# Some properties of the sequence space

## Abstract

In this paper we define the sequence space on a seminormed complex linear space by using an Orlicz function. We give various properties and some inclusion relations on this space.

MSC:40A05, 40C05, 40D05.

## 1 Introduction

Let ${\ell }_{\mathrm{\infty }}$ and c denote the Banach spaces of real bounded and convergent sequences $x=\left({x}_{n}\right)$ normed by $\parallel x\parallel ={sup}_{n}|{x}_{n}|$, respectively.

Let σ be a one-to-one mapping of the set of positive integers into itself such that ${\sigma }^{k}\left(n\right)=\sigma \left({\sigma }^{k-1}\left(n\right)\right)$, $k=1,2,\dots$ . A continuous linear functional φ on ${\ell }_{\mathrm{\infty }}$ is said to be an invariant mean or a σ-mean if and only if

1. (i)

$\phi \left(x\right)\ge 0$ when the sequence $x=\left({x}_{n}\right)$ has ${x}_{n}\ge 0$ for all n,

2. (ii)

$\phi \left(e\right)=1$, where $e=\left(1,1,1,\dots \right)$ and

3. (iii)

$\phi \left(\left\{{x}_{\sigma \left(n\right)}\right\}\right)=\phi \left(\left\{{x}_{n}\right\}\right)$ for all $x\in {\ell }_{\mathrm{\infty }}$.

If σ is the translation mapping $n\to n+1$, a σ-mean is often called a Banach limit [1], and ${V}_{\sigma }$, the set of σ-convergent sequences, that is, the set of bounded sequences all of whose invariant means are equal, is the set $\stackrel{ˆ}{f}$ of almost convergent sequences [2].

If $x=\left({x}_{n}\right)$, set $Tx=\left(T{x}_{n}\right)=\left({x}_{\sigma \left(n\right)}\right)$. It can be shown (see Schaefer [3]) that

(1.1)

where

${t}_{kn}\left(x\right)=\frac{1}{k+1}\sum _{j=0}^{k}{T}^{j}{x}_{n}.$

The special case of (1.1), in which $\sigma \left(n\right)=n+1$, was given by Lorentz [2].

Subsequently invariant means were studied by Ahmad and Mursaleen [4], Mursaleen [5], Raimi [6] and many others.

We may remark here that the concept of almost bounded variation was introduced and investigated by Nanda and Nayak [7] as follows:

where

${t}_{mn}\left(x\right)=\frac{1}{m\left(m+1\right)}\sum _{v=1}^{m}v\left({x}_{n+v}-{x}_{n+v-1}\right).$

By a lacunary sequence $\theta ={\left({k}_{r}\right)}_{r=0,1,2,\dots }^{\mathrm{\infty }}$, where ${k}_{0}=0$, we shall mean an increasing sequence of non-negative integers with ${k}_{r}-{k}_{r-1}\to \mathrm{\infty }$ as $r\to \mathrm{\infty }$. The intervals determined by θ will be denoted by ${I}_{r}=\left({k}_{r-1},{k}_{r}\right]$, and we let ${h}_{r}={k}_{r}-{k}_{r-1}$. The ratio $\frac{{k}_{r}}{{k}_{r-1}}$ will usually be denoted by ${q}_{r}$ (see [8]).

Karakaya and Savaş [9] defined the sequence spaces and as follows:

where

${\phi }_{r,n}\left(x\right)=\frac{1}{{h}_{r}+1}\sum _{j={k}_{r-1}+1}{x}_{j+n}-\frac{1}{{h}_{r}}\sum _{j={k}_{r-1}+1}^{{k}_{r}}{x}_{j+n},\phantom{\rule{1em}{0ex}}r>1.$

