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\title[The Generalized Non-absolute type of sequence spaces ] {The Generalized Non-absolute type of sequence spaces}
\vspace{15mm}

\author[N. Subramanian]{N. Subramanian$^3$  }
\address{$^3$Department of Mathematics,
\newline \indent SASTRA University,
\newline \indent Thanjavur-613 401, India}

\email{nsmaths@yahoo.com} 

\author[M.R.Bivin]{M.R.Bivin $^1$ }
\address{$^1$ Department of Mathematics,
\newline \indent Care Group of Institutions,
\newline \indent Trichirappalli-620 009, India}
 
\email{mrbivin@gmail.com} 


\author[N. Saivaraju]{N. Saivaraju $^2$ }
\address{$^2$ Department of Mathematics,
\newline \indent Sri Angalamman College of Engineering and Technology,
\newline \indent Trichirappalli-621 105, India}

\email{saivaraju@yahoo.com}


\keywords{ analytic sequence, double sequences, $\chi^{2}$ space, difference sequence space,Musielak - modulus function, $p-$ metric space, Ideal; ideal convergent; fuzzy number; multiplier
space; non-absolute type.   \\
\indent 2010 {\it Mathematics Subject Classification}. 40A05; 40C05; 46A45; 03E72; 46B20.\\
\indent {\it Received}: \\
\indent {\it Revised}: \\}



\begin{abstract}
In this paper we introduce the notion of $\lambda_{mn}-\chi^{2}$ and $\Lambda^{2}$ sequences. Further, we introduce the spaces $\left[\chi^{2q\lambda}_{f\mu },\left\|\left(d\left(x_{1},0\right),d\left(x_{2},0\right),\cdots, d\left(x_{n-1},0\right)\right)\right\|_{p}\right]^{\textit{I}\left(F\right)}$ and $\left[\Lambda^{2q\lambda}_{f\mu },\left\|\left(d\left(x_{1},0\right),d\left(x_{2},0\right),\cdots, d\left(x_{n-1},0\right)\right)\right\|_{p}\right]^{\textit{I}\left(F\right)},$ which are of non-absolute type and we prove that these spaces are linearly isomorphic to the spaces $\chi^{2}$ and $\Lambda^{2},$ respectively. Moreover, we establish some inclusion relations between 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 [1]. Later on, they were investigated by Hardy [2], Moricz [3], Moricz and Rhoades [4], Basarir and Solankan [5], Tripathy [6], Turkmenoglu [7], 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 [8,9] 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 [10] has essentially studied both the theory of topological double sequence spaces and the theory of summability of double sequences. Mursaleen and Edely [11] and Tripathy 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 [12] 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 [13] 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 [14] 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 [15] as an extension of the definition of strongly Ces$\grave{a}$ro summable sequences. Connor [16] 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 [17] the notion of convergence of double sequences was presented by A. Pringsheim. Also, in [18]-[19], and [20] 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 $\phi=\left\{finite\hspace{0.1cm} sequences\right\}.$ \\\\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 [21]]
\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 [22] 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 $t_{f}$ is defined as follows                                 
\begin{center}
$t_{f}=\left\{x\in w^{2}:M_{f}\left(\left|x_{mn}\right|\right)^{1/m+n}\rightarrow 0\hspace{0.05cm}as\hspace{0.05cm}m,n\rightarrow \infty\right\},$  
\end{center} 
where $M_{f}$ is a convex modular defined by 
\begin{center}
