Open Access

On certain generalized paranormed spaces

Journal of Inequalities and Applications20152015:37

https://doi.org/10.1186/s13660-015-0565-z

Received: 19 September 2014

Accepted: 14 January 2015

Published: 3 February 2015

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.

Keywords

Orlicz functionMusielak-Orlicz functionmodulus functionsequence space

MSC

40A0546A4546E30

1 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.

2 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. □

Declarations

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.

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.

Authors’ Affiliations

(1)
School of Mathematics, Shri Mata Vaishno Devi University
(2)
Department of Mathematics and Institute for Mathematical Research, University Putra Malaysia (UPM)

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© Raj and Kılıçman; licensee Springer. 2015