On generalized difference lacunary statistical convergence in a paranormed space

Abstract

In this article, we introduce the concept of ${\mathrm{\Delta }}^m$-lacunary statistical convergence and ${\mathrm{\Delta }}^m$-lacunary strong convergence in a paranormed space. Also, we establish some connections between these concepts.

Dedication

Dedicated to Professor Hari M Srivastava.

1 Introduction

In order to extend convergence of sequences, the notion of statistical convergence was introduced by Fast [1] and Steinhaus [2] and several generalizations and applications of this concept have been investigated by various authors [3, 4]. This notion was studied in normed spaces by Kolk [5], in locally convex Hausdorff topological spaces by Maddox [6], in topological Hausdorff groups by Çakallı [7] and in probabilistic normed space by Karakuş [8]. Recently, Alotaibi and Alroqi [9] extended this notion in paranormed spaces.

In this article, we study the concept of statistical convergence from difference sequence spaces which are defined over paranormed space.

2 Preliminaries and definitions

Let K be a subset of the set of natural numbers ℕ. Then the asymptotic density of K denoted by $\delta (K)={lim}_n\frac{1}{n}|\{k\le n:k\in K\}|$, where the vertical bars denote the cardinality of the enclosed set in [10].

A number sequence $x=(x_k)$ is said to be statistically convergent to the number L if for each $\epsilon >0$, the set $K(\epsilon )=\{k\le n:|x_k-L|\ge \epsilon \}$ has asymptotic density zero, i.e.,

$$\underset{n}{lim}\frac{1}{n}\left|\{k\le n:|x_k-L|\ge \epsilon \}\right|=0.$$

In this case we write $st\text{-}limx=L$. This concept was studied by [11, 12].

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 notion of difference sequence space $X(\mathrm{\Delta })$ was introduced by Kızmaz [13] as follows:

$$X(\mathrm{\Delta })=\{x=(x_k):(\mathrm{\Delta }{x}_k)\in X\}$$

for $X=l_{\mathrm{\infty }}$, c, $c_0$, where $\mathrm{\Delta }{x}_k=x_k-x_{k+1}$ for all $k\in \mathbb{N}$.

The notion of difference sequence spaces was further generalized by Et and Çolak [14] as follows:

$$X\left({\mathrm{\Delta }}^m\right)=\{x=(x_k)\in w:\left({\mathrm{\Delta }}^m{x}_k\right)\in X\}$$

for $X=l_{\mathrm{\infty }}$, c and $c_0$, where $m\in \mathbb{N}$, ${\mathrm{\Delta }}^m{x}_k={\mathrm{\Delta }}^{m-1}{x}_k-{\mathrm{\Delta }}^{m-1}{x}_{k+1}$, ${\mathrm{\Delta }}^0{x}_k=x_k$.

The sequence x is said to be ${\mathrm{\Delta }}^m$-statistically convergent to the number L provided that for each $\epsilon >0$,

$$\underset{n}{lim}\frac{1}{n}\left|\{k\le n:|{\mathrm{\Delta }}^m{x}_k-L|\ge \epsilon \}\right|=0.$$

The set of all ${\mathrm{\Delta }}^m$-statistically convergent sequences was denoted by $S({\mathrm{\Delta }}^m)$ in [15].

Furthermore, this notion was studied in [16, 17].

A paranorm is a function $g:X⟶\mathbb{R}$ defined on a linear space X such that for all $x,y,z\in X$,

(i) $g(x)=0$ if $x=\theta$;

(ii) $g(-x)=g(x)$;

(iii) $g(x+y)\le g(x)+g(y)$;

(iv) If $({\alpha }_n)$ is a sequence of scalars with ${\alpha }_n⟶{\alpha }_0$ ($n⟶\mathrm{\infty }$) and $x_n,a\in X$ with $x_n⟶a$ ($n⟶\mathrm{\infty }$) in the sense that $g(x_n-a)⟶0$ ($n⟶\mathrm{\infty }$), then ${\alpha }_n{x}_n⟶{\alpha }_{0}a$ ($n⟶\mathrm{\infty }$), in the sense that $g({\alpha }_n{x}_n-{\alpha }_{0}a)⟶0$ ($n⟶\mathrm{\infty }$).

