Some generalized difference statistically convergent sequence spaces in 2-normed space

Abstract

In this paper, we define a new generalized difference matrix $B_{(m)}^n$ and introduce some $B_{(m)}^n$-difference statistically convergent sequence spaces in a real linear 2-normed space. We also investigate some topological properties of these spaces.

MSC:40A05, 46A45, 46E30.

1 Introduction

We shall write w for the set of all real sequences $x=(x_k)={(x_k)}_{k=0}^{\mathrm{\infty }}$. Let c, $c_0$, $\overline{c}$, ${\overline{c}}_0$, $l_{\mathrm{\infty }}$, m and $m_0$ denote the sets of all convergent, null, statistically convergent, statistically null, bounded, bounded statistically convergent and bounded statistically null sequences, respectively. The difference sequence spaces $l_{\mathrm{\infty }}(\mathrm{\Delta })$, $c(\mathrm{\Delta })$ and $c_0(\mathrm{\Delta })$ were first defined by Kızmaz in [1]. The idea of difference sequences is generalized by Et and Çolak [2] as

$$Z\left({\mathrm{\Delta }}^n\right)=\{x=(x_k)\in w:\left({\mathrm{\Delta }}^n{x}_k\right)\in Z\}(n\in \mathbb{N})$$

for $Z=l_{\mathrm{\infty }},c,c_0$, where ${\mathrm{\Delta }}^n{x}_k={\mathrm{\Delta }}^{n-1}{x}_k-{\mathrm{\Delta }}^{n-1}{x}_{k+1}$ and ${\mathrm{\Delta }}^0{x}_k=x_k$ for all $k\in \mathbb{N}$, the difference operator is equivalent to the following binomial representation:

$${\mathrm{\Delta }}^n{x}_k=\sum _{v=0}^n{(-1)}^v\left(\genfrac{}{}{0ex}{}{n}{v}\right)x_{k+v}.$$

Et and Başarır [3] generalized these spaces to $E({\mathrm{\Delta }}^n)$, where $E=l_{\mathrm{\infty }}(p),c(p),c_0(p)$ are Maddox’s sequence spaces. Tripathy and Esi [4], who studied the spaces $l_{\mathrm{\infty }}({\mathrm{\Delta }}_m)$, $c({\mathrm{\Delta }}_m)$ and $c_0({\mathrm{\Delta }}_m)$, gave a new type of generalization of the difference sequence spaces, where ${\mathrm{\Delta }}_{m}x=({\mathrm{\Delta }}_m{x}_k)=(x_k-x_{k+m})$. Tripathy et al. [5] generalized this notion as follows:

$$Z\left({\mathrm{\Delta }}_m^n\right)=\{x=(x_k)\in w:\left({\mathrm{\Delta }}_m^n{x}_k\right)\in Z\}(n,m\in \mathbb{N}),$$

where ${\mathrm{\Delta }}_m^{n}x=({\mathrm{\Delta }}_m^n{x}_k)=({\mathrm{\Delta }}_m^{n-1}{x}_k-{\mathrm{\Delta }}_m^{n-1}{x}_{k+m})$ and ${\mathrm{\Delta }}_m^0{x}_k=x_k$ for all $k\in \mathbb{N}$, which is equivalent to the following binomial representation:

$${\mathrm{\Delta }}_m^n{x}_k=\sum _{v=0}^n{(-1)}^v\left(\genfrac{}{}{0ex}{}{n}{v}\right)x_{k+mv}.$$

The difference sequence spaces have been studied by several authors, [3, 6–25].

The concept of 2-normed spaces has been initially introduced by Gähler in the 1960s [26], as an interesting non-linear generalization of a normed linear space, which has been subsequently studied by many authors [27–29]. Since then, a lot of activities have been started to study summability, sequence spaces and related topics on 2-normed spaces [30–33]. Recently, some difference sequence spaces have been introduced in 2-normed spaces by several authors [30, 31, 34].

Dutta [34] introduced the sequence spaces $\overline{c}(\parallel \cdot ,\cdot \parallel ,{\mathrm{\Delta }}_{(m)}^n,p)$, ${\overline{c}}_0(\parallel \cdot ,\cdot \parallel ,{\mathrm{\Delta }}_{(m)}^n,p)$, $l_{\mathrm{\infty }}(\parallel \cdot ,\cdot \parallel ,{\mathrm{\Delta }}_{(m)}^n,p)$, $m(\parallel \cdot ,\cdot \parallel ,{\mathrm{\Delta }}_{(m)}^n,p)$ and $m_0(\parallel \cdot ,\cdot \parallel ,{\mathrm{\Delta }}_{(m)}^n,p)$, where $m,n\in \mathbb{N}$ and ${\mathrm{\Delta }}_{(m)}^{n}x=({\mathrm{\Delta }}_{(m)}^n{x}_k)=({\mathrm{\Delta }}_{(m)}^{n-1}{x}_k-{\mathrm{\Delta }}_{(m)}^{n-1}{x}_{k-m})$, and ${\mathrm{\Delta }}_{(m)}^0{x}_k=x_k$ for all $k\in \mathbb{N}$, which is equivalent to the following binomial representation:

$${\mathrm{\Delta }}_{(m)}^n{x}_k=\sum _{v=0}^n{(-1)}^v\left(\genfrac{}{}{0ex}{}{n}{v}\right)x_{k-mv}.$$

(1.1)

