Some new classes of paranorm ideal convergent double sequences of sigma-bounded variation over n-normed spaces

Abstract

The sequence space $B{V}_{\sigma }$, the space of all sequence of $\sigma$-bounded variation, was firstly defined and studied by Mursaleen. Later on, Vakeel and Tabassum developed the same space to double sequences. Recently, using the concept of I-convergence, Vakeel and Vakeel et al. and others introduced many sequence spaces related to the space we just mentioned above which are defined by different operators. In this article, we keep the same direction up introducing some new classes of I-convergent double sequences of $\sigma$-bounded variation over n-normed spaces. In addition, we study some basic topological and algebraic properties of these classes. Also, we prove some inclusion relations on these classes.

Keywords:

• Invariant mean
• sequence of $\sigma$-bounded variation
• n-normed space
• paranormed space
• Orlicz function
• ideal
• filter
• I-convergence double sequence over n-normed space
• I-null over n-normed space
• I-Cauchy over n-normed space
• I-bounded over n-normed space
• solid space
• sequence algebra
• convergence free space

AMS Subject Classifications:

• 40 A05
• 40 A35
• 40 C05
• 46 A45
• 46 E30
• 46 B20

Public Interest Statement

The notion of ideal convergence is the most important development of the notion of usual convergence which played a big role for modelling uncertainty and vagueness in so many various problems in the field of science and engineering. On the other hand, the concept of n-normed spaces was developed, so that it became the focus of the researchers, interest. Quite recently, Vakeel Khan defined the space of all ideal convergent sequence of $\sigma$-bounded variation in space of real numbers. It was also obvious to define some new n-normed spaces of ideal convergent sequence of $\sigma$-bounded variation by using Orlicz function and study some topological and algebraic properties and some inclusion relations of these spaces which is our aim in this article.

1. Introduction and preliminaries

Depending on the concept of natural density of a subset of the set of natural numbers $\mathbb{N}$, Fast (1951) and Steinhaus (1951) were the first ones who introduced the concept of statistical convergence independently for the real sequences. The idea of two-dimensional analogue of natural density is employed to extend the concept of statistical convergence to double sequences by (1983). The notion of I-convergence (I denotes the ideal of subsets of $\mathbb{N}$), which is the generalization of statistical convergence, was introduced by Kostyrko, Wilczynski and Salat (2000) and further studied by many other researchers. The concept of 2-normed spaces was initially developed by Gahler (1964), while that of n-normed spaces one can see in (Misiak, 1989). Since then, many others have studied this concept and obtained various results, (Gunawan, 2001, 2001) and many others. The notion of I-convergence in 2-normed spaces was initially introduced by Gurdal (2006). Later on, it was extended to n-normed spaces by Gurdal and Sahnier (2014).

Definition 1.1

Let $n\in \mathbb{N}$ and X be a linear space of dimension d,  where $d\ge n\ge 2$ over the field $\mathbb{K}$ ($\mathbb{K}$ is the field of real or complex numbers). A real valued function $\Vert ·,\cdots ,·\Vert$ on $X^n$ satisfying the following four conditions:

(i)	$\Vert x_1,x_2,\cdots ,x_n\Vert =0$ if and only if $x_1,x_2,\cdots ,x_n$ are linearly dependent in X,
(ii)	$\Vert x_1,x_2,\cdots ,x_n\Vert$ is invariant under permutation,
(iii)	$\Vert \alpha x_1,x_2,\cdots ,x_n\Vert =|\alpha |\Vert x_1,x_2,\cdots ,x_n\Vert$ for any $\alpha \in \mathbb{K},$
(iv)	$\Vert x+x^{{}^{\prime }},x_2,\cdots ,x_n\Vert \le \Vert x,x_2,\cdots ,x_n\Vert +\Vert x^{{}^{\prime }},x_2,\cdots ,x_n\Vert$ is called an n-norm on X and the pair $(X,\Vert ·,\cdots ,·\Vert )$ is called an n-normed space over the field $\mathbb{K}$.

Example 1.1

If we take $X={\mathbb{R}}^n$ equipped with Euclidean n-norm $\Vert x_1,x_2,\cdots ,x_n{\Vert }_E=$ volume of n-dimensional parallelepiped spanned by vectors $(x_1,x_2,\cdots ,x_n)$, then the n-norm may be given by the formula $\Vert x_1,x_2,\cdots ,x_n{\Vert }_E=|x_{\mathit{ij}}|$, where $x_{\mathit{ij}}=(x_{i1},x_{i2},\cdots ,x_{\mathit{in}})$ for each $i=1,2,3,\cdots ,n.$

Let $(X,\Vert ·,\cdots ,·\Vert )$ be an n-normed space of dimension $d\ge n\ge 2$ and $\{a_1,a_2,\cdots ,a_n\}$ a linearly independent set in X. Then, the function $\Vert ·,\cdots ,·\Vert$ on $X^{n-1}$ is defined by(1.1) $$\begin{array}{c}\hfill \Vert x_1,x_2,\cdots ,x_n{\Vert }_{\infty }=\underset{1\le i\le n}{max}\{\Vert x_1,x_2,\cdots ,x_{n-1},a_i\Vert \},\end{array}$$(1.1)

defines as $(n-1)$-norm on X with respect to $\{a_1,a_2,\cdots ,a_n\}$, and this is known as the derived $(n-1)$-norm ( see Gunawan & Mashadi, 2001 for details). The standard n-norm on X is defined as(1.2) $$\begin{array}{cc}\hfill \Vert x_1,x_2,\cdots ,x_n{\Vert }_E& ={\left|\begin{array}{ccc}⟨x_1,x_1⟩& \cdots & ⟨x_1,x_n⟩\\ ⋮& \dots & ⋮\\ ⟨x_n,x_1⟩& \cdots & ⟨x_n,x_n⟩\end{array}\right|}^{\frac{1}{2}},\hfill \end{array}$$(1.2)

where $⟨·,·⟩$ denote the inner product on X. If $X={\mathbb{R}}^n$, then this n-norm is exactly the same Euclidean n-norm $\Vert x_1,x_2,\cdots ,x_n{\Vert }_E$ mentioned earlier. For $n=1$, this n-norm is the usual norm $\Vert x\Vert ={⟨x,x⟩}^{\frac{1}{2}}.$

Definition 1.2

A double sequence $(x_{\mathit{ij}})$ in an n-normed space $(X,\Vert ·,\cdots ,·\Vert )$ is said to be convergent to some $L\in X$ if there exists an positive integer N such that$$\begin{array}{c}\hfill \underset{i,j\to \infty }{lim}\Vert x_{\mathit{ij}}-L,z_1,\cdots ,z_{n-1}\Vert =0\text{for all}i,j\ge N,\text{for every}{z}_1,z_2,\cdots ,z_{n-1}\in X.\end{array}$$

Definition 1.3

A double sequence $(x_{\mathit{ij}})$ in an n-normed space $(X,\Vert ·,\cdots ,·\Vert )$ is said to be Cauchy with respect to the n-norm if there exist two positive integers $s=s(ϵ),t=t(ϵ)$ such that$$\begin{array}{c}\hfill \underset{i,j\to \infty }{lim}\Vert x_{\mathit{ij}}-x_{\mathit{st}},z_1,\cdots ,z_{n-1}\Vert =0\text{for every}{z}_1,z_2,\cdots ,z_{n-1}\in X.\end{array}$$

If every Cauchy sequence in X converges to some number $L\in X,$ then X is said to be complete with respect to the n-norm. Any complete n-normed space is said to be an n-Banach space.