A paranorm g for which $g(x)=0$ implies $x=\theta$ is called a total paranorm on X and the pair $(X,g)$ is called a total paranormed space.

Note that each seminorm (norm) p on X is a paranorm (total) but converse need not be true.

The concept of paranorm is a generalization of absolute value [18].

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

(iii) f increasing;

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

Since $|f(x)-f(y)|\le f(|x-y|)$, it follows from condition (iv) that f is continuous on $[0,\mathrm{\infty })$. Furthermore, we have $f(nx)\le nf(x)$ for all $n\in \mathbb{N}$ from condition (ii) and so

$$f(x)=f\left(nx\frac{1}{n}\right)\le nf\left(\frac{x}{n}\right).$$

Hence, for all $n\in \mathbb{N}$,

$$\frac{1}{n}f(x)\le f\left(\frac{x}{n}\right).$$

A modulus may be bounded or unbounded. For example, $f(x)=x^p$ for $0<p\le 1$ is unbounded, but $f(x)=\frac{x}{1+x}$ is bounded. Ruckle [19] used the idea of a modulus function f to construct a class of FK spaces

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

In [9], the notion of statistical convergence was defined in a paranormed space.

Definition 2.1 A sequence $x=(x_k)$ is said to be statistically convergent to the number L in $(X,g)$ if for each $\epsilon >0$,

$$\underset{n}{lim}\frac{1}{n}\left|\{k\le n:g(x_k-L)\ge \epsilon \}\right|=0.$$

In this case, we write $g(st)\text{-}limx=L$. We denote the set of all $g(st)$-convergent sequences by $S_g$ [9].

Definition 2.2 A sequence $x=(x_k)$ is said to be strongly p-Cesaro summable ($0<p<\mathrm{\infty }$) to the limit L in $(X,g)$ if

$$\underset{n}{lim}\frac{1}{n}\sum _{j=1}^n{(g(x_j-L))}^p=0,$$

and we write it as $x_k⟶L({[C_1,g]}_p)$. In this case, L is called the ${[C_1,g]}_p\text{-}lim$ it of x [9].

In this article, we shall study the concept of ${\mathrm{\Delta }}^m$-lacunary statistical convergence, ${\mathrm{\Delta }}^m$-lacunary strong convergence and ${\mathrm{\Delta }}^m$-lacunary strong convergence with respect to a modulus function in a paranormed space.

3 Generalized difference statistical convergence in a paranormed space

Definition 3.1 A sequence $x=(x_k)$ is said to be ${\mathrm{\Delta }}^m$-statistically convergent to the number L in $(X,g)$ if for each $\epsilon >0$,

$$\underset{n}{lim}\frac{1}{n}\left|\{k\le n:g({\mathrm{\Delta }}^m{x}_k-L)\ge \epsilon \}\right|=0.$$

In this case, we write $S_g({\mathrm{\Delta }}^m)\text{-}limx=L$. We denote the set of all ${\mathrm{\Delta }}^m$-statistically convergent sequences in $(X,g)$ by $S_g({\mathrm{\Delta }}^m)$.

Definition 3.2 Let θ be a lacunary sequence. A sequence $x=(x_k)$ is said to be ${\mathrm{\Delta }}^m$-lacunary statistically convergent to the number L in $(X,g)$ if for each $\epsilon >0$,

$$\underset{r}{lim}\frac{1}{h_r}\left|\{k\in I_r:g({\mathrm{\Delta }}^m{x}_k-L)\ge \epsilon \}\right|=0.$$

In this case, we write $S_g^{\theta }({\mathrm{\Delta }}^m)\text{-}limx=L$. We denote the set of all ${\mathrm{\Delta }}^m$-lacunary statistically convergent sequences in $(X,g)$ by $S_g^{\theta }({\mathrm{\Delta }}^m)$.

