\sum_{m,n=1}^{\infty}x_{mn}$ is called a double series. The double series $\sum_{m,n=1}^{\infty}x_{mn}$ give one space is said to be  convergent if and only if the double sequence $(S_{mn})$is convergent, where
\begin{center}
$S_{mn}=\sum_{i,j=1}^{m,n}x_{ij}(m,n=1,2,3,. . . )$ . 
\end{center}
A double sequence $x=(x_{mn})$is said to be double analytic if
\begin{center}
$sup_{m,n}\left|x_{mn}\right|^{\frac{1}{m+n}}<\infty.$ 
\end{center}
The vector space of all double analytic sequences are usually denoted by $\Lambda^{2}$.
A sequence $x=(x_{mn})$ is called double entire sequence if 
\begin{center}
$\left|x_{mn}\right|^{\frac{1}{m+n}}\rightarrow 0$ as $m,n\rightarrow \infty.$ 
\end{center}
The vector space of all double entire sequences are usually denoted by $\Gamma^{2}.$ Let the set of sequences with this property be denoted by  $\Lambda^{2}$ and $\Gamma^{2}$ is a metric space with the metric
\begin{equation}
d(x,y)=sup_{m,n}\left\{\left|x_{mn}-y_{mn}\right|^{\frac{1}{m+n}}:m,n:1,2,3,. . . \right\},
\end{equation}
forall$\hspace{0.05cm}x=\left\{x_{mn}\right\}$\hspace{0.05cm}and\hspace{0.05cm}$y=\left\{y_{mn}\right\}in \hspace{0.05cm}\Gamma^{2}.$ 
Let $\phi=\left\{finite\hspace{0.1cm} sequences\right\}.$ 
\\\\\indent Consider a double sequence $x=(x_{mn}).$ The $(m,n)^{th}$ section $x^{[m,n]}$ of the sequence is defined by $x^{[m,n]}=\sum{_{i,j=0}^{m,n}}x_{ij}\delta_{ij}$ for all $m,n\in \mathbb{N},$
\begin{center}
$\delta_{mn}=
\begin{pmatrix}
  0 & 0 &  .  . . 0 & 0 & .  .  . \\
  0 & 0 &  .  . . 0 & 0 & .  .  . \\
  . & \\
  . & \\
  . & \\
  0 & 0 &  .  . . 1 & 0 & .  .  . \\
  0 & 0 &  .  . . 0 & 0 & .  .  . \\
\end{pmatrix}$
\end{center}
with 1 in the $(m,n)^{th}$ position and zero otherwise. 
\\\\\indent A double sequence $x=(x_{mn})$ is called double gai sequence if $\left((m+n)!\left|x_{mn}\right|\right)^{\frac{1}{m+n}}\rightarrow 0$ as $m,n\rightarrow \infty.$ The double gai sequences will be denoted by $\chi^{2}$.
\section{Definitions and Preliminaries}
A double sequence $x=\left(x_{mn}\right)$ has limit $0$ (denoted by $P-limx=0$) \\(i.e) $\left(\left(m+n\right)!\left|x_{mn}\right|\right)^{1/m+n} \rightarrow 0$ as $m,n \rightarrow \infty.$ We shall write more briefly as $P-convergent\hspace{0.2cm}to\hspace{0.2cm} 0.$
\\\indent An Orlicz function 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. An Orlicz function $M$ is said to satisfy $\Delta_{2}-$ condition for all values $u,$ if there exists $K>0$ such that $M\left(2u\right)\leq K M\left(u\right),u\geq 0.$
\subsection{Lemma} Let $M$ be an Orlicz function which satisfies $\Delta_{2}-$ condition and let $0<\delta<1.$ Then for each $t\geq \delta,$ we have $M\left(t\right)<K\delta^{-1}M\left(2\right)$ for some constant $K>0.$ 
\\\indent A double sequence $M=\left(M_{mn}\right)$ of Orlicz function is called a Musielak-Orlicz function [see [12]]. A double sequence $g=\left(g_{mn}\right)$ defined by 
\begin{center}
$g_{mn}\left(v\right)=sup\left\{\left|v\right|u-\left(M_{mn}\right)\left(u\right):u\geq 0\right\},m,n=1,2,\cdots$  
\end{center}
is called the complementary function of a sequence of Musielak-Orlicz $M$. For a given sequence of Musielak-Orlicz function $M,$ the Musielak-Orlicz sequence space $t_{M}$ is defined as follows                                 
\begin{center}
$t_{M}=\left\{x\in w^{2}:I_{M}\left(\left|x_{mn}\right|\right)^{1/m+n}\rightarrow 0\hspace{0.05cm}as\hspace{0.05cm}m,n,k\rightarrow \infty\right\},$  
\end{center} 
where $I_{M}$ is a convex modular defined by 
\begin{center}
$I_{M}\left(x\right)=\sum_{m=1}^{\infty}\sum_{n=1}^{\infty}M_{mn}\left(\left|x_{mn}\right|\right)^{1/m+n}.$
\end{center}
\subsection{Definition} A double sequence $x=\left(x_{mn}\right)$ of real numbers is called almost $P-$ convergent to a limit 0 if 
\begin{center}
$P-lim_{p,q\rightarrow \infty}sup_{r,s\geq 0}\frac{1}{pq}\sum_{m=r}^{r+p-1}\sum_{n=s}^{s+q-1}\left(\left(m+n\right)!\left|x_{mn}\right|\right)^{1/m+n} \rightarrow 0.$
\end{center}
that is, the average value of $\left(x_{mn}\right)$ taken over any rectangle \\$\left\{\left(m,n\right):r\leq m\leq r+p-1,s\leq n\leq s+q-1\right\}$ tends to 0 as both $p$ and $q$ to $\infty,$ and this $P-$ convergence is uniform in $r$ and $s.$ Let denote the set of sequences with this property as $\left[\widehat{\chi^{2}}\right].$
\subsection{Definition} Let $\lambda=\left(\lambda_{m}\right)$ and $\mu=\left(\mu_{n}\right)$ be two non-decreasing sequences of positive real numbers such that each tending to $\infty$ and \\$\lambda_{m+1}\leq \lambda_{m}+1,\lambda_{1}=1,\hspace{0.2cm} \mu_{n+1}\leq \mu_{n}+1,\mu_{1}=1.$
\\Let $I_{m}=\left[m-\lambda_{m}+1,m\right]$ and $I_{n}=\left[n-\mu_{n}+1,n\right].$
\\For any set $K\subseteq \mathbb{N}\times \mathbb{N},$ the number 
\\$\delta_{\lambda, \mu}\left(K\right)=lim_{m,n\rightarrow \infty}\frac{1}{\lambda_{m}\mu_{n}}\left|\left\{\left(i,j\right):i\in I_{m},j\in I_{n},\left(i,j\right)\in K\right\}\right|,$ is called the $\left(\lambda,\mu\right)-$ density of the set $K$ provided the limit exists. [See [31]].