Definition 1.4

A double sequence $(x_{\mathit{ij}})$ in an n-normed space $(X,\Vert ·,\cdots ,·\Vert )$ is said to be statistically convergent to $L\in X,$ if for each $ϵ>0,$ for every $z_1,\cdots ,z_{n-1}\in X,$ the double natural density of the set $\{i,j\in \mathbb{N}:\Vert x_{\mathit{ij}}-L,z_1,\cdots ,z_{n-1}\Vert \ge ϵ\}$ equal zero.

Definition 1.5

A function $M:[0,\infty )⟶[0,\infty )$ is said to be an Orlicz function if it satisfies the following conditions:

(i)	M is continuous, convex and non-decreasing.
(ii)	$M(0)=0,M(x)>0$ and $M(x)\to \infty$ as $x\to \infty .$

Remark 1.1

 

(i)	If the convexity of an Orlicz function is replaced by $M(x+y)\le M(x)+M(y)$ then this function is called Modulus function. (one may refer to Khan, Fatima, Abdullah & Khan, 2016)
(ii)	If M is an Orlicz function, then $M(\lambda x)\le \lambda M(x)$ for all $\lambda$ with $0<\lambda <1.$ (one may refer to Khan, Fatima, Abdullah & Khan, 2016)

An Orlicz function M is said to satisfy ${△}_2$-condition for all values of u if there exists a constant $K>0$ such that $M(Lu)\le KLM(u)$ for all values $L>1.$

Subsequently, Orlicz function was used to define sequence spaces by Parashar and Choudhary (1994).

The notation of paranormed sequence space was studied at the initial stage by Nakano (1951) and Simons (1965). Later on, it was further investigated by Maddox (1969), Lascarides (1971, 1983) and many others.

Definition 1.6

(see Maddox, 1989) Let X be a linear metric space. A function $g:X⟶\mathbb{R}$ is said to be paranorm, if for all $x,y\in X,$

(i)	$g(x)\ge 0$ for all $x\in X,$
(ii)	$g(-x)=g(x),$
(iii)	$g(x+y)\le g(x)+g(y)$ for all $x,y\in X,$
(iv)	If (${\lambda }_n$) is a sequence of scalars with ${\lambda }_n\to \lambda$ as $n\to \infty$ and $(x_n)$ is a sequence of vectors with $g(x_n-x)\to 0$ as $n\to \infty ,$ then $g({\lambda }_{n}x-\lambda x)\to 0$ as $n\to \infty .$

A paranorm g for which $g(x)=0$ implies that $x=0$ is called total paranorm and the pair (X, g) 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 [wilansky, 1984, Theorem 10.4.2, P-183]). For more details about sequence spaces, see Maddox (1969) and references therein.

Let ${\ell }_{\infty }$ and c denote the Banach spaces of bounded and convergent sequences $x=(x_k)$, respectively with the usual norm $\Vert x\Vert ={sup}_k|x_k|$. Let $\sigma$ be a one-to-one mapping from the set of positive integers into itself having no finite orbits for all positive integers and T be an operator on ${\ell }_{\infty }$ defined by $T_x=T(x_k)=(x_{\sigma (k)})$ for all $x=(x_k)\in {\ell }_{\infty }$. A continuous linear functional $\phi$ on ${\ell }_{\infty }$ is said to be an invariant mean or $\sigma$-mean if and only if:

(i)	$\phi (x)\ge 0$ where the sequence $x=(x_k)$ has $x_k\ge 0$ for all k.
(ii)	$\phi (e)=1$ where $e=\{1,1,1,\cdots \}.$
(iii)	$\phi (x_{\sigma (n)})=\phi (x)$ for all $x\in {\ell }_{\infty }.$

By $V_{\sigma }$, we denote the set of bounded sequences of all whose invariant means are equal. That is(1.3) $$\begin{array}{c}\hfill V_{\sigma }=\{x=(x_k):\sum _{m=1}^{\infty }{t}_{m,k}(x)=L\text{uniformly in}k,L=\sigma -limx\},\end{array}$$(1.3) (see Schaefer, 1972) where $m\ge 1,k>0$ and(1.4) $$\begin{array}{c}\hfill t_{m,k}(x)=\frac{x_k+x_{\sigma (k)}+\cdots +x_{{\sigma }^m(k)}}{m+1}\text{such that}{t}_{-1,k}=0,\end{array}$$(1.4)

where ${\sigma }^m(k)$ denotes $m^{\mathit{th}}$-iterate of $\sigma$ at k. Invariant means have recently been studied by Ahmad and Mursaleen (1988), Raimi (1988) and many others. Later on, the concept of invariant means for double sequences was defined in Mursaleen and Mohiuddine (2010).

Definition 1.7

Mursaleen (1983) A sequence $x\in {\ell }_{\infty }$ is of $\sigma$-bounded variation if and only if:

(i)	${\sum }_{m=1}^{\infty }|{\phi }_{m,k}(x)|$ converges uniformly in k,
(ii)	${\displaystyle \underset{m\to \infty }{lim}{t}_{m,k}(x)}$ which must exist, should take the same value for all k.

By $B{V}_{\sigma }$, we denote the space of all sequences of $\sigma$-bounded variation which was defined by Mursaleen (1983) as follow:(1.5) $$\begin{array}{c}\hfill B{V}_{\sigma }=\{x\in {\ell }_{\infty }:\sum _{m=1}^{\infty }|{\phi }_{m,k}(x)|<\infty ,\text{uniformly in}k\},\end{array}$$(1.5)

where(1.6) $$\begin{array}{c}\hfill {\phi }_{m,k}(x)=t_{m,k}(x)-t_{m-1,k}(x),\text{and}{t}_{-1,k}=0,\end{array}$$(1.6)

having the following properties, for any sequences x, y and scalar $\lambda$,

(i)	${\phi }_{m,k}(x+y)={\phi }_{m,k}(x)+{\phi }_{m,k}(y).$
(ii)	${\phi }_{m,k}(\lambda x)=\lambda {\phi }_{m,k}(x).$

Khan and Tabassum (2011) developed the same space we just mentioned above to double sequences. Later with the help of the concept of I-convergence, Khan, Esi and Shafiq (2014b) and many others have defined many different spaces related to the space $B{V}_{\sigma }$ that is being studied. For more details, see (Altinok, Altin & Isik, 2008; Isik, Altin & Et, 2013; Khan, 2008; Khan, Esi & Shafiq, 2014a; Khan, Fatima, Abdullah & Khan, 2016).

Definition 1.8

Let $\mathbb{N}\times \mathbb{N}$ be a non-empty set. A family of sets $I\subseteq 2^{\mathbb{N}\times \mathbb{N}}$ is said to be an ideal if :

(i)	$\mathrm{\varnothing }\in I$ ,
(ii)	I is additive i.e. for all $A,B\in I\Rightarrow A\cup B\in I,$
(iii)	I is hereditary i.e. for all $A\in I,B\subseteq A\Rightarrow B\in I$.

