Theory and Modern Applications

# On some new sequence spaces defined by infinite matrix and modulus

## Abstract

The goal of this paper is to introduce and study some properties of some sequence spaces that are defined using the φ-function and the generalized three parametric real matrix A. Also, we define A-statistical convergence.

MSC:40H05, 40C05.

## 1 Introduction and background

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:

There is a strong connection between ${N}_{\theta }$ and the space w of strongly Cesàro summable sequences which is defined by

In the special case where $\theta =\left({2}^{r}\right)$, we have ${N}_{\theta }=w$.

More results on lacunary strong convergence can be seen from [411].

Ruckle [12] 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 $x\ge 0$.

Maddox [13] introduced and examined some properties of the sequence spaces ${w}_{0}\left(f\right)$, $w\left(f\right)$ and ${w}_{\mathrm{\infty }}\left(f\right)$ defined using a modulus f, which generalized the well-known spaces ${w}_{0}$, w and ${w}_{\mathrm{\infty }}$ of strongly summable sequences.

Recently Savaş [14] generalized the concept of strong almost convergence by using a modulus f and examined some properties of the corresponding new sequence spaces. Waszak [15] defined the lacunary strong $\left(A,\phi \right)$-convergence with respect to a modulus function.

Following Ruckle, 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.

Since $|f\left(x\right)-f\left(y\right)|\le f\left(|x-y|\right)$, it follows from condition (iv) that f is continuous on $\left[0,\mathrm{\infty }\right)$.

By a φ-function we understood 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 no 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 $\psi \prec \phi$.

φ-functions φ and ψ are called equivalent and we write $\phi \sim \psi$ 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).

A φ-function φ is said to satisfy $\left({\mathrm{\Delta }}_{2}\right)$-condition (for all large u) if there exists a constant $K>1$ such that $\phi \left(2u\right)\le K\phi \left(u\right)$.

In the present paper, we introduce and study some properties of the following sequence space that is defined using the φ-function and the generalized three parametric real matrix.

## 2 Main results

Let φ and f be a given φ-function and a modulus function, respectively. Moreover, let $\mathbf{A}=\left({a}_{nk}\left(i\right)\right)$ be the generalized three parametric real matrix, and let a lacunary sequence θ be given. Then we define

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 $\mathbf{A}=I$ and $\phi \left(x\right)=x$ respectively, then we have [16]

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

then we have

then we have

We are now ready to write the following theorem.

Theorem 2.1 Let $\mathbf{A}=\left({a}_{nk}\left(i\right)\right)$ be the generalized three parametric real matrix, and let the φ-function $\phi \left(u\right)$ satisfy the condition $\left({\mathrm{\Delta }}_{2}\right)$. Then the following conditions are true.

1. (a)

If $x=\left({x}_{k}\right)\in w\left(\mathbf{A},\phi ,f\right)$ and α is an arbitrary number, then $\alpha x\in w\left(\mathbf{A},\phi ,f\right)$.

2. (b)

If $x,y\in w\left(\mathbf{A},\phi ,f\right)$, where $x=\left({x}_{k}\right)$, $y=\left({y}_{k}\right)$ and α, β are given numbers, then $\alpha x+\beta y\in w\left(\mathbf{A},\phi ,f\right)$.

The proof is a routine verification by using standard techniques and hence is omitted.

Theorem 2.2 Let f be any modulus function, and let the generalized three parametric real matrix A 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 $w\left(A,\phi ,f\right)\subseteq {N}_{\theta }^{0}\left(\mathbf{A},\phi ,f\right)$.

2. (b)

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

3. (c)

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

Proof (a) Let us suppose that $x\in w\left(A,\phi ,f\right)$. 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,

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

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

1. (b)

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

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

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,

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

where

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

It is easy to see that

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

Moreover, we have, for all i,

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

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

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

We now prove the following theorem.

Theorem 2.3 Let f be a modulus function. Then ${N}_{\theta }^{0}\left(A,\phi \right)\subset {N}_{\theta }^{0}\left(A,\phi ,f\right)$.

Proof Let $x\in {N}_{\theta }^{0}\left(A,\phi \right)$. Let $\epsilon >0$ be given and choose $0<\delta <1$ such that $f\left(x\right)<\epsilon$ for every $x\in \left[0,\delta \right]$. We can write

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

where ${S}_{1}=\frac{1}{{h}_{r}}{\sum }_{n\in {I}_{r}}f\left(|{\sum }_{k=1}^{\mathrm{\infty }}{a}_{nk}\left(i\right)\phi \left(|{x}_{k}|\right)|\right)$, and this sum is taken over

$|\sum _{k=1}^{\mathrm{\infty }}{a}_{nk}\left(i\right)\phi \left(|{x}_{k}|\right)|\le \delta$

and

${S}_{2}=\frac{1}{{h}_{r}}\sum _{n\in {I}_{r}}f\left(|\sum _{k=1}^{\mathrm{\infty }}{a}_{nk}\left(i\right)\phi \left(|{x}_{k}|\right)|\right),$

and this sum is taken over

$|\sum _{k=1}^{\mathrm{\infty }}{a}_{nk}\left(i\right)\phi \left(|{x}_{k}|\right)|>\delta .$

By the definition of the modulus f, we have ${S}_{1}=\frac{1}{{h}_{r}}{\sum }_{n\in {I}_{r}}f\left(\delta \right)=f\left(\delta \right)<\epsilon$ and further

${S}_{2}=f\left(1\right)\frac{1}{\delta }\frac{1}{{h}_{r}}\sum _{n\in {I}_{r}}\sum _{k=1}^{\mathrm{\infty }}{a}_{nk}\left(i\right)\phi \left(|{x}_{k}|\right).$