Straightforward calculation shows that

${\phi }_{r,n}\left(x\right)=\frac{1}{{h}_{r}\left({h}_{r}+1\right)}\sum _{u=1}^{{h}_{r}}u\left({x}_{{}_{{k}_{r-1}}+u+1+n}-{x}_{{}_{{k}_{r-1}}+u+n}\right)$

and

${\phi }_{r-1,n}\left(x\right)=\frac{1}{{h}_{r}\left({h}_{r}-1\right)}\sum _{u=1}^{{h}_{r}-1}\left({x}_{{}_{{k}_{r-1}}+u+1+n}-{x}_{{}_{{k}_{r-1}}+u+n}\right).$

Note that for any sequences x, y and scalar λ, we have

${\phi }_{r,n}\left(x+y\right)={\phi }_{r,n}\left(x\right)+{\phi }_{r,n}\left(y\right)\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}{\phi }_{r,n}\left(\lambda x\right)=\lambda {\phi }_{r,n}\left(x\right).$

An Orlicz function is a function $M:\left[0,\mathrm{\infty }\right)\to \left[0,\mathrm{\infty }\right)$, which is continuous, nondecreasing and convex with $M\left(0\right)=0$, $M\left(x\right)>0$ for $x>0$ and $M\left(x\right)\to \mathrm{\infty }$ as $x\to \mathrm{\infty }$. (For details, see Krasnoselskii and Rutickii [10].)

It is well known that if M is a convex function and $M\left(0\right)=0$, then $M\left(\lambda x\right)\le \lambda M\left(x\right)$ for all λ with $0<\lambda <1$.

Lindenstrauss and Tzafriri [11] used the idea of Orlicz function to construct the sequence space

The space ${\ell }_{M}$ is a Banach space with the norm

$\parallel x\parallel =inf\left\{\rho >0:\sum _{k=1}^{\mathrm{\infty }}M\left(\frac{|{x}_{k}|}{\rho }\right)\le 1\right\},$

and this space is called an Orlicz sequence space. For $M\left(t\right)={t}^{p}$, $1\le p<\mathrm{\infty }$, the space ${\ell }_{M}$ coincides with the classical sequence space ${\ell }_{p}$.

Definition 1.1 Any two Orlicz functions ${M}_{1}$ and ${M}_{2}$ are said to be equivalent if there are positive constants α and β, and ${x}_{0}$ such that ${M}_{1}\left(\alpha x\right)\le {M}_{2}\left(x\right)\le {M}_{1}\left(\beta x\right)$ for all x with $0\le x\le {x}_{0}$ (see Kamthan and Gupta [12]).

Later on, different types of sequence spaces were introduced by using an Orlicz function by Mursaleen et al. [13], Choudhary and Parashar [14], Tripathy and Mahanta [15] , Altinok et al. [16], Bhardwaj and Singh [17], Et et al. [18] and many others.

A sequence space E is said to be solid (or normal) if $\left({\alpha }_{k}{x}_{k}\right)\in E$ whenever $\left({x}_{k}\right)\in E$ for all sequences $\left({\alpha }_{k}\right)$ of scalars with $|{\alpha }_{k}|\le 1$.

It is well known that a sequence space E is normal implies that E is monotone.

Definition 1.2 Let ${q}_{1}$, ${q}_{2}$ be seminorms on a vector space X. Then ${q}_{1}$ is said to be stronger than ${q}_{2}$ if whenever $\left({x}_{n}\right)$ is a sequence such that ${q}_{1}\left({x}_{n}\right)\to 0$, then also ${q}_{2}\left({x}_{n}\right)\to 0$. If each is stronger than the others, ${q}_{1}$ and ${q}_{2}$ are said to be equivalent (one may refer to Wilansky [19]).

Lemma 1.3 Let ${q}_{1}$ and ${q}_{2}$ be seminorms on a linear space X. Then ${q}_{1}$ is stronger than ${q}_{2}$ if and only if there exists a constant T such that ${q}_{2}\left(x\right)\le T{q}_{1}\left(x\right)$ for all $x\in X$ (see, for instance, Wilansky [19]).