• An ideal $I\subseteq 2^{\mathbb{N}\times \mathbb{N}}$ is said to be non-trivial if $I\ne 2^{\mathbb{N}\times \mathbb{N}}.$

• A non-trivial ideal $I\subseteq 2^{\mathbb{N}\times \mathbb{N}}$ is said to be admissible if $I\supseteq \{\{x\}:x\in \mathbb{N}\times \mathbb{N}\}.$

• A non-trivial ideal $I\subseteq 2^{\mathbb{N}\times \mathbb{N}}$ is said to be maximal if there cannot exist any non-trivial ideal $J\ne I$ containing I as a subset.

Definition 1.9

Let $\mathbb{N}\times \mathbb{N}$ be a non-empty set. Then a family of sets $\mathcal{F}\subseteq 2^{\mathbb{N}\times \mathbb{N}}$ is said to be a filter on $\mathbb{N}\times \mathbb{N}$ if and only if

(i)	$\mathrm{\varnothing }\notin \mathcal{F},$
(ii)	For all $A,B\in \mathcal{F}\Rightarrow A\cap B\in \mathcal{F},$
(iii)	For all $A\in \mathcal{F}$ with $A\subseteq B\Rightarrow B\in \mathcal{F}$.

Remark 1.2

For each ideal I there is a filter $\mathcal{F}(I)$ which corresponding to I

( filter associate with ideal I), that is(1.7) $$\begin{array}{c}\hfill \mathcal{F}(I)=\{K\subseteq \mathbb{N}\times \mathbb{N}:K^c\in I\},\text{where}{K}^c=\mathbb{N}\times \mathbb{N}-K.\end{array}$$(1.7)

Definition 1.10

A double sequence $(x_{\mathit{ij}})$ in an n-normed space $(X,\Vert ·,\cdots ,·\Vert )$ is said to be I-convergent to a number $L\in \mathbb{R}$, if for every $ϵ>0$, the set(1.8) $$\begin{array}{c}\hfill \{(i,j)\in \mathbb{N}\times \mathbb{N}:\Vert (x_{\mathit{ij}})-L,z_1,z_2,\cdots ,z_{n-1}\Vert \ge ϵ\}\in I,\text{for every}{z}_1,z_2,\cdots ,z_{n-1}\in X.\end{array}$$(1.8)

and we write ${\displaystyle I-\underset{i,j}{lim}\Vert x_{\mathit{ij}},z_1,z_2,\cdots ,z_{n-1}\Vert =L.}$

Definition 1.11

A double sequence $(x_{\mathit{ij}})$ in an n-normed space $(X,\Vert ·,\cdots ,·\Vert )$ is said to be I-null if, for every $ϵ>0$, the set(1.9) $$\begin{array}{c}\hfill \{(i,j)\in \mathbb{N}\times \mathbb{N}:\Vert (x_{\mathit{ij}}),z_1,z_2,\cdots ,z_{n-1}\Vert \ge ϵ\}\in I,\text{for every}{z}_1,z_2,\cdots ,z_{n-1}\in X.\end{array}$$(1.9)

And we write ${\displaystyle I-\underset{i,j}{lim}\Vert x_{\mathit{ij}},z_1,z_2,\cdots ,z_{n-1}\Vert =0.}$

Definition 1.12

A double sequence $(x_{\mathit{ij}})$ in an n-normed space $(X,\Vert ·,\cdots ,·\Vert )$ is said to be I-Cauchy if, for each $ϵ>0$, there exists two numbers $s=s(ϵ)$ and $t=t(ϵ)$ such that the set$$\begin{array}{c}\hfill \{(i,j)\in \mathbb{N}\times \mathbb{N}:\Vert (x_{\mathit{ij}})-(x_{\mathit{st}}),z_1,z_2,\cdots ,z_{n-1}\Vert \ge ϵ\}\in I,\text{for every}{z}_1,z_2,\cdots ,z_1\in X.\end{array}$$

Definition 1.13

A double sequence $(x_{\mathit{ij}})$ in an n-normed space $(X,\Vert ·,\cdots ,·\Vert )$ is said to be I-bounded if there exists $M>0,$ such that, the set$$\begin{array}{c}\hfill \{(i,j)\in \mathbb{N}\times \mathbb{N}:\Vert x_{\mathit{ij}},z_1,z_2,\cdots ,z_{n-1}\Vert \ge M\}\in I,\text{for every}{z}_1,z_2,\cdots ,z_{n-1}\in X.\end{array}$$

Definition 1.14

A double sequence space E is said to be solid or normal, if $({\alpha }_{\mathit{ij}}{x}_{\mathit{ij}})\in E$ whenever $(x_{\mathit{ij}})\in E$ and for any double sequence of scalars $({\alpha }_{\mathit{ij}})$ with $|{\alpha }_{\mathit{ij}}|\le 1$, for all $(i,j)\in \mathbb{N}\times \mathbb{N}.$

Definition 1.15

A double sequence space E is said to be symmetric, if $(x_{\pi (i,j)})\in E$ whenever $(x_{\mathit{ij}})\in E$, where $\pi (i,j)$ is a permutation on $\mathbb{N}\times \mathbb{N}.$

Definition 1.16

A double sequence space E is said to be sequence algebra, if $(x_{\mathit{ij}})\ast (y_{\mathit{ij}})=(x_{\mathit{ij}}.y_{\mathit{ij}})\in E$ whenever $(x_{\mathit{ij}}),(y_{\mathit{ij}})\in E.$

Definition 1.17

A double sequence space E is said to be convergence free, if $(y_{\mathit{ij}})\in E$ whenever $(x_{\mathit{ij}})\in E$ and $x_{\mathit{ij}}=0$ implies that $y_{\mathit{ij}}=0$ for all $(i,j)\in \mathbb{N}\times \mathbb{N}.$

Definition 1.18

Let $K=\{(i_s,j_t)\in \mathbb{N}\times \mathbb{N}:i_1<i_2<\cdots \text{and}{j}_1<j_2<\cdots \}\subseteq \mathbb{N}\times \mathbb{N}$ and let E be a double sequence space. A K -step space of E is a double sequence space.$$\begin{array}{c}\hfill {\lambda }_K^E=\{x=(x_{i_s,j_t})\in {}_2\omega :x_{\mathit{st}}\in E\}.\end{array}$$

A canonical pre-image of a double sequence $(x_{i_s,j_t})\in {\lambda }_K^E$ is a double sequence $y=(y_{\mathit{ij}})\in {}_2\omega$ defined by$$\begin{array}{c}\hfill y_{\mathit{ij}}=\left\{\begin{array}{cc}{x}_{\mathit{ij}}\hfill & \text{if}i,j\in K\hfill \\ 0,\hfill & \text{otherwise}.\hfill \end{array}\right.\end{array}$$

A canonical pre-image of a step space is a set of canonical pre-images of all elements in ${\lambda }_K^E$, i.e. y is in the canonical pre-image of ${\lambda }_K^E$ iff y is a canonical pre-image of some element $x\in {\lambda }_K^E.$

Definition 1.19

A double sequence space E is said to be monotone, if it is contains the canonical pre-images of it is step space.