Definition 3.3 A sequence $x=(x_k)$ is said to be strongly ${\mathrm{\Delta }}^m$-Cesaro summable to the limit L in $(X,g)$ if

$$\underset{n}{lim}\frac{1}{n}\sum _{k=1}^n\left(g({\mathrm{\Delta }}^m{x}_k-L)\right)=0,$$

and we write it as $x_k⟶L(|{\sigma }_1{|}_g({\mathrm{\Delta }}^m))$. In this case L is called the $|{\sigma }_1{|}_g({\mathrm{\Delta }}^m)\text{-}lim$ of x.

Definition 3.4 A sequence $x=(x_k)$ is said to be strongly ${\mathrm{\Delta }}^m$-lacunary strongly summable to the limit L in $(X,g)$ if

$$\underset{r}{lim}\frac{1}{h_r}\sum _{k\in I_r}\left(g({\mathrm{\Delta }}^m{x}_k-L)\right)=0,$$

and we write it as $x_k⟶L(N_g^{\theta }({\mathrm{\Delta }}^m))$. In this case L is called the $N_g^{\theta }({\mathrm{\Delta }}^m)\text{-}lim$ of x.

Theorem 3.1 Let θ be a lacunary sequence and $(X,g)$ be a paranormed space. Then

(i) If $x_k⟶L(N_g^{\theta }({\mathrm{\Delta }}^m))$, then $x_k⟶L(S_g^{\theta }({\mathrm{\Delta }}^m))$ and the inclusion is strict;

(ii) If x is a ${\mathrm{\Delta }}^m$-bounded sequence and $x_k⟶L(S_g^{\theta }({\mathrm{\Delta }}^m))$, then $x_k⟶L(N_g^{\theta }({\mathrm{\Delta }}^m))$;

(iii) $l_g^{\mathrm{\infty }}({\mathrm{\Delta }}^m)\cap S_g^{\theta }({\mathrm{\Delta }}^m)=l_g^{\mathrm{\infty }}({\mathrm{\Delta }}^m)\cap N_g^{\theta }({\mathrm{\Delta }}^m)$.

Proof (i) If $\epsilon >0$ and $x_k\to L(N_g^{\theta }({\mathrm{\Delta }}^m))$, we can write

$$\sum _{k\in I_r}g({\mathrm{\Delta }}^m{x}_k-L)\ge \underset{g({\mathrm{\Delta }}^m{x}_k-L)\ge \epsilon }{\sum _{k\in I_r}}g({\mathrm{\Delta }}^m{x}_k-L)\ge \epsilon \left|\{k\in I_r:g({\mathrm{\Delta }}^m{x}_k-L)\ge \epsilon \}\right|,$$

which yields the result.

In order to prove that the inclusion $N_g^{\theta }({\mathrm{\Delta }}^m)\subset S_g^{\theta }({\mathrm{\Delta }}^m)$ is proper, let θ be given and $X=N_0^{\theta }(\mathrm{\Delta },\frac{1}{h_r})=\{x=(x_k):|\frac{1}{h_r}{\sum }_{k\in I_r}\mathrm{\Delta }{x}_k{|}^{\frac{1}{h_r}}\to 0,r\to \mathrm{\infty }\}$ with the paranorm $g(x)=|x_1|+{sup}_r|\frac{1}{h_r}{\sum }_{k\in I_r}\mathrm{\Delta }{x}_k{|}^{\frac{1}{h_r}}$. Define $x=(x_k)$ to be $2h_r{1}^{h_r}$ at the first term in $I_r$ for every $r\ge 1$, $x_k=h_r(1^{h_r}-2^{h_r}-\cdots -{(k-1)}^{h_r})$ between the second term and $([\sqrt{h_r}]+1)$th term in $I_r$, $x_k=h_r(1^{h_r}-2^{h_r}-\cdots -{([\sqrt{h_r}])}^{h_r})$ at the $([\sqrt{h_r}]+2)$th term in $I_r$ and $x_k=0$ otherwise.