In [35], Başar and Altay introduced the generalized difference matrix $B(r,s)=(b_{nk}(r,s))$ which is a generalization of ${\mathrm{\Delta }}_{(1)}^1$-difference operator as follows:

$$b_{nk}(r,s)=\{\begin{array}{cc}r\hfill & (k=n),\hfill \\ s\hfill & (k=n-1),\hfill \\ 0\hfill & (0\le k<n-1)\text{ or }(k>n)\hfill \end{array}$$

for all $k,n\in \mathbb{N}$, $r,s\in \mathbb{R}-\{0\}$. Recently, Başarır and Kayıkçı [36] have defined the generalized difference matrix $B^n$ of order n, which reduced the difference operator ${\mathrm{\Delta }}_{(1)}^n$ in case $r=1$, $s=-1$ and the binomial representation of this operator is

$$B^n{x}_k=\sum _{v=0}^n\left(\genfrac{}{}{0ex}{}{n}{v}\right)r^{n-v}{s}^v{x}_{k-v},$$

(1.2)

where $r,s\in \mathbb{R}-\{0\}$ and $n\in \mathbb{N}$.

Thus, for any sequence space Z, the space $Z(B^n)$ is more general and more comprehensive than the corresponding consequences of the space $Z({\mathrm{\Delta }}_{(1)}^n)$. For details, one may refer to [6, 15, 35–40].

The idea of statistical convergence was given by Zygmund [41] in 1935. The concept of statistical convergence was introduced by Fast [42] and Schoenberg [43], independently for the real sequences. Later on, it was further investigated from sequence point of view and linked with the summability theory by Fridy [44] and generalized to the concept of 2-normed space by Gürdal and Pehlivan [45]. The idea is based on the notion of natural density of subsets of ℕ, the set of positive integers, which is defined as follows: the natural density of a subset E of ℕ is denoted by

$$\delta (E)=\underset{n\to \mathrm{\infty }}{lim}\frac{1}{n}|\{k\in E:k⩽n\}|,$$

where the vertical bar denotes the cardinality of the enclosed set.

2 Definitions and preliminaries

A sequence space E is said to be solid (or normal) if $(x_k)\in E$ implies $({\alpha }_k{x}_k)\in E$ for all sequences of scalars $({\alpha }_k)$ with $|{\alpha }_k|\le 1$ for all $k\in \mathbb{N}$.

A linear topological space X over the real field R is said to be a paranormed space if there is a sub-additive function $g:X\to R$ such that $g(\theta )=0$, $g(x)=g(-x)$, $g(x+y)\le g(x)+g(y)$ and scalar multiplication is continuous, i.e. $|{\lambda }_n-\lambda |\to 0$ and $g(x_n-x)\to 0$ imply that $g({\lambda }_n{x}_n-\lambda x)\to 0$ for all λ’s in ℝ and all x’s in X, where θ is the zero vector in the linear space X.

The following inequality will be used throughout the paper:

Let $p=(p_k)$ be a positive sequence of real numbers with ${inf}_k{p}_k=h$, ${sup}_k{p}_k=H$ and $D=max\{1,2^{H-1}\}$. Then for all $a_k,b_k\in \mathbb{C}$ for all $k\in \mathbb{N}$, we have

$$|a_k+b_k{|}^{p_k}\le D\{|a_k{|}^{p_k}+|b_k{|}^{p_k}\}$$

and $|\lambda {|}^{p_k}\le max\{|\lambda {|}^h,|\lambda {|}^H\}$ for $\lambda \in \mathbb{C}$.

A 2-norm on a vector space X of d dimension, where $d\ge 2$, is a function $\parallel \cdot ,\cdot \parallel :X\times X\to \mathbb{R}$, which satisfies the following conditions:

• (1) $\parallel x_1,x_2\parallel =0$ if and only if $x_1$, $x_2$ are linearly dependent,

• (2) $\parallel x_1,x_2\parallel =\parallel x_2,x_1\parallel$,

• (3) $\parallel \alpha x_1,x_2\parallel =|\alpha |\parallel x_1,x_2\parallel$ for any $\alpha \in \mathbb{R}$,

• (4) $\parallel x+x^{\mathrm{\prime }},x_1\parallel \le \parallel x,x_1\parallel +\parallel x^{\mathrm{\prime }},x_1\parallel$.

The pair $(X,\parallel \cdot ,\cdot \parallel )$ is then called a 2-normed space. For example, standard and Euclidean 2-norms on ${\mathbb{R}}^2$ are respectively given by

$${\parallel x_1,x_2\parallel }_S={\left|\begin{array}{cc}〈x_1,x_1〉& 〈x_1,x_2〉\\ 〈x_2,x_1〉& 〈x_2,x_2〉\end{array}\right|}^{\frac{1}{2}}$$

and

$${\parallel x_1,x_2\parallel }_E=abs\left(\left|\begin{array}{cc}{x}_{11}& x_{12}\\ x_{21}& x_{22}\end{array}\right|\right),x_i=(x_{i1},x_{i2})\in {\mathbb{R}}^2(i=1,2),$$

(2.1)

where $〈\cdot ,\cdot 〉$ stands for the inner product on X [27].

Now we will give the following known example for 2-normed spaces.

Example 2.1 Consider the space Z for $l_{\mathrm{\infty }}$, c and $c_0$. Let us define:

$$\parallel x,y\parallel =\underset{i\in \mathbb{N}}{sup}\underset{j\in \mathbb{N}}{sup}|x_i{y}_j-x_j{y}_i|,$$

where $x=(x_1,x_2,\dots )$ and $y=(y_1,y_2,\dots )\in Z$. Then $\parallel \cdot ,\cdot \parallel$ is a 2-norm on Z.

