On generalized sequence spaces via modulus function

Abstract

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

MSC:40H05, 40C05.

1 Introduction

Let s denote the set of all real and complex sequences $x=(x_k)$. By $l_{\mathrm{\infty }}$ and c, we denote the Banach spaces of bounded and convergent sequences $x=(x_k)$ 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) $L(x)\ge 0$ if $n\ge 0$ (i.e. $x_n\ge 0$ for all n),

• (2) $L(e)=1$, where $e=(1,1,\dots )$,

• (3) $L(Dx)=L(x)$, where the shift operator D is defined by $D(x_n)=\{x_{n+1}\}$.

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 $\hat{c}$ denote the space of almost convergent sequences. Lorentz [2] has shown that

$$\hat{c}=\{x\in l_{\mathrm{\infty }}:\underset{m}{lim}{t}_{m,n}(x)\text{ exists uniformly in }n\},$$

where

$$t_{m,n}(x)=\frac{x_n+x_{n+1}+x_{n+2}+\cdots +x_{n+m}}{m+1}.$$

By a lacunary $\theta =(k_r)$; $r=0,1,2,\dots$ , 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 $I_r=(k_{r-1},k_r]$ 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:

$$N_{\theta }=\{x=(x_k):\underset{r}{lim}\frac{1}{h_r}\sum _{k\in I_r}|x_k-l|=0,\text{for some }l\}.$$

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

$$w=\{x=(x_k):\underset{n}{lim}\frac{1}{n}\sum _{k=0}^n|x_k-l|=0,\text{for some }l\}.$$

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:

$${\mathit{AC}}_{\theta }=\{x=(x_k):\underset{r}{lim}\frac{1}{h_r}\sum _{k\in I_r}(x_{k+n}-L)=0,\text{for some }L\text{ uniformly in }n\}$$

and

$$|{\mathit{AC}}_{\theta }|=\{x=(x_k):\underset{r}{lim}\frac{1}{h_r}\sum _{k\in I_r}|x_{k+n}-L|=0,\text{for some }L\text{ uniformly in }n\}.$$

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

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

The space $L(f)$ is closely related to the space $l_1$, which is an $L(f)$ space with $f(x)=x$ for all real $x\ge 0$.

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

$$[\hat{c}(f,p)]=\{x:\underset{n}{lim}\frac{1}{n}\sum _{k=1}^{n}f{(|x_{k+m}-L|)}^{p_k}=0,\text{for some }L,\text{uniformly in }m\}$$

and

$${[\hat{c}(f,p)]}_0=\{x:\underset{n}{lim}\frac{1}{n}\sum _{k=1}^{n}f{(|x_{k+m}|)}^{p_k}=0,\text{uniformly in }m\}.$$

More investigations in this direction and more applications of the modulus can be found in [6–12].

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

• (i) $f(x)=0$ if and only if $x=0$,

• (ii) $f(x+y)\le f(x)+f(x)$ for all $x,y\ge 0$,

• (iii) f increasing,

• (iv) f is continuous from the right at zero.

By a φ-function we understand a continuous non-decreasing function $\phi (u)$ defined for $u\ge 0$ and such that $\phi (0)=0$, $\phi (u)>0$, for $u>0$ and $\phi (u)\to \mathrm{\infty }$ 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 $c\psi (lu)\le b\phi (ku)$ (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 $b_1\phi (k_{1}u)\le c\psi (lu)\le b_2\phi (k_{2}u)$ (for all large u) and we write $\phi \sim \psi$.

A φ-function φ is said to satisfy the condition $({\mathrm{\Delta }}_2)$ (for all large u) if for some constant $k>1$ there is satisfied the inequality $\phi (2u)\le k\phi (u)$ (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 $p=(p_n)$ be a sequence of positive real numbers. Moreover, let $\mathbf{A}=({\mathbf{A}}_i)$ be the generalized three parametric real matrix with $A_i=(a_{n,k}(i))$ and a lacunary sequence θ be given. Then we define the following sequence spaces:

$$N_{\theta }^0(\mathbf{A},\phi ,f,p)=\{x=(x_k):\underset{r}{lim}\frac{1}{h_r}\sum _{n\in I_r}f{\left(\right|\sum _{k=1}^{\mathrm{\infty }}{a}_{nk}(i)\phi (|x_k|)\left|\right)}^{p_n}=0,\text{uniformly in }i\}.$$

