Generalized difference sequence spaces associated with a multiplier sequence on a real n-normed space

Abstract

The purpose of this paper is to introduce new sequence spaces associated with a multiplier sequence by using an infinite matrix, an Orlicz function and a generalized B-difference operator on a real n-normed space. Some topological properties of these spaces are examined. We also define a new concept, which will be called ${(B_{\mathrm{\Lambda }}^{\mu })}^n$-statistical A-convergence, and establish some inclusion connections between the sequence space $W(A,B_{\mathrm{\Lambda }}^{\mu },p,\parallel \cdot ,\dots ,\cdot \parallel )$ and the set of all ${(B_{\mathrm{\Lambda }}^{\mu })}^n$-statistically A-convergent sequences.

MSC:40A05, 40B50, 46A19, 46A45.

1 Introduction

Let w, $l_{\mathrm{\infty }}$, c and $c_0$ be the linear spaces of all, bounded, convergent and null sequences $x=(x_k)$ for all $k\in \mathbb{N}$, respectively.

Let X and Y be two subsets of w. By $(X,Y)$, we denote the class of all matrices of A such that $A_m(x)={\sum }_{k=1}^{\mathrm{\infty }}{a}_{mk}{x}_k$ converges for each $m\in \mathbb{N}$, the set of all natural numbers, and the sequence $Ax={(A_m(x))}_{m=1}^{\mathrm{\infty }}\in Y$ for all $x\in X$.

Let $A=(a_{mk})$ be an infinite matrix of complex numbers. Then A is said to be regular if and only if it satisfies the following well-known Silverman-Toeplitz conditions:

• (1) ${sup}_m{\sum }_{k=1}^{\mathrm{\infty }}|a_{mk}|<\mathrm{\infty }$,

• (2) ${lim}_{m\to \mathrm{\infty }}{a}_{mk}=0$ for each $k\in \mathbb{N}$,

• (3) ${lim}_{m\to \mathrm{\infty }}{\sum }_{k=1}^{\mathrm{\infty }}{a}_{mk}=1$.

The idea of statistical convergence was given by Zygmund [1] in 1935. The concept of statistical convergence was introduced by Fast [2] and Schoenberg [3] independently for the real sequences. Later on, it was further investigated from a sequence point of view and linked with the summability theory by Fridy [4] and many others. The natural density of a subset E of ℕ is denoted by

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

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

Spaces of strongly summable sequences were studied by Kuttner [5], Maddox [6] and others. The class of sequences that are strongly Cesaro summable with respect to a modulus was introduced by Maddox [7] as an extension of the definition of strongly Cesaro summable sequences. Connor [8] has further extended this definition to a definition of strong A-summability with respect to a modulus, where $A=(a_{mk})$ is a non-negative regular matrix, and established some connections between strong A-summability with respect to a modulus and A-statistical convergence.

Assume now that A is a non-negative regular summability matrix. Then a sequence $x=(x_k)$ is said to be A-statistically convergent to a number L if ${\delta }_A(K)={lim}_{m\to \mathrm{\infty }}{\sum }_{k=1}^{\mathrm{\infty }}{a}_{mk}{\chi }_K(k)=0$ or, equivalently, ${lim}_{m\to \mathrm{\infty }}{\sum }_{k\in K}{a}_{mk}=0$ for every $\epsilon >0$, where $K=\{k\in \mathbb{N}:|x_k-L|\ge \epsilon \}$ and ${\chi }_K(k)$ is the characteristic function of K. We denote this limit by ${\mathit{st}}_A\text{-}limx=L$ [9] (see also [8, 10, 11]).

For $A=C_1$, the Cesaro matrix, A-statistical convergence reduces to statistical convergence (see [2, 4]). Taking $A=I$, the identity matrix, A-statistical convergence coincides with ordinary convergence. We note that if $A=(a_{mk})$ is a regular summability matrix for which ${lim}_m{max}_k|a_{mk}|=0$, then A-statistical convergence is stronger than usual convergence [10]. It should be also noted that the concept of A-statistical convergence may also be given in normed spaces [12].

The notion of difference sequence space was introduced by Kızmaz [13]. It was further generalized by Et and Çolak [14] as follows: $Z({\mathrm{\Delta }}^{\mu })=\{x=(x_k)\in w:({\mathrm{\Delta }}^{\mu }{x}_k)\in Z\}$ for $Z=l_{\mathrm{\infty }},c\text{ and }{c}_0$, where μ is a non-negative integer, ${\mathrm{\Delta }}^{\mu }{x}_k={\mathrm{\Delta }}^{\mu -1}{x}_k-{\mathrm{\Delta }}^{\mu -1}{x}_{k+1}$, ${\mathrm{\Delta }}^0{x}_k=x_k$ for all $k\in \mathbb{N}$ or equivalent to the following binomial representation:

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

These sequence spaces were generalized by Et and Başarır [15] taking $Z=l_{\mathrm{\infty }}(p),c(p)\text{and }{c}_0(p)$.

Dutta [16] introduced the following difference sequence spaces using a new difference operator: $Z({\mathrm{\Delta }}_{(\eta )})=\{x=(x_k)\in w:{\mathrm{\Delta }}_{(\eta )}x\in Z\}$ for $Z=l_{\mathrm{\infty }},c\text{ and }{c}_0$, where ${\mathrm{\Delta }}_{(\eta )}x=({\mathrm{\Delta }}_{(\eta )}{x}_k)=(x_k-x_{k-\eta })$ for all $k,\eta \in \mathbb{N}$.

In [17], Dutta introduced the sequence spaces $\overline{c}(\parallel \cdot ,\cdot \parallel ,{\mathrm{\Delta }}_{(\eta )}^{\mu },p)$, $\overline{c_0}(\parallel \cdot ,\cdot \parallel ,{\mathrm{\Delta }}_{(\eta )}^{\mu },p)$, $l_{\mathrm{\infty }}(\parallel \cdot ,\cdot \parallel ,{\mathrm{\Delta }}_{(\eta )}^{\mu },p)$, $m(\parallel \cdot ,\cdot \parallel ,{\mathrm{\Delta }}_{(\eta )}^{\mu },p)$ and $m_0(\parallel \cdot ,\cdot \parallel ,{\mathrm{\Delta }}_{(\eta )}^n,p)$, where η, $\mu \in \mathbb{N}$ and ${\mathrm{\Delta }}_{(\eta )}^{\mu }x=({\mathrm{\Delta }}_{(\eta )}^{\mu }{x}_k)=({\mathrm{\Delta }}_{(\eta )}^{\mu -1}{x}_k-{\mathrm{\Delta }}_{(\eta )}^{\mu -1}{x}_{k-\eta })$ and ${\mathrm{\Delta }}_{(\eta )}^0{x}_k=x_k$ for all $k,\eta \in \mathbb{N}$, which is equivalent to the following binomial representation:

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

The difference sequence spaces have been studied by several authors, [15–34]. Başar and Altay [35] introduced the generalized difference matrix $B=(b_{mk})$ for all $k,m\in \mathbb{N}$, which is a generalization of ${\mathrm{\Delta }}_{(1)}$-difference operator, by

$$b_{mk}=\{\begin{array}{ll}r,& k=m,\\ s,& k=m-1,\\ 0& (k>m)\text{ or }(0\le k<m-1).\end{array}$$

Başarır and Kayıkçı [36] defined the matrix $B^{\mu }=(b_{mk}^{\mu })$ which reduced the difference matrix ${\mathrm{\Delta }}_{(1)}^{\mu }$ in case $r=1$, $s=-1$. The generalized $B^{\mu }$-difference operator is equivalent to the following binomial representation:

$$B^{\mu }x=B^{\mu }(x_k)=\sum _{v=0}^{\mu }\left(\genfrac{}{}{0ex}{}{\mu }{v}\right)r^{\mu -v}{s}^v{x}_{k-v}.$$

Related articles can be found in [35–41].

The concept of 2-normed space was initially introduced by Gähler [42] in the mid of 1960s, while that of n-normed spaces can be found in Misiak [43]. Since then, many others have used these concepts and obtained various results; see, for instance, Gunawan [44], Gunawan and Mashadi [45], Gunawan et al. [46] (see also [47–54]).

2 Definitions and preliminaries

Let n be a non-negative integer and let X be a real vector space of dimension $d\ge n\ge 2$. A real-valued function $\parallel \cdot ,\dots ,\cdot \parallel$ on $X^n$ satisfies the following conditions:

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

• (2) $\parallel x_1,\dots ,x_n\parallel$ is invariant under permutation,

• (3) $\parallel \alpha x_1,\dots ,x_{n-1},x_n\parallel =|\alpha |\parallel x_1,\dots ,x_{n-1},x_n\parallel$ for any $\alpha \in \mathbb{R}$,

• (4) $\parallel x_1,\dots ,x_{n-1},y+z\parallel ⩽\parallel x_1,\dots ,x_{n-1},y\parallel +\parallel x_1,\dots ,x_{n-1},z\parallel$.

Then it is called an n-norm on X and the pair $(X,\parallel \cdot ,\dots ,\cdot \parallel )$ is called an n-normed space. A trivial example of an n-normed space is $X={\mathbb{R}}^n$ equipped with the following Euclidean n-norm: ${\parallel x_1,\dots ,x_n\parallel }_E=|det(x_{ij})|$, where $x_i=(x_{i_1},\dots ,x_{i_n})\in {\mathbb{R}}^n$ for each $i=1,\dots ,n$. The standard n-norm on X, where X is a real inner product space of dimension $d⩾n$, is defined as

$${\parallel x_1,\dots ,x_n\parallel }_S:={\left|\begin{array}{ccc}〈x_1,x_1〉& \cdots & 〈x_1,x_n〉\\ ⋮& \ddots & ⋮\\ 〈x_n,x_1〉& \cdots & 〈x_n,x_n〉\end{array}\right|}^{\frac{1}{2}},$$

where $〈\cdot ,\cdot 〉$ denotes the inner product on X. If $X={\mathbb{R}}^n$, then this n-norm is exactly the same as the Euclidean n-norm ${\parallel x_1,\dots ,x_n\parallel }_E$ as mentioned earlier. Notice that for $n=1$, the n-norm above is the usual norm ${\parallel x_1\parallel }_S={〈x_1,x_1〉}^{\frac{1}{2}}$ which gives the length of $x_1$, while for $n=2$, it defines the standard 2-norm ${\parallel x_1,x_2\parallel }_S={({\parallel x_1\parallel }_S^2.{\parallel x_2\parallel }_S^2-{〈x_1,x_1〉}^2)}^{\frac{1}{2}}$ which represents the area of the parallelogram spanned by $x_1$ and $x_2$. Further, if $X={\mathbb{R}}^3$, then ${\parallel x_1,x_2,x_3\parallel }_S={\parallel x_1,x_2,x_3\parallel }_E$ represents the volume of the parallelograms spanned by $x_1$, $x_2$ and $x_3$. In general ${\parallel x_1,\dots ,x_n\parallel }_S$ represents the volume of the n-dimensional parallelepiped spanned by $x_1,\dots ,x_n$ in X.

A sequence $(x_k)$ in an n-normed space $(X,\parallel \cdot ,\dots ,\cdot \parallel )$ is said to converge to some $L\in X$ in the n-norm if for each $\epsilon >0$ there exists a positive integer $n_0=n_0(\epsilon )$ such that $\parallel x_k-L,z_1,\dots ,z_{n-1}\parallel <\epsilon$ for all $k⩾n_0$ and for every $z_1,\dots ,z_{n-1}\in X$ [45].

An Orlicz function is a function $M:[0,\mathrm{\infty })\to [0,\mathrm{\infty })$ which is continuous, non-decreasing and convex with $M(0)=0$, $M(x)>0$ for $x>0$ and $M(x)\to \mathrm{\infty }$ as $x\to \mathrm{\infty }$. It is well known that if M is a convex function, then $M(\alpha x)⩽\alpha M(x)$ with $0<\alpha <1$.

Let $\mathrm{\Lambda }=({\mathrm{\Lambda }}_k)$ be a sequence of nonzero scalars. Then, for a sequence space E, the multiplier sequence space $E_{\mathrm{\Lambda }}$, associated with the multiplier sequence Λ, is defined as

$$E_{\mathrm{\Lambda }}=\{x=(x_k)\in w:({\mathrm{\Lambda }}_k{x}_k)\in E\}.$$

The following well-known inequality will be used throughout the paper. Let $p=(p_k)$ be any sequence of positive real numbers with $0<h={inf}_k{p}_k\le p_k\le {sup}_k{p}_k=H$, $D=max\{1,2^{H-1}\}$. Then we have, for all $a_k,b_k\in \mathbb{C}$ and for all $k\in \mathbb{N}$,

$$|a_k+b_k{|}^{p_k}⩽D(|a_k{|}^{p_k}+|b_k{|}^{p_k}),$$

(2.1)

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

In this paper, we introduce some new sequence spaces on a real n-normed space by using an infinite matrix, an Orlicz function and a generalized $B_{\mathrm{\Lambda }}^{\mu }$-difference operator. Further, we examine some topological properties of these sequence spaces. We also introduce a new concept which will be called ${(B_{\mathrm{\Lambda }}^{\mu })}^n$-statistical A-convergence in an n-normed space.

