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:
which is called an Orlicz sequence space. The space \(\ell_{M}\) is a Banach space with the norm
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 \(u \geq0\).
$$\ell_{M} = \Biggl\{ x \in w : \sum^{\infty}_{k=1} M \biggl(\frac{|x_{k}|}{\rho} \biggr) < \infty, \mbox{for some } \rho>0 \Biggr\} , $$
$$\|x\| = \inf \Biggl\{ \rho> 0 : \sum^{\infty}_{k=1} M \biggl(\frac{|x_{k}|}{\rho} \biggr) \leq 1 \Biggr\} . $$
A sequence \(\mathcal{M} = (M_{k})\) of Orlicz functions is called a Musielak-Orlicz function (see [2‐4]). A sequence \(\mathcal{N} =(N_{k})\) is defined by
is called the complementary function of a Musielak-Orlicz function ℳ. For a given Musielak-Orlicz function ℳ, the Musielak-Orlicz sequence space \({t_{\mathcal{M}}}\) and its subspace \(h_{\mathcal{M}}\) are defined as follows:
where \(I_{\mathcal{M}}\) is a convex modular defined by
We consider \(t_{\mathcal{M}}\) equipped with the Luxemburg norm
or equipped with the Orlicz norm
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
holds for all \(k \in N\) and \(u \in R_{+}\) whenever \(M_{k}(u) \leq a\).
$${N}_{k}(v) = \sup\bigl\{ |v|u - M_{k}(u) :u\geq0\bigr\} , \quad k = 1,2,\ldots $$
$$\begin{aligned}& t_{\mathcal{M}} = \bigl\{ x \in w : I_{\mathcal{M}} (cx) < \infty \mbox{ for some } c > 0 \bigr\} , \\& h_{\mathcal{M}} = \bigl\{ x \in w : I_{\mathcal{M}} (cx) < \infty \mbox{ for all } c > 0 \bigr\} , \end{aligned}$$
$$I_{\mathcal{M}}(x) = \sum^{\infty}_{k=1} M_{k}(x_{k}),\quad x = (x_{k}) \in t_{\mathcal{M}}. $$
$$\|x\| = \inf \biggl\{ k > 0 : I_{\mathcal{M}} \biggl(\frac{x}{k} \biggr) \leq1 \biggr\} $$
$$\|x\|^{0} = \inf \biggl\{ \frac{1}{k} \bigl(1 + I_{\mathcal{M}}(kx) \bigr) : k > 0 \biggr\} . $$
$$M_{k}(2u) \leq K M_{k}(u) + c_{k} $$
Anzeige
A modulus function is a function \(f : [0, \infty) \rightarrow[0, \infty) \) such that 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, 5‐8] and references therein.
(1)
\(f(x) = 0 \) if and only if \(x =0 \),
(2)
\(f(x + y ) \leq f(x) + f(y)\) for all \(x \geq0\), \(y\geq0 \),
(3)
f is increasing,
(4)
f is continuous from right at 0.
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
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:
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, 11‐23] and references therein.
$$Z \bigl(\Delta^{n}_{m}\bigr) = \bigl\{ x = (x_{k})\in w : \bigl(\Delta^{n}_{m} x_{k}\bigr)\in Z \bigr\} $$
$$\Delta^{n}_{m} x_{k} = \sum ^{n}_{v=0}(-1)^{v}\left ( \begin{array}{@{}c@{}} n \\ v \end{array} \right )x_{k + mv}. $$
Anzeige
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:
and
If we take \(\mathcal{M}(x) = x\), we get
and
If we take \(p = (p_{k}) = 1\), ∀k, we get
and
If we take \(u = (u_{k}) = 1\), ∀k, we get
and
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
for all k and \(a_{k}, b_{k} \in\mathbb{C} \). Also \(|a|^{p_{k}}\leq\max (1, |a|^{K})\) for all \(a \in\mathbb{C}\).