$M_{f}\left(x\right)=\sum_{m=1}^{\infty}\sum_{n=1}^{\infty}f_{mn}\left(\left|x_{mn}\right|\right)^{1/m+n}, x=\left(x_{mn}\right)\in t_{f}.$
\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}
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 [1]. 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 $B^{\mu}$ difference operator is equivalent to the following binomial representation:
$B^{\mu}x_{mn}=\sum_{i=0}^{m}\sum_{j=0}^{m}\left(-1\right)^{i+j}\left(\stackrel{m} {i}\right)\left(\stackrel{m} {j}\right)x_{m+i,n+j}.$
\\\indent Let $n\in \mathbb{N}$ and $X$ be a real vector space of dimension $w,$ where $n\leq m.$ A real valued function $ d_p(x_1,\dots,x_n) = \|(d_1(x_1,0), \dots, d_n(x_n,0))\|_p$ on $X$ satisfying the following four conditions:
\\(i) $\|(d_1(x_1,0), \dots, d_n(x_n,0))\|_p=0$ if and only if  $d_1(x_1,0), \dots, d_n(x_n,0)$ are linearly dependent,
\\(ii) $\|(d_1(x_1,0), \dots, d_n(x_n,0))\|_p$ is invariant under permutation,
\\(iii) $\|(\alpha d_1(x_1,0), \dots, d_n(x_n,0))\|_p=\left|\alpha \right|\|(d_1(x_1,0), \dots, d_n(x_n,0))\|_p ,\alpha\in \mathbb{R}$  
\\(iv) $d_{p} \left( (x_{1}, y_{1}) , (x_{2}, y_{2})\cdots (x_{n}, y_{n}) \right) = \left( d_{X} (x_{1}, x_{2},\cdots x_{n})^{p} + d_{Y} (y_{1}, y_{2}, \cdots y_{n})^{p} \right)^{1/p} for 1 \leq p < \infty;$ (or)
\\(v) $d\left( (x_{1}, y_{1}) , (x_{2}, y_{2}),\cdots (x_{n}, y_{n}) \right) := \sup \left\{ d_{X} (x_{1}, x_{2}, \cdots x_{n}), d_{Y} (y_{1}, y_{2},\cdots y_{n}) \right\},$\\ for $x_{1}, x_{2},\cdots x_{n} \in X, y_{1}, y_{2},\cdots y_{n} \in Y$ is called the $p$ product metric of the Cartesian product of $n$ metric spaces 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_1,0), \dots, d_n(x_n,0))\|_E = sup \left(| det(d_{mn}\left(x_{mn},0\right))|\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},0\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.
\section{Notion of $\lambda_{mn}-$ double chi and double analytic sequences}
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 $B^{\mu}_{\eta}\left(x\right)\rightarrow 0\hspace{0.05cm}as\hspace{0.05cm}m,n\rightarrow \infty,$ where 
\\$B^{\mu}_{\eta}\left(x\right)=\frac{1}{\varphi_{rs}}\sum_{m\in I_{rs}}\sum_{n\in I_{rs} }
\left(\lambda_{m,n}-\lambda_{m,n+1}-\lambda_{m+1,n}+\lambda_{m+1,n+1}\right)\left(\left(m+n\right)!\left|\Delta^{m}x_{mn}\right|\right)^{1/m+n},$ where $\left(\left(m+n\right)!\left|\Delta^{m} x_{mn}\right|\right)^{1/m+n}=\\\left(m+n\right)!^{1/m+n}
\\\left(\Delta^{m-1}\lambda_{m,n}x_{mn}-\Delta^{m-1}\lambda_{m,n+1}x_{m,n+1}-\Delta^{m-1}\lambda_{m+1,n}x_{m+1,n}+\Delta^{m-1}\lambda_{m+1,n+1}x_{m+1,n+1}\right)^{1/m+n}.$ 
\\In particular, we say that $x$ is a $\lambda_{mn}-$ double gai sequence if $B_{\eta}^{\mu}\left(x\right)\rightarrow 0$ as $m,n\rightarrow \infty.$  Further we say that $x$ is $\lambda_{mn}-$ double analytic sequence if $sup_{mn}\left|B_{\eta}^{\mu}\left(x\right)\right|< \infty.$ We have 
\\$lim_{m,n\rightarrow \infty}\left|B_{\eta}^{\mu}\left(x\right)-a\right|=lim_{m,n\rightarrow \infty} 
\\\left|\frac{1}{\varphi_{rs}}\sum_{m\in I_{rs}}\sum_{n\in I_{rs} }
\left(\lambda_{m,n}-\lambda_{m,n+1}-\lambda_{m+1,n}+\lambda_{m+1,n+1}\right)\left(\left(m+n\right)!\left|\Delta^{m}x_{mn}\right|\right)^{1/m+n}\right|=0.$ So we can say that $lim_{m,n\rightarrow \infty}\left|B_{\eta}^{\mu}\left(x\right)\right|=a.$ Hence $x$ is $\lambda_{mn}-$ convergent to $a.$
\subsection{Lemma}Every convergent sequence is $\lambda_{mn}-$ convergent to the same ordinary limit.