3 Main results

In this section, we give some new sequence spaces on a real n-normed space and investigate some topological properties of these spaces. We also give some inclusion relations.

Let $A=(a_{mk})$ be an infinite matrix of non-negative real numbers, let $p=(p_k)$ be a bounded sequence of positive real numbers for all $k\in \mathbb{N}$, and let $\mathrm{\Lambda }=({\mathrm{\Lambda }}_k)$ be a sequence of nonzero scalars. Further, let M be an Orlicz function and $(X,\parallel \cdot ,\dots ,\cdot \parallel )$ be an n-normed space. We denote the space of all X-valued sequence spaces by $w(n-X)$ and $x=(x_k)\in w(n-X)$ by $x=(x_k)$ for brevity. We define the following sequence spaces for every nonzero $z_1,z_2,\dots ,z_{n-1}\in X$ and for some $\rho >0$:

$$\begin{array}{c}W(A,B_{\mathrm{\Lambda }}^{\mu },M,p,\parallel \cdot ,\dots ,\cdot \parallel )\hfill \\ =\{x=(x_k):\underset{m\to \mathrm{\infty }}{lim}\sum _{k=1}^{\mathrm{\infty }}{a}_{mk}{\left[M\left(\parallel \frac{B_{\mathrm{\Lambda }}^{\mu }{x}_k-L}{\rho },z_1,\dots ,z_{n-1}\parallel \right)\right]}^{p_k}=0\hfill \\ \text{for some }L\in X\},\hfill \\ W_0(A,B_{\mathrm{\Lambda }}^{\mu },M,p,\parallel \cdot ,\dots ,\cdot \parallel )\hfill \\ =\{x=(x_k):\underset{m\to \mathrm{\infty }}{lim}\sum _{k=1}^{\mathrm{\infty }}{a}_{mk}{\left[M\left(\parallel \frac{B_{\mathrm{\Lambda }}^{\mu }{x}_k}{\rho },z_1,\dots ,z_{n-1}\parallel \right)\right]}^{p_k}=0\},\hfill \\ W_{\mathrm{\infty }}(A,B_{\mathrm{\Lambda }}^{\mu },M,p,\parallel \cdot ,\dots ,\cdot \parallel )\hfill \\ =\{x=(x_k):\underset{m}{sup}\sum _{k=1}^{\mathrm{\infty }}{a}_{mk}{\left[M\left(\parallel \frac{B_{\mathrm{\Lambda }}^{\mu }{x}_k}{\rho },z_1,\dots ,z_{n-1}\parallel \right)\right]}^{p_k}<\mathrm{\infty }\},\hfill \end{array}$$

where and throughout the paper $B_{\mathrm{\Lambda }}^{\mu }{x}_k={\sum }_{v=0}^{\mu }\left(\begin{array}{c}\mu \\ v\end{array}\right)r^{\mu -v}{s}^v{x}_{k-v}{\mathrm{\Lambda }}_{k-v}$ and $\mu ,k\in \mathbb{N}$. If we consider some special cases of the spaces above, the following are obtained:

• (1) If we take $\mu =0$, then the spaces above are reduced to $W(A,\mathrm{\Lambda },M,p,\parallel \cdot ,\dots ,\cdot \parallel )$, $W_0(A,\mathrm{\Lambda },M,p,\parallel \cdot ,\dots ,\cdot \parallel )$, $W_{\mathrm{\infty }}(A,\mathrm{\Lambda },M,p,\parallel \cdot ,\dots ,\cdot \parallel )$, respectively.

• (2) If we take $r=1$, $s=-1$, then we get the spaces $W(A,{\mathrm{\Delta }}_{\mathrm{\Lambda }}^{\mu },M,p,\parallel \cdot ,\dots ,\cdot \parallel )$, $W_0(A,{\mathrm{\Delta }}_{\mathrm{\Lambda }}^{\mu },M,p,\parallel \cdot ,\dots ,\cdot \parallel )$, $W_{\mathrm{\infty }}(A,{\mathrm{\Delta }}_{\mathrm{\Lambda }}^{\mu },M,p,\parallel \cdot ,\dots ,\cdot \parallel )$.

• (3) If $M(x)=x$, then the spaces above are denoted by $W(A,B_{\mathrm{\Lambda }}^{\mu },p,\parallel \cdot ,\dots ,\cdot \parallel )$, $W_0(A,B_{\mathrm{\Lambda }}^{\mu },p,\parallel \cdot ,\dots ,\cdot \parallel )$, $W_{\mathrm{\infty }}(A,B_{\mathrm{\Lambda }}^{\mu },p,\parallel \cdot ,\dots ,\cdot \parallel )$, respectively.

• (4) If $p_k=1$ for all $k\in \mathbb{N}$ and $\mathrm{\Lambda }=({\mathrm{\Lambda }}_k)=(1,1,1,\dots )$, then the spaces above are reduced to the sequence spaces $W(A,B^{\mu },M,\parallel \cdot ,\dots ,\cdot \parallel )$, $W_0(A,B^{\mu },M,\parallel \cdot ,\dots ,\cdot \parallel )$, $W_{\mathrm{\infty }}(A,B^{\mu },M,\parallel \cdot ,\dots ,\cdot \parallel )$, respectively.

• (5) If $M(x)=x$ and $p_k=1$ for all $k\in \mathbb{N}$, then the spaces above are denoted by $W(A,B_{\mathrm{\Lambda }}^{\mu },\parallel \cdot ,\dots ,\cdot \parallel )$, $W_0(A,B_{\mathrm{\Lambda }}^{\mu },\parallel \cdot ,\dots ,\cdot \parallel )$, $W_{\mathrm{\infty }}(A,B_{\mathrm{\Lambda }}^{\mu },\parallel \cdot ,\dots ,\cdot \parallel )$, respectively.

• (6) If we take $A=C_1$, i.e., the Cesaro matrix, then the spaces above are reduced to the spaces $W(B_{\mathrm{\Lambda }}^{\mu },M,p,\parallel \cdot ,\dots ,\cdot \parallel )$, $W_0(B_{\mathrm{\Lambda }}^{\mu },M,p,\parallel \cdot ,\dots ,\cdot \parallel )$, $W_{\mathrm{\infty }}(B_{\mathrm{\Lambda }}^{\mu },M,p,\parallel \cdot ,\dots ,\cdot \parallel )$.