$$\begin{aligned}& \begin{aligned}[b] w_{0} \bigl(\mathcal{M},\Delta^{n}_{m},p,q,u \bigr) ={}& \Biggl\{ x = (x_{k}):\frac {1}{n}\sum ^{n}_{k=1} \biggl[M_{k} \biggl( \frac{q (u_{k}\Delta^{n}_{m}x_{k} )}{\rho} \biggr) \biggr]^{p_{k}} \rightarrow0 \mbox{ as } n \rightarrow\infty,\\ &{} \mbox{for some } \rho> 0 \Biggr\} , \end{aligned}\\& \begin{aligned}[b] w \bigl(\mathcal{M},\Delta^{n}_{m},p,q,u \bigr) ={}& \Biggl\{ x = (x_{k}):\frac {1}{n}\sum^{n}_{k=1} \biggl[M_{k} \biggl(\frac{q (u_{k}\Delta^{n}_{m}x_{k} - L )}{\rho} \biggr) \biggr]^{p_{k}} \rightarrow0 \mbox{ as } n \rightarrow\infty,\\ &{}\mbox{for some } L \in X, \rho> 0 \Biggr\} , \end{aligned} \end{aligned}$$
$$\begin{aligned}[b] &w_{\infty}\bigl(\mathcal{M},\Delta^{n}_{m},p,q,u \bigr) \\ &\quad= \Biggl\{ x = (x_{k}):\sup_{n} \frac{1}{n}\sum^{n}_{k=1} \biggl[M_{k} \biggl(\frac{q (u_{k}\Delta^{n}_{m}x_{k} )}{\rho} \biggr) \biggr]^{p_{k}} < \infty, \mbox{for some } \rho> 0 \Biggr\} . \end{aligned} $$
$$\begin{aligned}& \begin{aligned}[b] &w_{0} \bigl(\Delta^{n}_{m},p,q,u \bigr) \\ &\quad= \Biggl\{ x = (x_{k}):\frac{1}{n}\sum^{n}_{k=1} \biggl[ \biggl(\frac{q (u_{k}\Delta^{n}_{m}x_{k} )}{\rho} \biggr) \biggr]^{p_{k}} \rightarrow0 \mbox{ as } n \rightarrow\infty, \mbox{for some } \rho> 0 \Biggr\} , \end{aligned}\\& \begin{aligned}[b] &w \bigl(\Delta^{n}_{m},p,q,u \bigr) \\ &\quad= \Biggl\{ x = (x_{k}):\frac{1}{n}\sum^{n}_{k=1} \biggl[ \biggl(\frac{q (u_{k}\Delta^{n}_{m}x_{k} - L )}{\rho} \biggr) \biggr]^{p_{k}} \rightarrow0 \mbox{ as } n \rightarrow\infty, \mbox{for some } L \in X, \rho> 0 \Biggr\} , \end{aligned} \end{aligned}$$
$$w_{\infty}\bigl(\Delta^{n}_{m},p,q,u \bigr) = \Biggl\{ x = (x_{k}):\sup_{n}\frac {1}{n}\sum ^{n}_{k=1} \biggl[ \biggl(\frac{q (u_{k}\Delta^{n}_{m}x_{k} )}{\rho } \biggr) \biggr]^{p_{k}} < \infty, \mbox{for some } \rho> 0 \Biggr\} . $$
$$\begin{aligned}& \begin{aligned}[b] &w_{0} \bigl(\mathcal{M},\Delta^{n}_{m},q,u \bigr)\\ &\quad= \Biggl\{ x = (x_{k}):\frac {1}{n}\sum ^{n}_{k=1} \biggl[M_{k} \biggl( \frac{q (u_{k}\Delta^{n}_{m}x_{k} )}{\rho} \biggr) \biggr] \rightarrow0 \mbox{ as } n \rightarrow \infty, \mbox{for some } \rho> 0 \Biggr\} , \end{aligned}\\& \begin{aligned}[b] w \bigl(\mathcal{M},\Delta^{n}_{m},q,u \bigr) ={}& \Biggl\{ x = (x_{k}):\frac{1}{n}\sum^{n}_{k=1} \biggl[M_{k} \biggl(\frac{q (u_{k}\Delta^{n}_{m}x_{k} - L )}{\rho } \biggr) \biggr] \rightarrow0 \mbox{ as } n \rightarrow\infty,\\ &{}\mbox{for some } L \in X, \rho> 0 \Biggr\} , \end{aligned} \end{aligned}$$
