# On certain generalized paranormed spaces

## Article metrics

• 1046 Accesses

• 7 Citations

## Abstract

In the present paper we introduce and study some generalized paranormed sequence spaces defined by Musielak-Orlicz functions as well as by a sequence of modulus functions. We also study some topological properties and prove some inclusion relations between these spaces.

## Introduction and preliminaries

An Orlicz function $$M : [0, \infty) \rightarrow[0, \infty)$$ is convex and continuous such that $$M(0) = 0$$, $$M(x)>0$$ for $$x>0$$. Let w be the space of all real or complex sequences $$x = (x_{k})$$. Lindenstrauss and Tzafriri  used the idea of the Orlicz function to define the following sequence space:

$$\ell_{M} = \Biggl\{ x \in w : \sum^{\infty}_{k=1} M \biggl(\frac{|x_{k}|}{\rho} \biggr) < \infty, \mbox{for some } \rho>0 \Biggr\} ,$$

which is called an Orlicz sequence space. The space $$\ell_{M}$$ is a Banach space with the norm

$$\|x\| = \inf \Biggl\{ \rho> 0 : \sum^{\infty}_{k=1} M \biggl(\frac{|x_{k}|}{\rho} \biggr) \leq 1 \Biggr\} .$$

It is shown in  that every Orlicz sequence space $$\ell_{M}$$ contains a subspace isomorphic to $$\ell_{p}$$ ($$p \geq1$$). An Orlicz function M satisfies the $$\Delta_{2}$$-condition if and only if for any constant $$L > 1$$ there exists a constant $$K(L)$$ such that $$M(Lu) \leq K (L)M(u)$$ for all values of $$u \geq0$$.

A sequence $$\mathcal{M} = (M_{k})$$ of Orlicz functions is called a Musielak-Orlicz function (see ). A sequence $$\mathcal{N} =(N_{k})$$ is defined by

$${N}_{k}(v) = \sup\bigl\{ |v|u - M_{k}(u) :u\geq0\bigr\} , \quad k = 1,2,\ldots$$

is called the complementary function of a Musielak-Orlicz function . For a given Musielak-Orlicz function , the Musielak-Orlicz sequence space $${t_{\mathcal{M}}}$$ and its subspace $$h_{\mathcal{M}}$$ are defined as follows:

\begin{aligned}& t_{\mathcal{M}} = \bigl\{ x \in w : I_{\mathcal{M}} (cx) < \infty \mbox{ for some } c > 0 \bigr\} , \\& h_{\mathcal{M}} = \bigl\{ x \in w : I_{\mathcal{M}} (cx) < \infty \mbox{ for all } c > 0 \bigr\} , \end{aligned}

where $$I_{\mathcal{M}}$$ is a convex modular defined by

$$I_{\mathcal{M}}(x) = \sum^{\infty}_{k=1} M_{k}(x_{k}),\quad x = (x_{k}) \in t_{\mathcal{M}}.$$

We consider $$t_{\mathcal{M}}$$ equipped with the Luxemburg norm

$$\|x\| = \inf \biggl\{ k > 0 : I_{\mathcal{M}} \biggl(\frac{x}{k} \biggr) \leq1 \biggr\}$$

or equipped with the Orlicz norm

$$\|x\|^{0} = \inf \biggl\{ \frac{1}{k} \bigl(1 + I_{\mathcal{M}}(kx) \bigr) : k > 0 \biggr\} .$$

A Musielak-Orlicz function $$(M_{k})$$ is said to satisfy the $$\Delta _{2}$$-condition if there exist constants $$a, K > 0$$ and a sequence $$c = (c_{k})^{\infty}_{k = 1}\in\ell^{1}_{+}$$ (the positive cone of $$\ell^{1}$$) such that the inequality

$$M_{k}(2u) \leq K M_{k}(u) + c_{k}$$

holds for all $$k \in N$$ and $$u \in R_{+}$$ whenever $$M_{k}(u) \leq a$$.

A modulus function is a function $$f : [0, \infty) \rightarrow[0, \infty)$$ such that

1. (1)

$$f(x) = 0$$ if and only if $$x =0$$,

2. (2)

$$f(x + y ) \leq f(x) + f(y)$$ for all $$x \geq0$$, $$y\geq0$$,

3. (3)

f is increasing,

4. (4)

f is continuous from right at 0.

It follows that f must be continuous everywhere on $$[0, \infty)$$. The modulus function may be bounded or unbounded. For example, if we take $$f(x) = \frac{x}{x +1}$$, then $$f(x)$$ is bounded. If $$f(x)=x^{p}$$, $$0 < p < 1$$, then the modulus $$f(x)$$ is unbounded. Subsequently, modulus function has been discussed in [2, 58] and references therein.

Let $$l_{\infty}$$, c, and $$c_{0}$$ denote the spaces of all bounded, convergent, and null sequences $$x = (x_{k})$$ with complex terms, respectively. The zero sequence $$(0,0,\ldots)$$ is denoted by θ.

The notion of difference sequence spaces was introduced by Kızmaz , who studied the difference sequence spaces $$l_{\infty}(\Delta)$$, $$c(\Delta)$$, and $$c_{0}(\Delta)$$. The notion was further generalized by Et and Çolak  by introducing the spaces $$l_{\infty}(\Delta^{n})$$, $$c(\Delta^{n})$$, and $$c_{0}(\Delta^{n})$$. Another type of generalization of the difference sequence spaces is due to Tripathy and Esi  who studied the spaces $$l_{\infty}(\Delta^{m}_{n})$$, $$c(\Delta^{m}_{n})$$, and $$c_{0}(\Delta^{m}_{n})$$.

Let m, n be non-negative integers, then for Z a given sequence space, we have

$$Z \bigl(\Delta^{n}_{m}\bigr) = \bigl\{ x = (x_{k})\in w : \bigl(\Delta^{n}_{m} x_{k}\bigr)\in Z \bigr\}$$

for $$Z = c, c_{0}\mbox{ and }l_{\infty}$$ where $$\Delta^{n}_{m} x = (\Delta ^{n}_{m} x_{k}) = ( \Delta^{n-1}_{m} x_{k} - \Delta^{n-1}_{m} x_{k+m})$$ and $$\Delta^{0}_{m} x_{k} = x_{k}$$ for all $$k \in\mathbb{N}$$, which is equivalent to the following binomial representation:

$$\Delta^{n}_{m} x_{k} = \sum ^{n}_{v=0}(-1)^{v}\left ( \begin{array}{@{}c@{}} n \\ v \end{array} \right )x_{k + mv}.$$

Taking $$m = 1$$, we get the spaces $$l_{\infty}(\Delta^{n})$$, $$c(\Delta^{n})$$, and $$c_{0}(\Delta^{n})$$ studied by Et and Çolak . Taking $$m = n = 1$$, we get the spaces $$l_{\infty}(\Delta)$$, $$c(\Delta)$$, and $$c_{0}(\Delta)$$ introduced and studied by Kızmaz . For more details as regards sequence spaces, see [6, 1123] and references therein.

