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\title[The Generalized Difference of $\chi^{2}$ over $p-$ metric spaces defined by Musielak] {The Generalized Difference of $\chi^{2}$ over $p-$ metric spaces defined by Musielak}
\vspace{15mm}


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

\email{nsmaths@yahoo.com} 



\keywords{ analytic sequence, double sequences, $\chi^{2}$ space, difference sequence space,Musielak - modulus function, $p-$ metric space, Lacunary sequence, ideal.    \\
\indent 2010 {\it Mathematics Subject Classification}. 40A05,40C05,40D05.\\
\indent {\it Received}: \\
\indent {\it Revised}: \\}




\begin{abstract}
In this paper, we define the sequence spaces: $\chi^{2qu}_{f\mu}\left(\Delta\right)$ and $\Lambda^{2qu}_{f\mu}\left(\Delta\right),$ where for any sequence $x=\left(x_{mn}\right),$ the difference sequence $\Delta x$ is given by $\left(\Delta x_{mn}\right)_{m,n=1}^{\infty}=\left[\left(x_{mn}-x_{mn+1}\right)-\left(x_{m+1n}-x_{m+1n+1}\right)\right]_{m,n=1}^{\infty}.$ We also study some properties and theorems of these spaces.
\end{abstract}

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\maketitle

\section{Introduction}
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 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 Bromwich [2]. Later on, they were investigated by Hardy [3], Moricz [7], Moricz and Rhoades [8], Basarir and Solankan [1], Tripathy [11], Turkmenoglu [12], and many others. 
\\\\We procure the following sets of double sequences:
\begin{center}
$\mathcal{M}_{u}\left(t\right):=\left\{\left(x_{mn}\right)\in w^{2}: sup_{m,n\in N}\left|x_{mn}\right|^{t_{mn}}<\infty \right\},$
\end{center}
\begin{center}
$\mathcal{C}_{p}\left(t\right):=\left\{\left(x_{mn}\right)\in w^{2}: p-lim_{m,n\rightarrow \infty}\left|x_{mn}-\l\right|^{t_{mn}}=1\hspace{0.05cm}for\hspace{0.05cm}some\hspace{0.05cm}\l\in \mathbb{C}\right\},$
\end{center}
\begin{center}
$\mathcal{C}_{0p}\left(t\right):=\left\{\left(x_{mn}\right)\in w^{2}: p-lim_{m,n\rightarrow \infty}\left|x_{mn}\right|^{t_{mn}}=1\right\},$
\end{center}
\begin{center}
$\mathcal{L}_{u}\left(t\right):=\left\{\left(x_{mn}\right)\in w^{2}: \sum_{m=1}^{\infty}\sum_{n=1}^{\infty}\left|x_{mn}\right|^{t_{mn}}<\infty\right\},$
\end{center}
\begin{center}
$\mathcal{C}_{bp}\left(t\right):=\mathcal{C}_{p}\left(t\right)\bigcap \mathcal{M}_{u}\left(t\right)$ and $\mathcal{C}_{0bp}\left(t\right)=\mathcal{C}_{0p}\left(t\right)\bigcap \mathcal{M}_{u}\left(t\right)$;
\end{center}
where $t=\left(t_{mn}\right)$ is the sequence of strictly positive reals $t_{mn}$ for all $m,n\in \mathbb{N}$ and $p-lim_{m,n\rightarrow \infty}$ denotes the limit in the Pringsheim's sense. In the case $t_{mn}=1$ for all $m,n\in \mathbb{N};\mathcal{M}_{u}\left(t\right),\mathcal{C}_{p}\left(t\right),\mathcal{C}_{0p}\left(t\right),\mathcal{L}_{u}\left(t\right),\mathcal{C}_{bp}\left(t\right)$ and $\mathcal{C}_{0bp}\left(t\right)$ reduce to the sets $\mathcal{M}_{u},\mathcal{C}_{p},\mathcal{C}_{0p},\mathcal{L}_{u},\mathcal{C}_{bp}$ and $\mathcal{C}_{0bp},$ respectively. Now, we may summarize the knowledge given in some document related to the double sequence spaces. G$\ddot{o}$khan and Colak [14,15] have proved that $\mathcal{M}_{u}\left(t\right)$ and $\mathcal{C}_{p}\left(t\right),\mathcal{C}_{bp}\left(t\right)$ are complete paranormed spaces of double sequences and gave the $\alpha-,\beta-,\gamma-$ duals of the spaces $\mathcal{M}_{u}\left(t\right)$ and $\mathcal{C}_{bp}\left(t\right).$ Quite recently, in her PhD thesis, Zelter [16] has essentially studied both the theory of topological double sequence spaces and the theory of summability of double sequences. Mursaleen and Edely [17] and Tripathy [11] have independently introduced the statistical convergence and Cauchy for double sequences and given the relation between statistical convergent and strongly Ces$\grave{a}$ro summable double sequences. Altay and Basar [20] have defined the spaces $\mathcal{BS},\mathcal{BS}\left(t\right),\mathcal{CS}_{p},\mathcal{CS}_{bp},\mathcal{CS}_{r}$ and $\mathcal{BV}$ of double sequences consisting of all double series whose sequence of partial sums are in the spaces $\mathcal{M}_{u},\mathcal{M}_{u}\left(t\right),\mathcal{C}_{p},\mathcal{C}_{bp},\mathcal{C}_{r}$ and $\mathcal{L}_{u},$ respectively, and also examined some properties of those sequence spaces and determined the $\alpha-$ duals of the spaces $\mathcal{BS}, \mathcal{BV},\mathcal{CS}_{bp}$ and the $\beta\left(\vartheta \right)-$ duals of the spaces $\mathcal{CS}_{bp}$ and $\mathcal{CS}_{r}$ of double series. Basar and Sever [21] have introduced the Banach space $\mathcal{L}_{q}$ of double sequences corresponding to the well-known space $\ell_{q}$ of single 
sequences and examined some properties of the space $\mathcal{L}_{q}.$ Quite recently Subramanian and Misra [22] have studied the space $\chi^{2}_{M}\left(p,q,u\right)$ of double sequences and gave some inclusion relations.  
\\\indent The class of sequences which are strongly Ces$\grave{a}$ro summable with respect to a modulus was introduced by Maddox [6] as an extension of the definition of strongly Ces$\grave{a}$ro summable sequences. Connor [23] further extended this definition to a definition of strong $A-$ summability with respect to a modulus where $A=\left(a_{n,k}\right)$ is a nonnegative regular matrix and established some connections between strong $A-$ summability, strong $A-$ summability with respect to a modulus, and $A-$ statistical convergence. In [24] the notion of convergence of double sequences was presented by A. Pringsheim. Also, in [25]-[26], and [27] the four dimensional matrix transformation $\left(Ax\right)_{k,\ell}=\sum_{m=1}^{\infty}\sum_{n=1}^{\infty}a_{k \ell}^{mn}x_{mn}$ was studied extensively by Robison and Hamilton. 