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

$$N_{\theta }^0(\mathbf{A},f,p)=\{x=(x_k):\underset{r}{lim}\frac{1}{h_r}\sum _{n\in I_r}f{\left(\right|\sum _{k=1}^{\mathrm{\infty }}{a}_{nk}(i)x_k\left|\right)}^{p_n}=0,\text{uniformly in }i\}.$$

If we take $f(x)=x$, we write

$$N_{\theta }^0(\mathbf{A},\phi ,p)=\{x=(x_k):\underset{r}{lim}\frac{1}{h_r}\sum _{n\in I_r}|\sum _{k=1}^{\mathrm{\infty }}{a}_{nk}(i)\phi (|x_k|){|}^{p_n}=0,\text{uniformly in }i\}.$$

If we take $p_k=p$, for all k, we have

$$N_{\theta }^0(\mathbf{A},\phi ,f)=\{x=(x_k):\underset{r}{lim}\frac{1}{h_r}\sum _{n\in I_r}f{\left(\right|\sum _{k=1}^{\mathrm{\infty }}{a}_{nk}(i)\phi (|x_k|)\left|\right)}^p=0,\text{uniformly in }i\}.$$

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

$$N_{\theta }^0=\{x=(x_k):\underset{r}{lim}\frac{1}{h_r}\sum _{k\in I_r}f{(|x_k|)}^{p_n}=0\}.$$

If we define the matrix $A=(a_{nk}(i))$ as follows:

$$a_{nk}(i):=\{\begin{array}{ll}\frac{1}{n},& \text{if }n\ge k,\\ 0,& \text{otherwise},\end{array}$$

then we have

$$\begin{array}{c}{N}_{\theta }^0(\mathbf{C},\phi ,f,p)=\{x=(x_k):\underset{r}{lim}\frac{1}{h_r}\sum _{n\in I_r}f{\left(\right|\frac{1}{n}\sum _{k=1}^n\phi (|x_k|)\left|\right)}^{p_n}=0,\text{uniformly in }i\},\hfill \\ a_{nk}(i):=\{\begin{array}{ll}\frac{1}{n},& \text{if }i\le k\le i+n-1,\\ 0,& \text{otherwise},\end{array}\hfill \end{array}$$

then we have

$$N_{\theta }^0(\hat{c},\phi ,f,p)=\{x=(x_k):\underset{r}{lim}\frac{1}{h_r}\sum _{n\in I_r}f{\left(\right|\frac{1}{n}\sum _{k=i}^{i+n}\phi (|x_k|)\left|\right)}^{p_n}=0,\text{uniformly in }i\}.$$

If $x\in N_{\theta }^0(\hat{c},\phi ,f)$, 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 $w(A,\phi ,f,p)$ and $N_{\theta }^0(\mathbf{A},\phi ,f,p)$. 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 $p=(p_n)$ be a sequence of positive real numbers and the sequence θ be given. If

$$w(\mathbf{A},\phi ,f,p)=\{x=(x_k):\underset{m}{lim}\frac{1}{m}\sum _{n=1}^{m}f{\left(\right|\sum _{k=1}^{\mathrm{\infty }}{a}_{nk}(i)\phi (|x_k|)\left|\right)}^{p_n}=0,\mathit{\text{uniformly in }}i\},$$

then the following relations are true:

• (a) If ${lim\hspace{0.17em}inf}_r{q}_r>1$ then we have $w(A,\phi ,f,p)\subseteq N_{\theta }^0(\mathbf{A},\phi ,f,p)$.

• (b) If ${sup}_r{q}_r<\mathrm{\infty }$, then we have $N_{\theta }^0(\mathbf{A},\phi ,f,p)\subseteq w(A,\phi ,f,p)$.

• (c) $1<{lim\hspace{0.17em}inf}_r{q}_r\le {lim\hspace{0.17em}sup}_r{q}_r<\mathrm{\infty }$, then we have $N_{\theta }^0(\mathbf{A},\phi ,f,p)=w(A,\phi ,f,p)$.