$$w_{\infty}\bigl(\mathcal{M},\Delta^{n}_{m},q,u \bigr) = \Biggl\{ x = (x_{k}):\sup_{n}\frac{1}{n}\sum ^{n}_{k=1} \biggl[M_{k} \biggl( \frac{q (u_{k}\Delta ^{n}_{m}x_{k} )}{\rho} \biggr) \biggr] < \infty, \mbox{for some } \rho > 0 \Biggr\} . $$
$$\begin{aligned}& \begin{aligned}[b] w_{0} \bigl(\mathcal{M},\Delta^{n}_{m},p,q \bigr) ={}& \Biggl\{ x = (x_{k}):\frac {1}{n}\sum ^{n}_{k=1} \biggl[M_{k} \biggl( \frac{q (\Delta^{n}_{m}x_{k} )}{\rho } \biggr) \biggr]^{p_{k}} \rightarrow0 \mbox{ as } n \rightarrow \infty, \\ &{}\mbox{for some } \rho> 0 \Biggr\} , \end{aligned} \\& \begin{aligned}[b] w \bigl(\mathcal{M},\Delta^{n}_{m},p,q \bigr) ={}& \Biggl\{ x = (x_{k}):\frac{1}{n}\sum^{n}_{k=1} \biggl[M_{k} \biggl(\frac{q (\Delta^{n}_{m}x_{k} - L )}{\rho} \biggr) \biggr]^{p_{k}} \rightarrow0 \mbox{ as } n \rightarrow\infty,\\ &{}\mbox{for some } L \in X, \rho> 0 \Biggr\} , \end{aligned} \end{aligned}$$
$$w_{\infty}\bigl(\mathcal{M},\Delta^{n}_{m},p,q \bigr) = \Biggl\{ x = (x_{k}):\sup_{n}\frac{1}{n}\sum ^{n}_{k=1} \biggl[M_{k} \biggl( \frac{q (\Delta^{n}_{m}x_{k} )}{\rho} \biggr) \biggr]^{p_{k}} < \infty, \mbox{for some } \rho> 0 \Biggr\} . $$
$$ |a_{k} + b_{k}|^{p_{k}} \leq D \bigl\{ |a_{k}|^{p_{k}} + |b_{k}|^{p_{k}}\bigr\} $$
(1.1)
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 ℂ.
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
and
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
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. □
$$\sup_{n} \frac{1}{n}\sum^{n}_{k=1} \biggl[M_{k} \biggl(\frac{q (u_{k} \Delta ^{n}_{m} x_{k} )}{\rho_{1}} \biggr) \biggr]^{p_{k}} < \infty $$
$$\sup_{n}\frac{1}{n}\sum^{n}_{k=1} \biggl[M_{k} \biggl(\frac{q (u_{k} \Delta^{n}_{m} y_{k} )}{\rho_{2}} \biggr) \biggr]^{p_{k}} < \infty. $$