A sequence $(x_k)$ in a 2-normed space $(X,\parallel \cdot ,\cdot \parallel )$ is said to be convergent to some $L\in X$ in the 2-norm if

$$\underset{k\to \mathrm{\infty }}{lim}\parallel x_k-L,z\parallel =0\text{for every }z\in X\text{ [45]}.$$

A sequence $(x_k)$ in a 2-normed space $(X,\parallel \cdot ,\cdot \parallel )$ is said to be Cauchy sequence with respect to the 2-norm if

$$\underset{k,l\to \mathrm{\infty }}{lim}\parallel x_k-x_l,z\parallel =0\text{for every }z\in X\text{ [45]}.$$

If every Cauchy sequence in X converges to some $L\in X$, then X is said to be complete with respect to the 2-norm. Any complete 2-normed space is said to be a 2-Banach space [29].

Let recall that a sequence $(x_k)$ is said to be statistically convergent to L if for every $\epsilon >0$ the set $\{k\in \mathbb{N}:\parallel x_k-L,z\parallel \ge \epsilon \}$ has natural density zero for each nonzero z in X, in other words $(x_k)$ statistically converges to L in 2-normed space $(X,\parallel \cdot ,\cdot \parallel )$ if

$$\underset{k\to \mathrm{\infty }}{lim}\frac{1}{k}\left|\{k\in \mathbb{N}:\parallel x_k-L,z\parallel \ge \epsilon \}\right|=0,$$

for each nonzero z in X. For $L=0$, we say this is statistically null [45].

Firstly, we give the following lemma, which we need to establish our main results.

Lemma 2.2 [34]

Every closed linear subspace F of an arbitrary linear normed space E, different from E, is a nowhere dense set in E.

Throughout the paper $w(X)$, $c(X)$, $c_0(X)$, $\overline{c}(X)$, ${\overline{c}}_0(X)$, $l_{\mathrm{\infty }}(X)$, $m(X)$ and $m_0(X)$ denote the spaces of all, convergent, null, statistically convergent, statistically null, bounded, bounded statistically convergent and bounded statistically null X valued sequence spaces, where $(X,\parallel \cdot ,\cdot \parallel )$ is a real 2-normed space. By $\theta =(\theta ,\theta ,\theta ,\dots )$, we mean the zero element of X.

3 Main results

In this section, we define the generalized difference matrix $B_{(m)}^n$ and introduce difference sequence spaces $\overline{c}(B_{(m)}^n,p,\parallel \cdot ,\cdot \parallel )$, ${\overline{c}}_0(B_{(m)}^n,p,\parallel \cdot ,\cdot \parallel )$, $m(B_{(m)}^n,p,\parallel \cdot ,\cdot \parallel )$, $m_0(B_{(m)}^n,p,\parallel \cdot ,\cdot \parallel )$, $c(B_{(m)}^n,p,\parallel \cdot ,\cdot \parallel )$, $c_0(B_{(m)}^n,p,\parallel \cdot ,\cdot \parallel )$, $l_{\mathrm{\infty }}(B_{(m)}^n,p,\parallel \cdot ,\cdot \parallel )$, $W(B_{(m)}^n,p,\parallel \cdot ,\cdot \parallel )$, which are defined on a real linear 2-normed space. We investigate some topological properties of the spaces ${\overline{c}}_0(B_{(m)}^n,p,\parallel \cdot ,\cdot \parallel )$, $\overline{c}(B_{(m)}^n,p,\parallel \cdot ,\cdot \parallel )$, $m(B_{(m)}^n,p,\parallel \cdot ,\cdot \parallel )$ and $m_0(B_{(m)}^n,p,\parallel \cdot ,\cdot \parallel )$ including linearity, existence of paranorm and solidity. Further, we show that the sequence spaces $m(B_{(m)}^n,p,\parallel \cdot ,\cdot \parallel )$ and $m_0(B_{(m)}^n,p,\parallel \cdot ,\cdot \parallel )$ are complete paranormed spaces when the base space is a 2-Banach space. Moreover, we give some inclusion relations.

By the notation $x_k\stackrel{\mathrm{stat}}{\to }0$, we will mean that $x_k$ is statistically convergent to zero, throughout the paper. Let m, n be non-negative integers and $p=(p_k)$ be a sequence of strictly positive real numbers. Then we define new sequence spaces as follows:

$$\begin{array}{c}\overline{c}(B_{(m)}^n,p,\parallel \cdot ,\cdot \parallel )=\{x=(x_k)\in w(X):{\parallel B_{(m)}^n{x}_k-L,z\parallel }^{p_k}\stackrel{\mathrm{stat}}{\to }0,\hfill \\ \phantom{\overline{c}(B_{(m)}^n,p,\parallel \cdot ,\cdot \parallel )=}\text{for every nonzero }z\in X\text{ and some }L\in X\},\hfill \\ {\overline{c}}_0(B_{(m)}^n,p,\parallel \cdot ,\cdot \parallel )=\{x=(x_k)\in w(X):{\parallel B_{(m)}^n{x}_k,z\parallel }^{p_k}\stackrel{\mathrm{stat}}{\to }0,\text{ for every nonzero }z\in X\},\hfill \\ l_{\mathrm{\infty }}(B_{(m)}^n,p,\parallel \cdot ,\cdot \parallel )=\{x=(x_k)\in w(X):\underset{k\ge 1}{sup}\left({\parallel B_{(m)}^n{x}_k,z\parallel }^{p_k}\right)<\mathrm{\infty },\hfill \\ \phantom{l_{\mathrm{\infty }}(B_{(m)}^n,p,\parallel \cdot ,\cdot \parallel )=}\text{for every nonzero }z\in X\},\hfill \\ c(B_{(m)}^n,p,\parallel \cdot ,\cdot \parallel )=\{x=(x_k):\underset{k\to \mathrm{\infty }}{lim}{\parallel B_{(m)}^n{x}_k-L,z\parallel }^{p_k}=0,\hfill \\ \phantom{c(B_{(m)}^n,p,\parallel \cdot ,\cdot \parallel )=}\text{for every nonzero }z\in X\text{ and some }L\in X\},\hfill \\ c_0(B_{(m)}^n,p,\parallel \cdot ,\cdot \parallel )=\{x=(x_k):\underset{k\to \mathrm{\infty }}{lim}{\parallel B_{(m)}^n{x}_k,z\parallel }^{p_k}=0,\text{ for every nonzero }z\in X\},\hfill \\ W(B_{(m)}^n,p,\parallel \cdot ,\cdot \parallel )=\{x=(x_k)\in w(X):\underset{j\to \mathrm{\infty }}{lim}\frac{1}{j}\sum _{k=1}^j{\parallel B_{(m)}^n{x}_k-L,z\parallel }^{p_k}=0,\hfill \\ \phantom{W(B_{(m)}^n,p,\parallel \cdot ,\cdot \parallel )=}\text{for every nonzero }z\in X\text{ and some }L\in X\},\hfill \\ m(B_{(m)}^n,p,\parallel \cdot ,\cdot \parallel )=\overline{c}(B_{(m)}^n,p,\parallel \cdot ,\cdot \parallel )\cap l_{\mathrm{\infty }}(B_{(m)}^n,p,\parallel \cdot ,\cdot \parallel )\hfill \end{array}$$

and

$$m_0(B_{(m)}^n,p,\parallel \cdot ,\cdot \parallel )={\overline{c}}_0(B_{(m)}^n,p,\parallel \cdot ,\cdot \parallel )\cap l_{\mathrm{\infty }}(B_{(m)}^n,p,\parallel \cdot ,\cdot \parallel ),$$

where $B_{(m)}^{n}x=B_{(m)}^n{x}_k=r{B}_{(m)}^{n-1}{x}_k+s{B}_{(m)}^{n-1}{x}_{k-m}$ and $B_{(m)}^0{x}_k=x_k$ for all $k\in \mathbb{N}$, which is equivalent to the binomial representation as follows:

$$B_{(m)}^n{x}_k=\sum _{v=0}^n\left(\genfrac{}{}{0ex}{}{n}{v}\right)r^{n-v}{s}^v{x}_{k-mv}.$$

In this representation, we obtain the matrix $B_{(1)}^n$ defined in [36] for $n>1$ and in [35] for $n=1$.

• (1) If we take $n=0$ then the above sequence spaces are reduced to $\overline{c}(p,\parallel \cdot ,\cdot \parallel )$, ${\overline{c}}_0(p,\parallel \cdot ,\cdot \parallel )$, $l_{\mathrm{\infty }}(p,\parallel \cdot ,\cdot \parallel )$, $c(p,\parallel \cdot ,\cdot \parallel )$, $c_0(p,\parallel \cdot ,\cdot \parallel )$, $W(p,\parallel \cdot ,\cdot \parallel )$, $m(p,\parallel \cdot ,\cdot \parallel )$ and $m_0(p,\parallel \cdot ,\cdot \parallel )$, respectively.

• (2) If we take $r=1$, $s=-1$, then the sequence spaces $\overline{c}(B_{(m)}^n,p,\parallel \cdot ,\cdot \parallel )$, ${\overline{c}}_0(B_{(m)}^n,p,\parallel \cdot ,\cdot \parallel )$, $l_{\mathrm{\infty }}(B_{(m)}^n,p,\parallel \cdot ,\cdot \parallel )$, $W(B_{(m)}^n,p,\parallel \cdot ,\cdot \parallel )$, $m(B_{(m)}^n,p,\parallel \cdot ,\cdot \parallel )$, $m_0(B_{(m)}^n,p,\parallel \cdot ,\cdot \parallel )$ are reduced to $\overline{c}({\mathrm{\Delta }}_{(m)}^n,p,\parallel \cdot ,\cdot \parallel )$, ${\overline{c}}_0({\mathrm{\Delta }}_{(m)}^n,p,\parallel \cdot ,\cdot \parallel )$, $l_{\mathrm{\infty }}({\mathrm{\Delta }}_{(m)}^n,p,\parallel \cdot ,\cdot \parallel )$, $W({\mathrm{\Delta }}_{(m)}^n,p,\parallel \cdot ,\cdot \parallel )$, $m({\mathrm{\Delta }}_{(m)}^n,p,\parallel \cdot ,\cdot \parallel )$ and $m_0({\mathrm{\Delta }}_{(m)}^n,p,\parallel \cdot ,\cdot \parallel )$, respectively, which are studied in [34].

• (3) By taking $p_k=1$ for all $k\in \mathbb{N}$, then these sequence spaces are denoted by $\overline{c}(B_{(m)}^n,\parallel \cdot ,\cdot \parallel )$, ${\overline{c}}_0(B_{(m)}^n,\parallel \cdot ,\cdot \parallel )$, $l_{\mathrm{\infty }}(B_{(m)}^n,\parallel \cdot ,\cdot \parallel )$, $c(B_{(m)}^n,\parallel \cdot ,\cdot \parallel )$, $c_0(B_{(m)}^n,\parallel \cdot ,\cdot \parallel )$, $W(B_{(m)}^n,\parallel \cdot ,\cdot \parallel )$, $m(B_{(m)}^n,\parallel \cdot ,\cdot \parallel )$ and $m_0(B_{(m)}^n,\parallel \cdot ,\cdot \parallel )$, respectively.