Let $p=\left({p}_{r}\right)$ be a sequence of strictly positive real numbers, X be a seminormed space over the field of complex numbers with the seminorm q, M be an Orlicz function and $s\ge 0$ be a fixed real number. Then we define the sequence space as follows:

It is clear that $q\left(\frac{{\phi }_{rn}\left(x\right)}{\rho }\right)=\frac{q\left({\phi }_{rn}\left(x\right)\right)}{\rho }$ for any seminorm q and any $\rho >0$.

We get the following sequence spaces from $B{V}_{\theta }\left(M,p,q,s\right)$ by choosing some of the special p, M and s:

For $M\left(x\right)=x$ we get

for ${p}_{k}=1$, for all $r\in \mathbb{N}$, we get

for $s=0$ we get

for $M\left(x\right)=x$ and $s=0$ we get

for ${p}_{r}=1$, for all $r\in \mathbb{N}$, and $s=0$ we get

for $M\left(x\right)=x$, ${p}_{r}=1$, for all $r\in \mathbb{N}$, and $s=0$ we have

The following inequalities will be used throughout the paper. Let $p=\left({p}_{r}\right)$ be a bounded sequence of strictly positive real numbers with $0<{p}_{r}\le sup{p}_{r}=H$, $D=max\left(1,{2}^{H-1}\right)$, then

$|{a}_{r}+{b}_{r}{|}^{{p}_{r}}\le D\left\{|{a}_{r}{|}^{{p}_{r}}+|{b}_{r}{|}^{{p}_{r}}\right\},$
(1.2)

where ${a}_{r},{b}_{r}\in \mathbb{C}$.

## 2 Main results

In this section we prove the general results of this paper on the sequence space , those characterize the structure of this space.

Theorem 2.1 The sequence space is a linear space over the field of complex numbers.

Proof Omitted. □

Theorem 2.2 For any Orlicz function M and a bounded sequence $p=\left({p}_{r}\right)$ of strictly positive real numbers, is a paranormed space (not necessarily totally paranormed), paranormed by

$\begin{array}{rcl}g\left(x\right)& =& inf\left\{{\rho }^{{p}_{r}/H}:{\left(\sum _{r=1}^{\mathrm{\infty }}{r}^{-s}{\left[M\left(q\left(\frac{{\phi }_{rn}\left(x\right)}{\rho }\right)\right)\right]}^{{p}_{k}}\right)}^{\frac{1}{H}}\le 1,\\ r=1,2,3,\dots ,n=1,2,3,\dots \right\},\end{array}$

where $H=max\left(1,sup{p}_{r}\right)$.

Proof Clearly $g\left(x\right)=g\left(-x\right)$. By using Theorem 2.1 and then using Minkowski’s inequality, we get $g\left(x+y\right)\le g\left(x\right)+g\left(y\right)$.

Since $q\left(\overline{\theta }\right)=0$ and $M\left(0\right)=0$, we get $inf\left\{{\rho }^{{p}_{r}/H}\right\}=0$ for $x=\mathrm{\Theta }$, where $\overline{\mathrm{\Theta }}$ is the zero sequence of X.

Finally, we prove that scalar multiplication is continuous. Let λ be any numbers. By definition,

$\begin{array}{rcl}g\left(\lambda x\right)& =& inf\left\{{\rho }^{{p}_{r}/H}:{\left(\sum _{r}{r}^{-s}{\left[M\left(q\left(\frac{\lambda {\phi }_{rn}\left(x\right)}{\rho }\right)\right)\right]}^{{p}_{r}}\right)}^{\frac{1}{H}}\le 1,\\ r=1,2,3,\dots ,n=1,2,3,\dots \right\}.\end{array}$

Then

$\begin{array}{rcl}g\left(\lambda x\right)& =& inf\left\{{\left(\lambda r\right)}^{{p}_{r}/H}:{\left(\sum _{r=1}^{\mathrm{\infty }}{r}^{-s}{\left[M\left(q\left(\frac{{\phi }_{rn}\left(x\right)}{r}\right)\right)\right]}^{{p}_{r}}\right)}^{\frac{1}{H}}\le 1,\\ r=1,2,3,\dots ,n=1,2,3,\dots \right\},\end{array}$

where $r=\frac{\rho }{|\lambda |}$. Since $|\lambda {|}^{{p}_{r}}\le max\left(1,|\lambda {|}^{H}\right)$, it follows that $|\lambda {|}^{{p}_{r}/H}\le {\left(max\left(1,|\lambda {|}^{H}\right)\right)}^{\frac{1}{H}}$.