Let $$\mathcal{M} = (M_{k})$$ be a Musielak-Orlicz function, $$p=(p_{k})$$ be any bounded sequence of positive real numbers and $$u = (u_{k})$$ be a sequence of strictly positive real numbers. Let $$(X,q)$$ be a space seminormed by q. In the present paper we define the following sequence spaces:

\begin{aligned}& \begin{aligned}[b] w_{0} \bigl(\mathcal{M},\Delta^{n}_{m},p,q,u \bigr) ={}& \Biggl\{ x = (x_{k}):\frac {1}{n}\sum ^{n}_{k=1} \biggl[M_{k} \biggl( \frac{q (u_{k}\Delta^{n}_{m}x_{k} )}{\rho} \biggr) \biggr]^{p_{k}} \rightarrow0 \mbox{ as } n \rightarrow\infty,\\ &{} \mbox{for some } \rho> 0 \Biggr\} , \end{aligned}\\& \begin{aligned}[b] w \bigl(\mathcal{M},\Delta^{n}_{m},p,q,u \bigr) ={}& \Biggl\{ x = (x_{k}):\frac {1}{n}\sum^{n}_{k=1} \biggl[M_{k} \biggl(\frac{q (u_{k}\Delta^{n}_{m}x_{k} - L )}{\rho} \biggr) \biggr]^{p_{k}} \rightarrow0 \mbox{ as } n \rightarrow\infty,\\ &{}\mbox{for some } L \in X, \rho> 0 \Biggr\} , \end{aligned} \end{aligned}

and

\begin{aligned}[b] &w_{\infty}\bigl(\mathcal{M},\Delta^{n}_{m},p,q,u \bigr) \\ &\quad= \Biggl\{ x = (x_{k}):\sup_{n} \frac{1}{n}\sum^{n}_{k=1} \biggl[M_{k} \biggl(\frac{q (u_{k}\Delta^{n}_{m}x_{k} )}{\rho} \biggr) \biggr]^{p_{k}} < \infty, \mbox{for some } \rho> 0 \Biggr\} . \end{aligned}

If we take $$\mathcal{M}(x) = x$$, we get

\begin{aligned}& \begin{aligned}[b] &w_{0} \bigl(\Delta^{n}_{m},p,q,u \bigr) \\ &\quad= \Biggl\{ x = (x_{k}):\frac{1}{n}\sum^{n}_{k=1} \biggl[ \biggl(\frac{q (u_{k}\Delta^{n}_{m}x_{k} )}{\rho} \biggr) \biggr]^{p_{k}} \rightarrow0 \mbox{ as } n \rightarrow\infty, \mbox{for some } \rho> 0 \Biggr\} , \end{aligned}\\& \begin{aligned}[b] &w \bigl(\Delta^{n}_{m},p,q,u \bigr) \\ &\quad= \Biggl\{ x = (x_{k}):\frac{1}{n}\sum^{n}_{k=1} \biggl[ \biggl(\frac{q (u_{k}\Delta^{n}_{m}x_{k} - L )}{\rho} \biggr) \biggr]^{p_{k}} \rightarrow0 \mbox{ as } n \rightarrow\infty, \mbox{for some } L \in X, \rho> 0 \Biggr\} , \end{aligned} \end{aligned}

and

$$w_{\infty}\bigl(\Delta^{n}_{m},p,q,u \bigr) = \Biggl\{ x = (x_{k}):\sup_{n}\frac {1}{n}\sum ^{n}_{k=1} \biggl[ \biggl(\frac{q (u_{k}\Delta^{n}_{m}x_{k} )}{\rho } \biggr) \biggr]^{p_{k}} < \infty, \mbox{for some } \rho> 0 \Biggr\} .$$

If we take $$p = (p_{k}) = 1$$, k, we get

\begin{aligned}& \begin{aligned}[b] &w_{0} \bigl(\mathcal{M},\Delta^{n}_{m},q,u \bigr)\\ &\quad= \Biggl\{ x = (x_{k}):\frac {1}{n}\sum ^{n}_{k=1} \biggl[M_{k} \biggl( \frac{q (u_{k}\Delta^{n}_{m}x_{k} )}{\rho} \biggr) \biggr] \rightarrow0 \mbox{ as } n \rightarrow \infty, \mbox{for some } \rho> 0 \Biggr\} , \end{aligned}\\& \begin{aligned}[b] w \bigl(\mathcal{M},\Delta^{n}_{m},q,u \bigr) ={}& \Biggl\{ x = (x_{k}):\frac{1}{n}\sum^{n}_{k=1} \biggl[M_{k} \biggl(\frac{q (u_{k}\Delta^{n}_{m}x_{k} - L )}{\rho } \biggr) \biggr] \rightarrow0 \mbox{ as } n \rightarrow\infty,\\ &{}\mbox{for some } L \in X, \rho> 0 \Biggr\} , \end{aligned} \end{aligned}

and

$$w_{\infty}\bigl(\mathcal{M},\Delta^{n}_{m},q,u \bigr) = \Biggl\{ x = (x_{k}):\sup_{n}\frac{1}{n}\sum ^{n}_{k=1} \biggl[M_{k} \biggl( \frac{q (u_{k}\Delta ^{n}_{m}x_{k} )}{\rho} \biggr) \biggr] < \infty, \mbox{for some } \rho > 0 \Biggr\} .$$

If we take $$u = (u_{k}) = 1$$, k, we get

\begin{aligned}& \begin{aligned}[b] w_{0} \bigl(\mathcal{M},\Delta^{n}_{m},p,q \bigr) ={}& \Biggl\{ x = (x_{k}):\frac {1}{n}\sum ^{n}_{k=1} \biggl[M_{k} \biggl( \frac{q (\Delta^{n}_{m}x_{k} )}{\rho } \biggr) \biggr]^{p_{k}} \rightarrow0 \mbox{ as } n \rightarrow \infty, \\ &{}\mbox{for some } \rho> 0 \Biggr\} , \end{aligned} \\& \begin{aligned}[b] w \bigl(\mathcal{M},\Delta^{n}_{m},p,q \bigr) ={}& \Biggl\{ x = (x_{k}):\frac{1}{n}\sum^{n}_{k=1} \biggl[M_{k} \biggl(\frac{q (\Delta^{n}_{m}x_{k} - L )}{\rho} \biggr) \biggr]^{p_{k}} \rightarrow0 \mbox{ as } n \rightarrow\infty,\\ &{}\mbox{for some } L \in X, \rho> 0 \Biggr\} , \end{aligned} \end{aligned}

and

$$w_{\infty}\bigl(\mathcal{M},\Delta^{n}_{m},p,q \bigr) = \Biggl\{ x = (x_{k}):\sup_{n}\frac{1}{n}\sum ^{n}_{k=1} \biggl[M_{k} \biggl( \frac{q (\Delta^{n}_{m}x_{k} )}{\rho} \biggr) \biggr]^{p_{k}} < \infty, \mbox{for some } \rho> 0 \Biggr\} .$$

The following inequality will be used throughout the paper. If $$0 \leq p_{k} \leq\sup p_{k} = K$$, $$D = \max(1,2^{K-1})$$ then

$$|a_{k} + b_{k}|^{p_{k}} \leq D \bigl\{ |a_{k}|^{p_{k}} + |b_{k}|^{p_{k}}\bigr\}$$
(1.1)

for all k and $$a_{k}, b_{k} \in\mathbb{C}$$. Also $$|a|^{p_{k}}\leq\max (1, |a|^{K})$$ for all $$a \in\mathbb{C}$$.