\subsection{Definition} A double sequence $x=\left(x_{mn}\right)$ of numbers is said to be $\left(\lambda,\mu\right)-$ statistical convergent to a number $\xi$ provided that for each $\epsilon>0,$ 
\\$lim_{m,n\rightarrow \infty}\frac{1}{\lambda_{m}\mu_{n}}\left|\left\{\left(i,j\right):i\in I_{m},j\in I_{n},\left|x_{mn}-\xi\right|\geq \epsilon \right\}\right|=0,$
\\that is, the set $K\left(\epsilon\right)=\frac{1}{\lambda_{m}\mu_{n}}\left|\left\{\left(i,j\right):i\in I_{m},j\in I_{n},\left|x_{mn}-\xi\right|\geq \epsilon \right\}\right|$ has $\left(\lambda, \mu\right)-$ density zero. In this case the number $\xi$ is called the $\left(\lambda, \mu\right)-$ statistical limit of the sequence $x=\left(x_{mn}\right)$ and we write $St_{\left(\lambda, \mu\right)}lim_{m,n\rightarrow \infty}=\xi.$
\subsection{Definition} Let $M$ be an Orlicz function and $P=\left(p_{mn}\right)$ be any factorable double sequence of strictly positive real numbers, we define the following sequence space:
$\chi^{2}_{M}\left[AC_{\lambda_{m}\mu_{n}},P\right]=\\\left\{P-lim_{m,n}\frac{1}{\lambda_{m}\mu_{n}}\sum_{m\in I_{r,s}}\sum_{n\in I_{r,s}}\left[M\left(\left(m+n\right)!\left|x_{m+r,n+s}\right|\right)^{1/m+n}\right]^{p_{mn}}=0,\right\},$ uniformly in $r$ and $s$. 
\\\indent We shall denote $\chi^{2}_{M}\left[AC_{\lambda_{m}\mu_{n}},P\right]$ as $\chi^{2}\left[AC_{\lambda_{m}\mu_{n}}\right]$
respectively when $p_{mn}=1$ for all $m$ and $n.$ If $x$ is in $\chi^{2}\left[AC_{\lambda_{m}\mu_{n}},P\right],$
we shall say that $x$ is almost $\left(\lambda_{m}\mu_{n}\right)$ in $\chi^{2}$ strongly $P-$convergent with respect to the Orlicz function $M$. Also note if $M\left(x\right)=x,p_{mn}=1$ for all $m,n$ and $k$ then $\chi^{2}_{M}\left[AC_{\lambda_{m}\mu_{n}},P\right]=\chi^{2}\left[AC_{\lambda_{m}\mu_{n}},P\right],$
which are defined as follows:  
$\chi^{2}\left[AC_{\lambda_{m}\mu_{n}},P\right]
=\\\left\{P-lim_{m,n}\frac{1}{\lambda_{m} \mu_{n}}\sum_{m\in I_{r,s}}\sum_{n\in I_{r,s}}\left[M\left(\left(m+n\right)!\left|x_{m+r,n+s}\right|\right)^{1/m+n}\right]=0,\right\},$ uniformly in $r$ and $s$. 
\\Again note if $p_{mn}=1$ for all $m$ and $n$ then $\chi^{2}_{M}\left[AC_{\lambda_{m}\mu_{n}},P\right]
=\chi^{2}_{M}\left[AC_{\lambda_{m}\mu_{n}}\right].$ We define 
$\chi^{2}_{M}\left[AC_{\lambda_{m}\mu_{n}},P\right]
=\\\left\{P-lim_{m,n}\frac{1}{\lambda_{m}\mu_{n}}\sum_{m\in I_{r,s}}\sum_{n\in I_{r,s}}\left[M\left(\left(m+n\right)!\left|x_{m+r,n+s}\right|\right)^{1/m+n}\right]^{p_{mn}}=0,\right\},$ uniformly in $r$ and $s$.  
\subsection{Definition} Let $M$ be an Orlicz function and $P=\left(p_{mn}\right)$ be any factorable double sequence of strictly positive real numbers, we define the following sequence space: 
$\chi^{2}_{M}\left[P\right]
=\\\left\{P-lim_{p,q\rightarrow \infty}\frac{1}{pq}\sum_{m=1}^{p}\sum_{n=1}^{q}\left[M\left(\left(m+n\right)!\left|x_{m+r,n+s}\right|\right)^{1/m+n}\right]^{p_{mn}}= 0\right\},$ uniformly in $r$ and $s$.
\\\indent If we take $M\left(x\right)=x,p_{mn}=1$ for all $m$ and $n$ then $\chi^{2}_{M}\left[P\right]
=\chi^{2}.$
\subsection{Definition} The double number sequence $x$ is $\widehat{S_{\lambda_{m}\mu_{n}}}-P-$ convergent to 0 then
\\$P-lim_{m,n}\frac{1}{\lambda_{m}\mu_{n}}max_{r,s}\left|\left\{\left(m,n\right)\in I_{r,s}:M\left(\left(m+n\right)!\left|x_{m+r,n+s}-0\right|\right)^{1/m+n}\right\}\right|=0.$
\\In this case we write $\widehat{S_{\lambda_{m}\mu_{n}}}-lim\left(M\left(m+n\right)!\left|x_{m+r,n+s}-0\right|\right)^{1/m+n}=0.$
\section{The Backward operator of convergence of  double almost $\left(\lambda_{m}\mu_{n}\right)$ in $\chi^{2}$ Riesz space} 
Let $n\in \mathbb{N}$ and $X$ be a real vector space of dimension $m,$ where $n\leq m$ $\left(m\hspace{0.1cm}be\hspace{0.1cm}infinite\right),$ $\tau$ a triangle, and $F:\left(X\times X\right)\times \left(X\times X\right)\rightarrow D^{+}.$ Then $F$ is called a probabilistic Riesz space. A real valued function $ F\left(d_p(x_1,\dots,x_n),t\right) = F\left( \|(d_1(x_1,0), \dots, d_n(x_n,0))\|_p,t\right)$ on $X$ satisfying the following four conditions:
\\(i) $F\left(\|(d_1(x_1,0), \dots, d_n(x_n,0))\|_p,t\right)=0$ if and and only if  $F\left(d_1(x_1,0), \dots, d_n(x_n,0),t\right)$ are linearly dependent,
\\(ii) $F\left(\|(d_1(x_1,0), \dots, d_n(x_n,0))\|_p,t\right)$ is invariant under permutation,
\\(iii) $F\left(\|(\alpha d_1(x_1,0), \dots, \alpha d_n(x_n,0))\|_p,t\right)=F\left(\left|\alpha \right|\|(d_1(x_1,0), \dots, d_n(x_n,0))\|_p,t\right) ,\alpha\in \mathbb{R}$  
\\(iv) $F\left(d_{p} \left( (x_{1}, y_{1}) , (x_{2}, y_{2})\cdots (x_{n}, y_{n}),t \right) =F \left( d_{X} (x_{1}, x_{2},\cdots x_{n})^{p},t\right) +F\left( d_{Y} (y_{1}, y_{2}, \cdots y_{n})^{p} \right)^{1/p},t\right) $ \\ for $1 \leq p < \infty;$ (or)
\\(v) $F\left(d\left( (x_{1}, y_{1}) , (x_{2}, y_{2}),\cdots (x_{n}, y_{n}) \right),t\right) := \sup F\left(\left\{ d_{X} (x_{1}, x_{2}, \cdots x_{n}), d_{Y} (y_{1}, y_{2},\cdots y_{n}) \right\},t\right),$\\ for $\left(x_{1}, x_{2},\cdots x_{n} \in X, y_{1}, y_{2},\cdots y_{n} \in Y,F,*\right)$ is called the $p$ product metric of the Cartesian product of $n$ metric spaces is the $p$ norm of the $n$-vector of the norms of the $n$ subspaces. 