\\\indent We need the following inequality in the sequel of the paper. For $a,b,\geq 0$ and $0<p<1,$ we have 
\begin{equation}     
(a+b)^{p}\leq a^{p}+b^{p}
\end{equation}
\\The double series $\sum_{m,n=1}^{\infty}x_{mn}$ is called convergent if and only if the double sequence $(s_{mn})$ is  convergent, where
$s_{mn}=\sum_{i,j=1}^{m,n}x_{ij}(m,n\in \mathbb{N}).$ \\\\A sequence $x=(x_{mn})$is said to be double analytic if $sup_{mn}\left|x_{mn}\right|^{1/m+n}<\infty.$ The vector space of all double analytic sequences will be denoted by $\Lambda^{2}$. A sequence $x=(x_{mn})$ is called double gai sequence if $\left((m+n)!\left|x_{mn}\right|\right)^{1/m+n}\rightarrow 0$ as $m,n\rightarrow \infty.$ The double gai sequences will be denoted by $\chi^{2}$. Let $\phi=\left\{all finite sequences\right\}.$ \\\\Consider a double sequence $x=(x_{ij}).$ The $(m,n)^{th}$ section $x^{[m,n]}$ of the sequence is defined by $x^{[m,n]}=\sum{_{i,j=0}^{m,n}}x_{ij}\Im_{ij}$ for all $m,n\in \mathbb{N}\hspace{0.05cm} ;$ where $\Im_{ij}$ denotes the double sequence whose only non zero term is a $\frac{1}{\left(i+j\right)!}$ in the $\left(i,j\right)^{th}$ place for each $i,j\in \mathbb{N}.$ 
\\\\\indent An FK-space(or a metric space)$X$ is said to have AK property if $(\Im_{mn})$ is a Schauder basis for $X$. Or equivalently $x^{[m,n]}\rightarrow x$. 
\\\\\indent An FDK-space is a double sequence space endowed with a complete metrizable; locally convex topology under which the coordinate mappings $x=(x_{k})\rightarrow (x_{mn})(m,n\in \mathbb{N})$ are also continuous. 
\\\indent Let $M$ and $\Phi$ are mutually complementary modulus functions. Then, we have:
\\(i) For all $u,y\geq 0,$
\begin{equation}
uy\leq M\left(u\right)+\Phi\left(y\right), (Young's\hspace{0.05cm} inequality)[See [13]]
\end{equation} 
\\(ii) For all $u\geq 0,$
\begin{equation}
u\eta\left(u\right)=M\left(u\right)+\Phi\left(\eta\left(u\right)\right).
\end{equation}
\\(iii) For all $u\geq 0,$ and $0<\lambda<1,$
\begin{equation}
M\left(\lambda u\right)\leq \lambda M\left(u\right)
\end{equation}
Lindenstrauss and Tzafriri [5] used the idea of Orlicz function to construct Orlicz sequence space
\begin{center}
$\ell_{M}=\left\{x\in w: \sum_{k=1}^{\infty}M\left(\frac{\left|x_{k}\right|}{\rho}\right)< \infty,\hspace{0.05cm}for\hspace{0.05cm}some\hspace{0.05cm}\rho >0\right\},$
\end{center}
The space $\ell_{M}$ with the norm
\begin{center}
$\left\|x\right\|=inf\left\{\rho >0: \sum_{k=1}^{\infty}M\left(\frac{\left|x_{k}\right|}{\rho}\right)\leq 1 \right\},$ 
\end{center}
becomes a Banach space which is called an Orlicz sequence space. For $M\left(t\right)=t^{p}\left(1\leq p<\infty\right),$ the spaces $\ell_{M}$ coincide with the classical sequence space $\ell_{p}.$  
\\\indent A sequence $f=\left(f_{mn}\right)$ of modulus function is called a Musielak-modulus function. A sequence $g=\left(g_{mn}\right)$ defined by 
\begin{center}
$g_{mn}\left(v\right)=sup\left\{\left|v\right|u-\left(f_{mn}\right)\left(u\right):u\geq 0\right\},m,n=1,2,\cdots$  
\end{center}
is called the complementary function of a Musielak-modulus function $f$. For a given Musielak modulus function $f,$ the Musielak-modulus sequence space $t_{f}$ and its subspace $h_{f}$ are defined as follows                                 
\begin{center}
$t_{f}=\left\{x\in w^{2}:I_{f}\left(\left|x_{mn}\right|\right)^{1/m+n}\rightarrow 0\hspace{0.05cm}as\hspace{0.05cm}m,n\rightarrow \infty\right\},$  
\end{center} 
\begin{center}
$h_{f}=\left\{x\in w^{2}:I_{f}\left(\left|x_{mn}\right|\right)^{1/m+n}\rightarrow 0\hspace{0.05cm}as\hspace{0.05cm}m,n\rightarrow \infty\right\},$  
\end{center}
where $I_{f}$ is a convex modular defined by 
\begin{center}
$I_{f}\left(x\right)=\sum_{m=1}^{\infty}\sum_{n=1}^{\infty}f_{mn}\left(\left|x_{mn}\right|\right)^{1/m+n}, x=\left(x_{mn}\right)\in t_{f}.$
\end{center}
We consider $t_{f}$ equipped with the Luxemburg metric 
\begin{center}
$d\left(x,y\right)=sup_{mn}\left\{inf\left(\sum_{m=1}^{\infty}\sum_{n=1}^{\infty}f_{mn}\left(\frac{\left|x_{mn}\right|^{1/m+n}}{mn}\right)\right)\leq 1\right\}$
\end{center}
If $X$ is a sequence space, we give the following definitions:
\\\\(i)$X^{'}$= the continuous dual of $X;$
\\\\(ii)$X^{\alpha}=\left\{a=(a_{mn}):\sum{_{m,n=1}^{\infty}}\left|a_{mn}x_{mn}\right|<\infty,\hspace{.05cm}for\hspace{0.05cm} each\hspace{0.05cm} x\in X\right\};$
\\\\(iii)$X^{\beta}=\left\{a=(a_{mn}):\sum{_{m,n=1}^{\infty}}a_{mn}x_{mn}\hspace{0.05cm}is\hspace{0.05cm} convegent,\hspace{0.05cm}for each\hspace{0.05cm} x\in X\right\};$ 
\\\\(iv)$X^{\gamma}=\left\{a=(a_{mn}):sup_{mn}\geq 1\left|\sum_{m,n=1}^{M,N}a_{mn}x_{mn}\right|<\infty, for each x\in X \right\};$
\\\\(v)$let\hspace{0.05cm} X\hspace{0.05cm} be an FK-space\hspace{0.05cm}\supset\phi;\hspace{0.05cm}then\hspace{0.05cm}X^{f}=\left\{f(\Im_{mn}):f\in X^{'}\right\};$
\\\\(vi)$X^{\delta}=\left\{a=(a_{mn}):sup_{mn}\left|a_{mn}x_{mn}\right|^{1/m+n}<\infty,\hspace{0.05cm}for each \hspace{0.05cm}x\in X\right\};$
\\\\$X^{\alpha}.X^{\beta},X^{\gamma}$ are called $\alpha-\hspace{0.05cm}(or K\ddot{o}the-Toeplitz)$dual of $X,\beta-(or \hspace{0.05cm}generalized-K\ddot{o}the-Toeplitz)\hspace{0.05cm} dual\hspace{0.05cm} of X,\gamma-\hspace{0.05cm}dual \hspace{0.05cm}of\hspace{0.05cm} X,\hspace{0.05cm}\delta\hspace{0.05cm}-\hspace{0.05cm}dual\hspace{0.05cm} of X\hspace{0.05cm} respectively. X^{\alpha}$ is defined by Gupta and Kamptan [13]. It is clear that $X^{\alpha}\subset X^{\beta}$ and $X^{\alpha}\subset X^{\gamma},$ but $X^{\beta}\subset X^{\gamma}$ does not hold, since the sequence of partial sums of a double convergent series need not to be bounded.