\subsection{Lemma}If a $\lambda_{mn}-$ Musielak convergent sequence converges in the ordinary sense, then it
must Musielak converge to the same $\lambda_{mn}-$ limit.
\\\textbf{Proof: }Let $x=\left(x_{mn}\right)\in w^{2}$ and $m,n\geq 1.$ We have 
\\$\left(\left(m+n\right)!\left|\Delta^{m} x_{mn}\right|\right)^{1/m+n}-B_{\eta}^{\mu}\left(x\right)=\left(\left(m+n\right)!\left|\Delta^{m} x_{mn}\right|\right)^{1/m+n}-\\\frac{1}{\varphi_{rs}}\sum_{m\in I_{rs}}\sum_{n\in I_{rs} }
\left(\lambda_{m,n}-\lambda_{m,n+1}-\lambda_{m+1,n}+\lambda_{m+1,n+1}\right)\left(\left(m+n\right)!\left|\Delta^{m}x_{mn}\right|\right)^{1/m+n}$
\\=$\frac{1}{\varphi_{rs}}\sum_{m\in I_{rs}}\sum_{n\in I_{rs} }
\left(\lambda_{m,n}-\lambda_{m,n+1}-\lambda_{m+1,n}+\lambda_{m+1,n+1}\right)\\\sum_{m\in I_{rs}}\sum_{n\in I_{rs} }\left(\left(m+n\right)!\left(\Delta^{m-1}x_{mn}-\Delta^{m-1}x_{m,n+1}-\Delta^{m-1}x_{m+1,n}+\Delta^{m-1}x_{m+1,n+1}\right)\right)^{1/m+n}.$
\\=$\frac{1}{\varphi_{rs}}\sum_{m\in I_{rs}}\sum_{n\in I_{rs} }\left(\left(m+n\right)!\left(\Delta^{m-1}x_{mn}-\Delta^{m-1}x_{m,n+1}-\Delta^{m-1}x_{m+1,n}+\Delta^{m-1}x_{m+1,n+1}\right)\right)^{1/m+n}\\
\left(\lambda_{m,n}-\lambda_{m,n+1}-\lambda_{m+1,n}+\lambda_{m+1,n+1}\right).$
\\Therefore we have for every $x=\left(x_{mn}\right)\in w^{2}$ that $\left(\left(m+n\right)!\left|\Delta^{m} x_{mn}\right|\right)^{1/m+n}-B_{\eta}^{\mu}\left(x\right) =S_{mn}\left(x\right)\left(n,m\in \mathbb{N}\right).$ where the sequence $S\left(x\right)=\left(S_{mn}\left(x\right)\right)_{m,n=0}^{\infty}$ is defined by $S_{00}\left(x\right)=0$ and $S_{mn}\left(x\right)=\frac{1}{\varphi_{rs}}\sum_{m\in I_{rs}}\sum_{n\in I_{rs} }\\\left(\Delta^{m-1}x_{mn}-\Delta^{m-1}x_{m,n+1}-\Delta^{m-1}x_{m+1,n}+\Delta^{m-1}x_{m+1,n+1}\right)^{1/m+n}\\
\left(\lambda_{m,n}-\lambda_{m,n+1}-\lambda_{m+1,n}+\lambda_{m+1,n+1}\right),\left(n,m\geq 1\right).$ 
\subsection{Lemma}A $\lambda_{mn}-$ Musielak convergent sequence $x=\left(x_{mn}\right)$ converges if and only if $S\left(x\right)\in \left[\chi^{2}_{fB_{\eta}^{\mu}},\left\|\left(d\left(x_{1},0\right),d\left(x_{2},0\right),\cdots, d\left(x_{n-1},0\right)\right)\right\|_{p}\right]^{\textit{I}\left(F\right)}$ 
\\\textbf{Proof: } Let $x=\left(x_{mn}\right)$ be $\lambda_{mn}-$ Musielak convergent sequence. Then from Lemma 2.2 we have $x=\left(x_{mn}\right)$ converges to the same $\lambda_{mn}-$ limit. We obtain $S\left(x\right)\in \left[\chi^{2}_{fB_{\eta}^{\mu}},\left\|\left(d\left(x_{1},0\right),d\left(x_{2},0\right),\cdots, d\left(x_{n-1},0\right)\right)\right\|_{p}\right]^{\textit{I}\left(F\right)}.$ Conversely, \\let $S\left(x\right)\in \left[\chi^{2}_{fB_{\eta}^{\mu}},\left\|\left(d\left(x_{1},0\right),d\left(x_{2},0\right),\cdots, d\left(x_{n-1},0\right)\right)\right\|_{p}\right]^{\textit{I}\left(F\right)}.$ We have    