\\\indent A trivial example of $p$ product metric of $n$ metric space is the $p$ norm space is $X=\mathbb{R}$ equipped with the following Euclidean metric in the product space is the $p$ norm:
\begin{center}
$F\left(\|(d_1(x_1,0), \dots, d_n(x_n,0))\|_E,t\right) = sup\hspace{0.1cm}F\left(| det(d_{mn}\left(x_{mn},0\right))|,t\right)=sup
\begin{pmatrix}\begin{vmatrix}
  d_{11}\left(x_{11},0\right) & d_{12}\left(x_{12},0\right) &  .  . .  & d_{1n}\left(x_{1n},0\right) \\
  d_{21}\left(x_{21},0\right) & d_{22}\left(x_{22},0\right) &  .  . .  & d_{2n}\left(x_{1n},0\right)\\
 
  . & \\
  . & \\
  . & \\
  d_{n1}\left(x_{n1},0\right) & d_{n2}\left(x_{n2},0\right) &  .  . .  & d_{nn}\left(x_{nn},0\right)
  
\end{vmatrix}\end{pmatrix}$
\end{center}
where $x_{i}=\left(x_{{i1}},\cdots x_{in}\right)\in \mathbb{R}^{n}$ for each $i=1,2,\cdots n.$
\\If every Cauchy sequence in $X$ converges to some $L\in X,$ then $X$ is said to be complete with respect to the $p-$ metric. Any complete $p-$ metric space is said to be $p-$ Banach metric space.
\subsection{Definition} Let $L$ be a real vector space and let $\leq$ be a partial order on this space. $L$
is said to be an ordered vector space if it satisfies the following properties :
\\(i) If $x,y\in L$ and $y\leq x,$ then $y+z\leq x+z$ for each $z\in L.$
\\ (ii) If $x,y\in L$ and $y\leq x,$ then $\lambda y\leq \lambda x$ for each $\lambda\geq 0.$
\\If in addition $L$ is a lattice with respect to the partial ordering, then $L$ is said to be Riesz space. 
\\A subset $S$ of a Riesz space $X$ is said to be solid if $y\in S$ and $\left|x\right|\leq \left|y\right|$ implies $x\in S.$
\\A linear topology $\tau$ on a Riesz space $X$ is said to be locally solid if $\tau$ has a base at zero consisting of solid sets. 
\subsection{Definition} Let $\chi^{2\tau}_{M}\left[AC_{\lambda_{m}\mu_{n}},P,\left\|\left(d\left(x_{1},0\right),d\left(x_{2},0\right),\cdots, d\left(x_{n-1},0\right)\right)\right\|_{p}\right]$
be a Riesz space of Musielak-Orlicz function. A double sequence $\left(x_{mn}\right)$ of points in $\chi^{2}$ is said to be $S\left(\tau\right)-$ convergent in $\left(X,F,*\right)$ if for each $t>0,\hspace{0.1cm}\theta\in \left(0,1\right)$ and for non zero $z\in X$ such that 
\begin{center}
$\delta\left(\left\{m,n\in \mathbb{N}:F\left(M_{mn}\left(\left(\left(m+n\right)!\left|x_{mn}\right|\right)^{1/m+n},z;t \right)\leq 1- \theta \right\}\right)\right)=0$   
\end{center}
that is , \\$\left(P-lim_{m,n}\frac{1}{\lambda_{m}\mu_{n}}\left\{\sum_{m\in I_{r,s}}\sum_{n\in I_{r,s}}F\left(\left[M\left(\left(m+n\right)!\left|x_{m+r,n+s}\right|-0\right)^{1/m+n}\right]^{p_{mn}},z;t\right)\leq 1-\theta\right\}\right)=0.$
\\In this case we write \\$S\left(\tau\right)-\left(P-lim_{m,n}\frac{1}{\lambda_{m}\mu_{n}}\left\{\sum_{m\in I_{r,s}}\sum_{n\in I_{r,s}}F\left(\left[M\left(\left(m+n\right)!\left|x_{m,n}\right|-0\right)^{1/m+n}\right]^{p_{mn}},z;t\right)\right\}\right)=1.$
\subsection{Definition} Let $\chi^{2\tau}_{M}\left[AC_{\lambda_{m}\mu_{n}},P,\left\|\left(d\left(x_{1},0\right),d\left(x_{2},0\right),\cdots, d\left(x_{n-1},0\right)\right)\right\|_{p}\right]$
be a Riesz space of Musielak-Orlicz function. A double sequence $\left(x_{mn}\right)$ of points in $\chi^{2}$ is said to be $\nabla-$ convergent in $\left(X,F,*\right)$ if for each $t>0,\hspace{0.1cm}\beta\in \left(0,1\right)$ there exists an positive integer $n_{0}$ such that 
\begin{center}
$F\left(M_{mn}\left(\left(\left(m+n\right)!\left|x_{mn}\right|\right)^{1/m+n},z;t \right)\right\}>1-\beta.$   
\end{center}
whenever $m,n\geq n_{0}$ and for non zero $z\in X.$ 
\subsection{Definition} Let $\chi^{2\tau}_{M}\left[AC_{\lambda_{m}\mu_{n}},P,\left\|\left(d\left(x_{1},0\right),d\left(x_{2},0\right),\cdots, d\left(x_{n-1},0\right)\right)\right\|_{p}\right]$
be a Riesz space of Musielak-Orlicz function. A double sequence $\left(x_{mn}\right)$ of points in $\chi^{2}$ is said to be $\nabla-$ Cauchy in $\left(X,F,*\right)$ if for each $t>0,\hspace{0.1cm}\beta\in \left(0,1\right)$ there exists an positive integer $n_{0}=n_{0}\left(\epsilon\right)$ such that 
\begin{center}
$F\left(M_{mn}\left(\left(\left(m+n\right)!\left|x_{mn}-x_{rs}\right|\right)^{1/m+n},z;t \right)\right\}<1-\theta.$   
\end{center}
whenever $m,n,r,s\geq n_{0}$ and for non zero $z\in X.$ 
\subsection{Definition} Let $\chi^{2\tau}_{M}\left[AC_{\lambda_{m}\mu_{n}},P,\left\|\left(d\left(x_{1},0\right),d\left(x_{2},0\right),\cdots, d\left(x_{n-1},0\right)\right)\right\|_{p}\right]$