The aim of this paper is to study some topological and algebraic properties of the above sequence spaces.

## Main results

### Theorem 2.1

Suppose $$\mathcal{M}= (M_{k})$$ be a Musielak-Orlicz function, $$p=(p_{k})$$ be any bounded sequence of positive real numbers and $$u = (u_{k})$$ be a sequence of strictly positive real numbers. Then the spaces $$w_{0} (\mathcal{M}, \Delta^{n}_{m}, p, q, u )$$, $$w (\mathcal{M}, \Delta^{n}_{m}, p, q, u )$$ and $$w_{\infty}(\mathcal{M}, \Delta^{n}_{m}, p, q, u )$$ are linear spaces over the complex field .

### Proof

Let $$x =(x_{k}), y = (y_{k}) \in w_{\infty}(\mathcal{M}, \Delta^{n}_{m}, p, q, u )$$ and $$\alpha, \beta\in\mathbb{C}$$. Then there exist positive real numbers $$\rho_{1}$$ and $$\rho_{2}$$ such that

$$\sup_{n} \frac{1}{n}\sum^{n}_{k=1} \biggl[M_{k} \biggl(\frac{q (u_{k} \Delta ^{n}_{m} x_{k} )}{\rho_{1}} \biggr) \biggr]^{p_{k}} < \infty$$

and

$$\sup_{n}\frac{1}{n}\sum^{n}_{k=1} \biggl[M_{k} \biggl(\frac{q (u_{k} \Delta^{n}_{m} y_{k} )}{\rho_{2}} \biggr) \biggr]^{p_{k}} < \infty.$$

Define $$\rho_{3} = \max(2|\alpha|\rho_{1}, 2|\beta| \rho_{2})$$. Since $$(M_{k})$$ is non-decreasing, convex and so by using inequality (1.1), we have

\begin{aligned} &\sup_{n}\frac{1}{n}\sum^{n}_{k=1} \biggl[M_{k} \biggl(\frac {q(\alpha u_{k}\Delta^{n}_{m} x_{k} + \beta u_{k} \Delta^{n}_{m} y_{k}) }{\rho_{3}} \biggr) \biggr]^{p_{k}} \\ &\quad\leq \sup_{n}\frac{1}{n}\sum ^{n}_{k=1} \biggl[M_{k} \biggl( \frac{q(\alpha u_{k} \Delta^{n}_{m} x_{k} )}{\rho_{3}} + \frac{q(\beta u_{k} \Delta^{n}_{m} y_{k} )}{\rho _{3}} \biggr) \biggr]^{p_{k}} \\ &\quad \leq \sup_{n}\frac{1}{n}\sum ^{n}_{k=1} \frac{1}{2^{p_{k}}} \biggl[M_{k} \biggl(\frac{q(u_{k} \Delta^{n}_{m} x_{k} )}{\rho_{1}} \biggr) \biggr]^{p_{k}} + \sup _{n}\frac {1}{n}\sum^{n}_{k=1} \frac{1}{2^{p_{k}}} \biggl[M_{k} \biggl(\frac{q( u_{k} \Delta ^{n}_{m} y_{k} )}{\rho_{2}} \biggr) \biggr]^{p_{k}} \\ &\quad\leq D\sup_{n}\frac{1}{n}\sum ^{n}_{k=1} \biggl[M_{k} \biggl( \frac{q(u_{k} \Delta^{n}_{m} x_{k} )}{\rho_{1}} \biggr) \biggr]^{p_{k}} + D\sup_{n} \frac{1}{n}\sum^{n}_{k=1} \biggl[M_{k} \biggl(\frac{q(u_{k} \Delta^{n}_{m} y_{k} )}{\rho_{2}} \biggr) \biggr]^{p_{k}} \\ &\quad< \infty. \end{aligned}

Thus, $$\alpha x + \beta y \in w_{\infty}(\mathcal{M}, \Delta^{n}_{m}, p, q, u )$$. Hence $$w_{\infty}(\mathcal{M}, \Delta^{n}_{m}, p, q, u )$$ is a linear space. Similarly, we can prove $$w (\mathcal{M}, \Delta^{n}_{m}, p, q, u )$$ and $$w_{0} (\mathcal{M}, \Delta^{n}_{m}, p, q, u )$$ are linear spaces over the field of complex numbers. □

### Theorem 2.2

Suppose $$\mathcal{M} = (M_{k})$$ be a Musielak-Orlicz function, $$p=(p_{k})$$ be any bounded sequence of positive real numbers and $$u = (u_{k})$$ be a sequence of strictly positive real numbers. Then the space $$w_{\infty}(\mathcal{M},\Delta^{n}_{m}, p, q, u )$$ is a paranormed space with the paranorm defined by

$$g(x) = \inf \Biggl\{ \rho^{\frac{p_{k}}{H}}: \sup _{n} \Biggl(\frac{1}{n} \sum _{k = 1}^{n} \biggl[M_{k} \biggl( q \biggl( \frac{u_{k} \Delta^{n}_{m} x_{k}}{\rho} \biggr) \biggr) \biggr]^{p_{k}} \Biggr)^{\frac{1}{H}} \leq1, \rho> 0 \Biggr\} ,$$

where $$H = \max(1, \sup_{k} p_{k})$$.

### Proof

(i) Clearly, $$g(x) \geq0$$ for $$x = (x_{k}) \in w_{\infty}(\mathcal{M},\Delta^{n}_{m}, p, q, u )$$. Since $$M_{k}(0) = 0$$, we get $$g(\theta) = 0$$.

(ii) $$g(-x) = g(x)$$.