\\\indent The notion of difference sequence spaces (for single sequences) was introduced by Kizmaz as follows
\begin{center}
$Z\left(\Delta\right)=\left\{x=\left(x_{k}\right)\in w: \left(\Delta x_{k} \right)\in Z\right\}$ 
\end{center}
for $Z=c,c_{0}$ and $\ell_{\infty},$ where $\Delta x_{k}=x_{k}-x_{k+1}$ for all $k\in \mathbb{N}.$ \\
Here $c,c_{0}$ and $\ell_{\infty}$ denote the classes of convergent,null and bounded sclar valued single sequences respectively. The difference sequence space $bv_{p}$ of the classical space $\ell_{p}$ is introduced and studied in the case $1\leq p\leq \infty$ by Ba\c{s}ar and Altay and in the case $0<p<1$ by Altay and Ba\c{s}ar in [20]. The spaces $c\left(\Delta\right),c_{0}\left(\Delta\right),\ell_{\infty}\left(\Delta\right)$ and $bv_{p}$ are Banach spaces normed by 
\begin{center}
$\left\|x\right\|=\left|x_{1}\right|+sup_{k\geq 1}\left|\Delta x_{k}\right|$ and $\left\|x\right\|_{bv_{p}}=\left(\sum_{k=1}^{\infty}\left|x_{k}\right|^{p}\right)^{1/p},\left(1\leq p<\infty\right).$ 
\end{center}
Later on the notion was further investigated by many others. We now introduce the following difference double sequence spaces defined by 
\begin{center}
$Z\left(\Delta\right)=\left\{x=\left(x_{mn}\right)\in w^{2}:\left(\Delta x_{mn}\right)\in Z \right\}$ 
\end{center}
where $Z=\Lambda^{2},\chi^{2}$ and $\Delta x_{mn}=\left(x_{mn}-x_{mn+1}\right)-\left(x_{m+1n}-x_{m+1n+1}\right)=x_{mn}-x_{mn+1}-x_{m+1n}+x_{m+1n+1}$ for all $m,n\in \mathbb{N}.$ The generalized difference double notion has the following representation:
$\Delta^{m} x_{mn}=\Delta^{m-1}x_{mn}-\Delta^{m-1}x_{mn+1}-\Delta^{m-1}x_{m+1n}+\Delta^{m-1}x_{m+1n+1},$ and also this generalized difference double notion has the following binomial representation:
$\Delta^{m}x_{mn}=\sum_{i=0}^{m}\sum_{j=0}^{m}\left(-1\right)^{i+j}\left(\stackrel{m} {i}\right)\left(\stackrel{m} {j}\right)x_{m+i,n+j}.$
\section{Definition and Preliminaries}
Let $n\in \mathbb{N}$ and $X$ be a real vector space of dimension $w,$ where $n\leq w.$ A real valued function $ d_p(x_1,\dots,x_n) = \|(d_1(x_1), \dots, d_n(x_n))\|_p$ on $X$ satisfying the following four conditions:
\\(i) $\|(d_1(x_1), \dots, d_n(x_n))\|_p=0$ if and and only if  $d_1(x_1), \dots, d_n(x_n)$ are linearly dependent,
\\(ii) $\|(d_1(x_1), \dots, d_n(x_n))\|_p$ is invariant under permutation,
\\(iii) $\|(\alpha d_1(x_1), \dots, d_n(x_n))\|_p=\left|\alpha \right|\|(d_1(x_1), \dots, d_n(x_n))\|_p ,\alpha\in \mathbb{R}$  
\\(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 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}
$\|(d_1(x_1), \dots, d_n(x_n))\|_E = sup \left(| det(d_{mn}\left(x_{mn}\right))|\right)=sup
\begin{pmatrix}\begin{vmatrix}
  d_{11}\left(x_{11}\right) & d_{12}\left(x_{12}\right) &  .  . .  & d_{1n}\left(x_{1n}\right) \\
  d_{21}\left(x_{21}\right) & d_{22}\left(x_{22}\right) &  .  . .  & d_{2n}\left(x_{1n}\right)\\
 
  . & \\
  . & \\
  . & \\
  d_{n1}\left(x_{n1}\right) & d_{n2}\left(x_{n2}\right) &  .  . .  & d_{nn}\left(x_{nn}\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.
\\\indent Let $X$ be a linear metric space. A function $w:X\rightarrow \mathbb{R}$ is called paranorm, if 
\\(1) $w\left(x\right)\geq 0,$ for all $x\in X;$ 
\\(2) $w\left(-x\right)=w\left(x\right),$ for all $x\in X;$
\\(3) $w\left(x+y\right)\leq w\left(x\right)+w\left(y\right),$ for all $x,y\in X;$
\\(4) If $\left(\sigma_{mn}\right)$ is a sequence of scalars with $\sigma_{mn}\rightarrow \sigma$ as $m,n\rightarrow \infty$ and $\left(x_{mn}\right)$ is a sequence of vectors with $w\left(x_{mn}-x\right)\rightarrow 0$ as $m,n\rightarrow \infty,$  then $w\left(\sigma_{mn}x_{mn}-\sigma x\right)\rightarrow 0$ as $m,n\rightarrow \infty.$
\\A paranorm $w$ for which $w\left(x\right)=0$ implies $x=0$ is called total paranorm and the pair $\left(X,w\right)$ is called a total paranormed space. It is well known that the metric of any linear metric space is given by some total paranorm (see [32], Theorem 10.4.2, p.183).
\\\indent $\eta=\left(\varphi_{rs}\right)$ a nondecreasing sequence of positive reals tending to infinity and $\varphi_{11}=1$ and $\varphi_{r+1,s+1}\leq \varphi_{rs}+1.$
\\The generalized de la Vallee-Pussin means is defined by :
\begin{center}
$t_{rs}\left(x\right)=\frac{1}{\varphi_{rs}}\sum_{m\in I_{rs}}\sum_{n\in I_{rs}}x_{mn},$
\end{center}
where $I_{rs}=\left[rs-\lambda_{rs}+1,rs\right].$  For the set of sequences that are strongly summable to zero, strongly summable and strongly bounded by the de la Vallee-Poussin method. 
\\\indent The notion of $\lambda-$ double gai and double analytic sequences as follows: Let $\lambda=\left(\lambda_{mn}\right)_{m,n=0}^{\infty}$ be a strictly increasing sequences of positive real numbers tending to infinity, that is 
\begin{center}
$0<\lambda_{00}<\lambda_{11}<\cdots\hspace{0.05cm}and\hspace{0.05cm}\lambda_{mn}\rightarrow \infty\hspace{0.05cm}as\hspace{0.05cm}m,n\rightarrow \infty$ 
\end{center}
and said that a sequence $x=\left(x_{mn}\right)\in w^{2}$ is $\lambda-$ convergent to 0, called a the $\lambda-$ limit of $x,$ if $\mu_{mn}\left(x\right)\rightarrow 0\hspace{0.05cm}as\hspace{0.05cm}m,n\rightarrow \infty,$ where 
\begin{center}
$\mu_{mn}\left(x\right)=\frac{1}{\varphi_{rs}}\sum_{m\in I_{rs}}\sum_{n\in I_{rs} }\left(\Delta^{m-1}\lambda_{m,n}-\Delta^{m-1}\lambda_{m,n+1}-\Delta^{m-1}\lambda_{m+1,n}+\Delta^{m-1}\lambda_{m+1,n+1}\right)\left|x_{mn}\right|^{1/m+n}.$ 
\end{center}
The sequence $x=\left(x_{mn}\right)\in w^{2}$ is $\lambda-$ double analytic if $sup_{uv}\left|\mu_{mn}\left(x\right)\right|< \infty.$ If $lim_{mn}x_{mn}=0$ in the ordinary sense of convergence, then 
\\$lim_{mn}\\\left(\frac{1}{\varphi_{rs}}\sum_{m\in I_{rs}}\sum_{n\in  I_{rs}}\\\left(\Delta^{m-1}\lambda_{m,n}-\Delta^{m-1}\lambda_{m,n+1}-\Delta^{m-1}\lambda_{m+1,n}+\Delta^{m-1}\lambda_{m+1,n+1}\right)\\\left(\left(m+n\right)!\left|x_{mn}-0\right|\right)^{1/m+n}\right)=0.$
This implies that 
\\$lim_{mn}\left|\mu_{mn}\left(x\right)-0\right|=\lim_{mn}\\\left|\left(\frac{1}{\varphi_{rs}}\sum_{m\in I_{rs}}\sum_{n\in I_{rs}}\left(\Delta^{m-1}\lambda_{m,n}-\Delta^{m-1}\lambda_{m,n+1}-\Delta^{m-1}\lambda_{m+1,n}+\Delta^{m-1}\lambda_{m+1,n+1}\right)\\\left(\left(m+n\right)!\left|x_{mn}-0\right|\right)^{1/m+n}\right)\right|\\=0.$ 
\\which yields that $lim_{uv}\mu_{mn}\left(x\right)=0$ and hence $x=\left(x_{mn}\right)\in w^{2}$ is $\lambda-$ convergent to 0.