be a Riesz space of Musielak-Orlicz function. A double sequence $\left(x_{mn}\right)$ of points in $\chi^{2}$ is said to be $S\left(\tau\right)-$ convergent in $\left(X,F,*\right)$ if for each $t>0,\hspace{0.1cm}\beta\in \left(0,1\right)$ and for non zero $z\in X$ such that 
\begin{center}
$\delta_{\nabla}\left(\left\{m,n\in \mathbb{N}:F\left(M_{mn}\left(\left(\left(m+n\right)!\left|x_{mn}\right|\right)^{1/m+n},z;t \right)\leq 1- \beta \right\}\right)\right)=0$   
\end{center}
that is , \\$\left(P-lim_{m,n}\frac{1}{\lambda_{m}\mu_{n}}\left\{\sum_{m\in I_{r,s}}\sum_{n\in I_{r,s}}F\left(\left[M\left(\left(m+n\right)!\left|x_{m+r,n+s}\right|-0\right)^{1/m+n}\right]^{p_{mn}},z;t\right)\leq 1-\beta\right\}\right)=0.$
\\In this case we write \\$S\left(\tau\right)_{\nabla}-\left(P-lim_{m,n}\frac{1}{\lambda_{m}\mu_{n}}\left\{\sum_{m\in I_{r,s}}\sum_{n\in I_{r,s}}F\left(\left[M\left(\left(m+n\right)!\left|x_{m,n}\right|-0\right)^{1/m+n}\right]^{p_{mn}},z;t\right)\right\}\right)=1.$
\subsection{Definition} Let $\chi^{2\tau}_{M}\left[AC_{\lambda_{m}\mu_{n}},P,\left\|\left(d\left(x_{1},0\right),d\left(x_{2},0\right),\cdots, d\left(x_{n-1},0\right)\right)\right\|_{p}\right]$
be a Riesz space of Musielak-Orlicz function. A double sequence $\left(x_{mn}\right)$ of points in $\chi^{2}$ is said to be $\nabla-$ Cauchy in $\left(X,F,*\right)$ if for each $t>0,\hspace{0.1cm}\beta\in \left(0,1\right)$ there exists an positive integer $n_{0}=n_{0}\left(\epsilon\right)$ such that 
\begin{center}
$\delta_{\nabla}\left(\left\{m,n\in \mathbb{N}:F\left(M_{mn}\left(\left(\left(m+n\right)!\left|x_{mn}-x_{rs}\right|\right)^{1/m+n},z;t \right)\leq 1- \beta \right\}\right)\right)=0$   
\end{center}
or equivalently, 
\begin{center}
$\delta_{\nabla}\left(\left\{m,n\in \mathbb{N}:F\left(M_{mn}\left(\left(\left(m+n\right)!\left|x_{mn}-x_{rs}\right|\right)^{1/m+n},z;t \right)> 1- \beta \right\}\right)\right)=1$   
\end{center} 
\section{Main Results}
\subsection{Proposition}Let $\chi^{2\tau}_{M}\left[AC_{\lambda_{m}\mu_{n}},P,\left\|\left(d\left(x_{1},0\right),d\left(x_{2},0\right),\cdots, d\left(x_{n-1},0\right)\right)\right\|_{p}\right]$
be a Riesz space of Musielak-Orlicz function. A double sequence $\left(x_{mn}\right)$ of $\chi^{2}$ in  $\left(X,F,*\right)$ if for each $t>0,\hspace{0.1cm}\beta\in \left(0,1\right)$ and for non zero $z\in X,$ then the following statements are equivalent  
\\(i)$\delta_{\nabla}\left(\left\{m,n\in \mathbb{N}:F\left(M_{mn}\left(\left(\left(m+n\right)!\left|x_{mn}-x_{rs}\right|\right)^{1/m+n},z;t \right)\leq 1- \beta \right\}\right)\right)=0$
\\(ii)$\delta_{\nabla}\left(\left\{m,n\in \mathbb{N}:F\left(M_{mn}\left(\left(\left(m+n\right)!\left|x_{mn}-x_{rs}\right|\right)^{1/m+n},z;t \right)> 1- \beta \right\}\right)\right)=1$ 
\\(iii)$S\left(\tau\right)_{\nabla}-\left(P-lim_{m,n}\frac{1}{\lambda_{m}\mu_{n}}\left\{\sum_{m\in I_{r,s}}\sum_{n\in I_{r,s}}F\left(\left[M\left(\left(m+n\right)!\left|x_{m,n}\right|-0\right)^{1/m+n}\right]^{p_{mn}},z;t\right)\right\}\right)=1.$
\subsection{Theorem} Let $\chi^{2\tau}_{M}\left[AC_{\lambda_{m}\mu_{n}},P,\left\|\left(d\left(x_{1},0\right),d\left(x_{2},0\right),\cdots, d\left(x_{n-1},0\right)\right)\right\|_{p}\right]\in S\left(\tau\right)_{\nabla}$ and $c\in \mathbb{R}$ 
be a almost $\left(\lambda_{m}\mu_{n}\right)$ Riesz space of Musielak-Orlicz function. A double sequence $\left(x_{mn}\right)$ in  $\left(X,F,*\right)$ then 
\\(i)$S\left(\tau\right)_{\nabla}-\left(P-clim_{m,n}\frac{1}{\lambda_{m}\mu_{n}}\left\{\sum_{m\in I_{r,s}}\sum_{n\in I_{r,s}}F\left(\left[M\left(\left(m+n\right)!\left|x_{m,n}\right|-0\right)^{1/m+n}\right]^{p_{mn}},z;t\right)\right\}\right)=\\c\hspace{0.1cm}S\left(\tau\right)_{\nabla}-\left(P-lim_{m,n}\frac{1}{\lambda_{m}\mu_{n}}\left\{\sum_{m\in I_{r,s}}\sum_{n\in I_{r,s}}F\left(\left[M\left(\left(m+n\right)!\left|x_{m,n}\right|-0\right)^{1/m+n}\right]^{p_{mn}},z;t\right)\right\}\right)$
\\(ii)\\$S\left(\tau\right)_{\nabla}-\left(P-lim_{m,n}\frac{1}{\lambda_{m}\mu_{n}}\left\{\sum_{m\in I_{r,s}}\sum_{n\in I_{r,s}}F\left(\left[M\left(\left(m+n\right)!\left|x_{m,n}+y_{mn}\right|-0\right)^{1/m+n}\right]^{p_{mn}},z;t\right)\right\}\right)=\\S\left(\tau\right)_{\nabla}-\left(P-lim_{m,n}\frac{1}{\lambda_{m}\mu_{n}}\left\{\sum_{m\in I_{r,s}}\sum_{n\in I_{r,s}}F\left(\left[M\left(\left(m+n\right)!\left|x_{m,n}\right|-0\right)^{1/m+n}\right]^{p_{mn}},z;t\right)\right\}\right)+\\S\left(\tau\right)_{\nabla}-\left(P-lim_{m,n}\frac{1}{\lambda_{m}\mu_{n}}\left\{\sum_{m\in I_{r,s}}\sum_{n\in I_{r,s}}F\left(\left[M\left(\left(m+n\right)!\left|y_{m,n}\right|-0\right)^{1/m+n}\right]^{p_{mn}},z;t\right)\right\}\right)$
\\\textbf{Proof}: The proof of this theorem is straightforward, and thus will be omitted.