Proof (a) Let us suppose that $x\in w(A,\phi ,f,p)$. There exists $\delta >0$ such that $q_r>1+\delta$ for all $r\ge 1$ and we have $h_r/k_r\ge \delta /(1+\delta )$ for sufficiently large r. Then, for all i,

$$\begin{array}{c}\frac{1}{k_r}\sum _{n=1}^{k_r}f{\left(\right|\sum _{k=1}^{\mathrm{\infty }}{a}_{nk}(i)\phi (|x_k|)\left|\right)}^{p_n}\hfill \\ \ge \frac{1}{k_r}\sum _{n\in I_r}f{\left(\right|\sum _{k=1}^{\mathrm{\infty }}{a}_{nk}(i)\phi (|x_k|)\left|\right)}^{p_n}\hfill \\ =\frac{h_r}{k_r}\frac{1}{h_r}\sum _{n\in I_r}f{\left(\right|\sum _{k=1}^{\mathrm{\infty }}{a}_{nk}(i)\phi (|x_k|)\left|\right)}^{p_n}\hfill \\ \ge \frac{\delta }{1+\delta }\frac{1}{h_r}\sum _{n\in I_r}f{\left(\right|\sum _{k=1}^{\mathrm{\infty }}{a}_{nk}\phi (|x_k|)\left|\right)}^{p_n}.\hfill \end{array}$$

Hence, $x\in N_{\theta }^0(\mathbf{A},\phi ,f,p)$.

(b) If ${lim\hspace{0.17em}sup}_r{q}_r<\mathrm{\infty }$ then there exists $M>0$ such that $q_r<M$ for all $r\ge 1$. Let $x\in N_{\theta }^0(\mathbf{A},\phi ,f,p)$ 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,

$$R_j=\frac{1}{h_j}\sum _{n\in I_r}f{\left(\right|\sum _{k=1}^{\mathrm{\infty }}{a}_{nk}(i)\phi (|x_k|)\left|\right)}^{p_n}<\epsilon .$$

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,

$$I=\frac{1}{m}\sum _{n=1}^{m}f{\left(\right|\sum _{k=1}^{\mathrm{\infty }}{a}_{nk}(i)\phi (|x_k|)\left|\right)}^{p_n}\le \frac{1}{k_{r-1}}\sum _{n=1}^{k_r}f{\left(\right|\sum _{k=1}^{\mathrm{\infty }}{a}_{nk}(i)\phi (|x_k|)\left|\right)}^{p_n}=I_1+I_2,$$

where

$$\begin{array}{c}{I}_1=\frac{1}{k_{r-1}}\sum _{j=1}^{j_0}\sum _{n\in I_j}f{\left(\right|\sum _{k=1}^{\mathrm{\infty }}{a}_{nk}(i)\phi (|x_k|)\left|\right)}^{p_n},\hfill \\ I_2=\frac{1}{k_{r-1}}\sum _{j=j_0+1}^m\sum _{n\in I_j}f{\left(\right|\sum _{k=1}^{\mathrm{\infty }}{a}_{nk}(i)\phi (|x_k|)\left|\right)}^{p_n}.\hfill \end{array}$$

It is easy to see that

$$\begin{array}{rcl}{I}_1& =& \frac{1}{k_{r-1}}\sum _{j=1}^{j_0}\sum _{n\in I_j}f{\left(\right|\sum _{k=1}^{\mathrm{\infty }}{a}_{nk}(i)\phi (|x_k|)\left|\right)}^{p_n}\\ =& \frac{1}{k_{r-1}}(\sum _{n\in I_1}f{\left(\right|\sum _{k=1}^{\mathrm{\infty }}{a}_{nk}(i)\phi (|x_k|)\left|\right)}^{p_n}+\cdots +\sum _{n\in I_{j_0}}f{\left(\right|\sum _{k=1}^{\mathrm{\infty }}{a}_{nk}(i)\phi (|x_k|)\left|\right)}^{p_n})\\ \le & \frac{1}{k_{r-1}}(h_1{R}_1+\cdots +h_{j_0}{R}_{j_0})\\ \le & \frac{1}{k_{r-1}}{j}_0{k}_{j_0}\underset{1\le i\le j_0}{sup}{R}_i\\ \le & \frac{j_0{k}_{j_0}}{k_{r-1}}K.\end{array}$$

Moreover, we have for all i

$$\begin{array}{rcl}{I}_2& =& \frac{1}{k_{r-1}}\sum _{j=j_0+1}^m\sum _{n\in I_j}f{\left(\right|\sum _{k=1}^{\mathrm{\infty }}{a}_{nk}\phi (|x_k|)\left|\right)}^{p_n}\\ =& \frac{1}{k_{r-1}}\sum _{j=j_0+1}^m\frac{1}{h_j}\sum _{n\in I_j}f{\left(\right|\sum _{k=1}^{\mathrm{\infty }}{a}_{nk}\phi (|x_k|)\left|\right)}^{p_n}{h}_j\\ \le & \epsilon \frac{1}{k_{r-1}}\sum _{j=j_0+1}^m{h}_j\\ \le & \epsilon \frac{k_r}{k_{r-1}}\\ =& \epsilon q_r<\epsilon \cdot M.\end{array}$$

Thus $I\le \frac{j_0{k}_{j_0}}{k_{r-1}}K+\epsilon \cdot M$. Finally, $x\in w(A,\psi ,f,p)$.