(iii) Let $$x = (x_{k}), y = (y_{k}) \in w_{\infty}(\mathcal{M},\Delta ^{n}_{m}, p, q, u )$$ then there exist $$\rho_{1}, \rho_{2} > 0$$ such that

$$\frac{1}{n} \sum_{k = 1}^{n} \biggl[M_{k} \biggl(q \biggl(\frac{u_{k} \Delta^{n}_{m} x_{k}}{\rho_{1}} \biggr) \biggr) \biggr]^{p_{k}} \leq1$$

and

$$\frac{1}{n} \sum_{k = 1}^{n} \biggl[M_{k} \biggl(q \biggl(\frac{u_{k} \Delta^{n}_{m} y_{k}}{\rho_{2}} \biggr) \biggr) \biggr]^{p_{k}} \leq1 .$$

Let $$\rho= \rho_{1} + \rho_{2}$$, then by Minkowski’s inequality, we have

\begin{aligned} \frac{1}{n} \sum_{k = 1}^{n} \biggl[M_{k} \biggl(q \biggl(\frac{u_{k} \Delta^{n}_{m} x_{k}}{\rho} \biggr) \biggr) \biggr]^{p_{k}} \leq& \biggl(\frac{\rho _{1}}{\rho_{1} + \rho_{2}} \biggr) \frac{1}{n} \sum_{k = 1}^{n} \biggl[M_{k} \biggl( q \biggl(\frac{u_{k} \Delta^{n}_{m} x_{k}}{\rho_{1}} \biggr) \biggr) \biggr]^{p_{k}} \\ &{} + \biggl(\frac{\rho_{2}}{\rho_{1} + \rho_{2}} \biggr) \frac{1}{n} \sum _{k = 1}^{n} \biggl[M_{k} \biggl(q \biggl( \frac{u_{k} \Delta^{n}_{m} y_{k}}{\rho_{2}} \biggr) \biggr) \biggr]^{p_{k}} \end{aligned}

and thus

\begin{aligned} &g(x + y) \\ &\quad = \inf \Biggl\{ (\rho_{1} + \rho_{2})^{\frac{p_{k}}{H}}: \sup_{n} \Biggl(\frac{1}{n} \sum _{k = 1}^{n} \biggl[M_{k} \biggl(q \biggl( \frac{u_{k} \Delta^{n}_{m} x_{k} + u_{k} \Delta^{n}_{m} y_{k}}{\rho} \biggr) \biggr) \biggr]^{p_{k}} \Biggr)^{\frac{1}{H}} \leq1, \rho> 0 \Biggr\} \\ &\quad \leq g(x) + g(y). \end{aligned}

(iv) Finally we prove that scalar multiplication is continuous. Let λ be any complex number by definition

\begin{aligned} g(\lambda x) = & \inf \Biggl\{ (\rho)^{\frac{p_{k}}{H}}: \sup_{n} \Biggl(\frac {1}{n} \sum _{k = 1}^{n} \biggl[M_{k} \biggl(q \biggl( \frac{u_{k} \Delta^{n}_{m} \lambda x_{k}}{\rho} \biggr) \biggr) \biggr] ^{p_{k}} \Biggr)^{\frac{1}{H}} \leq1, \rho> 0 \Biggr\} \\ = & \inf \Biggl\{ \bigl(|\lambda| r\bigr)^{\frac {p_{k}}{H}}: \sup _{n} \Biggl(\frac{1}{n} \sum _{k = 1}^{n} \biggl[M_{k} \biggl(q \biggl( \frac{u_{k} \Delta^{n}_{m} x_{k}}{r} \biggr) \biggr) \biggr]^{p_{k}} \Biggr)^{\frac{1}{H}} \leq1, \rho> 0 \Biggr\} , \end{aligned}

where $$r = \frac{\rho}{|\lambda|}$$. Hence, $$w_{\infty}(\mathcal {M},\Delta^{n}_{m},p,q,u )$$ is a paranormed space. □

### Theorem 2.3

If $$0 < p_{k} \leq r_{k} < \infty$$ for each k, then $$Z (\mathcal{M},\Delta^{n}_{m}, p, q, u ) \subseteq Z (\mathcal{M},\Delta^{n}_{m}, r, q, u )$$ for $$Z = w_{0}, w, w_{\infty}$$.

### Proof

Let $$x = (x_{k}) \in w (\mathcal{M},\Delta^{n}_{m}, p, q, u )$$. Then there exist some $$\rho> 0$$ and $$L \in X$$ such that

$$\frac{1}{n} \sum_{k = 1}^{n} \biggl[M_{k} \biggl(q \biggl(\frac{u_{k} \Delta^{n}_{m} x_{k} - L}{\rho} \biggr) \biggr) \biggr]^{p_{k}} \rightarrow0 \quad\mbox{as } n \rightarrow\infty.$$

This implies that

$$\frac{1}{n} \sum_{k = 1}^{n} \biggl[M_{k} \biggl(q \biggl(\frac{u_{k} \Delta^{n}_{m} x_{k} - L}{\rho} \biggr) \biggr) \biggr]^{p_{k}} < \epsilon \quad (0 < \epsilon< 1)$$

for sufficiently large k. Hence we get

\begin{aligned} \frac{1}{n} \sum_{k = 1}^{n} \biggl[M_{k} \biggl(q \biggl(\frac {u_{k} \Delta^{n}_{m} x_{k} - L}{\rho} \biggr) \biggr) \biggr]^{r_{k}} \leq& \frac{1}{n} \sum_{k = 1}^{n} \biggl[M_{k} \biggl(q \biggl(\frac{u_{k} \Delta^{n}_{m} x_{k} - L}{\rho} \biggr) \biggr) \biggr]^{p_{k}} \\ \rightarrow& 0 \quad \mbox{as } n \rightarrow\infty. \end{aligned}

This implies that $$x = (x_{k}) \in w(\mathcal{M},\Delta^{n}_{m}, r, q, u )$$. This completes the proof. Similarly, we can prove for the cases $$Z = w_{0}, w_{\infty}$$. □

### Theorem 2.4

Suppose $$\mathcal{M'} = (M_{k}')$$ and $$\mathcal {M''} = (M_{k}'')$$ are Musielak-Orlicz functions satisfying the $$\Delta _{2}$$-condition, then we have the following results:

1. (i)

If $$p = (p_{k})$$ is a bounded sequence of positive real numbers then $$Z (\mathcal{M'},\Delta^{n}_{m}, p, q, u ) \subseteq Z (\mathcal {M''\circ M'},\Delta^{n}_{m}, p, q, u )$$ for $$Z = w_{0}, w,\textit{and }w_{\infty}$$.

2. (ii)

$$Z (\mathcal{M'},\Delta^{n}_{m}, p, q, u ) \cap Z (\mathcal{M}'',\Delta^{n}_{m}, p, q, u ) \subseteq Z (\mathcal{M' + M''},\Delta^{n}_{m},p,q,u )$$ for $$Z = w_{0}, w,\textit{and }w_{\infty}$$.