• (4) If we replace the base space X, which is a real linear 2-normed space by ℂ, complete normed linear space, and take $m=1$ and take $r=1$, $s=-1$, then the above sequence spaces are denoted by $\overline{c}({\mathrm{\Delta }}_{(1)}^n,p)$, ${\overline{c}}_0({\mathrm{\Delta }}_{(1)}^n,p)$, $l_{\mathrm{\infty }}({\mathrm{\Delta }}_{(1)}^n,p)$, $c({\mathrm{\Delta }}_{(1)}^n,p)$, $c_0({\mathrm{\Delta }}_{(1)}^n,p)$, $W({\mathrm{\Delta }}_{(1)}^n,p)$, $m({\mathrm{\Delta }}_{(1)}^n,p)$ and $m_0({\mathrm{\Delta }}_{(1)}^n,p)$, respectively.

• (5) If we take $r=1$, $s=-1$, $p_k=1$ for all $k\in \mathbb{N}$, then these sequence spaces are denoted by $\overline{c}({\mathrm{\Delta }}_{(m)}^n,\parallel \cdot ,\cdot \parallel )$, ${\overline{c}}_0({\mathrm{\Delta }}_{(m)}^n,\parallel \cdot ,\cdot \parallel )$, $l_{\mathrm{\infty }}({\mathrm{\Delta }}_{(m)}^n,\parallel \cdot ,\cdot \parallel )$, $c({\mathrm{\Delta }}_{(m)}^n,\parallel \cdot ,\cdot \parallel )$, $c_0({\mathrm{\Delta }}_{(m)}^n,\parallel \cdot ,\cdot \parallel )$, $W({\mathrm{\Delta }}_{(m)}^n,\parallel \cdot ,\cdot \parallel )$, $m({\mathrm{\Delta }}_{(m)}^n,\parallel \cdot ,\cdot \parallel )$ and $m_0({\mathrm{\Delta }}_{(m)}^n,p,\parallel \cdot ,\cdot \parallel )$, respectively.

• (6) If we replace the base space X, which is a real linear 2-normed space by ℂ, we obtain the spaces $\overline{c}(B_{(m)}^n,p)$, ${\overline{c}}_0(B_{(m)}^n,p)$, $l_{\mathrm{\infty }}(B_{(m)}^n,p)$, $c(B_{(m)}^n,p)$, $c_0(B_{(m)}^n,p)$, $W(B_{(m)}^n,p)$, $m(B_{(m)}^n,p)$ and $m_0(B_{(m)}^n,p)$, respectively.

• (7) Moreover, if we take $X=\mathbb{C}$, $n=0$ and $p_k=1$ for all $k\in \mathbb{N}$, we get the spaces $\overline{c}$, ${\overline{c}}_0$, $l_{\mathrm{\infty }}$, c, $c_0$, W, m and $m_0$, respectively.

Theorem 3.1 Let $p=(p_k)$ be a sequence of strictly positive real numbers. Then the sequence spaces $Z(B_{(m)}^n,p,\parallel \cdot ,\cdot \parallel )$ are linear spaces where $Z=\overline{c},{\overline{c}}_0,l_{\mathrm{\infty }},W,m,m_0$.

Proof The proof of the theorem can be obtained by similar techniques in [34]. □

Theorem 3.2 For any two sequences $p=(p_k)$ and $t=(t_k)$ of positive real numbers and for any two 2-norms ${\parallel \cdot ,\cdot \parallel }_1$ and ${\parallel \cdot ,\cdot \parallel }_2$ on X we have $Z(B_{(m)}^n,p,{\parallel \cdot ,\cdot \parallel }_1)\cap Z(B_{(m)}^n,p,{\parallel \cdot ,\cdot \parallel }_2)\ne \mathrm{\varnothing }$, where $Z=\overline{c},{\overline{c}}_0,m,m_0$.

Proof The proof follows from the fact that the zero element belongs to each of the sequence spaces involved in the intersection. □

Theorem 3.3 Let $(X,\parallel \cdot ,\cdot \parallel )$ be a 2-Banach space. Then the spaces $m(B_{(m)}^n,p,\parallel \cdot ,\cdot \parallel )$, $m_0(B_{(m)}^n,p,\parallel \cdot ,\cdot \parallel )$ are complete paranormed sequence spaces, paranormed by

$$g(x)=\underset{k\in \mathbb{N},z\in X}{sup}\left({\parallel B_{(m)}^n{x}_k,z\parallel }^{\frac{p_k}{M}}\right),$$

(3.1)

where $M=max\{1,H\}$ and $H={sup}_k{p}_k$, $h={inf}_k{p}_k$.

Proof We will prove the theorem for the sequence space $m_0(B_{(m)}^n,p,\parallel \cdot ,\cdot \parallel )$. It can be proved for the space $m(B_{(m)}^n,p,\parallel \cdot ,\cdot \parallel )$ similarly.