The following popular inequalities will be used throughout the article

Let $p=(p_{\mathit{ij}})$ be the bounded double sequence of positive real numbers. For any complex $\lambda ,$ with ${\displaystyle 0<p_{\mathit{ij}}\le H=\underset{i,j}{sup}(p_{\mathit{ij}})<\infty ,}$ we have$$\begin{array}{c}\hfill {|\lambda |}^{p_{\mathit{ij}}}\le {max(1,|\lambda |}^H).\end{array}$$

Let $D=max(1,2^{H-1}),$ then for the double sequences $(a_{\mathit{ij}})$ and $(b_{\mathit{ij}})$ in the complex plane, we have(1.10) $$\begin{array}{c}\hfill |a_{\mathit{ij}}+b_{\mathit{ij}}{|}^{p_{\mathit{ij}}}\le D\{|a_{\mathit{ij}}{|}^{p_{\mathit{ij}}}+|b_{\mathit{ij}}{|}^{p_{\mathit{ij}}}\},\end{array}$$(1.10) (see Maddox, 1989) for all i and j. Also $|a_{\mathit{ij}}{|}^{p_{\mathit{ij}}}\le max(1,|a_{\mathit{ij}}{|}^H)$ for all $a\in \mathbb{C}$.

We used the following lemmas to establish some results of this article

Lemma 1.1

Every solid space is monotone.

Lemma 1.2

Let $K\in \mathcal{F}(I)$ and $M\subseteq \mathbb{N}.$ If $M\notin I,$ then $M\cap K\notin I.$

Lemma 1.3

If $I\subset 2^{\mathbb{N}}$ and $M\subseteq \mathbb{N}.$ If $M\notin I,$ then $M\cap \mathbb{N}\notin I.$

Lemma 1.4

Gunawan and Mashadi (2001) Every n-normed space is an $(n-r)$-normed space for all $r=1,2,\cdots ,n-1$. In particular, every n-normed space is a normed space.

2. Main results

Let I be an admissible ideal of $\mathbb{N}\times \mathbb{N}$, let M be an Orlicz function, let $(X,\Vert ·,\cdots ,·\Vert )$ be an n-normed space, let $p=(p_{\mathit{ij}})$ be a factorable double sequence of strictly positive real numbers, let ${}_2\omega (n-X)$ be the space of all double sequences defined over the n-normed space $(X,\Vert ·,\cdots ,·\Vert )$, then for each $ϵ>0$, we define and introduce the following new classes of double sequences:(2.1) $$\begin{array}{cc}& {}_{2}B{V}_{\sigma }^I[M,p,\Vert ·,\cdots ,·\Vert ]=\{x=(x_{\mathit{ij}})\in {}_2\omega (n-X):\hfill \\ \hfill & \{(i,j)\in \mathbb{N}\times \mathbb{N}:[M(\Vert \frac{{\phi }_{\mathit{mkij}}(x)-L}{\rho },z_1,z_2,\cdots ,z_{n-1}\Vert ){]}^{p_{\mathit{ij}}}\ge ϵ\}\in I,\hfill \\ \hfill & \text{for some}\rho >0,L\in \mathbb{C}\text{and for every}{z}_1,z_2,\cdots ,z_{n-1}\in X\},\hfill \end{array}$$(2.1) (2.2) $$\begin{array}{cc}& {}_2{(}_{0}B{V}_{\sigma }^I[M,p,\Vert ·,\cdots ,·\Vert ])=\{x=(x_{\mathit{ij}})\in {}_2\omega (n-X):\hfill \\ \hfill & \{(i,j)\in \mathbb{N}\times \mathbb{N}:[M(\Vert \frac{{\phi }_{\mathit{mkij}}(x)}{\rho },z_1,z_2,\cdots ,z_{n-1}\Vert ){]}^{p_{\mathit{ij}}}\ge ϵ\}\in I,\hfill \\ \hfill & \text{for some}\rho >0\text{and for every}{z}_1,z_2,\cdots ,z_{n-1}\in X\},\hfill \end{array}$$(2.2) (2.3) $$\begin{array}{cc}& {}_2{(}_{\infty }B{V}_{\sigma }^I[M,p,\Vert ·,\cdots ,·\Vert ])=\{x=(x_{\mathit{ij}})\in {}_2\omega (n-X):\hfill \\ \hfill & \{(i,j)\in \mathbb{N}\times \mathbb{N}:\text{exists}K>0s.t[M(\Vert \frac{{\phi }_{\mathit{mkij}}(x)}{\rho },z_1,z_2,\cdots ,z_{n-1}\Vert ){]}^{p_{\mathit{ij}}}\ge K\}\in I,\hfill \\ \hfill & \text{for some}\rho >0\text{and for every}{z}_1,z_2,\cdots ,z_{n-1}\in X\},\hfill \end{array}$$(2.3) (2.4) $$\begin{array}{cc}& {}_2{(}_{\infty }B{V}_{\sigma }[M,p,\Vert ·,\cdots ,·\Vert ])=\hfill \\ \hfill & \{x=(x_{\mathit{ij}})\in {}_2\omega (n-X):\underset{i,j}{sup}[M(\Vert \frac{{\phi }_{\mathit{mkij}}(x)}{\rho },z_1,z_2,\cdots ,z_{n-1}\Vert ){]}^{p_{\mathit{ij}}}<\infty ,\hfill \\ \hfill & \text{for some}\rho >0\text{and for every}{z}_1,z_2,\cdots ,z_{n-1}\in X\}.\hfill \end{array}$$(2.4)

We denote$$\begin{array}{c}\hfill {}_2{\mathcal{M}}_{B{V}_{\sigma }}^I[M,P,\Vert ·,\cdots ,·\Vert ]={}_{2}B{V}_{\sigma }^I[M,p,\Vert ·,\cdots ,·\Vert ]\cap {}_2{(}_{\infty }B{V}_{\sigma }[M,p,\Vert ·,\cdots ,·\Vert ]),\end{array}$$

and$$\begin{array}{c}\hfill {}_2{(}_0{\mathcal{M}}_{B{V}_{\sigma }}^I[M,P,\Vert ·,\cdots ,·\Vert ])={}_2{(}_{0}B{V}_{\sigma }^I[M,p,\Vert ·,\cdots ,·\Vert ])\cap {}_2{(}_{\infty }B{V}_{\sigma }[M,p,\Vert ·,\cdots ,·\Vert ]).\end{array}$$

Theorem 2.1

The classes of double sequence ${}_{2}B{V}_{\sigma }^I[M,p,\Vert ·,\cdots ,·\Vert ],{}_2{(}_{0}B{V}_{\sigma }^I[M,p,\Vert ·,\cdots ,·\Vert ])$, ${}_2{\mathcal{M}}_{B{V}_{\sigma }}^I[M,P,\Vert ·,\cdots ,·\Vert ]$ and ${}_2{(}_0{\mathcal{M}}_{B{V}_{\sigma }}^I[M,P,\Vert ·,\cdots ,·\Vert ])$ are linear spaces over the field $\mathbb{K}$.

Proof

The proof of the result is a routine work and hence omitted.

Theorem 2.2

Let M be an Orlicz function and let $p=(p_{\mathit{ij}})$ be a bounded double sequence of positive real numbers. Then ${}_2{(}_{\infty }B{V}_{\sigma }^I[M,p,\Vert ·,\cdots ,·\Vert ])$ is a paranormed space with paranorm defined by$$\begin{array}{c}\hfill g(x)=inf\{{\rho }^{\frac{p_{\mathit{ij}}}{H}}:[sup[M(\Vert \frac{{\phi }_{\mathit{mkij}}(x)}{\rho },z_1,z_2,\cdots ,z_{n-1}\Vert ){]}^{p_{\mathit{ij}}}\le 1{]}^{\frac{1}{H}},\\ \hfill \text{for some}\rho >0,\forall z_1,z_2,\cdots ,z_{n-1}\in X\},\end{array}$$

where ${\displaystyle H=max\{1,\underset{i,j}{sup}{p}_{\mathit{ij}}\}}$.