We see that

$$\mathrm{\Delta }{x}_k=\{\begin{array}{cc}{h}_r{1}^{h_r},h_r{2}^{h_r},\dots ,h_r{[\sqrt{h_r}]}^{h_r},\hfill & \text{at the first }[\sqrt{h_r}]\text{ integers in }{I}_r,\hfill \\ 0,\hfill & \text{otherwise}.\hfill \end{array}$$

and

$$g(\mathrm{\Delta }{x}_k)=\{\begin{array}{cc}1,2,\dots ,[\sqrt{h_r}],\hfill & \text{at the first }[\sqrt{h_r}]\text{ integers in }{I}_r,\hfill \\ 0,\hfill & \text{otherwise}.\hfill \end{array}$$

Note that x is not Δ-bounded in $(X,g)$. We have, for every $\epsilon >0$,

$$\frac{1}{h_r}\left|\{k\in I_r:g(\mathrm{\Delta }{x}_k)\ge \epsilon )\}\right|=\frac{[\sqrt{h_r}]}{h_r}\to 0$$

as $r\to \mathrm{\infty }$, i.e., $x_k\to 0(S_g^{\theta }(\mathrm{\Delta }))$. On the other hand,

$$\frac{1}{h_r}\sum _{k\in I_r}g(\mathrm{\Delta }{x}_k)=\frac{1}{h_r}\frac{[\sqrt{h_r}]([\sqrt{h_r}]+1)}{2}\to \frac{1}{2}\ne 0;$$

hence $x_k\nrightarrow 0(N_g^{\theta }(\mathrm{\Delta }))$.

(ii) Suppose that $x_k\to L(S_g^{\theta }({\mathrm{\Delta }}^m))$ and say $g({\mathrm{\Delta }}^m{x}_k-L)\le M$ for all k. Given $\epsilon >0$, we get

$$\begin{array}{rl}\frac{1}{h_r}\sum _{k\in I_r}g({\mathrm{\Delta }}^m{x}_k-L)& =\frac{1}{h_r}\underset{g({\mathrm{\Delta }}^m{x}_k-L)\ge \epsilon }{\sum _{k\in I_r}}g({\mathrm{\Delta }}^m{x}_k-L)+\frac{1}{h_r}\underset{g({\mathrm{\Delta }}^m{x}_k-L)<\epsilon }{\sum _{k\in I_r}}g({\mathrm{\Delta }}^m{x}_k-L)\\ \le \frac{M}{h_r}\left|\{k\in I_r:g({\mathrm{\Delta }}^m{x}_k-L)\ge \epsilon \}\right|+\epsilon ,\end{array}$$

from which the result follows.

(iii) This is an immediate consequence of (i) and (ii). □

Corollary 3.1 If $x_k\to L(|{\sigma }_1{|}_g({\mathrm{\Delta }}^m))$, then $x_k\to L(S_g({\mathrm{\Delta }}^m))$. If $x\in l_g^{\mathrm{\infty }}({\mathrm{\Delta }}^m)$ and if $x_k\to L(S_g({\mathrm{\Delta }}^m))$, then $x_k\to L(|{\sigma }_1{|}_g({\mathrm{\Delta }}^m))$.

Theorem 3.2 Let θ be a lacunary sequence and $(X,g)$ be a paranormed space, then $S_g^{\theta }({\mathrm{\Delta }}^m)=S_g({\mathrm{\Delta }}^m)$ if and only if $1<{lim\hspace{0.17em}inf}_r{q}_r\le {lim\hspace{0.17em}\; sup}_r{q}_r<\mathrm{\infty }$.

