Lacunary Δ-statistical convergence in intuitionistic fuzzy n-normed space

Abstract

The concept of lacunary statistical convergence was introduced in intuitionistic fuzzy n-normed spaces in Sen and Debnath (Math. Comput. Model. 54:2978-2985, 2011). In this article, we introduce the notion of lacunary Δ-statistically convergent and lacunary Δ-statistically Cauchy sequences in an intuitionistic fuzzy n-normed space. Also, we give their properties using lacunary density and prove relation between these notions.

MSC:47H10, 54H25.

1 Introduction

Fuzzy set theory was introduced by Zadeh [1] in 1965. This theory has been applied not only in different branches of engineering such as in nonlinear dynamic systems [2], in the population dynamics [3], in the quantum physics [4], but also in many fields of mathematics such as in metric and topological spaces [5–7], in the theory of functions [8, 9], in the approximation theory [10]. 2-normed and n-normed linear spaces were initially introduced by Gähler [11, 12] and further studied by Kim and Cho [13], Malceski [14] and Gunawan and Mashadi [15]. Vijayabalaji and Narayanan [16] defined fuzzy n-normed linear space. After Saadati and Park [17] introduced the concept of intuitionistic fuzzy normed space, Vijayabalaji et al. [18] defined the notion of intuitionistic fuzzy n-normed space. The notion of statistical convergence was investigated by Steinhaus [19] and Fast [20]. Then a lot of authors applied this concept to probabilistic normed spaces [21, 22], random 2-normed spaces [23] and finally intuitionistic fuzzy normed spaces [24, 25]. Fridy and Orhan [26] introduced the idea of lacunary statistical convergence. Using this idea, Mursaleen and Mohiuddine [27], Sen and Debnath [28] investigated lacunary statistical convergence in intuitionistic fuzzy normed spaces and intuitionistic fuzzy n-normed spaces, respectively. The idea of difference sequences was introduced by Kızmaz [29] where $\mathrm{\Delta }x=(\mathrm{\Delta }{x}_k)=x_k-x_{k+1}$. Başarır [30] introduced the Δ-statistical convergence of sequences. Bilgin [31] introduced the definition of lacunary strongly Δ-convergence of fuzzy numbers. Hazarika [32] gave the definition of lacunary generalized difference statistical convergence in random 2-normed spaces. Also, the generalized difference sequence spaces were studied by various authors [33–35]. In this article, we shall introduce lacunary Δ-statistical convergence and lacunary Δ-statistically Cauchy sequences in IFnNLS.

2 Preliminaries, background and notation

In this section, we give the basic definitions.

Definition 2.1 ([27])

A binary operation $\ast :[0,1]\times [0,1]\to [0,1]$ is said to be a continuous t-norm if it satisfies the following conditions:

• (i) ∗ is associative and commutative,

• (ii) ∗ is continuous,

• (iii) $a\ast 1=a$ for all $a\in [0,1]$,

• (iv) $a\ast b\le c\ast d$ whenever $a\le c$ and $b\le d$ for each $a,b,c,d\in [0,1]$.

Definition 2.2 ([27])

A binary operation $\circ :[0,1]\times [0,1]\to [0,1]$ is said to be a continuous t-conorm if it satisfies the following conditions:

• (i) ∘ is associative and commutative,

• (ii) ∘ is continuous,

• (iii) $a\circ 0=a$ for all $a\in [0,1]$,

• (iv) $a\circ b\le c\circ d$ whenever $a\le c$ and $b\le d$ for each $a,b,c,d\in [0,1]$.

Definition 2.3 ([27])

Let $n\in \mathbb{N}$ and X be a real vector space of dimension $d\ge n$ (here we allow it to be infinite). A real-valued function $\parallel •,\dots ,•\parallel$ on $X\times \cdots \times X=X^n$ satisfying the following four properties:

• (i) $\parallel x_1,x_2,\dots ,x_n\parallel =0$ if and only if $x_1,x_2,\dots ,x_n$ are linearly dependent,

• (ii) $x_1,x_2,\dots ,x_n$ are invariant under any permutation,

• (iii) $\parallel x_1,x_2,\dots ,\alpha x_n\parallel =|\alpha |\parallel x_1,x_2,\dots ,x_n\parallel$ for any $\alpha \in \mathbb{R}$,

• (iv) $\parallel x_1,x_2,\dots ,x_{n-1},y+z\parallel \le \parallel x_1,x_2,\dots ,x_{n-1},y\parallel +\parallel x_1,x_2,\dots ,x_{n-1},z\parallel$,

is called an n-norm on X and the pair is called an n-normed space.