Clearly $g(-x)=g(x)$ and $g(\theta )=0$. From the following inequality, we have

$$\begin{array}{rcl}g(x+y)& =& \underset{k\in \mathbb{N},z\in X}{sup}\left({\parallel B_{(m)}^n(x_k+y_k),z\parallel }^{\frac{p_k}{M}}\right)\\ \le & \underset{k\in \mathbb{N},z\in X}{sup}\left({\parallel B_{(m)}^n{x}_k,z\parallel }^{\frac{p_k}{M}}\right)+\underset{k\in \mathbb{N},z\in X}{sup}\left({\parallel B_{(m)}^n{y}_k,z\parallel }^{\frac{p_k}{M}}\right).\end{array}$$

This implies that $g(x+y)\le g(x)+g(y)$.

To prove the continuity of scalar multiplication, assume that $(x^n)$ be any sequence of the points in $m_0(B_{(m)}^n,p,\parallel \cdot ,\cdot \parallel )$ such that $g(x^n-x)\to 0$ and $({\lambda }_n)$ be any sequence of scalars such that ${\lambda }_n\to \lambda$. Since the inequality

$$g\left(x^n\right)\le g(x)+g(x^n-x)$$

holds by subadditivity of g, $(g(x^n))$ is bounded. Thus, we have

$$\begin{array}{rcl}g({\lambda }_n{x}^n-\lambda x)& =& \underset{k\in \mathbb{N},z\in X}{sup}\left({\parallel B_{(m)}^n{\lambda }_n{x}_k^n-\lambda x_k,z\parallel }^{\frac{p_k}{M}}\right)\\ \le & {(max\{|{\lambda }_n-\lambda {|}^h,|{\lambda }_n-\lambda {|}^H\})}^{\frac{1}{M}}\underset{k\in \mathbb{N},z\in X}{sup}\left({\parallel B_{(m)}^n{x}_k,z\parallel }^{\frac{p_k}{M}}\right)\\ +{(max\{|\lambda {|}^h,|\lambda {|}^H\})}^{\frac{1}{M}}\underset{k\in \mathbb{N},z\in X}{sup}\left({\parallel B_{(m)}^n(x_k^n-x),z\parallel }^{\frac{p_k}{M}}\right)\\ =& {(max\{|{\lambda }_n-\lambda {|}^h,|{\lambda }_n-\lambda {|}^H\})}^{\frac{1}{M}}g\left(x^n\right)\\ +{(max\{|\lambda {|}^h,|\lambda {|}^H\})}^{\frac{1}{M}}g(x^n-x)\end{array}$$

which tends to zero as $n\to \mathrm{\infty }$. Hence, g is a paranorm on the sequence space $m_0(B_{(m)}^n,p,\parallel \cdot ,\cdot \parallel )$.

To prove that $m_0(B_{(m)}^n,p,\parallel \cdot ,\cdot \parallel )$ is complete, assume that $(x^i)$ is a Cauchy sequence in $m_0(B_{(m)}^n,p,\parallel \cdot ,\cdot \parallel )$. Then for a given ε ($0<\epsilon <1$), there exists a positive integer $N_0$ such that $g(x^i-x^j)<\epsilon$, for all $i,j\ge N_0$. This implies that

$$\underset{k\in \mathbb{N},z\in X}{sup}\left({\parallel B_{(m)}^n{x}_k^i-B_{(m)}^n{x}_k^j,z\parallel }^{\frac{p_k}{M}}\right)<\epsilon ,$$

for all $i,j\ge N_0$. It follows that for every nonzero $z\in X$,

$$\parallel B_{(m)}^n{x}_k^i-B_{(m)}^n{x}_k^j,z\parallel <\epsilon ,$$

for each $k\ge 1$ and for all $i,j\ge N_0$. Hence $(B_{(m)}^n{x}_k^i)$ is a Cauchy sequence in X for all $k\in \mathbb{N}$. Since X is a 2-Banach space, $(B_{(m)}^n{x}_k^i)$ is convergent in X for all $k\in \mathbb{N}$, so we write $(B_{(m)}^n{x}_k^i)\to (B_{(m)}^n{x}_k)$ as $i\to \mathrm{\infty }$. Now we have for all $i,j\ge N_0$,

$$\begin{array}{c}\underset{k\in \mathbb{N},z\in X}{sup}\left({\parallel B_{(m)}^n(x_k^i-x_k^j),z\parallel }^{\frac{p_k}{M}}\right)<\epsilon \hfill \\ \Rightarrow \underset{j\to \mathrm{\infty }}{lim}\left\{\underset{k\in \mathbb{N},z\in X}{sup}\left({\parallel B_{(m)}^n(x_k^i-x_k^j),z\parallel }^{\frac{p_k}{M}}\right)\right\}<\epsilon \hfill \\ \Rightarrow \underset{k\in \mathbb{N},z\in X}{sup}\left({\parallel B_{(m)}^n(x_k^i-x_k),z\parallel }^{\frac{p_k}{M}}\right)<\epsilon ,\hfill \end{array}$$

for all $i\ge N_0$. It follows that $(x^i-x)\in m_0(B_{(m)}^n,p,\parallel \cdot ,\cdot \parallel )$. Since $(x^i)\in m_0(B_{(m)}^n,p,\parallel \cdot ,\cdot \parallel )$ and $m_0(B_{(m)}^n,p,\parallel \cdot ,\cdot \parallel )$ is a linear space, so we have $x=x^i-(x^i-x)\in m_0(B_{(m)}^n,p,\parallel \cdot ,\cdot \parallel )$. This completes the proof. □

Theorem 3.4

• (1) If $Z_1\subset Z_2$, then $Z_1(B_{(m)}^n,p,\parallel \cdot ,\cdot \parallel )\subset Z_2(B_{(m)}^n,p,\parallel \cdot ,\cdot \parallel )$ and the inclusion is strict, where $Z_1,Z_2=c,c_0,l_{\mathrm{\infty }}$.