\\\indent Let $f=\left(f_{mn}\right)$ be a Musielak-modulus function and $\left(X,\left\|\left(d\left(x_{1}\right),d\left(x_{2}\right),\cdots, d\left(x_{n-1}\right)\right)\right\|_{p}\right)$ be a $p-$metric space, $q=\left(q_{mn}\right)$ be double analytic sequence of strictly positive real numbers and $u=\left(u_{mn}\right)$ be any sequence such that $u_{mn}\neq 0\left(m,n=1,2,\cdots\right).$ By $w^{2}\left(p-X\right)$ we denote the space of all sequences defined over \\$\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).$
The following inequality will be used throughout the paper. If $0\leq q_{mn}\leq supq_{mn}=H,K=max\left(1,2^{H-1}\right)$ then 
\begin{equation}
\left|a_{mn}+b_{mn}\right|^{q_{mn}}\leq K\left\{\left|a_{mn}\right|^{q_{mn}}+\left|b_{mn}\right|^{q_{mn}}\right\}
\end{equation}
for all $m,n$ and $a_{mn},b_{mn}\in \mathbb{C}.$ Also $\left|a\right|^{q_{mn}}\leq max\left(1,\left|a\right|^{H}\right)$ for all $a\in \mathbb{C}.$
\\In the present paper we define the following sequence spaces:
\\\\$\left[\chi^{2qu}_{f\mu },\left\|\left(d\left(x_{1}\right),d\left(x_{2}\right),\cdots, d\left(x_{n-1}\right)\right)\right\|_{p}^{\varphi}\right]^{V}=\\\left\{r,s\in I_{rs}:lim_{rs}u_{mn}\left[f_{mn}\left(\left\|\mu_{mn}\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]^{q_{mn}}=0\right\}$  ,  
\\\\$\left[\Lambda^{2qu}_{f\mu },\left\|\left(d\left(x_{1}\right),d\left(x_{2}\right),\cdots, d\left(x_{n-1}\right)\right)\right\|_{p}^{\varphi}\right]^{V}= 
\\\left\{r,s\in I_{rs}:sup_{rs}u_{mn}\left[f_{mn}\left(\left\|\mu_{mn}\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]^{q_{mn}}< \infty \right\}$  , 
\\\\If we take $f_{mn}\left(x\right)=x,$  we get 
\\\\$\left[\chi^{2qu}_{f\mu },\left\|\left(d\left(x_{1}\right),d\left(x_{2}\right),\cdots, d\left(x_{n-1}\right)\right)\right\|_{p}^{\varphi}\right]^{V}=\\\left\{r,s\in I_{rs}:lim_{rs}u_{mn}\left[\left(\left\|\mu_{mn}\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]^{q_{mn}}=0 \right\}$  ,  
\\\\$\left[\Lambda^{2qu}_{f\mu },\left\|\left(d\left(x_{1}\right),d\left(x_{2}\right),\cdots, d\left(x_{n-1}\right)\right)\right\|_{p}^{\varphi}\right]^{V}= 
\\\left\{r,s\in I_{rs}:sup_{rs}u_{mn}\left[\left(\left\|\mu_{mn}\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]^{q_{mn}}< \infty \right\}$  ,  
\\\\If we take $q=\left(q_{mn}\right)=1,$ we get
\\\\$\left[\chi^{2u}_{f\mu },\left\|\left(d\left(x_{1}\right),d\left(x_{2}\right),\cdots, d\left(x_{n-1}\right)\right)\right\|_{p}^{\varphi}\right]^{V}=\\\left\{r,s\in I_{rs}:u_{mn}\left[f_{mn}\left(\left\|\mu_{mn}\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]^{V}=0 \right\}$  ,  
\\\\$\left[\Lambda^{2u}_{f\mu },\left\|\left(d\left(x_{1}\right),d\left(x_{2}\right),\cdots, d\left(x_{n-1}\right)\right)\right\|_{p}^{\varphi}\right]^{V}= 
\\\left\{r,s\in I_{rs}:u_{mn}\left[f_{mn}\left(\left\|\mu_{mn}\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]<\infty \right\}$  , 
\\\\In the present paper we plan to study some topological properties and inclusion relation between the above defined sequence spaces. $\left[\chi^{2qu}_{f\mu },\left\|\left(d\left(x_{1}\right),d\left(x_{2}\right),\cdots, d\left(x_{n-1}\right)\right)\right\|_{p}^{\varphi}\right]^{V}$ and $\left[\Lambda^{2qu}_{f\mu },\left\|\left(d\left(x_{1}\right),d\left(x_{2}\right),\cdots, d\left(x_{n-1}\right)\right)\right\|_{p}^{\varphi}\right]^{V}$ which we shall discuss in this paper. 
\section{Main Results}
\subsection{Theorem} Let $f=\left(f_{mn}\right)$ be a Musielak-modulus function, $q=\left(q_{mn}\right)$ be a double analytic sequence of strictly positive real numbers, the sequence spaces
$\left[\chi^{2qu}_{f\mu },\left\|\left(d\left(x_{1}\right),d\left(x_{2}\right),\cdots, d\left(x_{n-1}\right)\right)\right\|_{p}^{\varphi}\right]^{V}$ and $\left[\Lambda^{2qu}_{f\mu },\left\|\left(d\left(x_{1}\right),d\left(x_{2}\right),\cdots, d\left(x_{n-1}\right)\right)\right\|_{p}^{\varphi}\right]^{V}$
are linear spaces.
\\\textbf{Proof:} It is routine verification. Therefore the proof is omitted.  