Hence

$\begin{array}{rcl}g\left(\lambda x\right)& =& {\left(max\left(1,|\lambda {|}^{H}\right)\right)}^{\frac{1}{H}}inf\left\{{r}^{{p}_{r}/H}:{\left(\sum _{r=1}^{\mathrm{\infty }}{r}^{-s}{\left[M\left(q\left(\frac{{\phi }_{rn}\left(x\right)}{r}\right)\right)\right]}^{{p}_{r}}\right)}^{\frac{1}{H}}\le 1,\\ r=1,2,3,\dots ,n=1,2,3,\dots \right\},\end{array}$

which converges to zero as $g\left(x\right)$ converges to zero in . Now suppose that ${\lambda }_{n}\to 0$ and x is in $B{V}_{\sigma }\left(M,p,q,s\right)$. For arbitrary $\epsilon >0$, let N be a positive integer such that

$\sum _{r=N+1}^{\mathrm{\infty }}{r}^{-s}{\left[M\left(q\left(\frac{{\phi }_{rn}\left(x\right)}{\rho }\right)\right)\right]}^{{p}_{r}}<\frac{\epsilon }{2}$

for some $\rho >0$, all n. This implies that

${\left(\sum _{r=N+1}^{\mathrm{\infty }}{r}^{-s}{\left[M\left(q\left(\frac{{\phi }_{rn}\left(x\right)}{\rho }\right)\right)\right]}^{{p}_{r}}\right)}^{\frac{1}{H}}\le \frac{\epsilon }{2}$

for some $\rho >0$, $r>N$ and all n.

Let $0<|\lambda |<1$, using convexity of M and all n, we get

$\sum _{r=N+1}^{\mathrm{\infty }}{r}^{-s}{\left[M\left(q\left(\frac{\lambda {\phi }_{rn}\left(x\right)}{\rho }\right)\right)\right]}^{{p}_{r}}<\sum _{r=N+1}^{\mathrm{\infty }}{r}^{-s}{\left[|\lambda |M\left(q\left(\frac{{\phi }_{rn}\left(x\right)}{\rho }\right)\right)\right]}^{{p}_{r}}<{\left(\frac{\epsilon }{2}\right)}^{H}.$

Since M is continuous everywhere in $\left[0,\mathrm{\infty }\right)$, then

$f\left(t\right)=\sum _{r=1}^{N}{r}^{-s}\left[M\left(q\left(\frac{t{\phi }_{rn}\left(x\right)}{\rho }\right)\right)\right]$

is continuous at 0. So there is $1>\delta >0$ such that $|f\left(t\right)|<\frac{\epsilon }{2}$ for $0. Let K be such that $|{\lambda }_{i}|<\delta$ for $i>K$, then for $i>K$, all n,

${\left(\sum _{r=1}^{N}{r}^{-s}{\left[M\left(q\left(\frac{{\lambda }_{i}{\phi }_{rn}\left(x\right)}{\rho }\right)\right)\right]}^{{p}_{r}}\right)}^{\frac{1}{H}}<\frac{\epsilon }{2}.$

Thus

${\left(\sum _{r=1}^{\mathrm{\infty }}{r}^{-s}{\left[M\left(q\left(\frac{{\lambda }_{i}{\phi }_{rn}\left(x\right)}{\rho }\right)\right)\right]}^{{p}_{r}}\right)}^{\frac{1}{H}}<\epsilon$

for $i>K$ and n, so that $g\left(\lambda x\right)\to 0$ ($\lambda \to 0$). □

Theorem 2.3 Let M, ${M}_{1}$, ${M}_{2}$ be Orlicz functions q, ${q}_{1}$, ${q}_{2}$ seminorms and $s,{s}_{1},{s}_{2}\ge 0$. Then

1. (i)

,

2. (ii)

If ${s}_{1}\le {s}_{2}$ then ,

3. (iii)

,

4. (iv)

If ${q}_{1}$ is stronger than ${q}_{2}$, then .