\begin{center}
$lim_{m,n\rightarrow \infty}\left(\left(m+n\right)!\left|\Delta^{m} x_{mn}\right|\right)^{1/m+n}=lim_{m,n\rightarrow \infty}B_{\eta}^{\mu}\left(x\right).$
\end{center}
From the above equation, we deduce that $\lambda_{mn}-$ convergent sequence $x=\left(x_{mn}\right)$ converges.
\subsection{Lemma}Every double analytic sequence is $\lambda_{mn}-$ double analytic.
\subsection{Lemma}A $\lambda_{mn}-$ Musielak analytic sequence $x=\left(x_{mn}\right)$ is analytic if and only if $S\left(x\right)\in \left[\Lambda^{2}_{fB_{\eta}^{\mu}},\left\|\left(d\left(x_{1},0\right),d\left(x_{2},0\right),\cdots, d\left(x_{n-1},0\right)\right)\right\|_{p}\right]^{\textit{I}\left(F\right)}$ 
\\\textbf{Proof: } From Lemma 2.4 and $S_{00}\left(x\right)=0$ and \\$S_{mn}\left(x\right)=\frac{1}{\varphi_{rs}}\sum_{m\in I_{rs}}\sum_{n\in I_{rs} }\\\left(\left(m+n\right)!\left(\Delta^{m-1}x_{mn}-\Delta^{m-1}x_{m,n+1}-\Delta^{m-1}x_{m+1,n}+\Delta^{m-1}x_{m+1,n+1}\right)\right)^{1/m+n}\\
\left(\lambda_{m,n}-\lambda_{m,n+1}-\lambda_{m+1,n}+\lambda_{m+1,n+1}\right),\left(n,m\geq 1\right).$ 
\section{The spaces of $\lambda_{mn}-$ double gai and double analytic sequences}
In this section we introduce the sequence space \\$\left[\chi^{2}_{f\Delta^{\lambda}_{mn}},\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]^{\textit{I}\left(F\right)}$ and \\$\left[\Lambda^{2}_{f\Delta^{\lambda}_{mn}},\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]^{\textit{I}\left(F\right)}$ as sets of $\lambda_{mn}$ double gai and  double analytic sequences:  
\\$\left[\chi^{2}_{f\Delta^{\lambda}_{mn}},\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]^{\textit{I}\left(F\right)}=\\lim_{m,n\rightarrow \infty}\left[B_{\eta}^{\mu},\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]^{\textit{I}\left(F\right)}=0$ 
\\$\left[\Lambda^{2}_{f\Delta^{\lambda}_{mn}},\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]^{\textit{I}\left(F\right)}=\\sup_{mn}\left[B_{\eta}^{\mu},\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]^{\textit{I}\left(F\right)}< \infty.$  