• (7) If we take $A=(a_{mk})$ is de la Vallee Poussin mean, i.e.,

  $$a_{mk}=\{\begin{array}{ll}\frac{1}{{\lambda }_m},& k\in I_m=[m-{\lambda }_m+1,m],\\ 0,& \text{otherwise},\end{array}$$

  (3.1)

where ${\lambda }_m$ is a non-decreasing sequence of positive numbers tending to ∞ and ${\lambda }_{m+1}\le {\lambda }_m+1$, ${\lambda }_1=1$, then the spaces above are denoted by $W(\lambda ,B_{\mathrm{\Lambda }}^{\mu },M,p,\parallel \cdot ,\dots ,\cdot \parallel )$, $W_0(\lambda ,B_{\mathrm{\Lambda }}^{\mu },M,p,\parallel \cdot ,\dots ,\cdot \parallel )$, $W_{\mathrm{\infty }}(\lambda ,B_{\mathrm{\Lambda }}^{\mu },M,p,\parallel \cdot ,\dots ,\cdot \parallel )$.

• (8) By a lacunary sequence $\theta =(k_m)$, $m=0,1,\dots$ , where $k_0=0$, we mean an increasing sequence of non-negative integers with $h_m=(k_m-k_{m-1})\to \mathrm{\infty }$ as $m\to \mathrm{\infty }$. The intervals determined by θ are denoted by $I_m=(k_{m-1},k_m]$. Let

  $$a_{mk}=\{\begin{array}{ll}\frac{1}{h_m},& k_{m-1}<k\le k_m,\\ 0,& \text{otherwise}.\end{array}$$

  (3.2)

Then we obtain the spaces $W(\theta ,B_{\mathrm{\Lambda }}^{\mu },M,p,\parallel \cdot ,\dots ,\cdot \parallel )$, $W_0(\theta ,B_{\mathrm{\Lambda }}^{\mu },M,p,\parallel \cdot ,\dots ,\cdot \parallel )$ and $W_{\mathrm{\infty }}(\theta ,B_{\mathrm{\Lambda }}^{\mu },M,p,\parallel \cdot ,\dots ,\cdot \parallel )$, respectively.

• (9) If we take $A=I$, where I is an identity matrix and $p_k=1$ for all $k\in \mathbb{N}$, then the spaces above are reduced to the sequence spaces $c(B_{\mathrm{\Lambda }}^{\mu },M,\parallel \cdot ,\dots ,\cdot \parallel )$, $c_0(B_{\mathrm{\Lambda }}^{\mu },M,\parallel \cdot ,\dots ,\cdot \parallel )$ and $l_{\mathrm{\infty }}(B_{\mathrm{\Lambda }}^{\mu },M,\parallel \cdot ,\dots ,\cdot \parallel )$, respectively.

• (10) If we take $A=I$, where I is an identity matrix, $M(x)=x$ and $p_k=1$ for all $k\in \mathbb{N}$, then we denote the spaces above by the sequence spaces $c(B_{\mathrm{\Lambda }}^{\mu },\parallel \cdot ,\dots ,\cdot \parallel )$, $c_0(B_{\mathrm{\Lambda }}^{\mu },\parallel \cdot ,\dots ,\cdot \parallel )$ and $l_{\mathrm{\infty }}(B_{\mathrm{\Lambda }}^{\mu },\parallel \cdot ,\dots ,\cdot \parallel )$.

Theorem 3.1 $W(A,B_{\mathrm{\Lambda }}^{\mu },M,p,\parallel \cdot ,\dots ,\cdot \parallel )$, $W_0(A,B_{\mathrm{\Lambda }}^{\mu },M,p,\parallel \cdot ,\dots ,\cdot \parallel )$ and $W_{\mathrm{\infty }}(A,B_{\mathrm{\Lambda }}^{\mu },M,p,\parallel \cdot ,\dots ,\cdot \parallel )$ are linear spaces.

Proof We consider only $W(A,B_{\mathrm{\Lambda }}^{\mu },M,p,\parallel \cdot ,\dots ,\cdot \parallel )$. Others can be treated similarly. Let $x,y\in W(A,B_{\mathrm{\Lambda }}^{\mu },M,p,\parallel \cdot ,\dots ,\cdot \parallel )$ and α, β be scalars. Suppose that $x\to L_1$ and $y\to L_2$. Then there exists $|\alpha |{\rho }_1+|\beta |{\rho }_2>0$ such that

$$\begin{array}{r}\sum _{k=1}^{\mathrm{\infty }}{a}_{mk}{\left[M\left(\parallel \frac{B_{\mathrm{\Lambda }}^{\mu }(\alpha x_k+\beta y_k)-(\alpha L_1+\beta L_2)}{|\alpha |{\rho }_1+|\beta |{\rho }_2},z_1,\dots ,z_{n-1}\parallel \right)\right]}^{p_k}\\ ⩽\sum _{k=1}^{\mathrm{\infty }}{a}_{mk}[M(\frac{|\alpha |{\rho }_1}{|\alpha |{\rho }_1+|\beta |{\rho }_2}\parallel \frac{B_{\mathrm{\Lambda }}^{\mu }{x}_k-L_1}{{\rho }_1},z_1,\dots ,z_{n-1}\parallel \\ {+\frac{|\beta |{\rho }_2}{|\alpha |{\rho }_1+|\beta |{\rho }_2}\parallel \frac{B_{\mathrm{\Lambda }}^{\mu }{y}_k-L_2}{{\rho }_2},z_1,\dots ,z_{n-1B_{\mathrm{\Lambda }}^{\mu }}\parallel )]}^{p_k}\\ ⩽\sum _{k=1}^{\mathrm{\infty }}{a}_{mk}[\frac{|\alpha |{\rho }_1}{|\alpha |{\rho }_1+|\beta |{\rho }_2}M\left(\parallel \frac{B_{\mathrm{\Lambda }}^{\mu }{x}_k-L_1}{{\rho }_1},z_1,\dots ,z_{n-1}\parallel \right)\\ {+\frac{|\beta |{\rho }_2}{|\alpha |{\rho }_1+|\beta |{\rho }_2}M\left(\parallel \frac{B_{\mathrm{\Lambda }}^{\mu }{y}_k-L_2}{{\rho }_2},z_1,\dots ,z_{n-1}\parallel \right)]}^{p_k}\\ ⩽D\sum _{k=1}^{\mathrm{\infty }}{a}_{mk}{\left[M\left(\parallel \frac{B_{\mathrm{\Lambda }}^{\mu }{x}_k-L_1}{{\rho }_1},z_1,\dots ,z_{n-1}\parallel \right)\right]}^{p_k}\\ +D\sum _{k=1}^{\mathrm{\infty }}{a}_{mk}{\left[M\left(\parallel \frac{B_{\mathrm{\Lambda }}^{\mu }{y}_k-L_2}{{\rho }_2},z_1,\dots ,z_{n-1}\parallel \right)\right]}^{p_k},\end{array}$$

which leads us, by taking limit as $m\to \mathrm{\infty }$, to the fact that we get $(\alpha x+\beta y)\in W(A,B_{\mathrm{\Lambda }}^{\mu },M,p,\parallel \cdot ,\dots ,\cdot \parallel )$. □

Theorem 3.2 For any two sequences $p=(p_k)$ and $q=(q_k)$ of positive real numbers and for any two n-norms ${\parallel \cdot ,\dots ,\cdot \parallel }_1$, ${\parallel \cdot ,\dots ,\cdot \parallel }_2$ on X, the following holds: $Z(A,B_{\mathrm{\Lambda }}^{\mu },M,p,{\parallel \cdot ,\dots ,\cdot \parallel }_1)\cap Z(A,B_{\mathrm{\Lambda }}^{\mu },M,q,{\parallel \cdot ,\dots ,\cdot \parallel }_2)\ne \mathrm{\varnothing }$, where $Z=W,W_0\mathit{\text{and}}{W}_{\mathrm{\infty }}$.