\subsection{Theorem} Let $f=\left(f_{mn}\right)$ be a Musielak-modulus function, $q=\left(q_{mn}\right)$ be a double analytic sequence of strictly positive real numbers, the sequence space
$\left[\chi^{2qu}_{f\mu },\left\|\left(d\left(x_{1}\right),d\left(x_{2}\right),\cdots, d\left(x_{n-1}\right)\right)\right\|_{p}^{\varphi}\right]^{V}$
is a paranormed space with respect to the paranorm defined by 
\\$g\left(x\right)=inf
\\\left\{u_{mn}\left[f_{mn}\left(\left\|\mu_{mn}\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]^{q_{mn}}\leq 1\right\}=0.$
\\\textbf{Proof: } Clearly $g\left(x\right)\geq 0$ for $x=\left(x_{mn}\right)\in
\left[\chi^{2qu}_{f\mu },\left\|\left(d\left(x_{1}\right),d\left(x_{2}\right),\cdots, d\left(x_{n-1}\right)\right)\right\|_{p}^{\varphi}\right]^{V}$
Since $f_{mn}\left(0\right)=0,$ we get $g\left(0\right)=0.$ 
\\\indent Conversely, suppose that $g\left(x\right)=0,$ then 
\\$inf\left\{u_{mn}\left[f_{mn}\left(\left\|\mu_{mn}\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]^{q_{mn}}\leq 1\right\}=0$  
\\Suppose that $\mu_{mn}\left(x\right)\neq 0$ for each $m,n\in \mathbb{N}.$ Then $\left\|\mu_{mn}\left(x\right),\left(d\left(x_{1}\right),d\left(x_{2}\right),\cdots, d\left(x_{n-1}\right)\right)\right\|_{p}^{\varphi}\rightarrow \infty.$ It follows that
$\left(u_{mn}\left[f_{mn}\left(\left\|\mu_{mn}\left(x\right),\left(d\left(x_{1}\right),d\left(x_{2}\right),\cdots, d\left(x_{n-1}\right)\right)\right\|_{p}\right)\right]^{q_{mn}}\right)^{1/H}\rightarrow \infty$ 
which is a contradiction. Therefore $\mu_{mn}\left(x\right)= 0.$  Let
\\$\left(u_{mn}\left[f_{mn}\left(\left\|\mu_{mn}\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]^{q_{mn}}\right)^{1/H}\leq 1$ \\\\and 
\\\\$\left(u_{mn}\left[f_{mn}\left(\left\|\mu_{mn}\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]^{q_{mn}}\right)^{1/H}\leq 1$
\\\\Then by using Minkowski's inequality, we have 
\\$\left(u_{mn}\left[f_{mn}\left(\left\|\mu_{mn}\left(x+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]^{q_{mn}}\right)^{1/H}\leq \\\left(u_{mn}\left[f_{mn}\left(\left\|\mu_{mn}\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]^{q_{mn}}\right)^{1/H}+\\\left(u_{mn}\left[f_{mn}\left(\left\|\mu_{mn}\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]^{q_{mn}}\right)^{1/H}.$
\\\\So we have 
\\$g\left(x+y\right)=inf
\left\{u_{mn}\left[f_{mn}\left(\left\|\mu_{mn}\left(x+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]^{q_{mn}}\leq 1\right\}\leq \\inf
\left\{u_{mn}\left[f_{mn}\left(\left\|\mu_{mn}\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]^{q_{mn}}\leq 1\right\}+\\inf
\left\{u_{mn}\left[f_{mn}\left(\left\|\mu_{mn}\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]^{q_{mn}}\leq 1\right\}$
\\Therefore, 
\begin{center}
$g\left(x+y\right)\leq g\left(x\right)+g\left(y\right).$ 
\end{center}
Finally, to prove that the scalar multiplication is continuous. Let $\lambda$ be any complex number. By definition,
\\$g\left(\lambda x\right)=inf
\left\{u_{mn}\left[f_{mn}\left(\left\|\mu_{mn}\left(\lambda 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]^{q_{mn}}\leq 1\right\}.$
\\Then 
\\$g\left(\lambda\hspace{0.05cm}x\right)=inf\left\{((\left|\lambda\right|t)^{q_{mn}/H}:
u_{mn}\left[f_{mn}\left(\left\|\mu_{mn}\left(\lambda 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]^{q_{mn}}\leq 1\right\}$
\\where $t=\frac{1}{\left|\lambda\right|}.$ Since $\left|\lambda\right|^{q_{mn}}\leq max\left(1,\left|\lambda\right|^{supp_{mn}}\right),$ we have  
\\$g\left(\lambda\hspace{0.05cm}x\right)\leq max\left(1,\left|\lambda\right|^{supp_{mn}}\right) 
inf\\\left\{t^{q_{mn}/H}:u_{mn}
\left[f_{mn}\left(\left\|\mu_{mn}\left(\lambda 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]^{q_{mn}}\leq 1\right\}$
\\This completes the proof.
\subsection{Theorem} (i) If the sequence $\left(f_{mn}\right)$ satisfies uniform $\Delta_{2}-$ condition, then \\$\left[\chi^{2qu}_{f\mu},\left\|\mu_{mn}\left(x\right),\left(d\left(x_{1}\right),d\left(x_{2}\right),\cdots, d\left(x_{n-1}\right)\right)\right\|_{p}^{\varphi}\right]^{V \alpha}=\\\left[\chi^{2qu\mu}_{g},\left\|\mu_{uv}\left(x\right),\left(d\left(x_{1}\right),d\left(x_{2}\right),\cdots, d\left(x_{n-1}\right)\right)\right\|_{p}^{\varphi}\right]^{V}.$
\\(ii) If the sequence $\left(g_{mn}\right)$ satisfies uniform $\Delta_{2}-$ condition, then \\$\left[\chi^{2qu\mu}_{g},\left\|\mu_{mn}\left(x\right),\left(d\left(x_{1}\right),d\left(x_{2}\right),\cdots, d\left(x_{n-1}\right)\right)\right\|_{p}^{\varphi}\right]^{V \alpha}= \\\left[\chi^{2qu}_{f\mu},\left\|\mu_{mn}\left(x\right),\left(d\left(x_{1}\right),d\left(x_{2}\right),\cdots, d\left(x_{n-1}\right)\right)\right\|_{p}^{\varphi}\right]^{V}$ 
\\\textbf{Proof:} Let the sequence $\left(f_{mn}\right)$ satisfies uniform $\Delta_{2}-$ condition, we get  
\begin{equation}
\left[\chi^{2qu\mu}_{g},\left\|\mu_{mn}\left(x\right),\left(d\left(x_{1}\right),d\left(x_{2}\right),\cdots, d\left(x_{n-1}\right)\right)\right\|_{p}^{\varphi}\right]^{V}
\subset \left[\chi^{2qu}_{f\mu},\left\|\mu_{mn}\left(x\right),\left(d\left(x_{1}\right),d\left(x_{2}\right),\cdots, d\left(x_{n-1}\right)\right)\right\|_{p}^{\varphi}\right]^{V\alpha}
\end{equation} 
To prove the inclusion \\$\left[\chi^{2qu}_{f\mu},\left\|\mu_{mn}\left(x\right),\left(d\left(x_{1}\right),d\left(x_{2}\right),\cdots, d\left(x_{n-1}\right)\right)\right\|_{p}^{\varphi}\right]^{V\alpha} \subset\\ \\\left[\chi^{2qu\mu}_{g},\left\|\mu_{mn}\left(x\right),\left(d\left(x_{1}\right),d\left(x_{2}\right),\cdots, d\left(x_{n-1}\right)\right)\right\|_{p}^{\varphi}\right]^{V},$ \\let $a\in \left[\chi^{2qu}_{f\mu},\left\|\mu_{mn}\left(x\right),\left(d\left(x_{1}\right),d\left(x_{2}\right),\cdots, d\left(x_{n-1}\right)\right)\right\|_{p}^{\varphi}\right]^{V}.$ Then for all $\left\{x_{mn}\right\}$ with $\left(x_{mn}\right)\in \left[\chi^{2qu}_{f\mu},\left\|\mu_{mn}\left(x\right),\left(d\left(x_{1}\right),d\left(x_{2}\right),\cdots, d\left(x_{n-1}\right)\right)\right\|_{p}^{\varphi}\right]^{V}$ we have 
\begin{equation}
\sum_{m=1}^{\infty}\sum_{n=1}^{\infty}\left|x_{mn}a_{mn}\right|< \infty.