Proof Suppose that $liminf{q}_r>1$, then there exists a $\delta >0$ such that $q_r⩾1+\delta$ for sufficiently large r, which implies that

$$\frac{h_r}{k_r}⩾\frac{\delta }{1+\delta }.$$

If $x_k\to L(S_g({\mathrm{\Delta }}^m))$, then for every $\epsilon >0$ and sufficiently large r, we have

$$\begin{array}{rl}\frac{1}{k_r}\left|\{k⩽k_r:g({\mathrm{\Delta }}^m{x}_k-L)⩾\epsilon \}\right|& ⩾\frac{1}{k_r}\left|\{k\in I_r:g({\mathrm{\Delta }}^m{x}_k-L)⩾\epsilon \}\right|\\ ⩾\frac{\delta }{1+\delta }\frac{1}{h_r}\left|\{k\in I_r:g({\mathrm{\Delta }}^m{x}_k-L)⩾\epsilon \}\right|,\end{array}$$

which proves the $S_g({\mathrm{\Delta }}^m)\subset S_g^{\theta }({\mathrm{\Delta }}^m)$.

Conversely, suppose that $liminf{q}_r=1$. Since θ is lacunary, we can select a subsequence $(k_{r_j})$ of θ satisfying $\frac{k_{r_j}}{k_{r_j-1}}<1+\frac{1}{j}$ and $\frac{k_{r_j-1}}{k_{r_{(j-1)}}}>j$, where $r_j⩾r_{j-1}+2$ and $X=N_0^{\theta }(\mathrm{\Delta },\frac{1}{h_r})=\{x=(x_k):|\frac{1}{h_r}{\sum }_{k\in I_r}\mathrm{\Delta }{x}_k{|}^{\frac{1}{h_r}}\to 0,r\to \mathrm{\infty }\}$ with the paranorm $g(x)=|x_1|+{sup}_r|\frac{1}{h_r}{\sum }_{k\in I_r}\mathrm{\Delta }{x}_k{|}^{\frac{1}{h_r}}$.

Now define a sequence by

$$\mathrm{\Delta }{x}_k=\{\begin{array}{cc}{h}_r+k,\hfill & k\in I_{r_{(j)}},j=1,2,\dots ,\hfill \\ 0,\hfill & \text{otherwise}.\hfill \end{array}$$

We can see that

$$g(\mathrm{\Delta }{x}_k)=\{\begin{array}{cc}1,\hfill & k\in I_{r_{(j)}},j=1,2,\dots ,\hfill \\ 0,\hfill & \text{otherwise}.\hfill \end{array}$$

and hence x is Δ-bounded in $(X,g)$.

We can see that $x\notin N_g^{\theta }({\mathrm{\Delta }}^m)$, but $x\in {\sigma }_g^1({\mathrm{\Delta }}^m)$. Theorem 3.1(ii) implies that $x\notin S_g^{\theta }({\mathrm{\Delta }}^m)$, but it follows from Corollary 3.1 that $x\in S_g({\mathrm{\Delta }}^m)$. Hence $S_g({\mathrm{\Delta }}^m)\not\subset S_g^{\theta }({\mathrm{\Delta }}^m)$ and $S_g({\mathrm{\Delta }}^m)\subset S_g^{\theta }({\mathrm{\Delta }}^m)$ implies that $liminf{q}_r>1$.