Definition 2.4 ([28])

An IFnNLS is the five-tuple $(X,\mu ,\upsilon ,\ast ,\circ )$ where X is a linear space over a field F, ∗ is a continuous t-norm, ∘ is a continuous t-conorm, μ, υ are fuzzy sets on $X^n\times (0,\mathrm{\infty })$, μ denotes the degree of membership and υ denotes the degree of nonmembership of $(x_1,x_2,\dots ,x_n,t)\in X^{n^{}}\times (0,\mathrm{\infty })$ satisfying the following conditions for every $(x_1,x_2,\dots ,x_n)\in X^{n^{}}$ and $s,t>0$:

• (i) $\mu (x_1,x_2,\dots ,x_n,t)+\upsilon (x_1,x_2,\dots ,x_n,t)\le 1$,

• (ii) $\mu (x_1,x_2,\dots ,x_n,t)>0$,

• (iii) $\mu (x_1,x_2,\dots ,x_n,t)=1$ if and only if $x_1,x_2,\dots ,x_n$ are linearly dependent,

• (iv) $\mu (x_1,x_2,\dots ,x_n,t)$ is invariant under any permutation of $x_1,x_2,\dots ,x_n$,

• (v) $\mu (x_1,x_2,\dots ,c{x}_n,t)=\mu (x_1,x_2,\dots ,x_n,\frac{t}{|c|})$ for all $c\ne 0$, $c\in F$,

• (vi) $\mu (x_1,x_2,\dots ,x_n,s)\ast \mu (x_1,x_2,\dots ,x_n^{\prime },t)\le \mu (x_1,x_2,\dots ,x_n+x_n^{\prime },s+t)$,

• (vii) $\mu (x_1,x_2,\dots ,x_n,t):(0,\mathrm{\infty })\to [0,1]$ is continuous in t,

• (viii) ${lim}_{t\to \mathrm{\infty }}\mu (x_1,x_2,\dots ,x_n,t)=1$ and ${lim}_{t\to 0}\mu (x_1,x_2,\dots ,x_n,t)=0$,

• (ix) $\upsilon (x_1,x_2,\dots ,x_n,t)<1$,

• (x) $\upsilon (x_1,x_2,\dots ,x_n,t)=0$ if and only if $x_1,x_2,\dots ,x_n$ are linearly dependent,

• (xi) $\upsilon (x_1,x_2,\dots ,x_n,t)$ is invariant under any permutation of $x_1,x_2,\dots ,x_n$,

• (xii) $\upsilon (x_1,x_2,\dots ,c{x}_n,t)=\upsilon (x_1,x_2,\dots ,x_n,\frac{t}{|c|})$ for all $c\ne 0$, $c\in F$,

• (xiii) $\upsilon (x_1,x_2,\dots ,x_n,s)\circ \upsilon (x_1,x_2,\dots ,x_n^{\prime },t)\ge \upsilon (x_1,x_2,\dots ,x_n+x_n^{\prime },s+t)$

• (xiv) $\upsilon (x_1,x_2,\dots ,x_n,t):(0,\mathrm{\infty })\to [0,1]$ is continuous in t,

• (xv) ${lim}_{t\to \mathrm{\infty }}\upsilon (x_1,x_2,\dots ,x_n,t)=0$ and ${lim}_{t\to 0}\upsilon (x_1,x_2,\dots ,x_n,t)=1$.

Example 2.1 ([28])

Let $(X,\parallel •,\dots ,•\parallel )$ be an n-normed linear space. Also let $a\ast b=ab$ and $a\circ b=min\{a+b,1\}$ for all $a,b\in [0,1]$,

$$\mu (x_1,x_2,\dots ,x_n,t)=\frac{t}{t+\parallel x_1,x_2,\dots ,x_n\parallel }\text{and}\upsilon (x_1,x_2,\dots ,x_n,t)=\frac{\parallel x_1,x_2,\dots ,x_n\parallel }{t+\parallel x_1,x_2,\dots ,x_n\parallel }.$$

Then $(X,\mu ,\upsilon ,\ast ,\circ )$ is an IFnNLS.

Definition 2.5 ([26])

A lacunary sequence is an increasing integer sequence $\theta =\{k_r\}$ such that $k_0=0$ and $h_r=k_r-k_{r-1}\to \mathrm{\infty }$ as $r\to \mathrm{\infty }$. The intervals determined by θ will be denoted by $I_r=(k_{r-1},k_r]$ and the ratio $\frac{k_r}{k_{r-1}}$ will be abbreviated as $q_r$. Let $K\subseteq \mathbb{N}$. The number

$${\delta }_{{\theta }_{}}(K)=\underset{r}{lim}\frac{1}{h_r}|\{k\in I_r:k\in K\}|$$

is said to be the θ-density of K, provided the limit exists.

Definition 2.6 ([28])

Let θ be a lacunary sequence. A sequence $x=\{x_k\}$ of numbers is said to be lacunary statistically convergent (or $S_{\theta }$-convergent) to the number L if for every $\epsilon >0$, the set $K(\epsilon )$ has θ-density zero, where

$$K(\epsilon )=\{k\in \mathbb{N}:|x_k-L|\ge \epsilon \}.$$

In this case, we write $S_{\theta }\text{-}limx=L$.

