# On generalized sequence spaces via modulus function

## Abstract

In this paper, we introduce and study the concept of lacunary strongly $\left(\mathbf{A},\phi \right)$-convergence with respect to a modulus function and lacunary $\left(\mathbf{A},\phi \right)$-statistical convergence and examine some properties of these sequence spaces. We establish some connections between lacunary strongly $\left(\mathbf{A},\phi \right)$-convergence and lacunary $\left(\mathbf{A},\phi \right)$-statistical convergence.

MSC:40H05, 40C05.

## 1 Introduction

Let s denote the set of all real and complex sequences $x=\left({x}_{k}\right)$. By ${l}_{\mathrm{\infty }}$ and c, we denote the Banach spaces of bounded and convergent sequences $x=\left({x}_{k}\right)$ normed by $\parallel x\parallel ={sup}_{n}|{x}_{n}|$, respectively. A linear functional L on ${l}_{\mathrm{\infty }}$ is said to be a Banach limit [1] if it has the following properties:

1. (1)

$L\left(x\right)\ge 0$ if $n\ge 0$ (i.e. ${x}_{n}\ge 0$ for all n),

2. (2)

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

3. (3)

$L\left(Dx\right)=L\left(x\right)$, where the shift operator D is defined by $D\left({x}_{n}\right)=\left\{{x}_{n+1}\right\}$.

Let B be the set of all Banach limits on ${l}_{\mathrm{\infty }}$. A sequence $x\in {\ell }_{\mathrm{\infty }}$ is said to be almost convergent if all Banach limits of x coincide. Let $\stackrel{ˆ}{c}$ denote the space of almost convergent sequences. Lorentz [2] has shown that

where

${t}_{m,n}\left(x\right)=\frac{{x}_{n}+{x}_{n+1}+{x}_{n+2}+\cdots +{x}_{n+m}}{m+1}.$

By a lacunary $\theta =\left({k}_{r}\right)$; $r=0,1,2,\dots$ , 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 ${h}_{r}={k}_{r}-{k}_{r-1}$. The ratio $\frac{{k}_{r}}{{k}_{r-1}}$ will be denoted by ${q}_{r}$. The space of lacunary strongly convergent sequences ${N}_{\theta }$ was defined by Freedman et al. [3] as follows:

In the special case where $\theta =\left({2}^{r}\right)$ (see [3]) we have ${N}_{\theta }=w$, which is defined by

Das and Mishra [4] have introduced the space ${\mathit{AC}}_{\theta }$ of lacunary almost convergent sequences and the space $|{\mathit{AC}}_{\theta }|$ of lacunary strongly almost convergent sequences as follows:

and

Ruckle used the idea of a modulus function f to construct a class of FK spaces,

$L\left(f\right)=\left\{x=\left({x}_{k}\right):\sum _{k=1}^{\mathrm{\infty }}f\left(|{x}_{k}|\right)<\mathrm{\infty }\right\}.$

The space $L\left(f\right)$ is closely related to the space ${l}_{1}$, which is an $L\left(f\right)$ space with $f\left(x\right)=x$ for all real $x\ge 0$.

In 1999, Savaş [5] generalized the concept of strong almost convergence by using a modulus f and $p=\left({p}_{k}\right)$ is a sequence of strictly positive real numbers as follows:

and

More investigations in this direction and more applications of the modulus can be found in [612].

Following Ruckle [13], a modulus function f is a function from $\left[0,\mathrm{\infty }\right)$ to $\left[0,\mathrm{\infty }\right)$ such that

1. (i)

$f\left(x\right)=0$ if and only if $x=0$,

2. (ii)

$f\left(x+y\right)\le f\left(x\right)+f\left(x\right)$ for all $x,y\ge 0$,

3. (iii)

f increasing,

4. (iv)

f is continuous from the right at zero.

By a φ-function we understand a continuous non-decreasing function $\phi \left(u\right)$ defined for $u\ge 0$ and such that $\phi \left(0\right)=0$, $\phi \left(u\right)>0$, for $u>0$ and $\phi \left(u\right)\to \mathrm{\infty }$ as $u\to \mathrm{\infty }$.

A φ-function φ is called non-weaker than a φ-function ψ if there are constants $c,b,k,l>0$ such that $c\psi \left(lu\right)\le b\phi \left(ku\right)$ (for all large u) and we write $\psi \prec \phi$.