\subsection{Theorem} The sequence spaces $\left[\chi^{2}_{f\Delta^{\lambda}_{mn}},\left\|\left(d\left(x_{1},0\right),d\left(x_{2},0\right),\cdots, d\left(x_{n-1},0\right)\right)\right\|_{p}\right]^{\textit{I}\left(F\right)}$ and $\left[\Lambda^{2}_{f\Delta^{\lambda}_{mn}},\left\|\left(d\left(x_{1},0\right),d\left(x_{2},0\right),\cdots, d\left(x_{n-1},0\right)\right)\right\|_{p}\right]^{\textit{I}\left(F\right)}$ are isomorphic to the spaces $\left[\chi^{2}_{f},\left\|\left(d\left(x_{1},0\right),d\left(x_{2},0\right),\cdots, d\left(x_{n-1},0\right)\right)\right\|_{p}\right]^{\textit{I}\left(F\right)}$ and \\ $\left[\Lambda^{2}_{f},\left\|\left(d\left(x_{1},0\right),d\left(x_{2},0\right),\cdots, d\left(x_{n-1},0\right)\right)\right\|_{p}\right]^{\textit{I}\left(F\right)}$ 
\\\textbf{Proof}: We only consider the case $\left[\chi^{2}_{f\Delta^{\lambda}_{mn}},\left\|\left(d\left(x_{1},0\right),d\left(x_{2},0\right),\cdots, d\left(x_{n-1},0\right)\right)\right\|_{p}\right]^{\textit{I}\left(F\right)}\cong \\\left[\chi^{2}_{f},\left\|\left(d\left(x_{1},0\right),d\left(x_{2},0\right),\cdots, d\left(x_{n-1},0\right)\right)\right\|_{p}\right]^{\textit{I}\left(F\right)}$ and  \\$\left[\Lambda^{2}_{f\Delta^{\lambda}_{mn}},\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]^{\textit{I}\left(F\right)}\cong \\\left[\Lambda^{2}_{f},\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]^{\textit{I}\left(F\right)}$ can be shown similarly.
\\Consider the transformation  $T$ defined, $Tx=B_{\eta}^{\mu}\in \left[\chi^{2}_{f},\left\|\left(d\left(x_{1},0\right),d\left(x_{2},0\right),\cdots, d\left(x_{n-1},0\right)\right)\right\|_{p}\right]^{\textit{I}\left(F\right)}$ for every $x\in \left[\chi^{2}_{f\Delta^{\lambda}_{mn}},\left\|\left(d\left(x_{1},0\right),d\left(x_{2},0\right),\cdots, d\left(x_{n-1},0\right)\right)\right\|_{p}\right]^{\textit{I}\left(F\right)}.$ The linearity of $T$ is obvious. It is trivial that $x=0$ whenever $Tx=0$ and hence $T$ is injective.
\\To show surjective we define the sequence $x=\left\{x_{mn}\left(\lambda\right)\right\}$ by 
\begin{equation} 
B^{\mu}_{\eta}\left(x\right)=\frac{1}{\varphi_{rs}}\sum_{m\in I_{rs}}\sum_{n\in I_{rs} }
\left(\lambda_{m,n}-\lambda_{m,n+1}-\lambda_{m+1,n}+\lambda_{m+1,n+1}\right)\left(\left(m+n\right)!\left|\Delta^{m}x_{mn}\right|\right)^{1/m+n}=y_{mn} 
\end{equation}
We can say that $B^{\mu}_{\eta}\left(x\right)=y_{mn}$ from (3.1) and $x\in \left[\chi^{2}_{f},\left\|\left(d\left(x_{1},0\right),d\left(x_{2},0\right),\cdots, d\left(x_{n-1},0\right)\right)\right\|_{p}\right]^{\textit{I}\left(F\right)},$ hence $B^{\mu}_{\eta}\left(x\right) \in \left[\chi^{2}_{f},\left\|\left(d\left(x_{1},0\right),d\left(x_{2},0\right),\cdots, d\left(x_{n-1},0\right)\right)\right\|_{p}\right]^{\textit{I}\left(F\right)}.$ We deduce from that $x\in \left[\chi^{2}_{f\Delta^{\lambda}_{mn}},\left\|\left(d\left(x_{1},0\right),d\left(x_{2},0\right),\cdots, d\left(x_{n-1},0\right)\right)\right\|_{p}\right]^{\textit{I}\left(F\right)}$ and $Tx=y.$ Hence $T$ is surjective. We have for every $x\in \left[\chi^{2}_{f\Delta^{\lambda}_{mn}},\left\|\left(d\left(x_{1},0\right),d\left(x_{2},0\right),\cdots, d\left(x_{n-1},0\right)\right)\right\|_{p}\right]^{\textit{I}\left(F\right)}$ that 