\subsection{Theorem} Let $\chi^{2\tau}_{M}\left[AC_{\lambda_{m}\mu_{n}},P,\left\|\left(d\left(x_{1},0\right),d\left(x_{2},0\right),\cdots, d\left(x_{n-1},0\right)\right)\right\|_{p}\right]$  
be a almost $\left(\lambda_{m}\mu_{n}\right)$ Riesz space of Musielak-Orlicz function. A double sequence $\left(x_{mn}\right)$ analytic  in  $\left(X,F,*\right)$ then
\\(a)$\chi^{2\tau}_{M}\left[AC_{\lambda_{m}\mu_{n}},P,\left\|\left(d\left(x_{1},0\right),d\left(x_{2},0\right),\cdots, d\left(x_{n-1},0\right)\right)\right\|_{p}\right]\rightarrow W\left(\tau\right)_{\nabla}$ implies 
\\$\chi^{2\tau}_{M}\left[AC_{\lambda_{m}\mu_{n}},P,\left\|\left(d\left(x_{1},0\right),d\left(x_{2},0\right),\cdots, d\left(x_{n-1},0\right)\right)\right\|_{p}\right]\rightarrow S\left(\tau\right)_{\nabla}.$
\\(b)$\Lambda^{2\tau}_{M}\left[AC_{\lambda_{m}\mu_{n}},P,\left\|\left(d\left(x_{1},0\right),d\left(x_{2},0\right),\cdots, d\left(x_{n-1},0\right)\right)\right\|_{p}\right]\rightarrow S\left(\tau\right)_{\nabla}$ imply
\\$\Lambda^{2\tau}_{M}\left[AC_{\lambda_{m}\mu_{n}},P,\left\|\left(d\left(x_{1},0\right),d\left(x_{2},0\right),\cdots, d\left(x_{n-1},0\right)\right)\right\|_{p}\right]\rightarrow W\left(\tau\right)_{\nabla}.$
\\(c)$S\left(\tau\right)_{\nabla}\bigcap \Lambda^{2\tau}_{M}\left[AC_{\lambda_{m}\mu_{n}},P,\left\|\left(d\left(x_{1},0\right),d\left(x_{2},0\right),\cdots, d\left(x_{n-1},0\right)\right)\right\|_{p}\right]=\\W\left(\tau\right)_{\nabla} \bigcap \Lambda^{2\tau}_{M}\left[AC_{\lambda_{m}\mu_{n}},P,\left\|\left(d\left(x_{1},0\right),d\left(x_{2},0\right),\cdots, d\left(x_{n-1},0\right)\right)\right\|_{p}\right].$
\\\textbf{Proof:} Let $\epsilon>0$ and $\chi^{2\tau}_{M}\left[AC_{\lambda_{m}\mu_{n}},P,\left\|\left(d\left(x_{1},0\right),d\left(x_{2},0\right),\cdots, d\left(x_{n-1},0\right)\right)\right\|_{p}\right]\rightarrow W\left(\tau\right)_{\nabla}$ for all $r,s\in \mathbb{N},$ we have \\$\left(lim_{m,n}\frac{1}{\lambda_{m}\mu_{n}}\left\{\sum_{m\in I_{r,s}}\sum_{n\in I_{r,s}}F\left(\left[M\left(\left(m+n\right)!\left|x_{m,n}\right|-0\right)^{1/m+n}\right]^{p_{mn}},z;t\right)\right\}\right)\geq \epsilon$
\\$\left\{\sum_{m\in I_{r,s}}\sum_{n\in I_{r,s}}F\left(\left[M\left(\left(m+n\right)!\left|x_{m,n}\right|-0\right)^{1/m+n}\right]^{p_{mn}},z;t\right)\right\}\geq \\\left|\left(lim_{m,n}\frac{1}{\lambda_{m}\mu_{n}}\left\{\sum_{m\in I_{r,s}}\sum_{n\in I_{r,s}}F\left(\left[M\left(\left(m+n\right)!\left|x_{m,n}\right|-0\right)^{1/m+n}\right]^{p_{mn}},z;t\right)\geq \epsilon\right\}\right)\right|\cdot min\left(\epsilon^{h},\epsilon^{H}\right).$
\\Hence $\chi^{2\tau}_{M}\left[AC_{\lambda_{m}\mu_{n}},P,\left\|\left(d\left(x_{1},0\right),d\left(x_{2},0\right),\cdots, d\left(x_{n-1},0\right)\right)\right\|_{p}\right]\rightarrow S\left(\tau\right)_{\nabla}.$ 
\\\textbf{Proof(b): } Suppose that $\chi^{2\tau}_{M}\left[AC_{\lambda_{m}\mu_{n}},P,\left\|\left(d\left(x_{1},0\right),d\left(x_{2},0\right),\cdots, d\left(x_{n-1},0\right)\right)\right\|_{p}\right]\in S\left(\tau\right)_{\nabla}\bigcap \Lambda^{2\tau}_{M}\left[AC_{\lambda_{m}\mu_{n}},P,\left\|\left(d\left(x_{1},0\right),d\left(x_{2},0\right),\cdots, d\left(x_{n-1},0\right)\right)\right\|_{p}\right].$ Since \\$\chi^{2\tau}_{M}\left[AC_{\lambda_{m}\mu_{n}},P,\left\|\left(d\left(x_{1},0\right),d\left(x_{2},0\right),\cdots, d\left(x_{n-1},0\right)\right)\right\|_{p}\right]\in \\\Lambda^{2\tau}_{M}\left[AC_{\lambda_{m}\mu_{n}},P,\left\|\left(d\left(x_{1},0\right),d\left(x_{2},0\right),\cdots, d\left(x_{n-1},0\right)\right)\right\|_{p}\right],$ we write
\\$\left\{\sum_{m\in I_{r,s}}\sum_{n\in I_{r,s}}F\left(\left[M\left(\left(m+n\right)!\left|x_{m,n}\right|-0\right)^{1/m+n}\right]^{p_{mn}},z;t\right)\right\}\leq T,$ for all $r,s\in \mathbb{N},$ let
\\$G_{rs}=\left|\left(\frac{1}{\lambda_{m}\mu_{n}}\left\{\sum_{m\in I_{r,s}}\sum_{n\in I_{r,s}}F\left(\left[M\left(\left(m+n\right)!\left|x_{m,n}\right|-0\right)^{1/m+n}\right]^{p_{mn}},z;t\right)\geq \epsilon\right\}\right)\right|$
\\and
\\$H_{rs}=\left|\left(\frac{1}{\lambda_{m}\mu_{n}}\left\{\sum_{m\in I_{r,s}}\sum_{n\in I_{r,s}}F\left(\left[M\left(\left(m+n\right)!\left|x_{m,n}\right|-0\right)^{1/m+n}\right]^{p_{mn}},z;t\right)< \epsilon\right\}\right)\right|.$
\\Then we have 
\\$\left(\frac{1}{\lambda_{m}\mu_{n}}\left\{\sum_{m\in I_{r,s}}\sum_{n\in I_{r,s}}F\left(\left[M\left(\left(m+n\right)!\left|x_{m,n}\right|-0\right)^{1/m+n}\right]^{p_{mn}},z;t\right)\right\}\right)=\\\left(\frac{1}{\lambda_{m}\mu_{n}}\left\{\sum_{m\in G_{r,s}}\sum_{n\in G_{r,s}}F\left(\left[M\left(\left(m+n\right)!\left|x_{m,n}\right|-0\right)^{1/m+n}\right]^{p_{mn}},z;t\right) \right\}\right)+\\\left(\frac{1}{\lambda_{m}\mu_{n}}\left\{\sum_{m\in H_{r,s}}\sum_{n\in H_{r,s}}F\left(\left[M\left(\left(m+n\right)!\left|x_{m,n}\right|-0\right)^{1/m+n}\right]^{p_{mn}},z;t\right) \right\}\right)\leq \\max\left(T^{h},T^{H}\right)G_{rs}+max\left(\epsilon^{h},\epsilon^{H}\right).$ Taking the limit as $\epsilon\rightarrow 0$ and $r,s\rightarrow \infty,$ it follows that 
$\chi^{2\tau}_{M}\left[AC_{\lambda_{m}\mu_{n}},P,\left\|\left(d\left(x_{1},0\right),d\left(x_{2},0\right),\cdots, d\left(x_{n-1},0\right)\right)\right\|_{p}\right]\in W\left(\tau\right)_{\nabla}.$ 
\\(c) Follows from (a) and (b).