A φ-function φ and ψ are called equivalent if there are positive constants ${b}_{1}$, ${b}_{2}$, c, ${k}_{1}$, ${k}_{2}$, l such that ${b}_{1}\phi \left({k}_{1}u\right)\le c\psi \left(lu\right)\le {b}_{2}\phi \left({k}_{2}u\right)$ (for all large u) and we write $\phi \sim \psi$.

A φ-function φ is said to satisfy the condition $\left({\mathrm{\Delta }}_{2}\right)$ (for all large u) if for some constant $k>1$ there is satisfied the inequality $\phi \left(2u\right)\le k\phi \left(u\right)$ (see [12, 14]).

In this paper, we introduce and study some properties of the following sequence space which is generalization of Savaş [14].

## 2 Main results

Let φ and f be a given φ-function and modulus function, respectively, and let $p=\left({p}_{n}\right)$ be a sequence of positive real numbers. Moreover, let $\mathbf{A}=\left({\mathbf{A}}_{i}\right)$ be the generalized three parametric real matrix with ${A}_{i}=\left({a}_{n,k}\left(i\right)\right)$ and a lacunary sequence θ be given. Then we define the following sequence spaces:

If $x\in {N}_{\theta }^{0}\left(\mathbf{A},\phi ,f\right)$, the sequence x is said to be lacunary strong $\left(\mathbf{A},\phi \right)$-convergent to zero with respect to a modulus f. When $\phi \left(x\right)=x$ for all x, we obtain

If we take $f\left(x\right)=x$, we write

If we take ${p}_{k}=p$, for all k, we have

If we take $\mathbf{A}=I$ and $\phi \left(x\right)=x$, respectively, then we have

${N}_{\theta }^{0}=\left\{x=\left({x}_{k}\right):\underset{r}{lim}\frac{1}{{h}_{r}}\sum _{k\in {I}_{r}}f{\left(|{x}_{k}|\right)}^{{p}_{n}}=0\right\}.$

If we define the matrix $A=\left({a}_{nk}\left(i\right)\right)$ as follows:

then we have

then we have

If $x\in {N}_{\theta }^{0}\left(\stackrel{ˆ}{c},\phi ,f\right)$, the sequence x is said to be almost lacunary strong φ-convergent to zero with respect to a modulus f. In the next theorem we establish inclusion relations between $w\left(A,\phi ,f,p\right)$ and ${N}_{\theta }^{0}\left(\mathbf{A},\phi ,f,p\right)$. We now have the following.

Theorem 2.1 Let f be any modulus function and let there be a φ-function φ and a generalized three parametric real matrix A; let $p=\left({p}_{n}\right)$ be a sequence of positive real numbers and the sequence θ be given. If

then the following relations are true:

1. (a)

If ${lim inf}_{r}{q}_{r}>1$ then we have $w\left(A,\phi ,f,p\right)\subseteq {N}_{\theta }^{0}\left(\mathbf{A},\phi ,f,p\right)$.

2. (b)

If ${sup}_{r}{q}_{r}<\mathrm{\infty }$, then we have ${N}_{\theta }^{0}\left(\mathbf{A},\phi ,f,p\right)\subseteq w\left(A,\phi ,f,p\right)$.

3. (c)

$1<{lim inf}_{r}{q}_{r}\le {lim sup}_{r}{q}_{r}<\mathrm{\infty }$, then we have ${N}_{\theta }^{0}\left(\mathbf{A},\phi ,f,p\right)=w\left(A,\phi ,f,p\right)$.

Proof (a) Let us suppose that $x\in w\left(A,\phi ,f,p\right)$. There exists $\delta >0$ such that ${q}_{r}>1+\delta$ for all $r\ge 1$ and we have ${h}_{r}/{k}_{r}\ge \delta /\left(1+\delta \right)$ for sufficiently large r. Then, for all i,

$\begin{array}{c}\frac{1}{{k}_{r}}\sum _{n=1}^{{k}_{r}}f{\left(|\sum _{k=1}^{\mathrm{\infty }}{a}_{nk}\left(i\right)\phi \left(|{x}_{k}|\right)|\right)}^{{p}_{n}}\hfill \\ \phantom{\rule{1em}{0ex}}\ge \frac{1}{{k}_{r}}\sum _{n\in {I}_{r}}f{\left(|\sum _{k=1}^{\mathrm{\infty }}{a}_{nk}\left(i\right)\phi \left(|{x}_{k}|\right)|\right)}^{{p}_{n}}\hfill \\ \phantom{\rule{1em}{0ex}}=\frac{{h}_{r}}{{k}_{r}}\frac{1}{{h}_{r}}\sum _{n\in {I}_{r}}f{\left(|\sum _{k=1}^{\mathrm{\infty }}{a}_{nk}\left(i\right)\phi \left(|{x}_{k}|\right)|\right)}^{{p}_{n}}\hfill \\ \phantom{\rule{1em}{0ex}}\ge \frac{\delta }{1+\delta }\frac{1}{{h}_{r}}\sum _{n\in {I}_{r}}f{\left(|\sum _{k=1}^{\mathrm{\infty }}{a}_{nk}\phi \left(|{x}_{k}|\right)|\right)}^{{p}_{n}}.\hfill \end{array}$