### Proof

(i) If $$x = (x_{k}) \in w_{0} (\mathcal{M'},\Delta^{n}_{m}, p, q, u)$$, then there exists some $$\rho> 0$$ such that

$$\frac{1}{n} \sum_{k = 1}^{n} \biggl[M_{k}' \biggl(q \biggl(\frac {u_{k} \Delta^{n}_{m} x_{k} }{\rho} \biggr) \biggr) \biggr]^{p_{k}} \rightarrow0 \quad\mbox{as } n \rightarrow\infty.$$

Suppose

$$y_{k} = M_{k}' \biggl(q \biggl( \frac{u_{k} \Delta^{n}_{m} x_{k} }{\rho} \biggr) \biggr)$$

for all $$k \in\mathbb{N}$$. Choose $$0 < \delta< 1$$, then for $$y_{k} \geq \delta$$ we have $$y_{k} < \frac{y_{k}}{\delta} < 1 + \frac{y_{k}}{\delta}$$. Now $$(M_{k}'')$$ satisfies the $$\Delta_{2}$$-condition so that there exists $$J \geq1$$ such that

$$M_{k}''(y_{k}) < \frac{J y_{k}}{2\delta} M_{k}'' (2) + \frac{J y_{k}}{2\delta} M_{k}'' (2) = \frac{J y_{k}}{\delta} M_{k}'' (2).$$

We obtain

\begin{aligned} \frac{1}{n} \sum_{k = 1}^{n} \biggl[M_{k}'' \circ M_{k}' \biggl(q \biggl(\frac{u_{k} \Delta^{n}_{m} x_{k} }{\rho} \biggr) \biggr) \biggr]^{p_{k}} = & \frac {1}{n} \sum_{k = 1}^{n} \biggl[M_{k}'' \biggl\{ M_{k}' \biggl(q \biggl(\frac {u_{k} \Delta^{n}_{m} x_{k} }{\rho} \biggr) \biggr)\biggr\} \biggr]^{p_{k}} \\ = & \frac{1}{n} \sum_{k = 1}^{n} \bigl[M_{k}''(y_{k}) \bigr]^{p_{k}} \\ \rightarrow& 0 \quad\mbox{as } n \rightarrow\infty. \end{aligned}

Similarly we can prove the other cases.

(ii) Suppose $$x = (x_{k} )\in w_{0} (M_{k}',\Delta^{n}_{m}, p, q, u ) \cap w_{0} (M_{k}'',\Delta^{n}_{m}, p, q, u )$$, then there exist $$\rho _{1}, \rho_{2} > 0$$ such that

$$\frac{1}{n} \sum_{k = 1}^{n} \biggl[M_{k}' \biggl(q \biggl(\frac {u_{k} \Delta^{n}_{m} x_{k} }{\rho_{1}} \biggr) \biggr) \biggr]^{p_{k}} \rightarrow0 , \quad\mbox{as } n \rightarrow\infty.$$

and

$$\frac{1}{n} \sum_{k = 1}^{n} \biggl[M_{k}'' \biggl(q \biggl( \frac {u_{k} \Delta^{n}_{m} x_{k} }{\rho_{2}} \biggr) \biggr) \biggr]^{p_{k}} \rightarrow0 , \quad \mbox{as } n \rightarrow\infty.$$

Let $$\rho= \max\{\rho_{1}, \rho_{2}\}$$. The remaining proof follows from the inequality

\begin{aligned}[b] \frac{1}{n} \sum_{k = 1}^{n} \biggl[ \bigl(M_{k}' + M_{k}'' \bigr) \biggl(q \biggl(\frac{u_{k} \Delta^{n}_{m} x_{k} }{\rho} \biggr) \biggr) \biggr]^{p_{k}} \leq{}& D \Biggl\{ \frac{1}{n} \sum_{k = 1}^{n} \biggl[M_{k}' \biggl(q \biggl(\frac {u_{k} \Delta^{n}_{m} x_{k} }{\rho_{1}} \biggr) \biggr) \biggr]^{p_{k}} \\ &{}+ \frac{1}{n} \sum_{k = 1}^{n} \biggl[M_{k}'' \biggl(q \biggl( \frac{u_{k} \Delta ^{n}_{m} x_{k} }{\rho_{2}} \biggr) \biggr) \biggr]^{p_{k}} \Biggr\} . \end{aligned}

Hence, $$w_{0} (M_{k}',\Delta^{n}_{m}, p, q, u ) \cap w_{0} (M_{k}'',\Delta ^{n}_{m}, p, q, u ) \subseteq w_{0} (M_{k}' + M_{k}'',\Delta^{n}_{m}, p, q, u )$$. Similarly we can prove the other cases. □

### Theorem 2.5

(i) If $$0 < \inf p_{k} \leq p_{k} < 1$$, then

$$w_{\infty}\bigl(\mathcal{M},\Delta^{n}_{m}, p, q, u \bigr) \subset w_{\infty}\bigl(\mathcal{M},\Delta^{n}_{m}, q, u \bigr).$$

(ii) If $$1 \leq p_{k} \leq\sup p_{k} < \infty$$, then

$$w_{\infty}\bigl(\mathcal{M},\Delta^{n}_{m}, q, u \bigr) \subset w_{\infty}\bigl(\mathcal{M},\Delta^{n}_{m}, p, q, u \bigr).$$

### Proof

(i) Let $$x = (x_{k}) \in w_{\infty}(\mathcal{M},\Delta ^{n}_{m}, p, q, u )$$. Since $$0 < \inf p_{k} \leq1$$, we have

$$\sup_{n} \Biggl\{ \frac{1}{n} \sum _{k = 1}^{n} \biggl[M_{k} \biggl(q \biggl( \frac{u_{k} \Delta^{n}_{m} x_{k} }{\rho} \biggr) \biggr) \biggr] \Biggr\} \leq\sup_{n} \Biggl\{ \frac{1}{n} \sum_{k = 1}^{n} \biggl[M_{k} \biggl(q \biggl(\frac{u_{k} \Delta^{n}_{m} x_{k} }{\rho} \biggr) \biggr) \biggr]^{p_{k}} \Biggr\}$$

and hence $$x = (x_{k}) \in w_{\infty}(\mathcal{M},\Delta^{n}_{m}, q, u )$$.