\subsection{Theorem} If $liminf_{rs}\left(\frac{\lambda_{r}\mu_{s}}{rs}\right)>0,$ then $S\left(\tau\right)\subset S\left(\tau\right)_{\nabla}$
\\\textbf{Proof:} Let $\chi^{2\tau}_{M}\left[AC_{\lambda_{m}\mu_{n}},P,\left\|\left(d\left(x_{1},0\right),d\left(x_{2},0\right),\cdots, d\left(x_{n-1},0\right)\right)\right\|_{p}\right]\in S\left(\tau\right).$ For given $\epsilon>0,$ we get 
\\$\left|\left(\frac{1}{\lambda_{m}\mu_{n}}\left\{\sum_{m\in I_{r,s}}\sum_{n\in I_{r,s}}\left(\left[M\left(\left(m+n\right)!\left|x_{m,n}\right|-0\right)^{1/m+n}\right]^{p_{mn}}\right)\geq \epsilon\right\}\right)\right|\supset G_{rs}$ where $G_{rs}$ is in the theorem of 4.3.(b). Thus,\\$\left|\left(\frac{1}{\lambda_{m}\mu_{n}}\left\{\sum_{m\in I_{r,s}}\sum_{n\in I_{r,s}}F\left(\left[M\left(\left(m+n\right)!\left|x_{m,n}\right|-0\right)^{1/m+n}\right]^{p_{mn}},z;t\right)\geq \epsilon\right\}\right)\right|\geq G_{rs}=\frac{\lambda_{r}\mu_{s}}{rs}.$ Taking limit as $r,s\rightarrow \infty$ and using  $liminf_{rs}\left(\frac{\lambda_{r}\mu_{s}}{rs}\right)>0,$ we get
\\$\chi^{2\tau}_{M}\left[AC_{\lambda_{m}\mu_{n}},P,\left\|\left(d\left(x_{1},0\right),d\left(x_{2},0\right),\cdots, d\left(x_{n-1},0\right)\right)\right\|_{p}\right]\in S\left(\tau\right)_{\nabla}.$
\subsection{Theorem} Let $0<u_{mn}\leq v_{mn}$ and $\left(u_{mn}v_{mn}^{-1}\right)$ be double analytic. Then $W\left(\tau,v\right)_{\nabla}\subset w\left(\tau,u\right)_{\nabla}$
\\\textbf{Proof: }Let $\chi^{2\tau}_{M}\left[AC_{\lambda_{m}\mu_{n}},P,\left\|\left(d\left(x_{1},0\right),d\left(x_{2},0\right),\cdots, d\left(x_{n-1},0\right)\right)\right\|_{p}\right]\in W\left(\tau,v\right)_{\nabla}.$ Let 
$W\left(\tau\right)_{\nabla}=\left(\frac{1}{\lambda_{m}\mu_{n}}\left\{\sum_{m\in I_{r,s}}\sum_{n\in I_{r,s}}F\left(\left[M\left(\left(m+n\right)!\left|x_{m,n}\right|-0\right)^{1/m+n}\right]^{p_{mn}},z;t\right) \right\}\right)$ for all $r,s\in \mathbb{N}$ and $\lambda_{m}\mu_{n}=u_{mn}v_{mn}^{-1}$ for all $m,n\in \mathbb{N}.$ Then $0<\lambda_{m}\mu_{n}\leq 1$ for all $m,n\in \mathbb{N}.$ Let $b$ be a constant such that $0<b\leq \lambda_{m}\mu_{n}\leq 1$ for all $m,n\in \mathbb{N}.$
\\\indent Define the double sequences $\left(k_{mn}\right)$ and $\left(\ell_{mn}\right)$ as follows:
\\For $W\left(\tau\right)_{\nabla}\geq 1,$ let $\left(k_{mn}\right)=\left(W\left(\tau\right)_{\nabla}\right)$ and $\ell_{mn}=0$ and for $W\left(\tau\right)_{\nabla}<1,$ let $k_{mn}=0$ and $\ell_{mn}=W\left(\tau\right)_{\nabla}.$ Then it is clear that for all $m,n\in \mathbb{N},$ we have $W\left(\tau\right)_{\nabla}=k_{mn}+\ell_{mn}$ and $W\left(\tau\right)_{\nabla}^{\lambda_{m}\mu_{n}}=k_{mn}^{\lambda_{m}\mu_{n}}+\ell_{mn}^{\lambda_{m}\mu_{n}}.$ Now it follows that $k_{mn}^{\lambda_{m}\mu_{n}}\leq k_{mn}\leq W\left(\tau\right)_{\nabla}$ and $\ell_{mn}^{\lambda_{m}\mu_{n}}\leq \ell_{mn}^{\lambda\mu}.$ Therefore
\\$\left(\frac{1}{\lambda_{m}\mu_{n}}\left\{\sum_{m\in I_{r,s}}\sum_{n\in I_{r,s}}F\left(\left[M\left(\left(m+n\right)!\left|W\left(\tau\right)_{\nabla}^{\lambda_{m}\mu_{n}}\right|-0\right)^{1/m+n}\right]^{p_{mn}},z;t\right) \right\}\right)=\\\left(\frac{1}{\lambda_{m}\mu_{n}}\left\{\sum_{m\in I_{r,s}}\sum_{n\in I_{r,s}}F\left(\left[M\left(\left(m+n\right)!\left|\left(k_{mn}+\ell_{mn}\right)^{\lambda_{m}\mu_{n}}\right|-0\right)^{1/m+n}\right]^{p_{mn}},z;t\right) \right\}\right)=\\\left(\frac{1}{\lambda_{m}\mu_{n}}\left\{\sum_{m\in I_{r,s}}\sum_{n\in I_{r,s}}F\left(\left[M\left(\left(m+n\right)!\left|W\left(\tau\right)_{\nabla}^{\lambda_{m}\mu_{n}}\right|-0\right)^{1/m+n}\right]^{p_{mn}},z;t\right) \right\}\right)+\\\left(\frac{1}{\lambda_{m}\mu_{n}}\left\{\sum_{m\in I_{r,s}}\sum_{n\in I_{r,s}}F\left(\left[M\left(\left(m+n\right)!\left|\left(\ell_{mn}\right)^{\lambda_{m}\mu_{n}}\right|-0\right)^{1/m+n}\right]^{p_{mn}},z;t\right) \right\}\right).$\\Now for each $r,s,$
\\$\left(\frac{1}{\lambda_{m}\mu_{n}}\left\{\sum_{m\in I_{r,s}}\sum_{n\in I_{r,s}}F\left(\left[M\left(\left(m+n\right)!\left|\left(\ell_{mn}\right)^{\lambda \mu}\right|-0\right)^{1/m+n}\right]^{p_{mn}},z;t\right) \right\}\right)=\\\left(\frac{1}{\lambda_{m}\mu_{n}}\left\{\sum_{m\in I_{r,s}}\sum_{n\in I_{r,s}}F\left(\left[M\left(\left(m+n\right)!\left|\left(\left(\ell_{mn}\right)^{\lambda \mu}\left(\frac{1}{\lambda_{m}\mu_{n}}\right)^{1-\lambda\mu}\right)\right|-0\right)^{1/m+n}\right]^{p_{mn}},z;t\right) \right\}\right)$ 
\\$\left(\frac{1}{\lambda_{m}\mu_{n}}\left\{\sum_{m\in I_{r,s}}\sum_{n\in I_{r,s}}F\left(\left[M\left(\left(m+n\right)!\left|\left(\left(\left(\ell_{mn}\right)^{\lambda \mu}\right)^{\lambda\mu}\right)^{1/\lambda \mu}\right|-0\right)^{1/m+n}\right]^{p_{mn}},z;t\right) \right\}\right)^{\lambda\mu}$ 