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

$$N_{\theta }^0(A,f_1,\phi ,p)\subset (A,\phi ,fo{f}_1,p).$$

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

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

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

$$K_{\theta }^r(A,\phi ,\epsilon )=\{n\in I_r:\sum _{k=1}^{\mathrm{\infty }}{a}_{nk}(i)\phi (|x_k|)\ge \epsilon \}.$$

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

$$\underset{r}{lim}\frac{1}{h_r}\mu (K_{\theta }^r(A,\phi ,\epsilon ))=0,\text{uniformly in }i,$$

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

$$S_{\theta }^0(\mathbf{A},\phi )=\{x=(x_k):\underset{r}{lim}\frac{1}{h_r}\mu (K_{\theta }^r(A,\phi ,\epsilon ))=0,\text{uniformly in }i\}.$$

More investigations in this direction can be found in [15–20].

We now establish inclusion relations between $N_{\theta }^0(\mathbf{A},\phi ,f,p)$ and $S_{\theta }^0(A,\phi )$.

In the following theorem we assume that $0<h=inf{p}_n\le p_n\le sup{p}_k\le H\le \mathrm{\infty }$.

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

$$N_{\theta }^0(A,\phi ,f,p)\subset S_{\theta }^0(A,\phi ).$$

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

$$S_{\theta }^0(A,\phi )\subset N_{\theta }^0(A,\phi ,f,p).$$

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

$$\begin{array}{c}\frac{1}{h_r}\sum _{n\in I_r}f{\left(\right|\sum _{k=1}^{\mathrm{\infty }}{a}_{nk}(i)\phi (|x_k|)\left|\right)}^{p_n}\hfill \\ =\frac{1}{h_r}\sum _{n\in I_r^1}f{\left(\right|\sum _{k=1}^{\mathrm{\infty }}{a}_{nk}(i)\phi (|x_k|)\left|\right)}^{p_n}\hfill \\ \ge \frac{1}{h_r}\sum _{n\in I_r^1}{[f(\epsilon )]}^{p_n}\hfill \\ \ge \frac{1}{h_r}\sum _{n\in I_r^1}min({[f(\epsilon )]}^{inf{p}_n},{[f(\epsilon )]}^H)\hfill \\ \ge \frac{1}{h_r}\mu (K_{\theta }^r(A,\phi ,\epsilon ))min({[f(\epsilon )]}^{inf{p}_n},{[f(\epsilon )]}^H),\hfill \end{array}$$

where

$$I_r^1=\{n\in I_r:\sum _{k=1}^{\mathrm{\infty }}{a}_{nk}(i)\phi (|x_k|)\ge \epsilon \}.$$

Finally, if $x\in N_{\theta }^0(A,\phi ,f,p)$ then $x\in S_{\theta }^0(A,\phi ,f)$.

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

  $$I_r^2=\{n\in I_r:\sum _{k=1}^{\mathrm{\infty }}{a}_{nk}(i)\phi (|x_k|)<\epsilon \}.$$

Thus we have, for all i,

$$\begin{array}{c}\frac{1}{h_r}\sum _{n\in I_r}f{\left(\right|\sum _{k=1}^{\mathrm{\infty }}{a}_{nk}(i)\phi (|x_k|)\left|\right)}^{p_n}\hfill \\ \le \frac{1}{h_r}\sum _{n\in I_r^1}f{\left(\right|\sum _{k=1}^{\mathrm{\infty }}{a}_{nk}(i)\phi (|x_k|)\left|\right)}^{p_n}\hfill \\ +\frac{1}{h_r}\sum _{n\in I_r^2}f{\left(\right|\sum _{k=1}^{\mathrm{\infty }}{a}_{nk}(i)\phi (|x_k|)\left|\right)}^{p_n}\hfill \\ \le \frac{1}{h_r}\sum _{n\in I_r^1}max(K^h,K^H)+\frac{1}{h_r}\sum _{n\in I_r^2}{[f(\epsilon )]}^{p_n}\hfill \\ \le max(K^h,K^H)\frac{1}{h_r}\mu (K_{\theta }^r(A,\phi ,\epsilon ))+max({[f(\epsilon )]}^h,{[f(\epsilon )]}^H).\hfill \end{array}$$

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

This completes the proof. □