\end{equation}
Since the sequence $\left(f_{mn}\right)$ satisfies uniform $\Delta_{2}-$ condition, then \\$\left(y_{mn}\right)\in 
\left[\chi^{2qu}_{f\mu},\left\|\mu_{mn}\left(x\right),\left(d\left(x_{1}\right),d\left(x_{2}\right),\cdots, d\left(x_{n-1}\right)\right)\right\|_{p}^{\varphi}\right]^{V}
,$ we get $\sum_{m=1}^{\infty}\sum_{n=1}^{\infty}\left|\frac{\varphi_{rs}y_{mn}a_{mn}}{\Delta^{m}\lambda_{mn} \left(m+n\right)!}\right|< \infty.$ by (3.2). Thus $\left(\varphi_{rs}a_{mn}\right)\in 
\left[\chi^{2qu}_{f\mu},\left\|\mu_{mn}\left(x\right),\left(d\left(x_{1}\right),d\left(x_{2}\right),\cdots, d\left(x_{n-1}\right)\right)\right\|_{p}^{\varphi
}\right]^{V}=\left[\chi^{2qu\mu}_{g},\left\|\mu_{mn}\left(x\right),\left(d\left(x_{1}\right),d\left(x_{2}\right),\cdots, d\left(x_{n-1}\right)\right)\right\|_{p}^{\varphi}\right]^{V}$
and hence \\$\left(a_{mn}\right)\in \left[\chi^{2qu\mu}_{g},\left\|\mu_{mn}\left(x\right),\left(d\left(x_{1}\right),d\left(x_{2}\right),\cdots, d\left(x_{n-1}\right)\right)\right\|_{p}^{\varphi}\right]^{V}.$ This gives that 
\begin{equation}
\left[\chi^{2qu}_{f\mu},\left\|\mu_{mn}\left(x\right),\left(d\left(x_{1}\right),d\left(x_{2}\right),\cdots, d\left(x_{n-1}\right)\right)\right\|_{p}^{\varphi}\right]^{V\alpha}\subset \left[\chi^{2qu\mu}_{g},\left\|\mu_{mn}\left(x\right),\left(d\left(x_{1}\right),d\left(x_{2}\right),\cdots, d\left(x_{n-1}\right)\right)\right\|_{p}^{\varphi}\right]^{V}
\end{equation} 
we are granted with (3.1) and (3.3) \\$\left[\chi^{2qu}_{f\mu},\left\|\mu_{mn}\left(x\right),\left(d\left(x_{1}\right),d\left(x_{2}\right),\cdots, d\left(x_{n-1}\right)\right)\right\|_{p}^{\varphi}\right]^{V\alpha}= \\\left[\chi^{2qu\mu}_{g},\left\|\mu_{mn}\left(x\right),\left(d\left(x_{1}\right),d\left(x_{2}\right),\cdots, d\left(x_{n-1}\right)\right)\right\|_{p}^{\varphi}\right]^{V}$
\\(ii) Similarly, one can prove that $\left[\chi^{2qu\mu}_{g},\left\|\mu_{mn}\left(x\right),\left(d\left(x_{1}\right),d\left(x_{2}\right),\cdots, d\left(x_{n-1}\right)\right)\right\|_{p}^{\varphi}\right]^{V \alpha}\subset \left[\chi^{2qu}_{f\mu},\left\|\mu_{mn}\left(x\right),\left(d\left(x_{1}\right),d\left(x_{2}\right),\cdots, d\left(x_{n-1}\right)\right)\right\|_{p}^{\varphi}\right]^{V}$ if the sequence $\left(g_{mn}\right)$ satisfies uniform $\Delta_{2}-$ condition.
\subsection{Proposition} If $0<q_{mn}<p_{mn}<\infty$ for each $m$ and $n,$ then 
\\$\left[\Lambda^{2qu}_{f\mu},\left\|\mu_{mn}\left(x\right),\left(d\left(x_{1}\right),d\left(x_{2}\right),\cdots, d\left(x_{n-1}\right)\right)\right\|_{p}^{\varphi}\right]^{V}\subseteq \\\left[\Lambda^{2pu}_{f\mu},\left\|\mu_{mn}\left(x\right),\left(d\left(x_{1}\right),d\left(x_{2}\right),\cdots, d\left(x_{n-1}\right)\right)\right\|_{p}^{\varphi}\right]^{V}$ 
\\\textbf{Proof:} The proof is standard, so we omit it.
\subsection{Proposition} (i) If $0<inf q_{mn}\leq q_{mn}<1$ then \\$\left[\Lambda^{2qu}_{f\mu},\left\|\mu_{mn}\left(x\right),\left(d\left(x_{1}\right),d\left(x_{2}\right),\cdots, d\left(x_{n-1}\right)\right)\right\|_{p}^{\varphi}\right]^{V}\subset \\\left[\Lambda^{2u}_{f\mu},\left\|\mu_{mn}\left(x\right),\left(d\left(x_{1}\right),d\left(x_{2}\right),\cdots, d\left(x_{n-1}\right)\right)\right\|_{p}^{\varphi}\right]^{V}.$
\\(ii) If $1\leq q_{mn}\leq supq_{mn}< \infty,$ then $\left[\Lambda^{2u}_{f\mu},\left\|\mu_{mn}\left(x\right),\left(d\left(x_{1}\right),d\left(x_{2}\right),\cdots, d\left(x_{n-1}\right)\right)\right\|_{p}^{\varphi}\right]^{V}\subset \left[\Lambda^{2qu}_{f\mu},\left\|\mu_{mn}\left(x\right),\left(d\left(x_{1}\right),d\left(x_{2}\right),\cdots, d\left(x_{n-1}\right)\right)\right\|_{p}^{\varphi}\right]^{V}$
\\\textbf{Proof:} The proof is standard, so we omit it. 
\subsection{Proposition} Let $f^{'}=\left(f^{'}_{mn}\right)$ and $f^{''}=\left(f^{''}_{mn}\right)$ are sequences of Musielak functions, we have 
\\$\left[\Lambda^{2qu}_{f^{'}\mu},\left\|\mu_{mn}\left(x\right),\left(d\left(x_{1}\right),d\left(x_{2}\right),\cdots, d\left(x_{n-1}\right)\right)\right\|_{p}^{\varphi}\right]^{V}\bigcap \\\left[\Lambda^{2qu}_{f^{''}\mu},\left\|\mu_{mn}\left(x\right),\left(d\left(x_{1}\right),d\left(x_{2}\right),\cdots, d\left(x_{n-1}\right)\right)\right\|_{p}^{\varphi}\right]^{V}\subseteq \\\left[\Lambda^{2qu}_{f^{'}+f^{''}\mu},\left\|\mu_{mn}\left(x\right),\left(d\left(x_{1}\right),d\left(x_{2}\right),\cdots, d\left(x_{n-1}\right)\right)\right\|_{p}^{\varphi}\right]^{V}$
\\\textbf{Proof:} The proof is easy so we omit it.
\subsection{Proposition} For any sequence of Musielak functions $f=\left(f_{mn}\right)$ and $q=\left(q_{mn}\right)$ be double analytic sequence of strictly positive real numbers. Then 
\\$\left[\chi^{2qu}_{f\mu},\left\|\mu_{mn}\left(x\right),\left(d\left(x_{1}\right),d\left(x_{2}\right),\cdots, d\left(x_{n-1}\right)\right)\right\|_{p}^{\varphi}\right]^{V}\subset \\\left[\Lambda^{2qu}_{f\mu},\left\|\mu_{mn}\left(x\right),\left(d\left(x_{1}\right),d\left(x_{2}\right),\cdots, d\left(x_{n-1}\right)\right)\right\|_{p}^{\varphi}\right]^{V}.$ 
\\\textbf{Proof:} The proof is easy so we omit it.