\subsection{Theorem} $\Lambda^{2\tau}_{M}\left[AC_{\lambda_{m}\mu_{n}},P,\left\|\left(d\left(x_{1},0\right),d\left(x_{2},0\right),\cdots, d\left(x_{n-1},0\right)\right)\right\|_{p}\right]=W\left(\tau,\Lambda^{2}\right)_{\nabla},$ where
\\$W\left(\tau,\Lambda^{2}\right)_{\nabla}=sup\left(\frac{1}{\lambda_{m}\mu_{n}}\left\{\sum_{m\in I_{r,s}}\sum_{n\in I_{r,s}}F\left(\left[M\left(\left(m+n\right)!\left|x_{m,n}\right|-0\right)^{1/m+n}\right]^{p_{mn}},z;t\right)< \infty \right\}\right)$
\\\textbf{Proof: } Let $x=\left(x_{mn}\right)\in W\left(\tau,\Lambda^{2}\right)_{\nabla}.$ Then there exists a constant $T_{1}>0$ such that
\\$\left(\frac{1}{\lambda_{m}\mu_{n}}\left\{\sum_{m\in I_{r,s}}\sum_{n\in I_{r,s}}F\left(\left[M\left(\left(m+n\right)!\left|x_{m,n}\right|-0\right)^{1/m+n}\right]^{p_{mn}},z;t\right) \right\}\right)\leq \\sup\left(\frac{1}{\lambda_{m}\mu_{n}}\left\{\sum_{m\in I_{r,s}}\sum_{n\in I_{r,s}}F\left(\left[M\left(\left(m+n\right)!\left|x_{m,n}\right|-0\right)^{1/m+n}\right]^{p_{mn}},z;t\right)\right\}\right)\leq T_{1}$ for all $r,s\in \mathbb{N}.$ Therefore we have \\$x=\left(x_{mn}\right)\in \Lambda^{2\tau}_{M}\left[AC_{\lambda_{m}\mu_{n}},P,\left\|\left(d\left(x_{1},0\right),d\left(x_{2},0\right),\cdots, d\left(x_{n-1},0\right)\right)\right\|_{p}\right].$ Conversely, let $x=\left(x_{mn}\right)\in \Lambda^{2\tau}_{M}\left[AC_{\lambda_{m}\mu_{n}},P,\left\|\left(d\left(x_{1},0\right),d\left(x_{2},0\right),\cdots, d\left(x_{n-1},0\right)\right)\right\|_{p}\right].$ Then there exists a constant $T_{2}>0$ such that 
\\$\left(\frac{1}{\lambda_{m}\mu_{n}}\left\{\sum_{m\in I_{r,s}}\sum_{n\in I_{r,s}}F\left(\left[M\left(\left(m+n\right)!\left|x_{m,n}\right|-0\right)^{1/m+n}\right]^{p_{mn}},z;t\right)\right\}\right)\leq T_{2}$  for all $m,n$ and $r,s.$ So,
\\$\left(\frac{1}{\lambda_{m}\mu_{n}}\left\{\sum_{m\in I_{r,s}}\sum_{n\in I_{r,s}}F\left(\left[M\left(\left(m+n\right)!\left|x_{m,n}\right|-0\right)^{1/m+n}\right]^{p_{mn}},z;t\right)\right\}\right)\leq \\T_{2}\hspace{0.1cm}\frac{1}{\lambda_{m}\mu_{n}}\sum_{m\in I_{r,s}}\sum_{n\in I_{r,s}}\hspace{0.1cm} 1\leq T_{2},$ for all $m,n$ and $r,s.$ Thus $x=\left(x_{mn}\right)\in W\left(\tau,\Lambda^{2}\right)_{\nabla}$ 
\subsection{Theorem}$\chi^{2\tau}_{M}\left[AC_{\lambda_{m}\mu_{n}},P,\left\|\left(d\left(x_{1},0\right),d\left(x_{2},0\right),\cdots, d\left(x_{n-1},0\right)\right)\right\|_{p}\right]$
be a almost $\left(\lambda_{m}\mu_{n}\right)$ Riesz space of Musielak-Orlicz function. A double sequence $\left(x_{mn}\right)$ in  $\left(X,F,*\right)$ is $\nabla-$ statistically convergent if and only if it is $\nabla-$statistically Cauchy
\\\textbf{Proof: } Let $x=\left(x_{mn}\right)$ be a $\nabla-$statistically convergent sequence in \\$\chi^{2\tau}_{M}\left[AC_{\lambda_{m}\mu_{n}},P,\left\|\left(d\left(x_{1},0\right),d\left(x_{2},0\right),\cdots, d\left(x_{n-1},0\right)\right)\right\|_{p}\right].$ Let $\epsilon>0$ be given. Choose $s>0$ such that
\begin{equation}
\left(1-s\right)* \left(1-s\right)>1-\epsilon
\end{equation}
is satisfied. For $t>0$ and non-zero $z\in \chi^{2\tau}_{M}\left[AC_{\lambda_{m}\mu_{n}},P,\left\|\left(d\left(x_{1},0\right),d\left(x_{2},0\right),\cdots, d\left(x_{n-1},0\right)\right)\right\|_{p}\right]$ define 
\\$A\left(s,t\right)=\left(\frac{1}{\lambda_{m}\mu_{n}}\left\{\sum_{m\in I_{r,s}}\sum_{n\in I_{r,s}}F\left(\left[M\left(\left(m+n\right)!\left|x_{m,n}\right|-0\right)^{1/m+n}\right]^{p_{mn}},z;\frac{t}{2}\right)\leq 1-s \right\}\right)$ and 
\\$A^{c}\left(s,t\right)=\left(\frac{1}{\lambda_{m}\mu_{n}}\left\{\sum_{m\in I_{r,s}}\sum_{n\in I_{r,s}}F\left(\left[M\left(\left(m+n\right)!\left|x_{m,n}\right|-0\right)^{1/m+n}\right]^{p_{mn}},z;\frac{t}{2}\right)> 1-s \right\}\right).$ It follows that $\delta_{\nabla}\left(A\left(s,t\right)\right)=0$ and consequently  $\delta_{\nabla}\left(A^{c}\left(s,t\right)\right)=1.$ Let $\eta\in A^{c}\left(s,t\right).$ Then 
\begin{equation}
F\left(\left[M\left(\left(m+n\right)!\left|x_{m,n}\right|-0\right)^{1/m+n}\right]^{p_{mn}},z;\frac{t}{2}\right)\leq 1-s
\end{equation}
\\$B\left(\epsilon,t\right)=\left(\frac{1}{\lambda_{m}\mu_{n}}\left\{\sum_{m\in I_{r,s}}\sum_{n\in I_{r,s}}F\left(\left[M\left(\left(m+n\right)!\left|x_{m,n}\right|-0\right)^{1/m+n}\right]^{p_{mn}},z;t\right)\leq 1-\epsilon \right\}\right).$ It is enough to prove that $B\left(\epsilon,t\right)\subseteq A\left(s,t\right).$ Let $a,b\in B\left(\epsilon,t\right),$ then for non-zero $z\in \chi^{2\tau}_{M}\left[AC_{\lambda_{m}\mu_{n}},P,\left\|\left(d\left(x_{1},0\right),d\left(x_{2},0\right),\cdots, d\left(x_{n-1},0\right)\right)\right\|_{p}\right].$ 
\begin{equation}
\frac{1}{\lambda_{m}\mu_{n}}\sum_{a\in I_{r,s}}\sum_{b\in I_{r,s}}F\left(\left[M\left(x_{a,b}-x_{c,d}\right)^{1/m+n}\right]^{p_{mn}},z;t\right)\leq 1-\epsilon.