Proof Since the zero element belongs to each of the above classes of sequences, thus the intersection is non-empty. □

Theorem 3.3 Let $A=(a_{mk})$ be a non-negative matrix, and let $p=(p_k)$ be a bounded sequence of positive real numbers. Then, for any fixed $m\in \mathbb{N}$, the sequence space $W_{\mathrm{\infty }}(A,B_{\mathrm{\Lambda }}^{\mu },M,p,\parallel \cdot ,\dots ,\cdot \parallel )$ is a paranormed space for every nonzero $z_1,\dots ,z_{n-1}\in X$ and for some $\rho >0$ with respect to the paranorm defined by

$$g_m(x)=inf\{{\rho }^{\frac{p_m}{H}}:{\left(\sum _{k=1}^{\mathrm{\infty }}{a}_{mk}{\left[M\left(\parallel \frac{B_{\mathrm{\Lambda }}^{\mu }{x}_k}{\rho },z_1,\dots ,z_{n-1}\parallel \right)\right]}^{p_k}\right)}^{\frac{1}{H}}<\mathrm{\infty }\}.$$

Proof That $g_m(\theta )=0$ and $g_m(-x)=g_m(x)$ are easy to prove. So, we omit them. Let us take $x=(x_k)$ and $y=(y_k)$ in $W_{\mathrm{\infty }}(A,B_{\mathrm{\Lambda }}^{\mu },M,p,\parallel \cdot ,\dots ,\cdot \parallel )$. Let

$$\begin{array}{c}A(x)=\{\rho >0:\sum _{k=1}^{\mathrm{\infty }}{a}_{mk}{\left[M\left(\parallel \frac{B_{\mathrm{\Lambda }}^{\mu }{x}_k}{\rho },z_1,\dots ,z_{n-1}\parallel \right)\right]}^{p_k}<\mathrm{\infty }\},\hfill \\ A(y)=\{\rho >0:\sum _{k=1}^{\mathrm{\infty }}{a}_{mk}{\left[M\left(\parallel \frac{B_{\mathrm{\Lambda }}^{\mu }{y}_k}{\rho },z_1,\dots ,z_{n-1}\parallel \right)\right]}^{p_k}<\mathrm{\infty }\}\hfill \end{array}$$

for every nonzero $z_1,\dots ,z_{n-1}\in X$. Let ${\rho }_1\in A(x)$ and ${\rho }_2\in A(y)$, then we have

$${\left(\sum _{k=1}^{\mathrm{\infty }}{a}_{mk}{\left[M\left(\parallel \frac{B_{\mathrm{\Lambda }}^{\mu }(x_k+y_k)}{({\rho }_1+{\rho }_2)},z_1,\dots ,z_{n-1}\parallel \right)\right]}^{p_k}\right)}^{\frac{1}{H}}<\mathrm{\infty }$$

by using Minkowski’s inequality for $p=(p_k)>1$. Thus,

$$\begin{array}{rl}{g}_m(x+y)& =inf\{{({\rho }_1+{\rho }_2)}^{\frac{p_m}{H}}:{\rho }_1\in A(x),{\rho }_2\in A(y)\}\\ ⩽inf\{{{\rho }_1}^{\frac{p_m}{H}}:{\rho }_1\in A(x)\}+inf\{{{\rho }_2}^{\frac{p_m}{H}}:{\rho }_2\in A(y)\}\\ =g_m(x)+g_m(y).\end{array}$$

We also get $g_m(x+y)\le g_m(x)+g_m(y)$ for $0<p_k\le 1$ by using (2.1). Hence, we complete the proof of this condition of the paranorm. Finally, we show that the scalar multiplication is continuous. Whenever $\alpha \to 0$ and x is fixed imply $g_m(\alpha x)\to 0$. Also, whenever $x\to \theta$ and α is any number imply $g_m(\alpha x)\to 0$. By using the definition of the paranorm, for every nonzero $z_1,\dots ,z_{n-1}\in X$, we have

$$g_m(\alpha x)=inf\{{\rho }^{\frac{p_m}{H}}:{\left(\sum _{k=1}^{\mathrm{\infty }}{a}_{mk}{\left[M\left(\parallel \frac{B_{\mathrm{\Lambda }}^{\mu }(\alpha x_k)}{\rho },z_1,\dots ,z_{n-1}\parallel \right)\right]}^{p_k}\right)}^{\frac{1}{H}}<\mathrm{\infty }\}.$$

Then

$$g_m(\alpha x)=inf\{{(\alpha \varrho )}^{\frac{p_m}{H}}:{\left(\sum _{k=1}^{\mathrm{\infty }}{a}_{mk}{\left[M\left(\parallel \frac{B_{\mathrm{\Lambda }}^{\mu }{x}_k}{\varrho },z_1,\dots ,z_{n-1}\parallel \right)\right]}^{p_k}\right)}^{\frac{1}{H}}<\mathrm{\infty }\},$$

where $\varrho =\frac{\rho }{\alpha }$. Since $|\alpha {|}^{p_k}\le max\{|\alpha {|}^h,|\alpha {|}^H\}$, therefore $|\alpha {|}^{\frac{p_k}{H}}\le {(max\{|\alpha {|}^h,|\alpha {|}^H\})}^{\frac{1}{H}}$. Then the required proof follows from the following inequality

$$\begin{array}{rl}{g}_m(\alpha x)⩽& {(max\{|\alpha {|}^h,|\alpha {|}^H\})}^{\frac{1}{H}}\\ \cdot inf\{{\varrho }^{\frac{p_m}{H}}:{\left(\sum _{k=1}^{\mathrm{\infty }}{a}_{mk}{\left[M\left(\parallel \frac{B_{\mathrm{\Lambda }}^{\mu }{x}_k}{\varrho },z_1,\dots ,z_{n-1}\parallel \right)\right]}^{p_k}\right)}^{\frac{1}{H}}<\mathrm{\infty }\}\\ =& {(max\{|\alpha {|}^h,|\alpha {|}^H\})}^{\frac{1}{H}}{g}_m(x).\end{array}$$

 □

Theorem 3.4 Let M, $M_1$, $M_2$ be Orlicz functions. Then the following hold:

• (1) Let $0<h\le p_k\le 1$. Then $Z(A,B_{\mathrm{\Lambda }}^{\mu },M,p,\parallel \cdot ,\dots ,\cdot \parallel )\subseteq Z(A,B_{\mathrm{\Lambda }}^{\mu },M,\parallel \cdot ,\dots ,\cdot \parallel )$, where $Z=W,W_0$.