3 Δ-Convergence and lacunary Δ-statistical convergence in IFnNLS

In this section, we define Δ-convergence and lacunary Δ-statistical convergence in intuitionistic fuzzy n-normed spaces.

Definition 3.1 Let $(X,\mu ,\upsilon ,\ast ,\circ )$ be an IFnNLS. A sequence $x=\{x_k\}$ in X is said to be Δ-convergent to $L\in X$ with respect to the intuitionistic fuzzy n-norm ${(\mu ,\upsilon )}^n$ if, for every $\epsilon >0$, $t>0$ and $y_1,y_2,\dots ,y_{n-1}\in X$, there exists $k_0\in \mathbb{N}$ such that $\mu (y_1,y_2,\dots ,y_{n-1},\mathrm{\Delta }{x}_k-L,t)>1-\epsilon$ and $\upsilon (y_1,y_2,\dots ,y_{n-1},\mathrm{\Delta }{x}_k-L,t)<\epsilon$ for all $k\ge k_0$, where $k\in \mathbb{N}$ and $\mathrm{\Delta }{x}_k=(x_k-x_{k+1})$. It is denoted by ${(\mu ,\upsilon )}^n\text{-}lim\mathrm{\Delta }x=L$ or $\mathrm{\Delta }{x}_k\to L$ as $k\to \mathrm{\infty }$.

Definition 3.2 Let $(X,\mu ,\upsilon ,\ast ,\circ )$ be an IFnNLS. A sequence $x=\{x_k\}$ in X is said to be lacunary Δ-statistically convergent or $S_{\theta }(\mathrm{\Delta })$-convergent to $L\in X$ with respect to the intuitionistic fuzzy n-norm ${(\mu ,\upsilon )}^n$ provided that for every $\epsilon >0$, $t>0$ and $y_1,y_2,\dots ,y_{n-1}\in \mathbb{X}$,

$$\begin{array}{c}{\delta }_{\theta }(\mathrm{\Delta })\left(\right\{k\in \mathbb{N}:\mu (y_1,y_2,\dots ,y_{n-1},\mathrm{\Delta }{x}_k-L,t)\le 1-\epsilon \hfill \\ \text{or }\upsilon (y_1,y_2,\dots ,y_{n-1},\mathrm{\Delta }{x}_k-L,t)\ge \epsilon \left\}\right)=0,\hfill \end{array}$$

or, equivalently,

$$\begin{array}{c}{\delta }_{\theta }(\mathrm{\Delta })\left(\right\{k\in \mathbb{N}:\mu (y_1,y_2,\dots ,y_{n-1},\mathrm{\Delta }{x}_k-L,t)>1-\epsilon \hfill \\ \text{and }\upsilon (y_1,y_2,\dots ,y_{n-1},\mathrm{\Delta }{x}_k-L,t)<\epsilon \left\}\right)=1.\hfill \end{array}$$

It is denoted by ${S_{\theta }}^{{(\mu ,\upsilon )}^n}(\mathrm{\Delta })\text{-}limx=L$ or $x_k\to L(S_{\theta }(\mathrm{\Delta }))$. Using Definition 3.2 and properties of the θ-density, we can easily obtain the following lemma.

Lemma 3.1 Let $(X,\mu ,\upsilon ,\ast ,\circ )$ be an IFnNLS and θ be a lacunary sequence. Then, for every $\epsilon >0$, $t>0$ and $y_1,y_2,\dots ,y_{n-1}\in \mathbb{X}$, the following statements are equivalent:

• (i) ${S_{\theta }}^{{(\mu ,\upsilon )}^n}(\mathrm{\Delta })\text{-}limx=L$,

• (ii) ${\delta }_{\theta }(\mathrm{\Delta })(\{k\in \mathbb{N}:\mu (y_1,y_2,\dots ,y_{n-1},\mathrm{\Delta }{x}_k-L,t)\le 1-\epsilon \})={\delta }_{\theta }(\mathrm{\Delta })(\{k\in \mathbb{N}:\upsilon (y_1,y_2,\dots ,y_{n-1},\mathrm{\Delta }{x}_k-L,t)\ge \epsilon \})=0$,

• (iii) ${\delta }_{\theta }(\mathrm{\Delta })(\{k\in \mathbb{N}:\mu (y_1,y_2,\dots ,y_{n-1},\mathrm{\Delta }{x}_k-L,t)>1-\epsilon \mathit{\text{ and }}\upsilon (y_1,y_2,\dots ,y_{n-1},\mathrm{\Delta }{x}_k-L,t)<\epsilon \})=1$,