\end{equation}
If 
\begin{center}
$\frac{1}{\lambda_{m}\mu_{n}}\sum_{a\in I_{r,s}}\sum_{b\in I_{r,s}}F\left(\left[M\left(x_{a,b}-x_{c,d}\right)^{1/m+n}\right]^{p_{mn}},z;t\right)\leq 1-\epsilon.$
\end{center}
then we have 
\begin{center}
$\frac{1}{\lambda_{m}\mu_{n}}\sum_{a\in I_{r,s}}\sum_{b\in I_{r,s}}F\left(\left[M\left(x_{a,b}-0\right)^{1/m+n}\right]^{p_{mn}},z;\frac{t}{2}\right)\leq 1-s$ 
\end{center}
and therefore $a,b\in A\left(s,t\right).$ As otherwise that is if 
\\$\left(\frac{1}{\lambda_{m}\mu_{n}}\left\{\sum_{a\in I_{r,s}}\sum_{b\in I_{r,s}}F\left(\left[M\left(\left(a+b\right)!\left|x_{a,b}\right|-0\right)^{1/a+b}\right]^{p_{ab}},z;\frac{t}{2}\right)> 1-s \right\}\right)$
\\then by (4.1),(4.2) and (4.3) we get 
\\\indent\hspace{0.5cm} $1-\epsilon\geq \frac{1}{\lambda_{m}\mu_{n}}\sum_{a\in I_{r,s}}\sum_{b\in I_{r,s}}F\left(\left[M\left(x_{a,b}-x_{c,d}\right)^{1/m+n}\right]^{p_{mn}},z;t\right)$ 
\\\indent\hspace{1.6cm }$\geq \left(\frac{1}{\lambda_{m}\mu_{n}}\left\{\sum_{a\in I_{r,s}}\sum_{b\in I_{r,s}}F\left(\left[M\left(\left(a+b\right)!\left|x_{a,b}\right|-0\right)^{1/a+b}\right]^{p_{ab}},z;\frac{t}{2}\right)> 1-s \right\}\right)*\\\indent \hspace{2cm}\left(\frac{1}{\lambda_{m}\mu_{n}}\left\{\sum_{c\in I_{r,s}}\sum_{c\in I_{r,s}}F\left(\left[M\left(\left(c+d\right)!\left|x_{c,d}\right|-0\right)^{1/c+d}\right]^{p_{cd}},z;\frac{t}{2}\right)> 1-s \right\}\right)$ 
\\\indent\hspace{1.6cm }$\geq \left(1-s\right)* \left(1-s\right)>1-\epsilon$
\\which is not possible. Thus $B\left(\epsilon,t\right)\subset A\left(s,t\right).$ Since $\delta_{\nabla}\left(A\left(s,t\right)\right)=0,$ it follows that $\delta_{\nabla}\left(B\left(\epsilon,t\right)\right)=0.$ This shows that $\left(x_{mn}\right)$ is $\nabla-$statistically Cauchy.
\\\indent Conversely, suppose $\left(x_{mn}\right)$ is $\nabla-$statistically Cauchy not in $\nabla-$statistically convergent. Then there exists positive integer $\eta$ and for non-zero \\$z\in \chi^{2\tau}_{M}\left[AC_{\lambda_{m}\mu_{n}},P,\left\|\left(d\left(x_{1},0\right),d\left(x_{2},0\right),\cdots, d\left(x_{n-1},0\right)\right)\right\|_{p}\right]$ such that if we take 
\begin{center}
$A\left(\epsilon,t\right)=\left(\frac{1}{\lambda_{m}\mu_{n}}\left\{\sum_{a\in I_{r,s}}\sum_{b\in I_{r,s}}F\left(\left[M\left(x_{a,b}-x_{cd}\right)^{1/a+b}\right]^{p_{ab}},z;t\right)\leq 1-\epsilon \right\}\right)$ 
\end{center}
and
\\$B\left(\epsilon,t\right)=\left(\frac{1}{\lambda_{m}\mu_{n}}\left\{\sum_{m\in I_{r,s}}\sum_{n\in I_{r,s}}F\left(\left[M\left(\left(m+n\right)!\left|x_{m,n}\right|-0\right)^{1/m+n}\right]^{p_{mn}},z;\frac{t}{2}\right)>1-\epsilon \right\}\right).$
\\then 
\begin{center}
$\delta_{\nabla}\left(A\left(\epsilon,t\right)\right)=0=\delta_{\nabla}\left(B\left(\epsilon,t\right)\right)$
\end{center}
consequently
\begin{equation}
\delta_{\nabla}\left(A^{c}\left(\epsilon,t\right)\right)=1=\delta_{\nabla}\left(B^{c}\left(\epsilon,t\right)\right).
\end{equation}
Since   
\\$\left(\frac{1}{\lambda_{m}\mu_{n}}\left\{\sum_{a\in I_{r,s}}\sum_{b\in I_{r,s}}F\left(\left[M\left(x_{a,b}-x_{cd}\right)^{1/a+b}\right]^{p_{ab}},z;t\right) \right\}\right)\geq \\2\hspace{0.1cm}\left(\frac{1}{\lambda_{m}\mu_{n}}\left\{\sum_{m\in I_{r,s}}\sum_{n\in I_{r,s}}F\left(\left[M\left(\left(m+n\right)!\left|x_{m,n}\right|-0\right)^{1/m+n}\right]^{p_{mn}},z;\frac{t}{2}\right) \right\}\right)>1-\epsilon,$
\\if
\\$\left(\frac{1}{\lambda_{m}\mu_{n}}\left\{\sum_{m\in I_{r,s}}\sum_{n\in I_{r,s}}F\left(\left[M\left(\left(m+n\right)!\left|x_{m,n}\right|-0\right)^{1/m+n}\right]^{p_{mn}},z;\frac{t}{2}\right)\right\}\right)>\frac{1-\epsilon}{2}$
\\then we have
\begin{center}
$\delta_{\nabla}\left(\frac{1}{\lambda_{m}\mu_{n}}\left\{\sum_{a\in I_{r,s}}\sum_{b\in I_{r,s}}F\left(\left[M\left(x_{a,b}-x_{cd}\right)^{1/a+b}\right]^{p_{ab}},z;t\right)>1-\epsilon \right\}\right)=0$  
\end{center}
that is $\delta_{\nabla}\left(A^{c}\left(\epsilon,t\right)\right)=0,$ which contradicts (4.4) as $\delta_{\nabla}\left(A^{c}\left(\epsilon,t\right)\right)=1.$ Hence $x=\left(x_{mn}\right)$ is $\nabla-$statistically convergent. 
\\\\\textbf{Competing Interests: } The authors declare that there is no conflict of interests regarding the publication of this research paper.
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