Proof

Omitted.

Theorem 2.3

Let $M_1$, $M_2$ be two Orlicz functions and satisfying ${△}_2$-condition, then

(i)	$Z[M_2,p,\Vert ·,\cdots ,·\Vert ])$$\subset$$Z[M_1{M}_2,p,\Vert ·,\cdots ,·\Vert ]$.
(ii)	$Z[M_1,p,\Vert ·,\cdots ,·\Vert ]$$\cap$$Z[M_2,p,\Vert ·,\cdots ,·\Vert ]$$\subset$$Z[M_1+M_2,p,\Vert ·,\cdots ,·\Vert ]$,

for $Z={}_{2}B{V}_{\sigma }^I,$${}_2{(}_{0}B{V}_{\sigma }^I),$${}_2{(}_0{\mathcal{M}}_{B{V}_{\sigma }}^I),$${}_2{\mathcal{M}}_{B{V}_{\sigma }}^I$ .

Proof

 

(i)

Let $x=(x_{\mathit{ij}})\in {}_2{(}_{0}B{V}_{\sigma }^I[M_2,p,\Vert ·,\cdots ,·\Vert ])$ be an arbitrary element, then there exist $\rho >0$ such that $$\begin{array}{c}\hfill I-\underset{i,j}{lim}[M_2(\Vert \frac{{\phi }_{\mathit{mkij}}(x)}{\rho },z_1,z_2,\cdots ,z_{n-1}\Vert ){]}^{p_{\mathit{ij}}}=0.\end{array}$$ Let $ϵ>0$ and choose $\delta$ with $0<\delta <1$ such that $M_1(t)<ϵ$ for $0<t<\delta$. Write $$\begin{array}{c}\hfill y_{i,j}=[M_2(\Vert \frac{{\phi }_{\mathit{mkij}}(x)}{\rho },z_1,z_2,\cdots ,z_{n-1}\Vert ){]}^{p_{\mathit{ij}}},\end{array}$$ and consider, (2.5) $$\begin{array}{c}\hfill \underset{i,j}{lim}{M}_1(y_{\mathit{ij}})=\underset{y_{\mathit{ij}}\le \delta ,i,j\in \mathbb{N}}{lim}{M}_1(y_{\mathit{ij}})+\underset{y_{\mathit{ij}}>\delta ,i,j\in \mathbb{N}}{lim}{M}_1(y_{\mathit{ij}}).\end{array}$$(2.5) Now, since $M_1$ is an Orlicz function, so we have $M_1(\lambda x)\le \lambda M_1(x)$ , $0<\lambda <1.$ Therefore we have, (2.6) $$\begin{array}{c}\hfill \underset{y_{\mathit{ij}}\le \delta ,i,j\in \mathbb{N}}{lim}{M}_1(y_{\mathit{ij}})\le M_1(2)\underset{y_{\mathit{ij}}\le \delta ,i,j\in \mathbb{N}}{lim}(y_{\mathit{ij}}).\end{array}$$(2.6) For $y_{\mathit{ij}}>\delta$, we have $y_{\mathit{ij}}<\frac{y_{\mathit{ij}}}{\delta }<1+\frac{y_{\mathit{ij}}}{\delta }.$ Now, since $M_1$ is non-decreasing and convex, it follows that, $$\begin{array}{c}\hfill M_1(y_{\mathit{ij}})<M_1(1+\frac{y_{\mathit{ij}}}{\delta })<\frac{1}{2}{M}_1(2)+\frac{1}{2}{M}_1(\frac{2y_{\mathit{ij}}}{\delta }).\end{array}$$ Since $M_1$ satisfies the ${△}_2$-condition we have, $$\begin{array}{c}\hfill \begin{array}{cc}\hfill M_1(y_{\mathit{ij}})& <\frac{1}{2}K\frac{y_{\mathit{ij}}}{\delta }{M}_1(2)+\frac{1}{2}K{M}_1(\frac{2y_{\mathit{ij}}}{\delta })\hfill \\ \hfill & <\frac{1}{2}K\frac{y_{\mathit{ij}}}{\delta }{M}_1(2)+\frac{1}{2}K\frac{y_{\mathit{ij}}}{\delta }{M}_1(2)=K\frac{y_{\mathit{ij}}}{\delta }{M}_1(2).\hfill \end{array}\end{array}$$ This implies that, $$\begin{array}{c}\hfill M_1(y_{\mathit{ij}})<K\frac{y_{\mathit{ij}}}{\delta }{M}_1(2).\end{array}$$ Hence, we have (2.7) $$\begin{array}{c}\hfill \underset{y_{\mathit{ij}}>\delta ,i,j\in \mathbb{N}}{lim}{M}_1(y_{\mathit{ij}})\le max\{1,K{\delta }^{-1}{M}_1(2)\underset{y_{\mathit{ij}}>\delta ,i,j\in \mathbb{N}}{lim}(y_{\mathit{ij}})\}.\end{array}$$(2.7) Then from Equations (2.5), (2.6) and (2.7), we have $$\begin{array}{c}\hfill I-\underset{i,j}{lim}{M}_1(y_{\mathit{ij}})=0,\end{array}$$$$\begin{array}{c}\hfill I-\underset{i,j}{lim}[M_1{M}_2(\Vert \frac{{\phi }_{\mathit{mkij}}(x)}{\rho },z_1,z_2,\cdots ,z_{n-1}\Vert ){]}^{p_{\mathit{ij}}}=0.\end{array}$$ This implies that $x=(x_{\mathit{ij}})\in {}_2{(}_{0}B{V}_{\sigma }^I[M_1{M}_2,p,\Vert ·,\cdots ,·\Vert ])$. Hence ${}_2{(}_{0}B{V}_{\sigma }^I[M_2,p,\Vert ·,\cdots ,·\Vert ])$$\subset$${}_2{(}_{0}B{V}_{\sigma }^I[M_1{M}_2,p,\Vert ·,\cdots ,·\Vert ])$. The other cases can be proved in similar way.

(ii)