To show for any lacunary sequence θ, $S_g^{\theta }({\mathrm{\Delta }}^m)\subset S_g({\mathrm{\Delta }}^m)$ implies $limsup{q}_r<\mathrm{\infty }$, the same technique of Lemma 3 of [20] can be used. Now suppose that $limsup{q}_r=\mathrm{\infty }$. Consider the same space defined above and the sequence defined by

$$\mathrm{\Delta }{x}_i=\{\begin{array}{cc}{h}_r+i,\hfill & k_{r_j-1}<i⩽2k_{r_j-1},j=1,2,\dots ,\hfill \\ 0,\hfill & \text{otherwise}.\hfill \end{array}$$

Then we get

$$g(\mathrm{\Delta }{x}_i)=\{\begin{array}{cc}1,\hfill & k_{r_j-1}<i⩽2k_{r_j-1},j=1,2,\dots ,\hfill \\ 0,\hfill & \text{otherwise}.\hfill \end{array}$$

Then $x\in N_g^{\theta }({\mathrm{\Delta }}^m)$, but $x\notin {\sigma }_g^1({\mathrm{\Delta }}^m)$. By Theorem 3.1(i) we conclude that $x\in S_g^{\theta }({\mathrm{\Delta }}^m)$, but by Corollary 3.1 that $x\notin S_g({\mathrm{\Delta }}^m)$. Hence, $S_g^{\theta }({\mathrm{\Delta }}^m)\not\subset S_g({\mathrm{\Delta }}^m)$. This completes the proof. □

Definition 3.5 Let f be a modulus function. Then a sequence $x=(x_k)$ is lacunary strongly p-Cesaro summable to L with respect to f in $(X,g)$ if

$$\underset{r}{lim}\frac{1}{h_r}\sum _{kϵI_r}{\left[f\left(g({\mathrm{\Delta }}^m{x}_k-L)\right)\right]}^{p_k}=0.$$

In this case, we write $x_k\to L(N_g^{\theta }(f,{\mathrm{\Delta }}^m,p))$. If we take $p_k=1$ for all $k\in \mathbb{N}$, we say $x_k\to L(N_g^{\theta }(f,{\mathrm{\Delta }}^m))$.

Lemma 3.1 Let f be a modulus function and let $0<\delta <1$. Then for each $x>\delta$ we have $f(x)⩽2f(1){\delta }^{-1}x$ [21].

Theorem 3.3 Let f be a modulus function and $(X,g)$ be a paranormed space. Then $N_g^{\theta }({\mathrm{\Delta }}^m)\subset N_g^{\theta }(f,{\mathrm{\Delta }}^m)$.

Proof Let $x\in N_g^{\theta }({\mathrm{\Delta }}^m)$. Then we have ${\tau }_r=\frac{1}{h_r}{\sum }_{k\in I_r}g({\mathrm{\Delta }}^m{x}_k-L)\to 0$ as $r\to \mathrm{\infty }$ for some L.

Let $\epsilon >0$ and choose δ with $0<\delta <1$ such that $f(u)<\epsilon$ for u with $0\le u\le \delta$. Then we can write

$$\begin{array}{rl}\frac{1}{h_r}\sum _{k\in I_r}f\left(g({\mathrm{\Delta }}^m{x}_k-L)\right)=& \frac{1}{h_r}\underset{g({\mathrm{\Delta }}^m{x}_k-L)\le \delta }{\sum _{k\in I_r}}f\left(g({\mathrm{\Delta }}^m{x}_k-L)\right)\\ +\frac{1}{h_r}\underset{g({\mathrm{\Delta }}^m{x}_k-L)>\delta }{\sum _{k\in I_r}}f\left(g({\mathrm{\Delta }}^m{x}_k-L)\right)\\ \le & \frac{1}{h_r}(h_r\delta )+\frac{1}{h_r}2f(1){\delta }^{-1}{h}_r{\tau }_r\end{array}$$

from Lemma 3.1. Therefore $x\in N_g^{\theta }(f,{\mathrm{\Delta }}^m)$. □

Theorem 3.4 Let $0<{inf}_k{p}_k\le p_k\le {sup}_k{p}_k<\mathrm{\infty }$. Then $S_g^{\theta }({\mathrm{\Delta }}^m)=N_g^{\theta }(f,{\mathrm{\Delta }}^m,p)$ if and only if f is bounded.

Proof Following the technique applied for establishing Theorem 3.16 of [22], we can prove the theorem. □