(ii) Let $$p_{k} \geq1$$ for each k and $$\sup_{k} p_{k} < \infty$$. Let $$x = (x_{k}) \in w_{\infty}(\mathcal{M},\Delta^{n}_{m}, q, u )$$, then for each $$\epsilon> 0$$ such that $$0 < \epsilon< 1$$, there exists a positive integer $$n \in\mathbb{N}$$ such that

$$\sup_{n} \Biggl\{ \frac{1}{n} \sum _{k = 1}^{n} \biggl[M_{k} \biggl(q \biggl( \frac{u_{k} \Delta^{n}_{m} x_{k} }{\rho} \biggr) \biggr) \biggr] \Biggr\} \leq \epsilon< 1.$$

This implies that

$$\sup_{n} \Biggl\{ \frac{1}{n} \sum _{k = 1}^{n} \biggl[M_{k} \biggl(q \biggl( \frac{u_{k} \Delta^{n}_{m} x_{k} }{\rho} \biggr) \biggr) \biggr]^{p_{k}} \Biggr\} \leq \sup _{n} \Biggl\{ \frac{1}{n} \sum _{k = 1}^{n} \biggl[M_{k} \biggl(q \biggl( \frac{u_{k} \Delta^{n}_{m} x_{k} }{\rho} \biggr) \biggr) \biggr] \Biggr\} .$$

Thus, $$x = (x_{k}) \in w_{\infty}(\mathcal{M},\Delta^{n}_{m}, p, q, u )$$ and this completes the proof. □

### Theorem 2.6

The sequence space $$w_{\infty}(\mathcal {M},\Delta^{n}_{m}, p, q, u )$$ is solid.

### Proof

Let $$x = (x_{k}) \in w_{\infty}(\mathcal{M},\Delta^{n}_{m}, p, q, u )$$, i.e.

$$\sup_{n} \frac{1}{n} \sum_{k = 1}^{n} \biggl[M_{k} \biggl(q \biggl(\frac{u_{k} \Delta^{n}_{m} x_{k}}{\rho} \biggr) \biggr) \biggr]^{p_{k}} < \infty.$$

Let $$(\alpha_{k})$$ be a sequence of scalars such that $$|\alpha_{k}| \leq1$$ for all $$k \in\mathbb{N}$$. Thus we have

$$\sup_{n} \frac{1}{n} \sum_{k = 1}^{n} \biggl[M_{k} \biggl(q \biggl(\frac{\alpha_{k} u_{k} \Delta^{n}_{m} x_{k}}{\rho} \biggr) \biggr) \biggr]^{p_{k}} \leq \sup_{n} \frac{1}{n} \sum _{k = 1}^{n} \biggl[M_{k} \biggl(q \biggl(\frac{u_{k} \Delta^{n}_{m} x_{k}}{\rho} \biggr) \biggr) \biggr]^{p_{k}} < \infty.$$

This shows that $$(\alpha_{k} x_{k}) \in w_{\infty}(\mathcal{M},\Delta ^{n}_{m}, p, q, u )$$ for all sequences of scalars $$(\alpha_{k})$$ with $$|\alpha_{k}| \leq1$$ for all $$k \in\mathbb{N}$$, whenever $$(x_{k}) \in w_{\infty}(\mathcal{M},\Delta^{n}_{m}, p, q, u )$$. Hence the space $$w_{\infty}(\mathcal{M},\Delta^{n}_{m}, p, q, u )$$ is a solid sequence space. □

### Theorem 2.7

The sequence space $$w_{\infty}(\mathcal {M},\Delta^{n}_{m}, p, q, u )$$ is monotone.

### Proof

The proof of the theorem is obvious and so we omit it. □

Let $$F = (f_{k})$$ be a sequence of modulus functions, $$p=(p_{k})$$ be any bounded sequence of positive real numbers and $$u = (u_{k})$$ be a sequence of strictly positive real numbers. Let $$(X,q)$$ be a space seminormed by q. Now, we define the following sequence spaces:

\begin{aligned}& \begin{aligned}[b] w_{0} \bigl(F,\Delta^{n}_{m},p,q,u \bigr) ={}& \Biggl\{ x = (x_{k}):\frac{1}{n}\sum^{n}_{k=1} \biggl[f_{k} \biggl(\frac{q (u_{k}\Delta^{n}_{m}x_{k} )}{\rho} \biggr) \biggr]^{p_{k}} \rightarrow0 \mbox{ as } n \rightarrow\infty,\\ &{}\mbox{for some } \rho> 0 \Biggr\} , \end{aligned} \\& \begin{aligned}[b] w \bigl(F,\Delta^{n}_{m},p,q,u \bigr) ={}& \Biggl\{ x = (x_{k}):\frac{1}{n}\sum^{n}_{k=1} \biggl[f_{k} \biggl(\frac{q (u_{k}\Delta^{n}_{m}x_{k} - L )}{\rho } \biggr) \biggr]^{p_{k}} \rightarrow0 \mbox{ as } n \rightarrow \infty, \\ &{}\mbox{for some } \rho> 0 \mbox{ and } L \in X \Biggr\} , \end{aligned} \end{aligned}

and

$$w_{\infty}\bigl(F,\Delta^{n}_{m},p,q,u \bigr) = \Biggl\{ x = (x_{k}):\sup_{n}\frac {1}{n}\sum ^{n}_{k=1} \biggl[f_{k} \biggl( \frac{q (u_{k}\Delta^{n}_{m}x_{k} )}{\rho} \biggr) \biggr]^{p_{k}} < \infty, \mbox{for some } \rho> 0 \Biggr\} .$$

### Theorem 2.8

Let $$F = (f_{k})$$ be a sequence of modulus functions, $$p=(p_{k})$$ be any bounded sequence of positive real numbers and $$u = (u_{k})$$ be a sequence of strictly positive real numbers. Then the spaces $$w_{0} (F, \Delta^{n}_{m}, p, q, u )$$, $$w (F, \Delta ^{n}_{m}, p, q, u )$$, and $$w_{\infty}(F, \Delta^{n}_{m}, p, q, u )$$ are linear spaces over the complex field .

### Proof

The proof of Theorem 2.1 holds along the same lines for this theorem and so we omit it. □

### Theorem 2.9

Let $$F = (f_{k})$$ be a sequence of modulus function, $$p=(p_{k})$$ be any bounded sequence of positive real numbers and $$u = (u_{k})$$ be a sequence of strictly positive real numbers. Then $$w_{\infty}(F,\Delta^{n}_{m}, p, q, u )$$ is a paranormed space with the paranorm defined by

$$g(x) = \inf \Biggl\{ \rho^{\frac{p_{k}}{H}}: \sup_{n} \Biggl(\frac {1}{n} \sum _{k = 1}^{n} \biggl[f_{k} \biggl( q \biggl( \frac{u_{k} \Delta^{n}_{m} x_{k}}{\rho} \biggr) \biggr) \biggr]^{p_{k}} \Biggr)^{\frac{1}{H}} \leq1, \rho> 0 \Biggr\} ,$$
(2.1)

where $$H = \max(1, \sup_{k} p_{k})$$.