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

This completes the proof. □

## 3 A-Statistical convergence

The idea of convergence of a real sequence was extended to statistical convergence by Fast [17] (see also Schoenberg [18]) as follows: If denotes the set of natural numbers and $K\subset \mathbb{N}$, then $K\left(m,n\right)$ denotes the cardinality of the set $K\cap \left[m,n\right]$. The upper and lower natural density of the subset K is defined by

$\overline{d}\left(K\right)=\underset{n\to \mathrm{\infty }}{lim}sup\frac{K\left(1,n\right)}{n}\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}\underline{d}\left(K\right)=\underset{n\to \mathrm{\infty }}{lim}inf\frac{K\left(1,n\right)}{n}.$

If $\overline{d}\left(K\right)=\underline{d}\left(K\right)$, then we say that the natural density of K exists and it is denoted simply by $d\left(K\right)$. Clearly, $d\left(K\right)={lim}_{n\to \mathrm{\infty }}\frac{K\left(1,n\right)}{n}$.

A sequence $\left({x}_{k}\right)$ of real numbers is said to be statistically convergent to L if for arbitrary $\epsilon >0$, the set $K\left(\epsilon \right)=\left\{k\in \mathbb{N}:|{x}_{k}-L|\ge \epsilon \right\}$ has natural density zero. Statistical convergence turned out to be one of the most active areas of research in summability theory after the work of Fridy [19] and Šalát [20].

In another direction, a new type of convergence, called lacunary statistical convergence, was introduced in [21] as follows.

A sequence ${\left({x}_{k}\right)}_{n\in \mathbb{N}}$ of real numbers is said to be lacunary statistically convergent to L (or ${S}_{\theta }$-convergent to L) if for any $\epsilon >0$,

$\underset{r\to \mathrm{\infty }}{lim}\frac{1}{{h}_{r}}|\left\{k\in {I}_{r}:|{x}_{k}-L|\ge \epsilon \right\}|=0,$

where $|A|$ denotes the cardinality of $A\subset \mathbb{N}$. In [21] the relation between lacunary statistical convergence and statistical convergence was established among other things. Moreover, Kolk [22] defined A-statistical convergence by using non-negative regular summability matrix.

In this section we define $\left(A,\phi \right)$-statistical convergence by using the generalized three parametric real matrix and the φ-function $\phi \left(u\right)$.

Let θ be a lacunary sequence, and let $\mathbf{A}=\left({a}_{nk}\left(i\right)\right)$ be the generalized three parametric real matrix; let 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,

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

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

where $\mu \left({K}_{\theta }^{r}\left(\left(A,\phi \right),\epsilon \right)\right)$ denotes the number of elements belonging to ${K}_{\theta }^{r}\left(\left(\mathbf{A},\phi \right),\epsilon \right)$. 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. We write

Theorem 3.1 If $\psi \prec \phi$, then ${S}_{\theta }^{0}\left(A,\psi \right)\subset {S}_{\theta }^{0}\left(A,\phi \right)$.

Proof By assumption we have $\psi \left(|{x}_{k}|\right)\le b\phi \left(c|{x}_{k}|\right)$ and we have, for all i,

$\sum _{k=1}^{\mathrm{\infty }}{a}_{nk}\left(i\right)\psi \left(|{x}_{k}|\right)\le b\sum _{k=1}^{\mathrm{\infty }}{a}_{nk}\left(i\right)\phi \left(c|{x}_{k}|\right)\le L\sum _{k=1}^{\mathrm{\infty }}{a}_{nk}\left(i\right)\phi \left(|{x}_{k}|\right)$

for $b,c>0$, where the constant L is connected with the properties of φ. Thus, the condition ${\sum }_{k=1}^{\mathrm{\infty }}{a}_{nk}\left(i\right)\phi \left(|{x}_{k}|\right)\ge 0$ implies the condition ${\sum }_{k=1}^{\mathrm{\infty }}{a}_{nk}\left(i\right)\phi \left(|{x}_{k}|\right)\ge \epsilon$, and finally we get

$\mu \left({K}_{\theta }^{r}\left(\left(A,\phi \right),\epsilon \right)\right)\subset \mu \left({K}_{\theta }^{r}\left(\left(A,\psi \right),\epsilon \right)\right)$

and

$\underset{r}{lim}\frac{1}{{h}_{r}}\mu \left({K}_{\theta }^{r}\left(\left(A,\phi \right),\epsilon \right)\right)\le \underset{r}{lim}\frac{1}{{h}_{r}}\mu \left({K}_{\theta }^{r}\left(\left(A,\psi \right),\epsilon \right)\right).$

This completes the proof. □

We finally prove the following theorem.

Theorem 3.2 (a) If the matrix A, the sequence θ and functions f and φ are given, then

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

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

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

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

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

Proof (a) Let f be a modulus function, and let ε be a positive number. We write the following inequalities:

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

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

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 L 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

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

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

The proof of (c) follows from (a) and (b).

This completes the proof. □

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## Acknowledgements

This paper was presented during the ‘International Conference on the Theory, Methods and Applications of Nonlinear Equations’ held on the campus of Texas A&M University-Kingsville, Kingsville, TX 78363, USA on December 17-21, 2012, and submitted for conference proceedings.

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Correspondence to Ekrem Savaş.

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Savaş, E. On some new sequence spaces defined by infinite matrix and modulus. Adv Differ Equ 2013, 274 (2013). https://doi.org/10.1186/1687-1847-2013-274