• (2) If $n_1<n_2$, then $Z(B_{(m)}^{n_1},p,\parallel \cdot ,\cdot \parallel )\subset Z(B_{(m)}^{n_2},p,\parallel \cdot ,\cdot \parallel )$ and the inclusion is strict, where $Z=c,c_0,l_{\mathrm{\infty }}$.

Proof The parts of proof $Z_1(B_{(m)}^n,p,\parallel \cdot ,\cdot \parallel )\subset Z_2(B_{(m)}^n,p,\parallel \cdot ,\cdot \parallel )$ and $Z_1(B_{(m)}^{n_1},p,\parallel \cdot ,\cdot \parallel )\subset Z_2(B_{(m)}^{n_2},p,\parallel \cdot ,\cdot \parallel )$ are easy. To show the inclusions are strict, choose $Z_1=c_0$, $Z_2=c$, $x=(x_k)=(k^2,k^2)$ and consider the 2-norm as defined in (2.1), let $p_k=1$ for all $k\in \mathbb{N}$, $m=1$, $n=2$, $r=1$, $s=-1$, then $x\in c(B_{(1)}^2,\parallel \cdot ,\cdot \parallel )$ but $x\notin c_0(B_{(1)}^2,\parallel \cdot ,\cdot \parallel )$. If we choose $Z=c$, $x=(x_k)=(k^2,k^2)$ and $p_k=1$ for all $k\in \mathbb{N}$, $m=1$, $n=2$, $r=1$, $s=-1$, then $x\in c(B_{(1)}^2,\parallel \cdot ,\cdot \parallel )$ but $x\notin c(B_{(1)}^1,\parallel \cdot ,\cdot \parallel )$. These complete the proofs of parts (1) and (2) of the theorem, respectively. □

Theorem 3.5

• (1) $c(B_{(m)}^n,\parallel \cdot ,\cdot \parallel )\subset \overline{c}(B_{(m)}^n,\parallel \cdot ,\cdot \parallel )$ and the inclusion is strict.

• (2) $\overline{c}(\parallel \cdot ,\cdot \parallel )\subset \overline{c}(B_{(m)}^n,\parallel \cdot ,\cdot \parallel )$ and the inclusion is strict.

• (3) $\overline{c}(B_{(m)}^n,\parallel \cdot ,\cdot \parallel )$ and $l_{\mathrm{\infty }}(B_{(m)}^n,\parallel \cdot ,\cdot \parallel )$ overlap but neither one contains the other.

Proof

• (1) It is clear that $c(B_{(m)}^n,\parallel \cdot ,\cdot \parallel )\subset \overline{c}(B_{(m)}^n,\parallel \cdot ,\cdot \parallel )$. To show that the inclusion is strict, choose the sequence $x=(x_k)$ such that,

  $$B_{(m)}^n{x}_k=\{\begin{array}{cc}(0,\sqrt{k}),\hfill & k=n^2,\hfill \\ (0,0),\hfill & k\ne n^2,\hfill \end{array}$$

  (3.2)

where $n\in \mathbb{N}-\{0\}$, and consider the 2-norm as defined in (2.1). Then we obtain $B_{(m)}^n{x}_k\in \overline{c}(\parallel \cdot ,\cdot \parallel )$, but $B_{(m)}^n{x}_k\notin c(\parallel \cdot ,\cdot \parallel )$. That is, $x_k\in \overline{c}(B_{(m)}^n,p,\parallel \cdot ,\cdot \parallel )$, but $x_k\notin c(B_{(m)}^n,p,\parallel \cdot ,\cdot \parallel )$.

• (2) It is easy to see that $\overline{c}(\parallel \cdot ,\cdot \parallel )\subset \overline{c}(B_{(m)}^n,\parallel \cdot ,\cdot \parallel )$. To show that the inclusion is strict, let us take $x=(x_k)=(k,k)$ and consider the 2-norm as defined in (2.1), $m=1$, $n=1$, $r=1$, $s=-1$, then $x\in \overline{c}(B_{(1)}^1,\parallel \cdot ,\cdot \parallel )$ but $x\notin \overline{c}(\parallel \cdot ,\cdot \parallel )$.

• (3) Since the sequence $x=\theta$ belongs to each of the sequence spaces, the overlapping part of the proof is obvious. For the other part of the proof, consider the sequence defined by (3.2) and the 2-norm as defined in (2.1). Then $x\in \overline{c}(B_{(m)}^n,\parallel \cdot ,\cdot \parallel )$, but $x\notin l_{\mathrm{\infty }}(B_{(m)}^n,\parallel \cdot ,\cdot \parallel )$. Conversely if we choose $(B_{(m)}^n{x}_k)=(\overline{1},\overline{0},\overline{1},\overline{0},\dots )$ where $\overline{k}=(k,k)$ for all $k=0,1$, then $B_{(m)}^n{x}_k\in l_{\mathrm{\infty }}(\parallel \cdot ,\cdot \parallel )$ but $B_{(m)}^n{x}_k\notin \overline{c}(\parallel \cdot ,\cdot \parallel )$. That is, $x\in l_{\mathrm{\infty }}(B_{(m)}^n,\parallel \cdot ,\cdot \parallel )$ but $x\notin \overline{c}(B_{(m)}^n,\parallel \cdot ,\cdot \parallel )$.