Hence, $x\in {N}_{\theta }^{0}\left(\mathbf{A},\phi ,f,p\right)$.

(b) If ${lim sup}_{r}{q}_{r}<\mathrm{\infty }$ then there exists $M>0$ such that ${q}_{r} for all $r\ge 1$. Let $x\in {N}_{\theta }^{0}\left(\mathbf{A},\phi ,f,p\right)$ and ε is an arbitrary positive number, then there exists an index ${j}_{0}$ such that for every $j\ge {j}_{0}$ and all i,

${R}_{j}=\frac{1}{{h}_{j}}\sum _{n\in {I}_{r}}f{\left(|\sum _{k=1}^{\mathrm{\infty }}{a}_{nk}\left(i\right)\phi \left(|{x}_{k}|\right)|\right)}^{{p}_{n}}<\epsilon .$

Thus, we can also find $K>0$ such that ${R}_{j}\le K$ for all $j=1,2,\dots$ . Now let m be any integer with ${k}_{r-1}\le m\le {k}_{r}$, then we obtain, for all i,

$I=\frac{1}{m}\sum _{n=1}^{m}f{\left(|\sum _{k=1}^{\mathrm{\infty }}{a}_{nk}\left(i\right)\phi \left(|{x}_{k}|\right)|\right)}^{{p}_{n}}\le \frac{1}{{k}_{r-1}}\sum _{n=1}^{{k}_{r}}f{\left(|\sum _{k=1}^{\mathrm{\infty }}{a}_{nk}\left(i\right)\phi \left(|{x}_{k}|\right)|\right)}^{{p}_{n}}={I}_{1}+{I}_{2},$

where

$\begin{array}{c}{I}_{1}=\frac{1}{{k}_{r-1}}\sum _{j=1}^{{j}_{0}}\sum _{n\in {I}_{j}}f{\left(|\sum _{k=1}^{\mathrm{\infty }}{a}_{nk}\left(i\right)\phi \left(|{x}_{k}|\right)|\right)}^{{p}_{n}},\hfill \\ {I}_{2}=\frac{1}{{k}_{r-1}}\sum _{j={j}_{0}+1}^{m}\sum _{n\in {I}_{j}}f{\left(|\sum _{k=1}^{\mathrm{\infty }}{a}_{nk}\left(i\right)\phi \left(|{x}_{k}|\right)|\right)}^{{p}_{n}}.\hfill \end{array}$

It is easy to see that

$\begin{array}{rcl}{I}_{1}& =& \frac{1}{{k}_{r-1}}\sum _{j=1}^{{j}_{0}}\sum _{n\in {I}_{j}}f{\left(|\sum _{k=1}^{\mathrm{\infty }}{a}_{nk}\left(i\right)\phi \left(|{x}_{k}|\right)|\right)}^{{p}_{n}}\\ =& \frac{1}{{k}_{r-1}}\left(\sum _{n\in {I}_{1}}f{\left(|\sum _{k=1}^{\mathrm{\infty }}{a}_{nk}\left(i\right)\phi \left(|{x}_{k}|\right)|\right)}^{{p}_{n}}+\cdots +\sum _{n\in {I}_{{j}_{0}}}f{\left(|\sum _{k=1}^{\mathrm{\infty }}{a}_{nk}\left(i\right)\phi \left(|{x}_{k}|\right)|\right)}^{{p}_{n}}\right)\\ \le & \frac{1}{{k}_{r-1}}\left({h}_{1}{R}_{1}+\cdots +{h}_{{j}_{0}}{R}_{{j}_{0}}\right)\\ \le & \frac{1}{{k}_{r-1}}{j}_{0}{k}_{{j}_{0}}\underset{1\le i\le {j}_{0}}{sup}{R}_{i}\\ \le & \frac{{j}_{0}{k}_{{j}_{0}}}{{k}_{r-1}}K.\end{array}$