• (2) Let $1<p_k\le H<\mathrm{\infty }$. Then $Z(A,B_{\mathrm{\Lambda }}^{\mu },M,\parallel \cdot ,\dots ,\cdot \parallel )\subseteq Z(A,B_{\mathrm{\Lambda }}^{\mu },M,p,\parallel \cdot ,\dots ,\cdot \parallel )$, where $Z=W,W_0$.

• (3) $W_0(A,B_{\mathrm{\Lambda }}^{\mu },M_1,p,\parallel \cdot ,\dots ,\cdot \parallel )\cap W_0(A,B_{\mathrm{\Lambda }}^{\mu },M_2,p,\parallel \cdot ,\dots ,\cdot \parallel )\subseteq W_0(A,B_{\mathrm{\Lambda }}^{\mu },M_1+M_2,p,\parallel \cdot ,\dots ,\cdot \parallel )$.

Proof (1) We give the proof for the sequence space $W_0(A,B_{\mathrm{\Lambda }}^{\mu },M,p,\parallel \cdot ,\dots ,\cdot \parallel )$ only. The other can be proved by a similar argument. Let $(x_k)\in W_0(A,B_{\mathrm{\Lambda }}^{\mu },M,p,\parallel \cdot ,\dots ,\cdot \parallel )$ and $0<h\le p_k\le 1$, then

$$\sum _{k=1}^{\mathrm{\infty }}{a}_{mk}\left[M\left(\parallel \frac{B_{\mathrm{\Lambda }}^{\mu }{x}_k}{\rho },z_1,\dots ,z_{n-1}\parallel \right)\right]\le \sum _{k=1}^{\mathrm{\infty }}{a}_{mk}{\left[M\left(\parallel \frac{B_{\mathrm{\Lambda }}^{\mu }{x}_k}{\rho },z_1,\dots ,z_{n-1}\parallel \right)\right]}^{p_k}.$$

Hence, we have the result by taking the limit as $m\to \mathrm{\infty }$. This completes the proof.

• (2) Let $1<p_k\le H<\mathrm{\infty }$ and $(x_k)\in W_0(A,B_{\mathrm{\Lambda }}^{\mu },M,\parallel \cdot ,\dots ,\cdot \parallel )$. Then, for each $0<\epsilon <1$, there exists a positive integer $M_0$ such that

  $$\sum _{k=1}^{\mathrm{\infty }}{a}_{mk}\left[M\left(\parallel \frac{B_{\mathrm{\Lambda }}^{\mu }{x}_k}{\rho },z_1,\dots ,z_{n-1}\parallel \right)\right]<\epsilon <1$$

for all $m>M_0$. This implies that

$$\sum _{k=1}^{\mathrm{\infty }}{a}_{mk}{\left[M\left(\parallel \frac{B_{\mathrm{\Lambda }}^{\mu }{x}_k}{\rho },z_1,\dots ,z_{n-1}\parallel \right)\right]}^{p_k}\le \sum _{k=1}^{\mathrm{\infty }}{a}_{mk}\left[M\left(\parallel \frac{B_{\mathrm{\Lambda }}^{\mu }{x}_k}{\rho },z_1,\dots ,z_{n-1}\parallel \right)\right].$$

Hence we have the result.

• (3) Let $x=(x_k)\in W_0(A,B_{\mathrm{\Lambda }}^{\mu },M_1,p,\parallel \cdot ,\dots ,\cdot \parallel )\cap W_0(A,B_{\mathrm{\Lambda }}^{\mu },M_2,p,\parallel \cdot ,\dots ,\cdot \parallel )$. Then, by the following inequality, the result follows

  $$\begin{array}{r}\sum _{k=1}^{\mathrm{\infty }}{a}_{mk}{[(M_1+M_2)\left(\parallel \frac{B_{\mathrm{\Lambda }}^{\mu }{x}_k}{\rho },z_1,\dots ,z_{n-1}\parallel \right)]}^{p_k}\\ ⩽D\sum _{k=1}^{\mathrm{\infty }}{a}_{mk}{\left[M_1\left(\parallel \frac{B_{\mathrm{\Lambda }}^{\mu }{x}_k}{\rho },z_1,\dots ,z_{n-1}\parallel \right)\right]}^{p_k}\\ +D\sum _{k=1}^{\mathrm{\infty }}{a}_{mk}{\left[M_2\left(\parallel \frac{B_{\mathrm{\Lambda }}^{\mu }{x}_k}{\rho },z_1,\dots ,z_{n-1}\parallel \right)\right]}^{p_k}.\end{array}$$

If we take the limit as $m\to \mathrm{\infty }$, then we get $(x_k)\in W_0(A,B_{\mathrm{\Lambda }}^{\mu },M_1+M_2,p,\parallel \cdot ,\dots ,\cdot \parallel )$. This completes the proof. □

Theorem 3.5 $Z(A,B_{\mathrm{\Lambda }}^{\mu -1},M,p,\parallel \cdot ,\dots ,\cdot \parallel )\subset Z(A,B_{\mathrm{\Lambda }}^{\mu },M,p,\parallel \cdot ,\dots ,\cdot \parallel )$ and the inclusion is strict for $\mu \ge 1$. In general, $Z(A,B_{\mathrm{\Lambda }}^j,M,p,{\parallel \cdot ,\dots ,\cdot \parallel }_1)\subset Z(A,B_{\mathrm{\Lambda }}^{\mu },M,p,\parallel \cdot ,\dots ,\cdot \parallel )$ for $j=0,1,2,\dots ,\mu -1$ and the inclusions are strict, where $Z=W,W_0\mathit{\text{and}}{W}_{\mathrm{\infty }}$.

