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 $x=(x_n)$ 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 ${\sigma }^k(n)=\sigma ({\sigma }^{k-1}(n))$, $k=1,2,\dots$ . A continuous linear functional φ on ${\ell }_{\mathrm{\infty }}$ is said to be an invariant mean or a σ-mean if and only if

• (i) $\phi (x)\ge 0$ when the sequence $x=(x_n)$ has $x_n\ge 0$ for all n,

• (ii) $\phi (e)=1$, where $e=(1,1,1,\dots )$ and

• (iii) $\phi (\{x_{\sigma (n)}\})=\phi (\{x_n\})$ 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 $\hat{f}$ of almost convergent sequences [2].

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

$$V_{\sigma }=\{x=(x_n):\underset{k}{lim}{t}_{kn}(x)=Le\text{ uniformly in }n,L=\sigma -limx\},$$

(1.1)

where

$$t_{kn}(x)=\frac{1}{k+1}\sum _{j=0}^k{T}^j{x}_n.$$

The special case of (1.1), in which $\sigma (n)=n+1$, 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

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

By a lacunary sequence $\theta ={(k_r)}_{r=0,1,2,\dots }^{\mathrm{\infty }}$, 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 $I_r=(k_{r-1},k_r]$, 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

$${\phi }_{r,n}(x)=\frac{1}{h_r+1}\sum _{j=k_{r-1}+1}{x}_{j+n}-\frac{1}{h_r}\sum _{j=k_{r-1}+1}^{k_r}{x}_{j+n},r>1.$$

Straightforward calculation shows that

$${\phi }_{r,n}(x)=\frac{1}{h_r(h_r+1)}\sum _{u=1}^{h_r}u(x_{{}_{k_{r-1}}+u+1+n}-x_{{}_{k_{r-1}}+u+n})$$

and

$${\phi }_{r-1,n}(x)=\frac{1}{h_r(h_r-1)}\sum _{u=1}^{h_r-1}(x_{{}_{k_{r-1}}+u+1+n}-x_{{}_{k_{r-1}}+u+n}).$$

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

$${\phi }_{r,n}(x+y)={\phi }_{r,n}(x)+{\phi }_{r,n}(y)\text{and}{\phi }_{r,n}(\lambda x)=\lambda {\phi }_{r,n}(x).$$

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

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

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

$${\ell }_M=\{x\in w:\sum _{k=1}^{\mathrm{\infty }}M\left(\frac{|x_k|}{\rho }\right)<\mathrm{\infty }\text{ for some }\rho >0\}.$$

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

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

and this space is called an Orlicz sequence space. For $M(t)=t^p$, $1\le p<\mathrm{\infty }$, 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 $M_1(\alpha x)\le M_2(x)\le M_1(\beta x)$ 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 $({\alpha }_k{x}_k)\in E$ whenever $(x_k)\in E$ for all sequences $({\alpha }_k)$ 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 $(x_n)$ is a sequence such that $q_1(x_n)\to 0$, then also $q_2(x_n)\to 0$. 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 $q_2(x)\le T{q}_1(x)$ for all $x\in X$ (see, for instance, Wilansky [19]).

Let $p=(p_r)$ be a sequence of strictly positive real numbers, X be a seminormed space over the field ℂ 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 $q(\frac{{\phi }_{rn}(x)}{\rho })=\frac{q({\phi }_{rn}(x))}{\rho }$ for any seminorm q and any $\rho >0$.

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

For $M(x)=x$ we get

for $p_k=1$, for all $r\in \mathbb{N}$, we get

for $s=0$ we get

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

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

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

$$B{V}_{\theta }(q)=\{x=(x_k)\in X:\sum _{r=1}^{\mathrm{\infty }}q({\phi }_{rn}(x))<\mathrm{\infty },\text{ uniformly in }n\}.$$

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

$$|a_r+b_r{|}^{p_r}\le D\{|a_r{|}^{p_r}+|b_r{|}^{p_r}\},$$

(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 ℂ of complex numbers.

Proof Omitted. □

Theorem 2.2 For any Orlicz function M and a bounded sequence $p=(p_r)$ of strictly positive real numbers, is a paranormed space (not necessarily totally paranormed), paranormed by

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

where $H=max(1,sup{p}_r)$.

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

Since $q(\overline{\theta })=0$ and $M(0)=0$, we get $inf\{{\rho }^{p_r/H}\}=0$ 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,

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

Then

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

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

Hence

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

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

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

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

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

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

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

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

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

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

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

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

Thus

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

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

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

• (i) ,

• (ii) If $s_1\le s_2$ then ,

• (iii) ,

• (iv) If $q_1$ is stronger than $q_2$, then .

Proof Omitted □

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

• (i) If $q_1\cong$(equivalent to) $q_2$, then ,

• (ii) ,

• (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

$$\sum _{r=1}^{\mathrm{\infty }}{\left[M\left(q\left(\frac{{\phi }_{rn}(x)}{\rho }\right)\right)\right]}^{m_k}<\mathrm{\infty }\text{uniformly in }n.$$

This implies that $M(q(\frac{{\phi }_{rn}(x)}{\rho }))\le 1$ 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

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

for all $k\ge k_0$, and therefore

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

Hence we have

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

so . This completes the proof. □

The following result is a consequence of the above result.

Corollary 2.6

• (i) If $0<p_r\le 1$ for each r, then ,

• (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.,

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

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

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

 □

Corollary 2.9 The sequence space is monotone.