Moreover, we have for all i

$\begin{array}{rcl}{I}_{2}& =& \frac{1}{{k}_{r-1}}\sum _{j={j}_{0}+1}^{m}\sum _{n\in {I}_{j}}f{\left(|\sum _{k=1}^{\mathrm{\infty }}{a}_{nk}\phi \left(|{x}_{k}|\right)|\right)}^{{p}_{n}}\\ =& \frac{1}{{k}_{r-1}}\sum _{j={j}_{0}+1}^{m}\frac{1}{{h}_{j}}\sum _{n\in {I}_{j}}f{\left(|\sum _{k=1}^{\mathrm{\infty }}{a}_{nk}\phi \left(|{x}_{k}|\right)|\right)}^{{p}_{n}}{h}_{j}\\ \le & \epsilon \frac{1}{{k}_{r-1}}\sum _{j={j}_{0}+1}^{m}{h}_{j}\\ \le & \epsilon \frac{{k}_{r}}{{k}_{r-1}}\\ =& \epsilon {q}_{r}<\epsilon \cdot M.\end{array}$

Thus $I\le \frac{{j}_{0}{k}_{{j}_{0}}}{{k}_{r-1}}K+\epsilon \cdot M$. Finally, $x\in w\left(A,\psi ,f,p\right)$.

The proof of (c) follows from (a) and (b). This completes the proof. □

Theorem 2.2 Let f, ${f}_{1}$, be modulus functions. Then we have

${N}_{\theta }^{0}\left(A,{f}_{1},\phi ,p\right)\subset \left(A,\phi ,fo{f}_{1},p\right).$

Proof This can be proved by using techniques similar to those used in the theorem of Savaş [14]. □

Recently Savaş [14] defined $\left(\mathbf{A},\phi \right)$-statistical convergence as follows.

Let θ be a lacunary sequence, and let $\mathbf{A}=\left({a}_{nk}\left(i\right)\right)$ be the generalized three parametric real matrix, the sequence $x=\left({x}_{k}\right)$, the φ-function $\phi \left(u\right)$ and a positive number $\epsilon >0$ be given. We write, for all i,

${K}_{\theta }^{r}\left(A,\phi ,\epsilon \right)=\left\{n\in {I}_{r}:\sum _{k=1}^{\mathrm{\infty }}{a}_{nk}\left(i\right)\phi \left(|{x}_{k}|\right)\ge \epsilon \right\}.$

The sequence x is said to be $\left(\mathbf{A},\phi \right)$-statistically convergent to a number zero if for every $\epsilon >0$

where $\mu \left({K}_{\theta }^{r}\left(A,\phi ,\epsilon \right)\right)$ denotes the number of elements belonging to ${K}_{\theta }^{r}\left(\mathbf{A},\phi ,\epsilon \right)$. We denote by ${S}_{\theta }^{0}\left(\mathbf{A},\phi \right)$, the set of sequences $x=\left({x}_{k}\right)$ which are lacunary $\left(\mathbf{A},\phi \right)$-statistical convergent to zero and we write

More investigations in this direction can be found in [1520].

We now establish inclusion relations between ${N}_{\theta }^{0}\left(\mathbf{A},\phi ,f,p\right)$ and ${S}_{\theta }^{0}\left(A,\phi \right)$.

In the following theorem we assume that $0.

Theorem 2.3 (a) If the matrix A and the sequence θ and functions f and φ are given, then

${N}_{\theta }^{0}\left(A,\phi ,f,p\right)\subset {S}_{\theta }^{0}\left(A,\phi \right).$

(b) If the φ-function $\phi \left(u\right)$ and the matrix A are given, and if the modulus function f is bounded, then

${S}_{\theta }^{0}\left(A,\phi \right)\subset {N}_{\theta }^{0}\left(A,\phi ,f,p\right).$

Proof (a) Let f be a modulus function and let ε be a positive numbers. We write the following inequalities, for all i,