$$\begin{aligned} &\sup_{n}\frac{1}{n}\sum^{n}_{k=1} \biggl[M_{k} \biggl(\frac {q(\alpha u_{k}\Delta^{n}_{m} x_{k} + \beta u_{k} \Delta^{n}_{m} y_{k}) }{\rho_{3}} \biggr) \biggr]^{p_{k}} \\ &\quad\leq \sup_{n}\frac{1}{n}\sum ^{n}_{k=1} \biggl[M_{k} \biggl( \frac{q(\alpha u_{k} \Delta^{n}_{m} x_{k} )}{\rho_{3}} + \frac{q(\beta u_{k} \Delta^{n}_{m} y_{k} )}{\rho _{3}} \biggr) \biggr]^{p_{k}} \\ &\quad \leq \sup_{n}\frac{1}{n}\sum ^{n}_{k=1} \frac{1}{2^{p_{k}}} \biggl[M_{k} \biggl(\frac{q(u_{k} \Delta^{n}_{m} x_{k} )}{\rho_{1}} \biggr) \biggr]^{p_{k}} + \sup _{n}\frac {1}{n}\sum^{n}_{k=1} \frac{1}{2^{p_{k}}} \biggl[M_{k} \biggl(\frac{q( u_{k} \Delta ^{n}_{m} y_{k} )}{\rho_{2}} \biggr) \biggr]^{p_{k}} \\ &\quad\leq D\sup_{n}\frac{1}{n}\sum ^{n}_{k=1} \biggl[M_{k} \biggl( \frac{q(u_{k} \Delta^{n}_{m} x_{k} )}{\rho_{1}} \biggr) \biggr]^{p_{k}} + D\sup_{n} \frac{1}{n}\sum^{n}_{k=1} \biggl[M_{k} \biggl(\frac{q(u_{k} \Delta^{n}_{m} y_{k} )}{\rho_{2}} \biggr) \biggr]^{p_{k}} \\ &\quad< \infty. \end{aligned}$$
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
where
\(H = \max(1, \sup_{k} p_{k})\).
$$g(x) = \inf \Biggl\{ \rho^{\frac{p_{k}}{H}}: \sup _{n} \Biggl(\frac{1}{n} \sum _{k = 1}^{n} \biggl[M_{k} \biggl( q \biggl( \frac{u_{k} \Delta^{n}_{m} x_{k}}{\rho} \biggr) \biggr) \biggr]^{p_{k}} \Biggr)^{\frac{1}{H}} \leq1, \rho> 0 \Biggr\} , $$
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
and
Let \(\rho= \rho_{1} + \rho_{2}\), then by Minkowski’s inequality, we have
and thus
$$\frac{1}{n} \sum_{k = 1}^{n} \biggl[M_{k} \biggl(q \biggl(\frac{u_{k} \Delta^{n}_{m} x_{k}}{\rho_{1}} \biggr) \biggr) \biggr]^{p_{k}} \leq1 $$
$$\frac{1}{n} \sum_{k = 1}^{n} \biggl[M_{k} \biggl(q \biggl(\frac{u_{k} \Delta^{n}_{m} y_{k}}{\rho_{2}} \biggr) \biggr) \biggr]^{p_{k}} \leq1 . $$
$$\begin{aligned} \frac{1}{n} \sum_{k = 1}^{n} \biggl[M_{k} \biggl(q \biggl(\frac{u_{k} \Delta^{n}_{m} x_{k}}{\rho} \biggr) \biggr) \biggr]^{p_{k}} \leq& \biggl(\frac{\rho _{1}}{\rho_{1} + \rho_{2}} \biggr) \frac{1}{n} \sum_{k = 1}^{n} \biggl[M_{k} \biggl( q \biggl(\frac{u_{k} \Delta^{n}_{m} x_{k}}{\rho_{1}} \biggr) \biggr) \biggr]^{p_{k}} \\ &{} + \biggl(\frac{\rho_{2}}{\rho_{1} + \rho_{2}} \biggr) \frac{1}{n} \sum _{k = 1}^{n} \biggl[M_{k} \biggl(q \biggl( \frac{u_{k} \Delta^{n}_{m} y_{k}}{\rho_{2}} \biggr) \biggr) \biggr]^{p_{k}} \end{aligned}$$