Proof We give the proof for $W_0(A,B_{\mathrm{\Lambda }}^{\mu -1},M,p,\parallel \cdot ,\dots ,\cdot \parallel )$ only. The others can be proved by a similar argument. Let $x=(x_k)$ be any element in the space $W_0(A,B_{\mathrm{\Lambda }}^{\mu -1},M,p,\parallel \cdot ,\dots ,\cdot \parallel )$, then there exists $\rho =|r|{\rho }_1+|s|{\rho }_2>0$ such that

$$\underset{m\to \mathrm{\infty }}{lim}\sum _{k=1}^{\mathrm{\infty }}{a}_{mk}{\left[M\left(\parallel \frac{B_{\mathrm{\Lambda }}^{\mu -1}{x}_k}{\rho },z_1,\dots ,z_{n-1}\parallel \right)\right]}^{p_k}=0.$$

Since M is non-decreasing and convex, it follows that

$$\begin{array}{r}\sum _{k=1}^{\mathrm{\infty }}{a}_{mk}{\left[M\left(\parallel \frac{B_{\mathrm{\Lambda }}^{\mu }{x}_k}{|r|{\rho }_1+|s|{\rho }_2},z_1,\dots ,z_{n-1}\parallel \right)\right]}^{p_k}\\ =\sum _{k=1}^{\mathrm{\infty }}{a}_{mk}{\left[M\left(\parallel \frac{r{B}_{\mathrm{\Lambda }}^{\mu -1}{x}_k+s{B}_{\mathrm{\Lambda }}^{\mu -1}{x}_{k-1}}{|r|{\rho }_1+|s|{\rho }_2},z_1,\dots ,z_{n-1}\parallel \right)\right]}^{p_k}\\ ⩽D\sum _{k=1}^{\mathrm{\infty }}{a}_{mk}{\left[M\left(\parallel \frac{B_{\mathrm{\Lambda }}^{\mu -1}{x}_k}{{\rho }_1},z_1,\dots ,z_{n-1}\parallel \right)\right]}^{p_k}\\ +D\sum _{k=1}^{\mathrm{\infty }}{a}_{mk}{\left[M\left(\parallel \frac{B_{\mathrm{\Lambda }}^{\mu -1}{x}_{k-1}}{{\rho }_2},z_1,\dots ,z_{n-1}\parallel \right)\right]}^{p_k}.\end{array}$$

The result holds by taking the limit as $m\to \mathrm{\infty }$. □

In the following example we show that the inclusion given in the theorem above is strict.

Example 3.6 Let $M(x)=x$, $p_k=1$ for all $k\in \mathbb{N}$, $\mathrm{\Lambda }=({\mathrm{\Lambda }}_k)=(1,1,\dots )$, $A=C_1$, i.e., the Cesaro matrix, $r=1$, $s=-1$, where $B_{\mathrm{\Lambda }}^{\mu }{x}_k={\sum }_{v=0}^{\mu }\left(\begin{array}{c}\mu \\ v\end{array}\right)r^{\mu -v}{s}^v{x}_{k-v}{\mathrm{\Lambda }}_{k-v}$ for all $r,s\in \mathbb{R}-\{0\}$. Consider the sequence $x=(x_k)=(k^{\mu -1})$. Then $x=(x_k)$ belongs to $W_0(B^{\mu },M,p,\parallel \cdot ,\dots ,\cdot \parallel )$ but does not belong to $W_0(B^{\mu -2},M,p,\parallel \cdot ,\dots ,\cdot \parallel )$.

Theorem 3.7 Let $A=(a_{mk})$ be a non-negative regular matrix and $p=(p_k)$ be such that $0<h\le p_k\le H<\mathrm{\infty }$. Then

$$l_{\mathrm{\infty }}(B_{\mathrm{\Lambda }}^{\mu },M,\parallel \cdot ,\dots ,\cdot \parallel )\subseteq W_{\mathrm{\infty }}(A,B_{\mathrm{\Lambda }}^{\mu },M,p,\parallel \cdot ,\dots ,\cdot \parallel ).$$

Proof Let $l_{\mathrm{\infty }}(B_{\mathrm{\Lambda }}^{\mu },M,\parallel \cdot ,\dots ,\cdot \parallel )$. Then there exists $T_0>0$ such that $[M(\parallel \frac{B_{\mathrm{\Lambda }}^{\mu }{x}_k}{\rho },z_1,\dots ,z_{n-1}\parallel )]⩽T_0$ for all $k\in \mathbb{N}$ and for every nonzero $z_1,\dots ,z_{n-1}\in X$. Since $A=(a_{mk})$ is a non-negative regular matrix, we have the following inequality by (1) of Silverman-Toeplitz conditions:

$$\underset{m}{sup}\sum _{k=1}^{\mathrm{\infty }}{a}_{mk}{\left[M\left(\parallel \frac{B_{\mathrm{\Lambda }}^{\mu }{x}_k}{\rho },z_1,\dots ,z_{n-1}\parallel \right)\right]}^{p_k}⩽max\{T_0^h,T_0^H\}\underset{m}{sup}\sum _{k=1}^{\mathrm{\infty }}{a}_{mk}<\mathrm{\infty }.$$

Hence $l_{\mathrm{\infty }}(B_{\mathrm{\Lambda }}^{\mu },M,\parallel \cdot ,\dots ,\cdot \parallel )\subseteq W_{\mathrm{\infty }}(A,B_{\mathrm{\Lambda }}^{\mu },M,p,\parallel \cdot ,\dots ,\cdot \parallel )$. □

4 ${(B_{\mathrm{\Lambda }}^{\mu })}^n$-statistically A-convergent sequences

In this section we introduce and study a new concept of ${(B_{\mathrm{\Lambda }}^{\mu })}^n$-statistical A-convergence in an n-normed space as follows.

Definition 4.1 Let $(X,\parallel \cdot ,\dots ,\cdot \parallel )$ be an n-normed space and let $A=(a_{mk})$ be a non-negative regular matrix. A real sequence $x=(x_k)$ is said to be ${(B_{\mathrm{\Lambda }}^{\mu })}^n$-statistically A-convergent to a number L if ${\delta }_{A{(B_{\mathrm{\Lambda }}^{\mu })}^n}(K)={lim}_{m\to \mathrm{\infty }}{\sum }_{k=1}^{\mathrm{\infty }}{a}_{mk}{\chi }_K(k)=0$ or, equivalently, ${lim}_{m\to \mathrm{\infty }}{\sum }_{k\in K}{a}_{mk}=0$ for each $\epsilon >0$ and for every nonzero $z_1,\dots ,z_{n-1}\in X$, where $K=\{k\in \mathbb{N}:\parallel B_{\mathrm{\Lambda }}^{\mu }{x}_k-L,z_1,\dots ,z_{n-1}\parallel ⩾\epsilon \}$ and ${\chi }_K$ is the characteristic function of K.

In this case, we write ${(B_{\mathrm{\Lambda }}^{\mu })}^n\mathit{stat}\text{-}A\text{-}limx=L$. $S(A{(B_{\mathrm{\Lambda }}^{\mu })}^n)$ denotes the set of all ${(B_{\mathrm{\Lambda }}^{\mu })}^n$-statistically A-convergent sequences.