$d\left(Tx,0\right)_{\chi^{2}}=d\left(Tx,0\right)_{\Lambda^{2}}=d\left(x,0\right)_{\left[\chi^{2}_{f\Delta^{\lambda}_{mn}},\left\|\left(d\left(x_{1},0\right),d\left(x_{2},0\right),\cdots, d\left(x_{n-1},0\right)\right)\right\|_{p}\right]^{\textit{I}\left(F\right)}.}.$ \\Hence $\left[\chi^{2}_{f\Delta^{\lambda}_{mn}},\left\|\left(d\left(x_{1},0\right),d\left(x_{2},0\right),\cdots, d\left(x_{n-1},0\right)\right)\right\|_{p}\right]^{\textit{I}\left(F\right)}$ and \\$\left[\chi^{2}_{f},\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]^{\textit{I}\left(F\right)}$ are ismorphic. Similarly obtain other sequence spaces.
\section{Some Inclusion and Relations}  
\subsection{Theorem} The inclusion $\left[\chi^{2}_{f\Delta_{mn}},\left\|\left(d\left(x_{1},0\right),d\left(x_{2},0\right),\cdots, d\left(x_{n-1},0\right)\right)\right\|_{p}\right]^{\textit{I}\left(F\right)}\subset \left[\chi^{2}_{f\Delta^{\lambda}_{mn}},\left\|\left(d\left(x_{1},0\right),d\left(x_{2},0\right),\cdots, d\left(x_{n-1},0\right)\right)\right\|_{p}\right]^{\textit{I}\left(F\right)}$ holds
\\\textbf{Proof:} Let $\left[\chi^{2}_{f\Delta_{mn}},\left\|\left(d\left(x_{1},0\right),d\left(x_{2},0\right),\cdots, d\left(x_{n-1},0\right)\right)\right\|_{p}\right]^{\textit{I}\left(F\right)}.$  Then we deduce that 
$\frac{1}{\varphi_{rs}}\sum_{m\in I_{rs}}\sum_{n\in I_{rs} }
\left(\lambda_{m,n}-\lambda_{m,n+1}-\lambda_{m+1,n}+\lambda_{m+1,n+1}\right)\left(\left(m+n\right)!\left|\Delta^{m}x_{mn}\right|\right)^{1/m+n}\leq \\\frac{1}{\varphi_{rs}}lim_{m,n\rightarrow \infty}\sum_{m\in I_{rs}}\sum_{n\in I_{rs} }
\left(\lambda_{m,n}-\lambda_{m,n+1}-\lambda_{m+1,n}+\lambda_{m+1,n+1}\right)\left(\left(m+n\right)!\left|\Delta^{m}x_{mn}\right|\right)^{1/m+n}=lim_{m,n\rightarrow \infty}\left(\left(m+n\right)!\left|\Delta^{m}x_{mn}\right|\right)^{1/m+n}=0.$ Hence \\$x\in \left[\chi^{2}_{f\Delta^{\lambda}_{mn}},\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]^{\textit{I}\left(F\right)}.$
\subsection{Theorem} The inclusion $\left[\Lambda^{2}_{f\Delta_{mn}},\left\|\left(d\left(x_{1},0\right),d\left(x_{2},0\right),\cdots, d\left(x_{n-1},0\right)\right)\right\|_{p}\right]^{\textit{I}\left(F\right)}\subset \left[\Lambda^{2}_{f\Delta^{\lambda}_{mn}},\left\|\left(d\left(x_{1},0\right),d\left(x_{2},0\right),\cdots, d\left(x_{n-1},0\right)\right)\right\|_{p}\right]^{\textit{I}\left(F\right)}$ holds.