Let $x=(x_{\mathit{ij}})\in {}_2{(}_{0}B{V}_{\sigma }^I[M_1,p,\Vert ·,\cdots ,·\Vert ])\cap {}_2{(}_{0}B{V}_{\sigma }^I[M_2,p,\Vert ·,\cdots ,·\Vert ]).$ Let $ϵ>0$ be given, then there exist $\rho >0$ such that (2.8) $$\begin{array}{c}\hfill I-\underset{i,j}{lim}[M_1(\Vert \frac{{\phi }_{\mathit{mkij}}(x)}{\rho },z_1,z_2,\cdots ,z_{n-1}\Vert ){]}^{p_{\mathit{ij}}}=0,\end{array}$$(2.8) and (2.9) $$\begin{array}{c}\hfill I-\underset{i,j}{lim}[M_2(\Vert \frac{{\phi }_{\mathit{mkij}}(x)}{\rho },z_1,z_2,\cdots ,z_{n-1}\Vert ){]}^{p_{\mathit{ij}}}=0.\end{array}$$(2.9) Therefore $$\begin{array}{c}\hfill \begin{array}{cc}\hfill I-\underset{i,j}{lim}[M_1+M_2(\Vert \frac{{\phi }_{\mathit{mkij}}(x)}{\rho },& z_1,z_2,\cdots ,z_{n-1}\Vert ){]}^{p_{\mathit{ij}}}=\hfill \\ \hfill I-\underset{i,j}{lim}[M_1(\Vert \frac{{\phi }_{\mathit{mkij}}(x)}{\rho },z_1,z_2,\cdots ,z_{n-1}\Vert ){]}^{p_{\mathit{ij}}}& +I-\underset{i,j}{lim}[M_2(\Vert \frac{{\phi }_{\mathit{mkij}}(x)}{\rho },z_1,z_2,\cdots ,z_{n-1}\Vert ){]}^{p_{\mathit{ij}}},\hfill \end{array}\end{array}$$ from Equations (2.8) and (2.9), we have $$\begin{array}{c}\hfill I-\underset{i,j}{lim}[M_1+M_2(\Vert \frac{{\phi }_{\mathit{mkij}}(x)}{\rho },z_1,z_2,\cdots ,z_{n-1}\Vert ){]}^{p_{\mathit{ij}}}=0.\end{array}$$ Thus, $x=(x_{\mathit{ij}})\in {}_2{(}_{0}B{V}_{\sigma }^I[M_1+M_2,p,\Vert ·,\cdots ,·\Vert ])$. Hence ${}_2{(}_{0}B{V}_{\sigma }^I[M_1,p,\Vert ·,\cdots ,·\Vert ])$$\cap$${}_2{(}_{0}B{V}_{\sigma }^I[M_2,p,\Vert ·,\cdots ,·\Vert ])$  $\subset$${}_2{(}_{0}B{V}_{\sigma }^I[M_1+M_2,p,\Vert ·,\cdots ,·\Vert ])$. For $Z={}_{2}B{V}_{\sigma }^I$, ${}_2{(}_0{\mathcal{M}}_{B{V}_{\sigma }}^I)$, ${}_2{\mathcal{M}}_{B{V}_{\sigma }}^I$ the inclusion are similar.

Theorem 2.4

 

(i)	Let $0<inf{p}_{\mathit{ij}}\le p_{\mathit{ij}}\le 1$. Then ${}_{2}B{V}_{\sigma }^I[M,p,\Vert ·,\cdots ,·\Vert ]$$\subset$${}_{2}B{V}_{\sigma }^I[M,\Vert ·,\cdots ,·\Vert ].$
(ii)	$1<p_{\mathit{ij}}\le sup{p}_{\mathit{ij}}<\infty .$ Then ${}_{2}B{V}_{\sigma }^I[M,\Vert ·,\cdots ,·\Vert ]$$\subset$${}_{2}B{V}_{\sigma }^I[M,p,\Vert ·,\cdots ,·\Vert ]$.

Proof

 

(i)

Let $x=(x_{\mathit{ij}})\in {}_{2}B{V}_{\sigma }^I[M,p,\Vert ·,\cdots ,·\Vert ]$. Since $0<inf{p}_{\mathit{ij}}\le p_{\mathit{ij}}\le 1$, we have $$\begin{array}{c}\hfill [M(\Vert \frac{{\phi }_{\mathit{mkij}}(x)}{\rho },z_1,z_2,\cdots ,z_{n-1}\Vert )]\le [M(\Vert \frac{{\phi }_{\mathit{mkij}}(x)}{\rho },z_1,z_2,\cdots ,z_{n-1}\Vert ){]}^{p_{\mathit{ij}}}.\end{array}$$ This implies that $$\begin{array}{c}\hfill \{(i,j)\in \mathbb{N}\times \mathbb{N}:[M(\Vert \frac{{\phi }_{\mathit{mkij}}(x)}{\rho },z_1,z_2,\cdots ,z_{n-1}\Vert )]\ge ϵ\}\\ \hfill \subseteq \{(i,j)\in \mathbb{N}\times \mathbb{N}:[M(\Vert \frac{{\phi }_{\mathit{mkij}}(x)}{\rho },z_1,z_2,\cdots ,z_{n-1}\Vert ){]}^{p_{\mathit{ij}}}\ge ϵ\}\in I.\end{array}$$

(ii)

Let $p_{\mathit{ij}}\ge 1$ for each i, j and ${\displaystyle \underset{i,j}{sup}{p}_{\mathit{ij}}<\infty }$. Let $x=(x_{\mathit{ij}})\in {}_{2}B{V}_{\sigma }^I[M,p,\Vert ·,\cdots ,·\Vert ]$. Then for each $0<ϵ<1$, there exists a positive integer N such that $$\begin{array}{c}\hfill [M(\Vert \frac{{\phi }_{\mathit{mkij}}(x)}{\rho },z_1,z_2,\cdots ,z_{n-1}\Vert )]\le ϵ<1,\end{array}$$ for all $i,j\ge N$. This implies that $$\begin{array}{c}\hfill [M(\Vert \frac{{\phi }_{\mathit{mkij}}(x)}{\rho },z_1,z_2,\cdots ,z_{n-1}\Vert ){]}^{p_{\mathit{ij}}}\le [M(\Vert \frac{{\phi }_{\mathit{mkij}}(x)}{\rho },z_1,z_2,\cdots ,z_{n-1}\Vert )].\end{array}$$ So we have $$\begin{array}{c}\hfill \{(i,j)\in \mathbb{N}\times \mathbb{N}:[M(\Vert \frac{{\phi }_{\mathit{mkij}}(x)}{\rho },z_1,z_2,\cdots ,z_{n-1}\Vert ){]}^{p_{\mathit{ij}}}\ge ϵ\}\\ \hfill \subseteq \{(i,j)\in \mathbb{N}\times \mathbb{N}:[M(\Vert \frac{{\phi }_{\mathit{mkij}}(x)}{\rho },z_1,z_2,\cdots ,z_{n-1}\Vert )]\ge ϵ\}\in I.\end{array}$$

This completes the proof.

Theorem 2.5

The inclusions$$\begin{array}{c}\hfill {}_2{(}_{0}B{V}_{\sigma }^I[M,p,\Vert ·,\cdots ,·\Vert ])\subset {}_{2}B{V}_{\sigma }^I[M,p,\Vert ·,\cdots ,·\Vert ]\subset {}_2{(}_{\infty }B{V}_{\sigma }^I[M,p,\Vert ·,\cdots ,·\Vert ])\text{holds}.\end{array}$$

Proof

For this, let us consider $x=(x_{\mathit{ij}})\in {}_2{(}_{0}B{V}_{\sigma }^I[M,p,\Vert ·,\cdots ,·\Vert ])$. It is obvious that it must belong to ${}_{2}B{V}_{\sigma }^I[M,p,\Vert ·,\cdots ,·\Vert ]$. Now it remains to show that ${}_{2}B{V}_{\sigma }^I[M,p,\Vert ·,\cdots ,·\Vert ]\subset {}_2{(}_{\infty }B{V}_{\sigma }^I[M,p,\Vert ·,\cdots ,·\Vert ])$. For this let us consider $x=(x_{\mathit{ij}})\in {}_{2}B{V}_{\sigma }^I[M,p,\Vert ·,\cdots ,·\Vert ]$, then there exist $L\in \mathbb{C}$ and $\rho >0$ such that$$\begin{array}{c}\hfill I-\underset{i,j}{lim}[M(\Vert \frac{{\phi }_{\mathit{mkij}}(x)-L}{\rho },z_1,z_2,\cdots ,z_{n-1}\Vert ){]}^{p_{\mathit{ij}}}=0.\end{array}$$