\subsection{Proposition} The sequence space  $\left[\Lambda^{2qu}_{f\mu},\left\|\mu_{mn}\left(x\right),\left(d\left(x_{1}\right),d\left(x_{2}\right),\cdots, d\left(x_{n-1}\right)\right)\right\|_{p}^{\varphi}\right]^{V}c$ is solid
\\\textbf{Proof:} Let $x=\left(x_{mn}\right)\in \left[\Lambda^{2qu}_{f\mu},\left\|\mu_{mn}\left(x\right),\left(d\left(x_{1}\right),d\left(x_{2}\right),\cdots, d\left(x_{n-1}\right)\right)\right\|_{p}^{\varphi}\right]^{V},$ (i.e)
\begin{center}
$sup_{mn}\left[\Lambda^{2qu}_{f\mu},\left\|\mu_{mn}\left(x\right),\left(d\left(x_{1}\right),d\left(x_{2}\right),\cdots, d\left(x_{n-1}\right)\right)\right\|_{p}^{\varphi}\right]^{V}< \infty.$
\end{center}
Let $\left(\alpha_{mn}\right)$ be double sequence of scalars such that $\left|\alpha_{mn}\right|\leq 1$ for all $m,n\in \mathbb{N}\times \mathbb{N}.$ Then we get 
\\$sup_{mn} \left[\Lambda^{2qu}_{f\mu},\left\|\mu_{mn}\left(\alpha  x\right),\left(d\left(x_{1}\right),d\left(x_{2}\right),\cdots, d\left(x_{n-1}\right)\right)\right\|_{p}^{\varphi}\right]^{V}\leq  \\
\sup_{mn} \left[\Lambda^{2qu}_{f\mu},\left\|\mu_{mn}\left(x\right),\left(d\left(x_{1}\right),d\left(x_{2}\right),\cdots, d\left(x_{n-1}\right)\right)\right\|_{p}^{\varphi}\right]^{V}.$ This completes the proof.
\subsection{Proposition} The sequence space $\left[\Lambda^{2qu}_{f\mu},\left\|\mu_{mn}\left(x\right),\left(d\left(x_{1}\right),d\left(x_{2}\right),\cdots, d\left(x_{n-1}\right)\right)\right\|_{p}^{\varphi}\right]^{V}$ is monotone
\\\textbf{Proof:} The proof follows from Proposition 3.8.
\subsection{Proposition} If $f=\left(f_{mn}\right)$ be any  Musielak function. Then  
\\$\left[\Lambda^{2qu}_{f\mu},\left\|\mu_{mn}\left(x\right),\left(d\left(x_{1}\right),d\left(x_{2}\right),\cdots, d\left(x_{n-1}\right)\right)\right\|_{p}^{\varphi^{*}}\right]^{V}\subset\\ \left[\Lambda^{2qu}_{f\mu},\left\|\mu_{mn}\left(x\right),\left(d\left(x_{1}\right),d\left(x_{2}\right),\cdots, d\left(x_{n-1}\right)\right)\right\|_{p}^{\varphi^{**}}\right]^{V}$ if and only if $sup_{r,s\geq 1}\frac{\varphi_{rs}^{*}}{\varphi_{rs}^{**}}< \infty.$
\\\textbf{Proof:} Let $x\in \left[\Lambda^{2qu}_{f\mu},\left\|\mu_{mn}\left(x\right),\left(d\left(x_{1}\right),d\left(x_{2}\right),\cdots, d\left(x_{n-1}\right)\right)\right\|_{p}^{\varphi^{*}}\right]^{V}$ and $N=sup_{r,s \geq1}\frac{\varphi_{rs}^{*}}{\varphi_{rs}^{**}}< \infty.$ Then we get 
\\$\left[\Lambda^{2qu}_{f\mu},\left\|\mu_{mn}\left(x\right),\left(d\left(x_{1}\right),d\left(x_{2}\right),\cdots, d\left(x_{n-1}\right)\right)\right\|_{p}^{\varphi_{rs}^{**}}\right]^{V}=\\\\N \left[\Lambda^{2qu}_{f\mu},\left\|\mu_{mn}\left(x\right),\left(d\left(x_{1}\right),d\left(x_{2}\right),\cdots, d\left(x_{n-1}\right)\right)\right\|_{p}^{\varphi_{rs}^{*}}\right]^{V}=0.$
\\Thus $x\in \left[\Lambda^{2qu}_{f\mu},\left\|\mu_{mn}\left(x\right),\left(d\left(x_{1}\right),d\left(x_{2}\right),\cdots, d\left(x_{n-1}\right)\right)\right\|_{p}^{\varphi^{**}}\right]^{V}.$ Conversely, suppose that 
\\$\left[\Lambda^{2qu}_{f\mu},\left\|\mu_{mn}\left(x\right),\left(d\left(x_{1}\right),d\left(x_{2}\right),\cdots, d\left(x_{n-1}\right)\right)\right\|_{p}^{\varphi^{*}}\right]^{V}\subset \\\\\left[\Lambda^{2qu}_{f\mu},\left\|\mu_{mn}\left(x\right),\left(d\left(x_{1}\right),d\left(x_{2}\right),\cdots, d\left(x_{n-1}\right)\right)\right\|_{p}^{\varphi^{**}}\right]^{V}$ and \\$x\in \left[\Lambda^{2qu}_{f\mu},\left\|\mu_{mn}\left(x\right),\left(d\left(x_{1}\right),d\left(x_{2}\right),\cdots, d\left(x_{n-1}\right)\right)\right\|_{p}^{\varphi^{*}}\right]^{V}.$ Then \\$\left[\Lambda^{2qu}_{f\mu},\left\|\mu_{mn}\left(x\right),\left(d\left(x_{1}\right),d\left(x_{2}\right),\cdots, d\left(x_{n-1}\right)\right)\right\|_{p}^{\varphi^{*}}\right]^{V}< \epsilon,$ for every $\epsilon>0.$ Suppose that $sup_{r,s\geq 1}\frac{\varphi_{rs}^{*}}{\varphi_{rs}^{**}}=\infty,$ then there exists a sequence of members $\left(rs_{jk}\right)$ such that $lim_{j,k\rightarrow \infty}\frac{\varphi_{jk}^{*}}{\varphi_{jk}^{**}}=\infty.$ Hence, we have \\$\left[\Lambda^{2qu}_{f\mu},\left\|\mu_{mn}\left(x\right),\left(d\left(x_{1}\right),d\left(x_{2}\right),\cdots, d\left(x_{n-1}\right)\right)\right\|_{p}^{\varphi_{rs}^{*}}\right]^{V}=\infty.$ Therefore 
\\$x\notin \left[\Lambda^{2qu}_{f\mu},\left\|\mu_{mn}\left(x\right),\left(d\left(x_{1}\right),d\left(x_{2}\right),\cdots, d\left(x_{n-1}\right)\right)\right\|_{p}^{\varphi^{**}}\right]^{V},$ which is a contradiction. This completes the proof.
\subsection{Proposition} If $f=\left(f_{mn}\right)$ be any  Musielak function. Then  
\\$\left[\Lambda^{2qu}_{f\mu},\left\|\mu_{mn}\left(x\right),\left(d\left(x_{1}\right),d\left(x_{2}\right),\cdots, d\left(x_{n-1}\right)\right)\right\|_{p}^{\varphi^{*}}\right]^{V} =\\ \left[\Lambda^{2qu}_{f\mu},\left\|\mu_{mn}\left(x\right),\left(d\left(x_{1}\right),d\left(x_{2}\right),\cdots, d\left(x_{n-1}\right)\right)\right\|_{p}^{\varphi^{**}}\right]^{V}$ if and only if $sup_{r,s\geq 1}\frac{\varphi_{rs}^{*}}{\varphi_{rs}^{**}}< \infty, sup_{r,s\geq1}\frac{\varphi_{rs}^{**}}{\varphi_{rs}^{*}}> \infty.$
\\\textbf{Proof: } It is easy to prove so we omit.
\subsection{Proposition} The sequence space $\left[\chi^{2qu}_{f\mu},\left\|\mu_{mn}\left(x\right),\left(d\left(x_{1}\right),d\left(x_{2}\right),\cdots, d\left(x_{n-1}\right)\right)\right\|_{p}^{\varphi}\right]^{V}$ is not solid
\\\textbf{Proof:} The result follows from the following example.