\\Proof: It is obvious. Therefore omit the proof.
\subsection{Theorem} The inclusion $\left[\chi^{2}_{f},\left\|\left(d\left(x_{1},0\right),d\left(x_{2},0\right),\cdots, d\left(x_{n-1},0\right)\right)\right\|_{p}\right]^{\textit{I}\left(F\right)}\subset\\ \left[\chi^{2}_{f\Delta^{\lambda}_{mn}},\left\|\left(d\left(x_{1},0\right),d\left(x_{2},0\right),\cdots, d\left(x_{n-1},0\right)\right)\right\|_{p}\right]^{\textit{I}\left(F\right)}$ hold. Furthermore, the equalities hold if and only if $S\left(x\right)\in \left[\chi^{2}_{f},\left\|\left(d\left(x_{1},0\right),d\left(x_{2},0\right),\cdots, d\left(x_{n-1},0\right)\right)\right\|_{p}\right]^{\textit{I}\left(F\right)}$ for every sequence $x$ in the space $\left[\chi^{2}_{f\Delta^{\lambda}_{mn}},\left\|\left(d\left(x_{1},0\right),d\left(x_{2},0\right),\cdots, d\left(x_{n-1},0\right)\right)\right\|_{p}\right]^{\textit{I}\left(F\right)}$
\\\textbf{Proof:} Consider 
\begin{equation}
\left[\chi^{2}_{f},\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]^{\textit{I}\left(F\right)}\subset\\ \left[\chi^{2}_{f\Delta^{\lambda}_{mn}},\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]^{\textit{I}\left(F\right)}
\end{equation}
is obvious from Lemma 2.1. Then, we have for every \\$x\in \left[\chi^{2}_{f\Delta^{\lambda}_{mn}},\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]^{\textit{I}\left(F\right)}$ that \\$x\in \left[\chi^{2}_{f},\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]^{\textit{I}\left(F\right)}$ and hence \\$S\left(x\right)\in \left[\chi^{2}_{f},\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]^{\textit{I}\left(F\right)}$ by Lemma 2.3. Conversely, let $x\in \left[\chi^{2}_{f\Delta^{\lambda}_{mn}},\left\|\left(d\left(x_{1},0\right),d\left(x_{2},0\right),\cdots, d\left(x_{n-1},0\right)\right)\right\|_{p}\right]^{\textit{I}\left(F\right)}.$ Then, we have that $S\left(x\right)\in \left[\chi^{2}_{f},\left\|\left(d\left(x_{1},0\right),d\left(x_{2},0\right),\cdots, d\left(x_{n-1},0\right)\right)\right\|_{p}\right]^{\textit{I}\left(F\right)}.$ Thus, it follows by Lemma 2.3 and then Lemma 2.2, that $x\in \left[\chi^{2}_{f},\left\|\left(d\left(x_{1},0\right),d\left(x_{2},0\right),\cdots, d\left(x_{n-1},0\right)\right)\right\|_{p}\right]^{\textit{I}\left(F\right)}.$ We get 
\begin{equation} 
\left[\chi^{2}_{f\Delta^{\lambda}_{mn}},\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]^{\textit{I}\left(F\right)}\subset\\\left[\chi^{2}_{f},\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]^{\textit{I}\left(F\right)}
\end{equation}
From the equation (4.1) and (4.2) we get $\left[\chi^{2}_{f\Delta^{\lambda}_{mn}},\left\|\left(d\left(x_{1},0\right),d\left(x_{2},0\right),\cdots, d\left(x_{n-1},0\right)\right)\right\|_{p}\right]^{\textit{I}\left(F\right)}=\\\left[\chi^{2}_{f},\left\|\left(d\left(x_{1},0\right),d\left(x_{2},0\right),\cdots, d\left(x_{n-1},0\right)\right)\right\|_{p}\right]^{\textit{I}\left(F\right)}.$       
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