We have$$\begin{array}{c}\hfill \begin{array}{cc}\hfill [M(\Vert \frac{{\phi }_{\mathit{mkij}}(x)}{2\rho },z_1,z_2,\cdots ,z_{n-1}\Vert ){]}^{p_{\mathit{ij}}}\le & [\frac{1}{2}M(\Vert \frac{{\phi }_{\mathit{mkij}}(x)-L}{\rho },z_1,z_2,\cdots ,z_{n-1}\Vert ){]}^{p_{\mathit{ij}}}\hfill \\ \hfill & +[\frac{1}{2}M(\Vert \frac{L}{\rho },z_1,z_2,\cdots ,z_{n-1}\Vert ){]}^{p_{\mathit{ij}}}.\hfill \end{array}\end{array}$$

Now taking the supremum over i and j on both sides, we get$$\begin{array}{c}\hfill \underset{i,j}{sup}[M(\Vert \frac{{\phi }_{\mathit{mkij}}(x)}{\rho },z_1,z_2,\cdots ,z_{n-1}\Vert ){]}^{p_{\mathit{ij}}}<\infty .\end{array}$$

Hence $x=(x_{\mathit{ij}})\in {}_2{(}_{\infty }B{V}_{\sigma }^I[M,p,\Vert ·,\cdots ,·\Vert ])$. Thus,$$\begin{array}{c}\hfill {}_2{(}_{0}B{V}_{\sigma }^I[M,p,\Vert ·,\cdots ,·\Vert ])\subset {}_{2}B{V}_{\sigma }^I[M,p,\Vert ·,\cdots ,·\Vert ]\subset {}_2{(}_{\infty }B{V}_{\sigma }^I[M,p,\Vert ·,\cdots ,·\Vert ]).\end{array}$$

This completes the proof of the theorem.

Theorem 2.6

For any Orlicz function M and a double sequence of strictly positive real numbers $p=(p_{\mathit{ij}})$, the spaces ${}_2{(}_{0}B{V}_{\sigma }^I[M,p,\Vert ·,\cdots ,·\Vert ])$ and ${}_2{(}_0{\mathcal{M}}_{B{V}_{\sigma }}^I[M,P,\Vert ·,\cdots ,·\Vert ])$ are solid and monotone.

Proof

Here we consider ${}_2{(}_{0}B{V}_{\sigma }^I[M,p,\Vert ·,\cdots ,·\Vert ])$ and for ${}_2{(}_0{\mathcal{M}}_{B{V}_{\sigma }}^I[M,P,\Vert ·,\cdots ,·\Vert ])$ the proof shall be similar. Let $x=(x_{\mathit{ij}})\in {}_2{(}_{0}B{V}_{\sigma }^I[M,p,\Vert ·,\cdots ,·\Vert ])$ be an arbitrary element, then there exist $\rho >0$ such that$$\begin{array}{c}\hfill I-lim[M(\Vert \frac{{\phi }_{\mathit{mkij}}(x)}{\rho },z_1,z_2,\cdots ,z_{n-1}\Vert ){]}^{p_{\mathit{ij}}}=0.\end{array}$$

Let $\alpha =({\alpha }_{\mathit{ij}})$ be a double sequence of scalars with $|{\alpha }_{\mathit{ij}}|\le 1$ for $i,j\in \mathbb{N}.$ Now, M is an Orlicz function. Therefore$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \{(i,j)\in \mathbb{N}\times \mathbb{N}:& [M(\Vert \frac{{\alpha }_{\mathit{ij}}{\phi }_{\mathit{mkij}}(x)}{\rho },z_1,z_2,\cdots ,z_{n-1}\Vert ){]}^{p_{\mathit{ij}}}\ge ϵ\}\hfill \\ \hfill & \subseteq \{(i,j)\in \mathbb{N}\times \mathbb{N}:E[M(\Vert \frac{{\phi }_{\mathit{mkij}}(x)}{\rho },z_1,z_2,\cdots ,z_{n-1}\Vert ){]}^{p_{\mathit{ij}}}\ge ϵ\}\in I.\hfill \end{array}\end{array}$$

where $E=max\{1,|{\alpha }_{\mathit{ij}}^H|\}$. Thus $({\alpha }_{\mathit{ij}}{x}_{\mathit{ij}})\in {}_2{(}_{0}B{V}_{\sigma }^I[M,p,\Vert ·,\cdots ,·\Vert ])$. for all double sequence of scalars $({\alpha }_{\mathit{ij}})$ with $|{\alpha }_{\mathit{ij}}|\le 1$ for all $i,j\in \mathbb{N}$ whenever $(x_{\mathit{ij}})\in {}_2{(}_{0}B{V}_{\sigma }^I[M,p,\Vert ·,\cdots ,·\Vert ])$. By definition 1.14 the space ${}_2{(}_{0}B{V}_{\sigma }^I[M,p,\Vert ·,\cdots ,·\Vert ])$ is solid. Therefore by lemma 1.1 the space ${}_2{(}_{0}B{V}_{\sigma }^I[M,p,\Vert ·,\cdots ,·\Vert ])$ is monotone.

Theorem 2.7

For any Orlicz function M and a factorable double sequence of strictly positive real numbers $p=(p_{\mathit{ij}})$, the spaces ${}_{2}B{V}_{\sigma }^I[M,p,\Vert ·,\cdots ,·\Vert ]$ and ${}_2{\mathcal{M}}_{B{V}_{\sigma }}^I[M,P,\Vert ·,\cdots ,·\Vert ]$ are neither solid nor monotone in general.

Proof

Here we give counter example for establishment of this result. Let $X={}_{2}B{V}_{\sigma }^I$ and ${}_2{\mathcal{M}}_{B{V}_{\sigma }}^I$. Let us consider $I=I_f$ and $M(x)=x$, for all $x=(x_{\mathit{ij}})\in [0,\infty )$. Consider, the K-step space $X_K[M,p,\Vert ·,\cdots ,·\Vert ]$ of $X[M,p,\Vert ·,\cdots ,·\Vert ]$ defined as follows:

Let $x=(x_{\mathit{ij}})\in X[M,p,\Vert ·,\cdots ,·\Vert ]$ and $y=(y_{\mathit{ij}})\in X_K[M,p,\Vert ·,\cdots ,·\Vert ]$ be such that$$\begin{array}{c}\hfill y_{\mathit{ij}}=\begin{array}{cc}{x}_{\mathit{ij}},\hfill & \text{if}i,j\text{is even}\hfill \\ 0,\hfill & \text{otherwise}.\hfill \end{array}\end{array}$$

Consider the double sequence $(x_{\mathit{ij}})$ defined by $(x_{\mathit{ij}})=1$ for all $i,j\in \mathbb{N}$. Then $x=(x_{\mathit{ij}})\in {}_{2}B{V}_{\sigma }^I[M,p,\Vert ·,\cdots ,·\Vert ]$ and ${}_2{\mathcal{M}}_{B{V}_{\sigma }}^I[M,p,\Vert ·,\cdots ,·\Vert ]$, but K-step space pre-image does not belong to ${}_{2}B{V}_{\sigma }^I[M,p,\Vert ·,\cdots ,·\Vert ]$ and ${}_2{\mathcal{M}}_{B{V}_{\sigma }}^I[M,p,\Vert ·,\cdots ,·\Vert ]$. Thus, ${}_{2}B{V}_{\sigma }^I[M,p,\Vert ·,\cdots ,·\Vert ]$ and ${}_2{\mathcal{M}}_{B{V}_{\sigma }}^I[M,p,\Vert ·,\cdots ,·\Vert ]$ are not monotone and hence they are not solid.