### Proof

The proof follows from Theorem 2.2 and so we omit it. □

### Theorem 2.10

Let $$F= (f_{k})$$ be a sequence of modulus functions, $$p=(p_{k})$$ be any bounded sequence of positive real numbers and $$u = (u_{k})$$ be a sequence of strictly positive real numbers. Then

$$w_{0} \bigl(F, \Delta^{n}_{m}, p, q, u \bigr) \subset w \bigl(F, \Delta^{n}_{m}, p, q, u \bigr)\subset w_{\infty}\bigl(F, \Delta^{n}_{m}, p, q, u \bigr),$$

and the inclusions are strict.

### Proof

The proof is obvious. □

### Theorem 2.11

Let $$F = (f_{k})$$ and $$G = (g_{k})$$ be any two sequences of modulus functions. For any bounded sequences $$p = (p_{k})$$ of positive real numbers and for any two seminorms q and r. Then

1. (i)

$$w_{Z} (F, \Delta^{n}_{m}, q, u ) \subset w_{Z} (F\circ G, \Delta ^{n}_{m}, q, u )$$,

2. (ii)

$$w_{Z} (F, \Delta^{n}_{m}, p, q, u )\cap w_{Z} (F, \Delta ^{n}_{m}, p, r, u ) \subset w_{Z} (F, \Delta^{n}_{m}, p, q + r, u )$$,

3. (iii)

$$w_{Z} (F, \Delta^{n}_{m}, p, q, u )\cap w_{Z} (G, \Delta ^{n}_{m}, p, q, u )\subset w_{Z} (F+G, \Delta^{n}_{m}, p, q, u )$$, where $$Z= 0,1, \infty$$.

### Proof

(i) We shall prove it for the relation $$w_{0} (F, \Delta ^{n}_{m}, q, u ) \subset w_{0} (F\circ G, \Delta^{n}_{m}, q, u )$$. For $$\epsilon> 0$$, we choose δ, $$0<\delta< 1$$, such that $$f_{k}(t) < \epsilon$$ for $$0\leq t \leq\delta$$ and all $$k\in\mathbb {N}$$. We write $$y_{k} = g_{k} (\frac{q (\Delta_{n}^{m} u_{k} x_{k} )}{\rho} )$$ and consider

$$\sum^{n}_{k=1}\bigl[f_{k}(y_{k}) \bigr] = \sum_{1}\bigl[f_{k}(y_{k}) \bigr] + \sum_{2}\bigl[f_{k}(y_{k}) \bigr],$$

where the first summation is over $$y_{k} \leq\delta$$ and the second summation is over $$y_{k} > \delta$$. Since F is continuous, we have

$$\sum_{1} \bigl[f_{k}(y_{k}) \bigr] < n\epsilon.$$
(2.2)

By the definition of F, we have the following relation for $$y_{k} > \delta$$:

$$f_{k}(y_{k})< 2 f_{k}(1)\frac{y_{k}}{\delta}.$$

Hence,

$$\frac{1}{n} \sum_{2} \bigl[f_{k}(y_{k})\bigr] \leq2\delta^{-1} f_{k}(1)\frac{1}{n}\sum^{n}_{k=1}y_{k}.$$
(2.3)

It follows from (2.2) and (2.3) that $$w_{0} (F, \Delta^{n}_{m}, q, u ) \subset w_{0} (F\circ G, \Delta^{n}_{m}, q, u )$$. Similarly, we can prove $$w (F, \Delta^{n}_{m}, q, u ) \subset w (F\circ G, \Delta ^{n}_{m}, q, u )$$ and $$w_{\infty}(F, \Delta^{n}_{m}, q, u ) \subset w_{\infty}(F\circ G, \Delta^{n}_{m}, q, u )$$.

The proof of (ii) and (iii) follows from (i). □

### Corollary 2.12

Let f be a modulus function. Then

$$w_{Z} \bigl(\Delta^{n}_{m}, q,u \bigr)\subset w_{Z} \bigl(f, \Delta^{n}_{m}, q, u \bigr), \quad \textit{for } Z= 0,1, \infty.$$

### Theorem 2.13

Let $$F = (f_{k})$$ be a sequence of modulus functions, $$p=(p_{k})$$ be any bounded sequence of positive real numbers and $$u = (u_{k})$$ be a sequence of strictly positive real numbers. Then $$w_{\infty}(F,\Delta^{n}_{m}, p, q, u )$$ is complete and seminormed by (2.1).

### Proof

Suppose $$(x^{n})$$ is a Cauchy sequence in $$w_{\infty}(F,\Delta^{n}_{m}, p, q, u )$$, where $$x^{n} = (x^{n}_{k})^{\infty}_{k = 1}$$ for all $$n\in\mathbb{N}$$. So that $$g(x^{i} - x^{j}) \rightarrow0$$ as $$i, j \rightarrow\infty$$. Suppose $$\epsilon> 0$$ is given and let s and $$x_{0}$$ be such that $$\frac{\epsilon}{s x_{0}} > 0$$ and $$f_{k} (\frac {s x_{0}}{2} ) \geq \sup_{k\geq1}(p_{k})$$. Since $$g(x^{i} - x^{j}) \rightarrow0$$, as $$i, j \rightarrow\infty$$, which means that there exists $$n_{0} \in\mathbb{N}$$ such that

$$g\bigl(x^{i} - x^{j}\bigr) < \frac{\epsilon}{s x_{0}}, \quad \mbox{for all } i,j \geq n_{0}.$$

This gives $$g(x^{i}_{1} - x^{j}_{1}) < \frac{\epsilon}{s x_{0}}$$ and

$$\inf \biggl\{ \rho^{\frac{p_{k}}{H}}: \sup_{k\geq1} \biggl(f_{k} \biggl(\frac{q(u_{k} \Delta^{n}_{m} x^{i}_{k} - u_{k}\Delta^{n}_{m} x^{j}_{k})}{\rho } \biggr) \biggr)\leq1, \rho> 0 \biggr\} < \frac{\epsilon}{s x_{0}}.$$
(2.4)

It shows that $$(x^{i}_{1})$$ is a Cauchy sequence in X. Thus, $$(x^{i}_{1})$$ is convergent in X because X is complete. Suppose $$\lim_{i\rightarrow\infty} x^{i}_{1} = x_{1}$$ then $$\lim_{j\rightarrow\infty} g(x^{i}_{1} - x^{j}_{1}) < \frac{\epsilon}{ s x_{0}}$$, we get

$$g\bigl(x^{i}_{1} - x_{1}\bigr) < \frac{\epsilon}{s x_{0}}.$$

Thus, we have

$$f_{k} \biggl(\frac{q(u_{k}\Delta^{n}_{m} x^{i}_{k} - u_{k}\Delta^{n}_{m} x^{j}_{k})}{g(x^{i} - x^{j})} \biggr) \leq1.$$