$$\begin{aligned} &g(x + y) \\ &\quad = \inf \Biggl\{ (\rho_{1} + \rho_{2})^{\frac{p_{k}}{H}}: \sup_{n} \Biggl(\frac{1}{n} \sum _{k = 1}^{n} \biggl[M_{k} \biggl(q \biggl( \frac{u_{k} \Delta^{n}_{m} x_{k} + u_{k} \Delta^{n}_{m} y_{k}}{\rho} \biggr) \biggr) \biggr]^{p_{k}} \Biggr)^{\frac{1}{H}} \leq1, \rho> 0 \Biggr\} \\ &\quad \leq g(x) + g(y). \end{aligned}$$
(iv) Finally we prove that scalar multiplication is continuous. Let λ be any complex number by definition
where \(r = \frac{\rho}{|\lambda|}\). Hence, \(w_{\infty}(\mathcal {M},\Delta^{n}_{m},p,q,u )\) is a paranormed space. □
$$\begin{aligned} g(\lambda x) = & \inf \Biggl\{ (\rho)^{\frac{p_{k}}{H}}: \sup_{n} \Biggl(\frac {1}{n} \sum _{k = 1}^{n} \biggl[M_{k} \biggl(q \biggl( \frac{u_{k} \Delta^{n}_{m} \lambda x_{k}}{\rho} \biggr) \biggr) \biggr] ^{p_{k}} \Biggr)^{\frac{1}{H}} \leq1, \rho> 0 \Biggr\} \\ = & \inf \Biggl\{ \bigl(|\lambda| r\bigr)^{\frac {p_{k}}{H}}: \sup _{n} \Biggl(\frac{1}{n} \sum _{k = 1}^{n} \biggl[M_{k} \biggl(q \biggl( \frac{u_{k} \Delta^{n}_{m} x_{k}}{r} \biggr) \biggr) \biggr]^{p_{k}} \Biggr)^{\frac{1}{H}} \leq1, \rho> 0 \Biggr\} , \end{aligned}$$
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
This implies that
for sufficiently large k. Hence we get
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 \(Z = w_{0}, w_{\infty}\). □
$$\frac{1}{n} \sum_{k = 1}^{n} \biggl[M_{k} \biggl(q \biggl(\frac{u_{k} \Delta^{n}_{m} x_{k} - L}{\rho} \biggr) \biggr) \biggr]^{p_{k}} \rightarrow0 \quad\mbox{as } n \rightarrow\infty. $$
$$\frac{1}{n} \sum_{k = 1}^{n} \biggl[M_{k} \biggl(q \biggl(\frac{u_{k} \Delta^{n}_{m} x_{k} - L}{\rho} \biggr) \biggr) \biggr]^{p_{k}} < \epsilon \quad (0 < \epsilon< 1) $$
$$\begin{aligned} \frac{1}{n} \sum_{k = 1}^{n} \biggl[M_{k} \biggl(q \biggl(\frac {u_{k} \Delta^{n}_{m} x_{k} - L}{\rho} \biggr) \biggr) \biggr]^{r_{k}} \leq& \frac{1}{n} \sum_{k = 1}^{n} \biggl[M_{k} \biggl(q \biggl(\frac{u_{k} \Delta^{n}_{m} x_{k} - L}{\rho} \biggr) \biggr) \biggr]^{p_{k}} \\ \rightarrow& 0 \quad \mbox{as } n \rightarrow\infty. \end{aligned}$$
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:
(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
\(Z = w_{0}, w,\textit{and }w_{\infty}\).
(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
Suppose
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
We obtain
Similarly we can prove the other cases.