$\begin{array}{c}\frac{1}{{h}_{r}}\sum _{n\in {I}_{r}}f{\left(|\sum _{k=1}^{\mathrm{\infty }}{a}_{nk}\left(i\right)\phi \left(|{x}_{k}|\right)|\right)}^{{p}_{n}}\hfill \\ \phantom{\rule{1em}{0ex}}=\frac{1}{{h}_{r}}\sum _{n\in {I}_{r}^{1}}f{\left(|\sum _{k=1}^{\mathrm{\infty }}{a}_{nk}\left(i\right)\phi \left(|{x}_{k}|\right)|\right)}^{{p}_{n}}\hfill \\ \phantom{\rule{1em}{0ex}}\ge \frac{1}{{h}_{r}}\sum _{n\in {I}_{r}^{1}}{\left[f\left(\epsilon \right)\right]}^{{p}_{n}}\hfill \\ \phantom{\rule{1em}{0ex}}\ge \frac{1}{{h}_{r}}\sum _{n\in {I}_{r}^{1}}min\left({\left[f\left(\epsilon \right)\right]}^{inf{p}_{n}},{\left[f\left(\epsilon \right)\right]}^{H}\right)\hfill \\ \phantom{\rule{1em}{0ex}}\ge \frac{1}{{h}_{r}}\mu \left({K}_{\theta }^{r}\left(A,\phi ,\epsilon \right)\right)min\left({\left[f\left(\epsilon \right)\right]}^{inf{p}_{n}},{\left[f\left(\epsilon \right)\right]}^{H}\right),\hfill \end{array}$

where

${I}_{r}^{1}=\left\{n\in {I}_{r}:\sum _{k=1}^{\mathrm{\infty }}{a}_{nk}\left(i\right)\phi \left(|{x}_{k}|\right)\ge \epsilon \right\}.$

Finally, if $x\in {N}_{\theta }^{0}\left(A,\phi ,f,p\right)$ then $x\in {S}_{\theta }^{0}\left(A,\phi ,f\right)$.

1. (b)

Let us suppose that $x\in {S}_{\theta }^{0}\left(A,\phi \right)$. If the modulus function f is a bounded function, then there exists an integer K such that $f\left(x\right) for $x\ge 0$. Let us take

${I}_{r}^{2}=\left\{n\in {I}_{r}:\sum _{k=1}^{\mathrm{\infty }}{a}_{nk}\left(i\right)\phi \left(|{x}_{k}|\right)<\epsilon \right\}.$

Thus we have, for all i,

$\begin{array}{c}\frac{1}{{h}_{r}}\sum _{n\in {I}_{r}}f{\left(|\sum _{k=1}^{\mathrm{\infty }}{a}_{nk}\left(i\right)\phi \left(|{x}_{k}|\right)|\right)}^{{p}_{n}}\hfill \\ \phantom{\rule{1em}{0ex}}\le \frac{1}{{h}_{r}}\sum _{n\in {I}_{r}^{1}}f{\left(|\sum _{k=1}^{\mathrm{\infty }}{a}_{nk}\left(i\right)\phi \left(|{x}_{k}|\right)|\right)}^{{p}_{n}}\hfill \\ \phantom{\rule{2em}{0ex}}+\frac{1}{{h}_{r}}\sum _{n\in {I}_{r}^{2}}f{\left(|\sum _{k=1}^{\mathrm{\infty }}{a}_{nk}\left(i\right)\phi \left(|{x}_{k}|\right)|\right)}^{{p}_{n}}\hfill \\ \phantom{\rule{1em}{0ex}}\le \frac{1}{{h}_{r}}\sum _{n\in {I}_{r}^{1}}max\left({K}^{h},{K}^{H}\right)+\frac{1}{{h}_{r}}\sum _{n\in {I}_{r}^{2}}{\left[f\left(\epsilon \right)\right]}^{{p}_{n}}\hfill \\ \phantom{\rule{1em}{0ex}}\le max\left({K}^{h},{K}^{H}\right)\frac{1}{{h}_{r}}\mu \left({K}_{\theta }^{r}\left(A,\phi ,\epsilon \right)\right)+max\left({\left[f\left(\epsilon \right)\right]}^{h},{\left[f\left(\epsilon \right)\right]}^{H}\right).\hfill \end{array}$

Taking the limit as $\epsilon \to 0$, we observe that $x\in {N}_{\theta }^{0}\left(A,\phi ,f,p\right)$.

This completes the proof. □

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## Acknowledgements

I would like to express my gratitude to the referee of the paper for his useful comments and suggestions towards the quality improvement of the paper. This paper was presented during the ‘2nd International Eurasian Conference on Mathematical Sciences and Applications’ held on Sarajevo, Bosnia and Herzegovina on 26th to 29th August 2013, and it was submitted for the conference proceedings.

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Correspondence to Ekrem Savaş.

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Savaş, E. On generalized sequence spaces via modulus function. J Inequal Appl 2014, 101 (2014). https://doi.org/10.1186/1029-242X-2014-101