Proof Omitted □

Corollary 2.4 Let M be an Orlicz function, then we have

1. (i)

If ${q}_{1}\cong$(equivalent to) ${q}_{2}$, then ,

2. (ii)

,

3. (iii)

.

Theorem 2.5 Suppose that $0<{m}_{k}\le {t}_{k}<\mathrm{\infty }$ for each $k\in \mathbb{N}$. Then .

Proof Let . Then there exists some $\rho >0$ such that

This implies that $M\left(q\left(\frac{{\phi }_{rn}\left(x\right)}{\rho }\right)\right)\le 1$ for sufficiently large values of k, say $k\ge {k}_{0}$ for some fixed ${k}_{0}\in \mathbb{N}$. Since ${m}_{k}\le {t}_{k}$, for each $k\in \mathbb{N}$ we get

${\left[M\left(q\left(\frac{{\phi }_{rn}\left(x\right)}{\rho }\right)\right)\right]}^{{t}_{k}}\le {\left[M\left(q\left(\frac{{\phi }_{rn}\left(x\right)}{\rho }\right)\right)\right]}^{{m}_{k}}$

for all $k\ge {k}_{0}$, and therefore

$\sum _{r=1}^{\mathrm{\infty }}{\left[M\left(q\left(\frac{{\phi }_{rn}\left(x\right)}{\rho }\right)\right)\right]}^{{t}_{r}}\le \sum _{r=1}^{\mathrm{\infty }}{\left[M\left(q\left(\frac{{\phi }_{m}\left(x\right)}{\rho }\right)\right)\right]}^{{m}_{k}}.$

Hence we have

$\sum _{r=1}^{\mathrm{\infty }}{\left[M\left(q\left(\frac{{\phi }_{rn}\left(x\right)}{\rho }\right)\right)\right]}^{{t}_{r}}<\mathrm{\infty },$

so . This completes the proof. □

The following result is a consequence of the above result.

Corollary 2.6

1. (i)

If $0<{p}_{r}\le 1$ for each r, then ,

2. (ii)

If ${p}_{r}\ge 1$ for all r, then .

Theorem 2.7 Let ${M}_{1}$ and ${M}_{2}$ be any two of Orlicz functions. If ${M}_{1}$ and ${M}_{2}$ are equivalent, then .

Proof Proof follows from Definition 1.1. □

Theorem 2.8 The sequence space is solid.

Proof Let , i.e.,

$\sum _{r=1}^{\mathrm{\infty }}{r}^{-s}{\left[M\left(q\left(\frac{{\phi }_{rn}\left(x\right)}{\rho }\right)\right)\right]}^{{p}_{r}}<\mathrm{\infty }.$

Let $\left({\alpha }_{r}\right)$ be sequence of scalars such that $|{\alpha }_{r}|\le 1$ for all $r\in \mathbb{N}$. Then the result follows from the following inequality:

$\sum _{r=1}^{\mathrm{\infty }}{r}^{-s}{\left[M\left(q\left(\frac{{\alpha }_{r}{\phi }_{rn}\left(x\right)}{\rho }\right)\right)\right]}^{{p}_{r}}\le \sum _{r=1}^{\mathrm{\infty }}{r}^{-s}{\left[M\left(q\left(\frac{{\phi }_{rn}\left(x\right)}{\rho }\right)\right)\right]}^{{p}_{r}}.$

□

Corollary 2.9 The sequence space is monotone.

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Correspondence to Yavuz Altin.

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The authors declare that they have no competing interest.

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MI, YA and ME have contributed to all parts of the article. All authors read and approved the final manuscript.

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Işik, M., Altin, Y. & Et, M. Some properties of the sequence space . J Inequal Appl 2013, 305 (2013). https://doi.org/10.1186/1029-242X-2013-305