If we consider some special cases of the matrix, then we have the following:

• (1) If $A=C_1$, the Cesaro matrix, then the definition reduces to ${(B_{\mathrm{\Lambda }}^{\mu })}^n$-statistical convergence.

• (2) If $A=(a_{mk})$ is de la Vallee Poussin mean, which is given by (3.1), then the definition reduces to ${(B_{\mathrm{\Lambda }}^{\mu })}^n$-statistical λ-convergence.

• (3) If we take $A=(a_{mk})$ as in (3.2), then the definition reduces to ${(B_{\mathrm{\Lambda }}^{\mu })}^n$-statistical lacunary convergence.

Theorem 4.2 Let $p=(p_k)$ be a sequence of non-negative bounded real numbers such that ${inf}_k{p}_k>0$. Then $W(A,B_{\mathrm{\Lambda }}^{\mu },p,\parallel \cdot ,\dots ,\cdot \parallel )\subset S(A{(B_{\mathrm{\Lambda }}^{\mu })}^n)$.

Proof Assume that $x=(x_k)\in W(A,B_{\mathrm{\Lambda }}^{\mu },p,\parallel \cdot ,\dots ,\cdot \parallel )$. So, we have for every nonzero $z_1,\dots ,z_{n-1}\in X$

$$\underset{m\to \mathrm{\infty }}{lim}\sum _{k=1}^{\mathrm{\infty }}{a}_{mk}{\parallel B_{\mathrm{\Lambda }}^{\mu }{x}_k-L,z_1,\dots ,z_{n-1}\parallel }^{p_k}=0.$$

Let $\epsilon >0$ and $K=\{k\in \mathbb{N}:\parallel B_{\mathrm{\Lambda }}^{\mu }{x}_k-L,z_1,\dots ,z_{n-1}\parallel ⩾\epsilon \}$. We obtain the following:

$$\begin{array}{r}\sum _{k=1}^{\mathrm{\infty }}{a}_{mk}{\parallel B_{\mathrm{\Lambda }}^{\mu }{x}_k-L,z_1,\dots ,z_{n-1}\parallel }^{p_k}\\ =\sum _{k\in K}{a}_{mk}{\parallel B_{\mathrm{\Lambda }}^{\mu }{x}_k-L,z_1,\dots ,z_{n-1}\parallel }^{p_k}+\sum _{k\notin K}{a}_{mk}{\parallel B_{\mathrm{\Lambda }}^{\mu }{x}_k-L,z_1,\dots ,z_{n-1}\parallel }^{p_k}\\ ⩾min\{{\epsilon }^h,{\epsilon }^H\}\sum _{k\in K}{a}_{mk}.\end{array}$$

If we take the limit as $m\to \mathrm{\infty }$, then we get $x\in S(A{(B_{\mathrm{\Lambda }}^{\mu })}^n)$. This completes the proof. □

Theorem 4.3 Let $p=(p_k)$ be a sequence of non-negative bounded real numbers such that ${inf}_k{p}_k>0$. Then

$$l_{\mathrm{\infty }}(B_{\mathrm{\Lambda }}^{\mu },\parallel \cdot ,\dots ,\cdot \parallel )\cap S\left(A{\left(B_{\mathrm{\Lambda }}^{\mu }\right)}^n\right)\subset W(A,B_{\mathrm{\Lambda }}^{\mu },p,\parallel \cdot ,\dots ,\cdot \parallel ).$$

Proof Suppose that $x=(x_k)\in l_{\mathrm{\infty }}(B_{\mathrm{\Lambda }}^{\mu },\parallel \cdot ,\dots ,\cdot \parallel )\cap S(A{(B_{\mathrm{\Lambda }}^{\mu })}^n)$. Then there exists an integer T such that $\parallel B_{\mathrm{\Lambda }}^{\mu }{x}_k-L,z_1,\dots ,z_{n-1}\parallel ⩽T$ for all $k>0$ and for every nonzero $z_1,\dots ,z_{n-1}\in X$, and ${lim}_{m\to \mathrm{\infty }}{\sum }_{k\in K}{a}_{mk}=0$, where $K=\{k\in \mathbb{N}:\parallel B_{\mathrm{\Lambda }}^{\mu }{x}_k-L,z_1,\dots ,z_{n-1}\parallel ⩾\epsilon \}$. Then we can write

$$\begin{array}{r}\sum _{k=1}^{\mathrm{\infty }}{a}_{mk}{\parallel B_{\mathrm{\Lambda }}^{\mu }{x}_k-L,z_1,\dots ,z_{n-1}\parallel }^{p_k}\\ =\sum _{k\notin K}{a}_{mk}{\parallel B_{\mathrm{\Lambda }}^{\mu }{x}_k-L,z_1,\dots ,z_{n-1}\parallel }^{p_k}+\sum _{k\in K}{a}_{mk}{\parallel B_{\mathrm{\Lambda }}^{\mu }{x}_k-L,z_1,\dots ,z_{n-1}\parallel }^{p_k}\\ <max\{{\epsilon }^h,{\epsilon }^H\}\sum _{k\notin K}{a}_{mk}+max\{T^h,T^H\}\sum _{k\in K}{a}_{mk}.\end{array}$$

Since $A=(a_{mk})$ is a non-negative regular matrix, then we have

$$\begin{array}{rl}1& =\underset{m\to \mathrm{\infty }}{lim}\sum _{k=1}^{\mathrm{\infty }}{a}_{mk}\\ =\underset{m\to \mathrm{\infty }}{lim}\sum _{k\notin K}{a}_{mk}+\underset{m\to \mathrm{\infty }}{lim}\sum _{k\in K}{a}_{mk}.\end{array}$$

Hence, ${lim}_{m\to \mathrm{\infty }}{\sum }_{k\notin K}{a}_{mk}=1$. Thus

$$\begin{array}{r}\underset{m\to \mathrm{\infty }}{lim}\sum _{k=1}^{\mathrm{\infty }}{a}_{mk}{\parallel B_{\mathrm{\Lambda }}^{\mu }{x}_k-L,z_1,\dots ,z_{n-1}\parallel }^{p_k}\\ <{\epsilon }^{\prime }\underset{m\to \mathrm{\infty }}{lim}\sum _{k\notin K}{a}_{mk}+T^{\prime }\underset{m\to \mathrm{\infty }}{lim}\sum _{k\in K}{a}_{mk}\\ <{\epsilon }^{\prime },\end{array}$$

where $max\{{\epsilon }^h,{\epsilon }^H\}={\epsilon }^{\prime }$ and $max\{T^h,T^H\}=T^{\prime }$ .

Hence, $x_k\in W(A,B_{\mathrm{\Lambda }}^{\mu },p,\parallel \cdot ,\dots ,\cdot \parallel )$. □