$$\frac{1}{n} \sum_{k = 1}^{n} \biggl[M_{k}' \biggl(q \biggl(\frac {u_{k} \Delta^{n}_{m} x_{k} }{\rho} \biggr) \biggr) \biggr]^{p_{k}} \rightarrow0 \quad\mbox{as } n \rightarrow\infty. $$
$$y_{k} = M_{k}' \biggl(q \biggl( \frac{u_{k} \Delta^{n}_{m} x_{k} }{\rho} \biggr) \biggr) $$
$$M_{k}''(y_{k}) < \frac{J y_{k}}{2\delta} M_{k}'' (2) + \frac{J y_{k}}{2\delta} M_{k}'' (2) = \frac{J y_{k}}{\delta} M_{k}'' (2). $$
$$\begin{aligned} \frac{1}{n} \sum_{k = 1}^{n} \biggl[M_{k}'' \circ M_{k}' \biggl(q \biggl(\frac{u_{k} \Delta^{n}_{m} x_{k} }{\rho} \biggr) \biggr) \biggr]^{p_{k}} = & \frac {1}{n} \sum_{k = 1}^{n} \biggl[M_{k}'' \biggl\{ M_{k}' \biggl(q \biggl(\frac {u_{k} \Delta^{n}_{m} x_{k} }{\rho} \biggr) \biggr)\biggr\} \biggr]^{p_{k}} \\ = & \frac{1}{n} \sum_{k = 1}^{n} \bigl[M_{k}''(y_{k}) \bigr]^{p_{k}} \\ \rightarrow& 0 \quad\mbox{as } n \rightarrow\infty. \end{aligned}$$
(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
and
Let \(\rho= \max\{\rho_{1}, \rho_{2}\}\). The remaining proof follows from the inequality
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. □
$$\frac{1}{n} \sum_{k = 1}^{n} \biggl[M_{k}' \biggl(q \biggl(\frac {u_{k} \Delta^{n}_{m} x_{k} }{\rho_{1}} \biggr) \biggr) \biggr]^{p_{k}} \rightarrow0 , \quad\mbox{as } n \rightarrow\infty. $$
$$\frac{1}{n} \sum_{k = 1}^{n} \biggl[M_{k}'' \biggl(q \biggl( \frac {u_{k} \Delta^{n}_{m} x_{k} }{\rho_{2}} \biggr) \biggr) \biggr]^{p_{k}} \rightarrow0 , \quad \mbox{as } n \rightarrow\infty. $$
$$\begin{aligned}[b] \frac{1}{n} \sum_{k = 1}^{n} \biggl[ \bigl(M_{k}' + M_{k}'' \bigr) \biggl(q \biggl(\frac{u_{k} \Delta^{n}_{m} x_{k} }{\rho} \biggr) \biggr) \biggr]^{p_{k}} \leq{}& D \Biggl\{ \frac{1}{n} \sum_{k = 1}^{n} \biggl[M_{k}' \biggl(q \biggl(\frac {u_{k} \Delta^{n}_{m} x_{k} }{\rho_{1}} \biggr) \biggr) \biggr]^{p_{k}} \\ &{}+ \frac{1}{n} \sum_{k = 1}^{n} \biggl[M_{k}'' \biggl(q \biggl( \frac{u_{k} \Delta ^{n}_{m} x_{k} }{\rho_{2}} \biggr) \biggr) \biggr]^{p_{k}} \Biggr\} . \end{aligned} $$
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
and hence \(x = (x_{k}) \in w_{\infty}(\mathcal{M},\Delta^{n}_{m}, q, u )\).
$$\sup_{n} \Biggl\{ \frac{1}{n} \sum _{k = 1}^{n} \biggl[M_{k} \biggl(q \biggl( \frac{u_{k} \Delta^{n}_{m} x_{k} }{\rho} \biggr) \biggr) \biggr] \Biggr\} \leq\sup_{n} \Biggl\{ \frac{1}{n} \sum_{k = 1}^{n} \biggl[M_{k} \biggl(q \biggl(\frac{u_{k} \Delta^{n}_{m} x_{k} }{\rho} \biggr) \biggr) \biggr]^{p_{k}} \Biggr\} $$
(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
This implies that
Thus, \(x = (x_{k}) \in w_{\infty}(\mathcal{M},\Delta^{n}_{m}, p, q, u )\) and this completes the proof. □
$$\sup_{n} \Biggl\{ \frac{1}{n} \sum _{k = 1}^{n} \biggl[M_{k} \biggl(q \biggl( \frac{u_{k} \Delta^{n}_{m} x_{k} }{\rho} \biggr) \biggr) \biggr] \Biggr\} \leq \epsilon< 1. $$
$$\sup_{n} \Biggl\{ \frac{1}{n} \sum _{k = 1}^{n} \biggl[M_{k} \biggl(q \biggl( \frac{u_{k} \Delta^{n}_{m} x_{k} }{\rho} \biggr) \biggr) \biggr]^{p_{k}} \Biggr\} \leq \sup _{n} \Biggl\{ \frac{1}{n} \sum _{k = 1}^{n} \biggl[M_{k} \biggl(q \biggl( \frac{u_{k} \Delta^{n}_{m} x_{k} }{\rho} \biggr) \biggr) \biggr] \Biggr\} . $$
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.