Theorem 2.8

For any Orlicz function M and a factorable double sequence of strictly positive real numbers $p=(p_{\mathit{ij}})$, the spaces ${}_2{(}_{0}B{V}_{\sigma }^I[M,p,\Vert ·,\cdots ,·\Vert ])$ and ${}_{2}B{V}_{\sigma }^I[M,p,\Vert ·,\cdots ,·\Vert ]$ are not convergence free.

Proof

To show this, let $I=I_f$ and $M(x)=x$ for all $x\in [0,\infty ).$ Now consider the double sequences $(x_{\mathit{ij}})$ and $(y_{\mathit{ij}})$ which defined as follows:$$\begin{array}{c}\hfill x_{\mathit{ij}}=\frac{1}{i+j}\text{and}{y}_{\mathit{ij}}=i+j,\text{for all}i,j\in \mathbb{N}.\end{array}$$

Then we have $(x_{\mathit{ij}})$ belong to both ${}_2{(}_{0}B{V}_{\sigma }^I[M,p,\Vert ·,\cdots ,·\Vert ])$ and ${}_{2}B{V}_{\sigma }^I[M,p,\Vert ·,\cdots ,·\Vert ]$, but $(y_{i,j})$ does not belong to ${}_2{(}_{0}B{V}_{\sigma }^I[M,p,\Vert ·,\cdots ,·\Vert ])$ and ${}_{2}B{V}_{\sigma }^I[M,p,\Vert ·,\cdots ,·\Vert ]$. Hence, the spaces ${}_2{(}_{0}B{V}_{\sigma }^I[M,p,\Vert ·,\cdots ,·\Vert ])$ and ${}_{2}B{V}_{\sigma }^I[M,p,\Vert ·,\cdots ,·\Vert ]$ are not convergence free.

Theorem 2.9

The spaces ${}_2{(}_{0}B{V}_{\sigma }^I[M,p,\Vert ·,\cdots ,·\Vert ])$ and ${}_{2}B{V}_{\sigma }^I[M,p,\Vert ·,\cdots ,·\Vert ]$ are sequence algebra.

Proof

let $x=(x_{\mathit{ij}}),y=(y_{\mathit{ij}})\in {}_2{(}_{0}B{V}_{\sigma }^I[M,p,\Vert ·,\cdots ,·\Vert ])$ be any two arbitrary elements, then there exist ${\rho }_1,{\rho }_2>0$ such that$$\begin{array}{c}\hfill A=\{(i,j)\in \mathbb{N}\times \mathbb{N}:[M(\Vert \frac{{\phi }_{\mathit{mkij}}(x)}{{\rho }_1},z_1,z_2,\cdots ,z_{n-1}\Vert ){]}^{p_{\mathit{ij}}}\ge ϵ\}\end{array}$$

and$$\begin{array}{c}\hfill B=\{(i,j)\in \mathbb{N}\times \mathbb{N}:[M(\Vert \frac{{\phi }_{\mathit{mkij}}(y)}{{\rho }_2},z_1,z_2,\cdots ,z_{n-1}\Vert ){]}^{p_{\mathit{ij}}}\ge ϵ\}\end{array}$$

belongs to I. Let $\rho ={\rho }_1{\rho }_2>0$, the following inclusion can be checked.$$\begin{array}{c}\hfill \{(i,j)\in \mathbb{N}\times \mathbb{N}:[M(\Vert \frac{{\phi }_{\mathit{mkij}}(x·y)}{\rho },z_1,z_2,\cdots ,z_{n-1}\Vert ){]}^{p_{\mathit{ij}}}\ge ϵ\}\subseteq \{A\cup B\}.\end{array}$$

From the definition 1.8, it easily follows that the set on the left-hand side of the above inclusion belongs to I. Therefore, we have $(x_{\mathit{ij}})·(y_{\mathit{ij}})\in {}_2{(}_{0}B{V}_{\sigma }^I[M,p,\Vert ·,\cdots ,·\Vert ])$. Hence ${}_2{(}_{0}B{V}_{\sigma }^I[M,p,\Vert ·,\cdots ,·\Vert ])$ is sequence algebra. The rest result can be proved similarly.

The following theorem expresses a relation between the notions of I-convergence and I-Cauchy for double sequences in our settings.

Theorem 2.10

A double sequence $x=(x_{\mathit{ij}})\in {}_2{\mathcal{M}}_{B{V}_{\sigma }}^I[M,P,\Vert ·,\cdots ,·\Vert ]$ is I-converges if and only if for every $ϵ>0$, there exists $s=s(ϵ),t=t(ϵ)\in \mathbb{N}$ such that the set$$\begin{array}{c}\hfill \{(i,j)\in \mathbb{N}\times \mathbb{N}:[M(\Vert \frac{{\phi }_{\mathit{mkij}}(x)-{\phi }_{\mathit{mkst}}(x)}{\rho },z_1,z_2,\cdots ,z_{n-1}\Vert ){]}^{p_{\mathit{ij}}}<ϵ\}\in \mathcal{F}(I)\end{array}$$

Proof

The proof of the result is easy. So, it can be left to readers.

3. Conclusion

In this present paper, we have defined and introduced some new spaces of I-convergent double sequences of $\sigma$-bounded variation, that are ${}_{2}B{V}_{\sigma }^I[M,p,\Vert ·,\cdots ,·\Vert ],$${}_2{(}_{0}B{V}_{\sigma }^I[M,p,\Vert ·,\cdots ,·\Vert ]),$${}_2{(}_{\infty }B{V}_{\sigma }^I[M,p,\Vert ·,\cdots ,·\Vert ])$

and ${}_2{(}_{\infty }B{V}_{\sigma }[M,p,\Vert ·,\cdots ,·\Vert ]).$ In addition, we studied some basic topological and algebraic properties of these spaces. These definitions and results provide new tools to deal with the convergence problems of double sequences occurring in many branches of science and engineering.

Acknowledgements

I would like to thank the referees and the editor for their careful reading and their valuable comments.

Additional information

Funding

This work was supported by Department of Mathematics, Amman Arab University, Amman, Jordan.

Notes on contributors

Vakeel A. Khan

Vakeel. A. Khan received his MPhil and PhD degrees in Mathematics from Aligarh Muslim University, Aligarh, India. Currently, he is an associate professor at Aligarh Muslim University, Aligarh, India. A vigorous researcher in the area of sequence spaces, he has published a number of research papers in reputed national and international journals, including Numerical Functional Analysis and Optimization (Taylors and Francis), Information Sciences (Elsevier), Applied Mathematics Letters (Elsevier), A Journal of Chinese Universities (Springer- Verlag, China).

Kamal M.A.S. Alshlool

Kamal M.A.S. Alshlool received MSc, from Aligarh Muslim University, and is currently a PhD, scholar at Aligarh Muslim University.

Sameera A.A. Abdullah

Sameera A.A. Abdullah received MSc, from Aligarh Muslim University, and is currently a PhD scholar at Aligarh Muslim University..

Rami K.A. Rababah

Rami K.A. Rababah is working as an assistant professor in the Department of Mathematics, Amman Arab University, Jordan.

Ayaz Ahmad

Ayaz Ahmad is working as an assistant professor in the National Institute Technology, Patna, India.