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On certain generalized paranormed spaces

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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 [1] 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 [1] 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 [24]). 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 [9], who studied the difference sequence spaces \(l_{\infty}(\Delta)\), \(c(\Delta)\), and \(c_{0}(\Delta)\). The notion was further generalized by Et and Çolak [10] 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 [8] 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 [10]. Taking \(m = n = 1 \), we get the spaces \(l_{\infty}(\Delta)\), \(c(\Delta)\), and \(c_{0}(\Delta)\) introduced and studied by Kızmaz [9]. 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. □

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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.

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Correspondence to Adem Kılıçman.

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

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Raj, K., Kılıçman, A. On certain generalized paranormed spaces. J Inequal Appl 2015, 37 (2015) doi:10.1186/s13660-015-0565-z

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MSC

  • 40A05
  • 46A45
  • 46E30

Keywords

  • Orlicz function
  • Musielak-Orlicz function
  • modulus function
  • sequence space