Let \((\alpha_{k})\) be a sequence of scalars such that \(|\alpha_{k}| \leq1\) for all \(k \in\mathbb{N}\). Thus we have
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. □
$$\sup_{n} \frac{1}{n} \sum_{k = 1}^{n} \biggl[M_{k} \biggl(q \biggl(\frac{u_{k} \Delta^{n}_{m} x_{k}}{\rho} \biggr) \biggr) \biggr]^{p_{k}} < \infty. $$
$$\sup_{n} \frac{1}{n} \sum_{k = 1}^{n} \biggl[M_{k} \biggl(q \biggl(\frac{\alpha_{k} u_{k} \Delta^{n}_{m} x_{k}}{\rho} \biggr) \biggr) \biggr]^{p_{k}} \leq \sup_{n} \frac{1}{n} \sum _{k = 1}^{n} \biggl[M_{k} \biggl(q \biggl(\frac{u_{k} \Delta^{n}_{m} x_{k}}{\rho} \biggr) \biggr) \biggr]^{p_{k}} < \infty. $$
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:
and
$$\begin{aligned}& \begin{aligned}[b] w_{0} \bigl(F,\Delta^{n}_{m},p,q,u \bigr) ={}& \Biggl\{ x = (x_{k}):\frac{1}{n}\sum^{n}_{k=1} \biggl[f_{k} \biggl(\frac{q (u_{k}\Delta^{n}_{m}x_{k} )}{\rho} \biggr) \biggr]^{p_{k}} \rightarrow0 \mbox{ as } n \rightarrow\infty,\\ &{}\mbox{for some } \rho> 0 \Biggr\} , \end{aligned} \\& \begin{aligned}[b] w \bigl(F,\Delta^{n}_{m},p,q,u \bigr) ={}& \Biggl\{ x = (x_{k}):\frac{1}{n}\sum^{n}_{k=1} \biggl[f_{k} \biggl(\frac{q (u_{k}\Delta^{n}_{m}x_{k} - L )}{\rho } \biggr) \biggr]^{p_{k}} \rightarrow0 \mbox{ as } n \rightarrow \infty, \\ &{}\mbox{for some } \rho> 0 \mbox{ and } L \in X \Biggr\} , \end{aligned} \end{aligned}$$
$$w_{\infty}\bigl(F,\Delta^{n}_{m},p,q,u \bigr) = \Biggl\{ x = (x_{k}):\sup_{n}\frac {1}{n}\sum ^{n}_{k=1} \biggl[f_{k} \biggl( \frac{q (u_{k}\Delta^{n}_{m}x_{k} )}{\rho} \biggr) \biggr]^{p_{k}} < \infty, \mbox{for some } \rho> 0 \Biggr\} . $$
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 ℂ.
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
where
\(H = \max(1, \sup_{k} p_{k})\).
$$ g(x) = \inf \Biggl\{ \rho^{\frac{p_{k}}{H}}: \sup_{n} \Biggl(\frac {1}{n} \sum _{k = 1}^{n} \biggl[f_{k} \biggl( q \biggl( \frac{u_{k} \Delta^{n}_{m} x_{k}}{\rho} \biggr) \biggr) \biggr]^{p_{k}} \Biggr)^{\frac{1}{H}} \leq1, \rho> 0 \Biggr\} , $$
(2.1)
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
and the inclusions are strict.