 □

Theorem 3.6 The space $Z(B_{(m)}^n,p,\parallel \cdot ,\cdot \parallel )$ is not solid in general, where $Z=\overline{c},{\overline{c}}_0,m,m_0$.

Proof To show that the space is not solid in general, consider the following examples. □

Example 3.7 Let $m=3$, $n=1$, $r=1$, $s=-1$ and consider the 2-normed space as defined in Example 2.1. Let $p_k=5$ for all $k\in \mathbb{N}$. Consider the sequence $(x_k)$, where $x_k=(x_k^i)$ is defined by $(x_k^i)=(k,k,k,\dots )$ for each fixed $k\in \mathbb{N}$. Then $x_k\in Z(B_{(3)}^1,p,\parallel \cdot ,\cdot \parallel )$ for $Z=\overline{c},m$. Let ${\alpha }_k={(-1)}^k$, then $({\alpha }_k{x}_k)\notin Z(B_{(3)}^1,p,\parallel \cdot ,\cdot \parallel )$ for $Z=\overline{c},m$. Thus $Z(B_{(3)}^1,p,\parallel \cdot ,\cdot \parallel )$ for $Z=\overline{c},m$ is not solid in general.

Example 3.8 Let $m=3$, $n=1$, $r=1$, $s=-1$ and consider the 2-normed space as defined in Example 2.1. Let $p_k=1$ for all odd k and $p_k=2$ for all even k. Consider the sequence $(x_k)$, where $x_k=(x_k^i)$ is defined by $(x_k^i)=(3,3,3,\dots )$ for each fixed $k\in \mathbb{N}$. Then $x_k\in Z(B_{(3)}^1,p,\parallel \cdot ,\cdot \parallel )$ for $Z={\overline{c}}_0,{\overline{m}}_0$. Let ${\alpha }_k={(-1)}^k$, then $({\alpha }_k{x}_k)\notin Z(B_{(3)}^1,p,\parallel \cdot ,\cdot \parallel )$ for $Z={\overline{c}}_0,{\overline{m}}_0$. Thus $Z(B_{(3)}^1,p,\parallel \cdot ,\cdot \parallel )$ for $Z={\overline{c}}_0,{\overline{m}}_0$ is not solid in general.

Theorem 3.9 The spaces $m_0(B_{(m)}^n,p,\parallel \cdot ,\cdot \parallel )$ and $m(B_{(m)}^n,p,\parallel \cdot ,\cdot \parallel )$ are nowhere dense subsets of $l_{\mathrm{\infty }}(B_{(m)}^n,p,\parallel \cdot ,\cdot \parallel )$.

Proof From Theorem 3.3, it follows that $m_0(B_{(m)}^n,p,\parallel \cdot ,\cdot \parallel )$ and $m(B_{(m)}^n,p,\parallel \cdot ,\cdot \parallel )$ are closed subspaces of $l_{\mathrm{\infty }}(B_{(m)}^n,p,\parallel \cdot ,\cdot \parallel )$. Since the inclusion relations

$$m_0(B_{(m)}^n,p,\parallel \cdot ,\cdot \parallel )\subset l_{\mathrm{\infty }}(B_{(m)}^n,p,\parallel \cdot ,\cdot \parallel ),m(B_{(m)}^n,p,\parallel \cdot ,\cdot \parallel )\subset l_{\mathrm{\infty }}(B_{(m)}^n,p,\parallel \cdot ,\cdot \parallel )$$

are strict, the spaces $m_0(B_{(m)}^n,p,\parallel \cdot ,\cdot \parallel )$ and $m(B_{(m)}^n,p,\parallel \cdot ,\cdot \parallel )$ are nowhere dense subsets of $l_{\mathrm{\infty }}(B_{(m)}^n,p,\parallel \cdot ,\cdot \parallel )$ by Lemma 2.2. □

Theorem 3.10 Let $p=(p_k)$ be a sequence of non-negative bounded real numbers such that ${inf}_k{p}_k>0$. Then

$$W(B_{(m)}^n,p,\parallel \cdot ,\cdot \parallel )\cap l_{\mathrm{\infty }}(B_{(m)}^n,p,\parallel \cdot ,\cdot \parallel )\subset m(B_{(m)}^n,p,\parallel \cdot ,\cdot \parallel ).$$

Proof Let $(x_k)\in W(B_{(m)}^n,p,\parallel \cdot ,\cdot \parallel )\cap l_{\mathrm{\infty }}(B_{(m)}^n,p,\parallel \cdot ,\cdot \parallel )$. Then for a given $\epsilon >0$, we have

$$\begin{array}{rcl}\frac{1}{j}\sum _{k=1}^j{\parallel B_{(m)}^n{x}_k-L,z\parallel }^{p_k}& \ge & \frac{1}{j}\underset{{\parallel B_{(m)}^n{x}_k-L,z\parallel }^{p_k}\ge \epsilon }{\sum _{k=1}^j}{\parallel B_{(m)}^n{x}_k-L,z\parallel }^{p_k}\\ \ge & \epsilon \frac{1}{j}\left|\{k\le j:{\parallel B_{(m)}^n{x}_k-L,z\parallel }^{p_k}\ge \epsilon \}\right|.\end{array}$$

If we take the limit for $j\to \mathrm{\infty }$, it follows that $(x_k)\in c(B_{(m)}^n,p,\parallel \cdot ,\cdot \parallel )$ from the inequality above. Since $(x_k)\in l_{\mathrm{\infty }}(B_{(m)}^n,p,\parallel \cdot ,\cdot \parallel )$, we have the result. □