This implies that

$$f_{k} \biggl(\frac{q(u_{k}\Delta^{n}_{m} x^{i}_{k} - u_{k}\Delta^{n}_{m} x^{j}_{k})}{g(x^{i} - x^{j})} \biggr) \leq f_{k}\biggl( \frac{s x_{0}}{2}\biggr)$$

and thus

$$q \bigl(u_{k}\Delta^{n}_{m} x^{i}_{k} - u_{k}\Delta^{n}_{m} x^{j}_{k} \bigr) < \frac{s x_{0}}{2}\cdot\frac{\epsilon}{s x_{0}} < \frac{\epsilon}{2},$$

which shows that $$(u_{k}\Delta^{n}_{m} x^{i}_{k})$$ is a Cauchy sequence in X for all $$k\in\mathbb{N}$$. Therefore, $$(u_{k}\Delta^{n}_{m} x^{i}_{k})$$ converges in X. Suppose $$\lim_{i\rightarrow\infty}\Delta^{n}_{m} x^{i}_{k} = y_{k}$$ for all $$k\in\mathbb{N}$$. Also, we have $$\lim_{i\rightarrow\infty}u_{k}\Delta^{n}_{m} x^{i}_{2} =y_{1}- x_{1}$$. On repeating the same procedure, we obtain $$\lim_{i\rightarrow\infty }u_{k}\Delta^{n}_{m} x^{i}_{k+1} =y_{k}- x_{k}$$ for all $$k \in\mathbb{N}$$. Therefore by continuity of $$f_{k}$$, we get

$$\lim_{ j\rightarrow\infty}\sup_{k\geq1} f_{k} \biggl( \frac{q(u_{k}\Delta^{n}_{m} x^{i}_{k} - u_{k}\Delta^{n}_{m} x^{j}_{k})}{\rho} \biggr) \leq1,$$

so that

$$\sup_{k\geq1} f_{k} \biggl(\frac{q(u_{k}\Delta^{n}_{m} x^{i}_{k} - u_{k}\Delta^{n}_{m} x_{k})}{\rho} \biggr) \leq1.$$

Let $$i\geq n_{0}$$ and taking the infimum of each ρ, we have

$$g\bigl(x^{i} - x\bigr) < \epsilon.$$

So $$(x^{i} - x)\in w_{\infty}(F,\Delta^{n}_{m}, p, q, u )$$. Hence $$x= x^{i} - (x^{i} - x)\in w_{\infty}(F,\Delta^{n}_{m}, p, q, u )$$, since $$w_{\infty}(F,\Delta^{n}_{m}, p, q, u )$$ is a linear space. Hence, $$w_{\infty}(F,\Delta ^{n}_{m}, p, q, u )$$ is a complete paranormed space. □

## References

1. 1.

Lindenstrauss, J, Tzafriri, L: On Orlicz sequence spaces. Isr. J. Math. 10, 379-390 (1971)

2. 2.

Maligranda, L: Orlicz Spaces and Interpolation. Seminars in Mathematics, vol. 5. Polish Academy of Science, Warsaw (1989)

3. 3.

Malkowsky, E, Savaş, E: Some λ-sequence spaces defined by a modulus. Arch. Math. 36, 219-228 (2000)

4. 4.

Musielak, J: Orlicz Spaces and Modular Spaces. Lecture Notes in Mathematics, vol. 1034 (1983)

5. 5.

Bilgen, T: On statistical convergence. An. Univ. Vest. Timiş., Ser. Mat.-Inform. 32, 3-7 (1994)

6. 6.

Savaş, E: On some generalized sequence spaces defined by a modulus. Indian J. Pure Appl. Math. 30, 459-464 (1999)

7. 7.

Savaş, E, Kılıçman, A: A note on some strongly sequence spaces. Abstr. Appl. Anal. 2011, Article ID 598393 (2011)

8. 8.

Tripathy, BC, Esi, A: A new type of difference sequences spaces. Int. J. Sci. Technol. 1, 11-14 (2006)

9. 9.

Kızmaz, H: On certain sequence spaces. Can. Math. Bull. 24, 169-176 (1981)

10. 10.

Et, M, Çolak, R: On some generalized difference sequence spaces. Soochow J. Math. 21, 377-386 (1995)

11. 11.

Et, M, Gokhan, A, Altınok, H: On statistical convergence of vector-valued sequences associated with multiplier sequences. Ukr. Math. J. 58, 139-146 (2006)

12. 12.

Gunawan, H: On n-inner product, n-norms and the Cauchy-Schwarz inequality. Sci. Math. Jpn. 5, 47-54 (2001)

13. 13.

Gunawan, H: The space of p-summable sequence and its natural n-norm. Bull. Aust. Math. Soc. 64, 137-147 (2001)

14. 14.

Gunawan, H, Mashadi, M: On n-normed spaces. Int. J. Math. Math. Sci. 27, 631-639 (2001)

15. 15.

Kılıçman, A, Borgohain, S: Some new classes of generalized difference strongly summable n-normed sequence spaces defined by ideal convergence and Orlicz function. Abstr. Appl. Anal. 2014, Article ID 621383 (2014)

16. 16.

Kılıçman, A, Borgohain, S: On generalized difference Hahn sequence spaces. Sci. World J. 2014, Article ID 398203 (2014)

17. 17.

Mohiuddine, SA, Alotaibi, A, Mursaleen, M: A new variant of statistical convergence. J. Inequal. Appl. 2013, Article ID 309 (2013)

18. 18.

Mursaleen, M, Mohiuddine, SA: Some matrix transformations of convex and paranormed sequence spaces into the spaces of invariant means. J. Funct. Spaces Appl. 2012, Article ID 612671 (2012)

19. 19.

Mursaleen, M, Noman, AK: On some difference sequence spaces of non-absolute type. Math. Comput. Model. 52, 603-617 (2010)

20. 20.

Raj, K, Kılıçman, A: On generalized difference Hahn sequence spaces. Sci. World J. 2014, Article ID 398203 (2014)

21. 21.

Raj, K, Sharma, SK: Some sequence spaces in 2-normed spaces defined by a Musielak-Orlicz function. Acta Univ. Sapientiae Math. 3, 97-109 (2011)

22. 22.

Mursaleen, M, Sharma, SK, Kılıçman, A: Sequence spaces defined by Musielak-Orlicz function over α-normed spaces. Abstr. Appl. Anal. 2013, Article ID 364743 (2013)

23. 23.

Raj, K, Sharma, SK, Sharma, AK: Some difference sequence spaces in n-normed spaces defined by Musielak-Orlicz function. Armen. J. Math. 3, 127-141 (2010)

## Acknowledgements

The authors express their sincere thanks to the referees for the careful and details reading of the manuscript and suggestions on the current literature.

## Author information

### Competing interests

The authors declare that they have no competing interests.

### Authors’ contributions

Both authors contributed equally during the development of manuscript and the authors read and approved the final manuscript.

## Rights and permissions

Open Access This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.

Reprints and Permissions 