$$w_{0} \bigl(F, \Delta^{n}_{m}, p, q, u \bigr) \subset w \bigl(F, \Delta^{n}_{m}, p, q, u \bigr)\subset w_{\infty}\bigl(F, \Delta^{n}_{m}, p, q, u \bigr), $$
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
(i)
\(w_{Z} (F, \Delta^{n}_{m}, q, u ) \subset w_{Z} (F\circ G, \Delta ^{n}_{m}, q, u )\),
(ii)
\(w_{Z} (F, \Delta^{n}_{m}, p, q, u )\cap w_{Z} (F, \Delta ^{n}_{m}, p, r, u ) \subset w_{Z} (F, \Delta^{n}_{m}, p, q + r, u ) \),
(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
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
By the definition of F, we have the following relation for \(y_{k} > \delta\):
Hence,
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 \(w_{\infty}(F, \Delta^{n}_{m}, q, u ) \subset w_{\infty}(F\circ G, \Delta^{n}_{m}, q, u )\).
$$\sum^{n}_{k=1}\bigl[f_{k}(y_{k}) \bigr] = \sum_{1}\bigl[f_{k}(y_{k}) \bigr] + \sum_{2}\bigl[f_{k}(y_{k}) \bigr], $$
$$ \sum_{1} \bigl[f_{k}(y_{k}) \bigr] < n\epsilon. $$
(2.2)
$$f_{k}(y_{k})< 2 f_{k}(1)\frac{y_{k}}{\delta}. $$
$$ \frac{1}{n} \sum_{2} \bigl[f_{k}(y_{k})\bigr] \leq2\delta^{-1} f_{k}(1)\frac{1}{n}\sum^{n}_{k=1}y_{k}. $$
(2.3)
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
This gives \(g(x^{i}_{1} - x^{j}_{1}) < \frac{\epsilon}{s x_{0}}\) and
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
Thus, we have
This implies that
and thus
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
so that
Let \(i\geq n_{0}\) and taking the infimum of each ρ, we have
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. □
$$g\bigl(x^{i} - x^{j}\bigr) < \frac{\epsilon}{s x_{0}}, \quad \mbox{for all } i,j \geq n_{0}. $$
$$ \inf \biggl\{ \rho^{\frac{p_{k}}{H}}: \sup_{k\geq1} \biggl(f_{k} \biggl(\frac{q(u_{k} \Delta^{n}_{m} x^{i}_{k} - u_{k}\Delta^{n}_{m} x^{j}_{k})}{\rho } \biggr) \biggr)\leq1, \rho> 0 \biggr\} < \frac{\epsilon}{s x_{0}}. $$
(2.4)
$$g\bigl(x^{i}_{1} - x_{1}\bigr) < \frac{\epsilon}{s x_{0}}. $$
$$f_{k} \biggl(\frac{q(u_{k}\Delta^{n}_{m} x^{i}_{k} - u_{k}\Delta^{n}_{m} x^{j}_{k})}{g(x^{i} - x^{j})} \biggr) \leq1. $$
$$f_{k} \biggl(\frac{q(u_{k}\Delta^{n}_{m} x^{i}_{k} - u_{k}\Delta^{n}_{m} x^{j}_{k})}{g(x^{i} - x^{j})} \biggr) \leq f_{k}\biggl( \frac{s x_{0}}{2}\biggr) $$
$$q \bigl(u_{k}\Delta^{n}_{m} x^{i}_{k} - u_{k}\Delta^{n}_{m} x^{j}_{k} \bigr) < \frac{s x_{0}}{2}\cdot\frac{\epsilon}{s x_{0}} < \frac{\epsilon}{2}, $$
$$ \lim_{ j\rightarrow\infty}\sup_{k\geq1} f_{k} \biggl( \frac{q(u_{k}\Delta^{n}_{m} x^{i}_{k} - u_{k}\Delta^{n}_{m} x^{j}_{k})}{\rho} \biggr) \leq1, $$
$$\sup_{k\geq1} f_{k} \biggl(\frac{q(u_{k}\Delta^{n}_{m} x^{i}_{k} - u_{k}\Delta^{n}_{m} x_{k})}{\rho} \biggr) \leq1. $$
$$g\bigl(x^{i} - x\bigr) < \epsilon. $$
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.
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Authors’ contributions
Both authors contributed equally during the development of manuscript and the authors read and approved the final manuscript.