\\\textbf{Example:} Consider 
\\$x=\left(x_{mn}\right)=\begin{pmatrix}
  1 & 1 &  .  . .  & 1 \\
  1 & 1 &  .  . .  & 1 \\
  . & \\
  . & \\
  . & \\
  1 & 1 & .  . .  & 1
  
\end{pmatrix}\in \left[\chi^{2qu}_{f\mu},\left\|\mu_{mn}\left(x\right),\left(d\left(x_{1}\right),d\left(x_{2}\right),\cdots, d\left(x_{n-1}\right)\right)\right\|_{p}^{\varphi}\right]^{V}.$ Let 
\\\\$\alpha_{mn}=\begin{pmatrix}
  -1^{m+n} & -1^{m+n} &  .  . .  & -1^{m+n} \\
  -1^{m+n} & -1^{m+n} &  .  . .  & -1^{m+n} \\
  . & \\
  . & \\
  . & \\
  -1^{m+n} & -1^{m+n} & .  . .  & -1^{m+n}
  
\end{pmatrix},$ for all $m,n\in \mathbb{N}.$ 
\\\\Then $\alpha_{mn} x_{mn}\notin  \left[\chi^{2qu}_{f\mu},\left\|\mu_{mn}\left(x\right),\left(d\left(x_{1}\right),d\left(x_{2}\right),\cdots, d\left(x_{n-1}\right)\right)\right\|_{p}^{\varphi}\right]^{V}.$ Hence 
\\ $\left[\chi^{2qu}_{f\mu},\left\|\mu_{mn}\left(x\right),\left(d\left(x_{1}\right),d\left(x_{2}\right),\cdots, d\left(x_{n-1}\right)\right)\right\|_{p}^{\varphi}\right]^{V}$ is not solid.
\subsection{Proposition} The sequence space $\left[\chi^{2qu}_{f\mu},\left\|\mu_{mn}\left(x\right),\left(d\left(x_{1}\right),d\left(x_{2}\right),\cdots, d\left(x_{n-1}\right)\right)\right\|_{p}^{\varphi}\right]^{V}$ is not monotone
\\\textbf{Proof:} The proof follows from Proposition 3.12.
\\\indent A sequence $x=\left(x_{mn}\right)$ is said to be $\varphi-$ statistically convergent or $s_{\varphi}-$ statistically convergent to 0 if for every $\epsilon >0,$
\begin{center}
$lim_{rs}\left|\left\{u_{mn}\left[f_{mn}\left(\left\|\mu_{mn}\left(x\right),\left(d\left(x_{1}\right),d\left(x_{2}\right),\cdots, d\left(x_{n-1}\right)\right)\right\|_{p}\right)\right]^{q_{mn}}\right|\geq \epsilon \right\}=0$
\end{center}
where the vertical bars indicates the number of elements in the enclosed set. In this case we write $s_{\varphi}-limx=0$ or $x_{mn}\rightarrow 0\left(s_{\varphi}\right)$ and $s_{\varphi}=\left\{x:\exists 0\in \mathbb{R}:s_{\varphi}-lim x=0 \right\}.$
\subsection{Proposition} For any sequence of Musielak functions $f=\left(f_{mn}\right)$ and $q=\left(q_{mn}\right)$ be double analytic sequence of strictly positive real numbers. Then 
\\$\left[\chi^{2qu}_{f\mu},\left\|\mu_{mn}\left(x\right),\left(d\left(x_{1}\right),d\left(x_{2}\right),\cdots, d\left(x_{n-1}\right)\right)\right\|_{p}^{\varphi}\right]^{V}\subset \\\left[s^{2qu}_{\varphi f\mu},\left\|\mu_{mn}\left(x\right),\left(d\left(x_{1}\right),d\left(x_{2}\right),\cdots, d\left(x_{n-1}\right)\right)\right\|_{p}^{\varphi}\right]^{V}.$  
\\\textbf{Proof:}Let $x\in \left[\chi^{2qu}_{f\mu},\left\|\mu_{mn}\left(x\right),\left(d\left(x_{1}\right),d\left(x_{2}\right),\cdots, d\left(x_{n-1}\right)\right)\right\|_{p}^{\varphi}\right]^{V}$ and $\epsilon>0.$ Then 
\\$u_{mn}\left[f_{mn}\left(\left\|\mu_{mn}\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]^{q_{mn}}\geq \\\left|\left\{u_{mn}\left[f_{mn}\left(\left\|\mu_{mn}\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]^{q_{mn}}\right|\geq \epsilon \right\}$ 
\\from which it follows that $x\in \left[s^{2qu}_{\varphi f\mu},\left\|\mu_{mn}\left(x\right),\left(d\left(x_{1}\right),d\left(x_{2}\right),\cdots, d\left(x_{n-1}\right)\right)\right\|_{p}^{\varphi}\right]^{V}.$
\\\indent To show that $\left[s^{2qu}_{\varphi f\mu},\left\|\mu_{mn}\left(x\right),\left(d\left(x_{1}\right),d\left(x_{2}\right),\cdots, d\left(x_{n-1}\right)\right)\right\|_{p}^{\varphi}\right]^{V}$ strictly contain \\$\left[\chi^{2qu}_{f\mu},\left\|\mu_{mn}\left(x\right),\left(d\left(x_{1}\right),d\left(x_{2}\right),\cdots, d\left(x_{n-1}\right)\right)\right\|_{p}^{\varphi}\right]^{V}.$ We define $x=\left(x_{mn}\right)$ by $\left(x_{mn}\right)=mn$ if $rs-\left[\sqrt{\varphi_{rs}}\right]+\leq mn\leq rs$ and $\left(x_{mn}\right)=0$ otherwise. Then \\$x\notin \left[\Lambda^{2qu}_{f\mu},\left\|\mu_{mn}\left(x\right),\left(d\left(x_{1}\right),d\left(x_{2}\right),\cdots, d\left(x_{n-1}\right)\right)\right\|_{p}^{\varphi}\right]^{V}$ and for every $\epsilon \left(0<\epsilon\leq 1\right),$
\begin{center}
$\left|\left\{u_{mn}\left[f_{mn}\left(\left\|\mu_{mn}\left(x\right),\left(d\left(x_{1}\right),d\left(x_{2}\right),\cdots, d\left(x_{n-1}\right)\right)\right\|_{p}\right)\right]^{q_{mn}}\right|\geq \epsilon \right\}=\frac{\left[\sqrt{\varphi_{rs}}\right]}{\varphi_{rs}}\rightarrow 0$ as $r,s\rightarrow \infty$
\end{center}
i.e $x\rightarrow 0\left(\left[s^{2qu}_{\varphi f\mu},\left\|\mu_{mn}\left(x\right),\left(d\left(x_{1}\right),d\left(x_{2}\right),\cdots, d\left(x_{n-1}\right)\right)\right\|_{p}^{\varphi}\right]^{V}\right),$ where $\left[\right]$ denotes the greatest integer function. On the other hand, 
\begin{center}
$u_{mn}\left[f_{mn}\left(\left\|\mu_{mn}\left(x\right),\left(d\left(x_{1}\right),d\left(x_{2}\right),\cdots, d\left(x_{n-1}\right)\right)\right\|_{p}\right)\right]^{q_{mn}}\rightarrow \infty$ as $r,s\rightarrow \infty$
\end{center}
i.e $x_{mn}\not\rightarrow 0\left[\chi^{2qu}_{f\mu},\left\|\mu_{mn}\left(x\right),\left(d\left(x_{1}\right),d\left(x_{2}\right),\cdots, d\left(x_{n-1}\right)\right)\right\|_{p}^{\varphi}\right]^{V}.$ This completes the proof.
\\\\\textbf{Competing Interests: } Authors have declared that no competing interests exist.
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