• (iv) ${\delta }_{\theta }(\mathrm{\Delta })(\{k\in \mathbb{N}:\mu (y_1,y_2,\dots ,y_{n-1},\mathrm{\Delta }{x}_k-L,t)>1-\epsilon \})={\delta }_{\theta }(\mathrm{\Delta })(\{k\in \mathbb{N}:\upsilon (y_1,y_2,\dots ,y_{n-1},\mathrm{\Delta }{x}_k-L,t)<\epsilon \})=1$,

• (v) $S_{\theta }\text{-}lim\mu (y_1,y_2,\dots ,y_{n-1},\mathrm{\Delta }{x}_k-L,t)=1$ and $S_{\theta }\text{-}lim\upsilon (y_1,y_2,\dots ,y_{n-1},\mathrm{\Delta }{x}_k-L,t)=0$.

Proceeding exactly in a similar way as in [36], the following theorem can be proved.

Theorem 3.1 Let $(X,\mu ,\upsilon ,\ast ,\circ )$ be an IFnNLS and θ be a lacunary sequence. If a sequence $x=\{x_k\}$ in X is lacunary Δ-statistically convergent or $S_{\theta }(\mathrm{\Delta })$-convergent to $L\in X$ with respect to the intuitionistic fuzzy n-norm ${(\mu ,\upsilon )}^n$, ${S_{\theta }}^{{(\mu ,\upsilon )}^n}(\mathrm{\Delta })\text{-}limx$ is unique.

Theorem 3.2 Let $(X,\mu ,\upsilon ,\ast ,\circ )$ be an IFnNLS and θ be a lacunary sequence. If ${(\mu ,\upsilon )}^n\text{-}lim\mathrm{\Delta }x=L$, then ${S_{\theta }}^{{(\mu ,\upsilon )}^n}(\mathrm{\Delta })\text{-}limx=L$.

Proof Let ${(\mu ,\upsilon )}^n\text{-}lim\mathrm{\Delta }x=L$. Then, for every $\epsilon >0$, $t>0$ and $y_1,y_2,\dots ,y_{n-1}\in \mathbb{X}$, there exists $k_0\in \mathbb{N}$ such that $\mu (y_1,y_2,\dots ,y_{n-1},\mathrm{\Delta }{x}_k-L,t)>1-\epsilon$ and $\upsilon (y_1,y_2,\dots ,y_{n-1},\mathrm{\Delta }{x}_k-L,t)<\epsilon$ for all $k\ge k_0$. Hence the set

$$\begin{array}{c}\{k\in \mathbb{N}:\mu (y_1,y_2,\dots ,y_{n-1},\mathrm{\Delta }{x}_k-L,t)\le 1-\epsilon \hfill \\ \text{or }\upsilon (y_1,y_2,\dots ,y_{n-1},\mathrm{\Delta }{x}_k-L,t)\ge \epsilon \}\hfill \end{array}$$

has a finite number of terms. Since every finite subset of ℕ has lacunary density zero,

$$\begin{array}{c}{\delta }_{\theta }(\mathrm{\Delta })\left(\right\{k\in \mathbb{N}:\mu (y_1,y_2,\dots ,y_{n-1},\mathrm{\Delta }{x}_k-L,t)\le 1-\epsilon \hfill \\ \text{or }\upsilon (y_1,y_2,\dots ,y_{n-1},\mathrm{\Delta }{x}_k-L,t)\ge \epsilon \left\}\right)=0,\hfill \end{array}$$

that is, ${S_{\theta }}^{{(\mu ,\upsilon )}^n}(\mathrm{\Delta })\text{-}limx=L$.

It follows from the following example that the converse of Theorem 3.2 is not true in general.

Example 3.1 Consider $X={\mathbb{R}}^n$ with

$$\parallel x_1,x_2,\dots ,x_n\parallel =abs\left(\left|\begin{array}{lll}{x}_{11}& \cdots & x_{1n}\\ ⋮& \ddots & ⋮\\ x_{n1}& \cdots & x_{nn}\end{array}\right|\right),$$

where $x_i=(x_{i1},x_{i2},\dots ,x_{in})\in {\mathbb{R}}^n$ for each $i=1,2,\dots ,n$, and let $a\ast b=ab$, $a\circ b=min\{a+b,1\}$ for all $a,b\in [0,1]$. Now, for all $y_1,y_2,\dots ,y_{n-1},x\in {\mathbb{R}}^n$ and $t>0$, $\mu (y_1,y_2,\dots ,y_{n-1},x,t)=\frac{t}{t+\parallel y_1,y_2,\dots ,y_{n-1},x\parallel }$ and $\upsilon (y_1,y_2,\dots ,y_{n-1},x,t)=\frac{\parallel y_1,y_2,\dots ,y_{n-1},x\parallel }{t+\parallel y_1,y_2,\dots ,y_{n-1},x\parallel }$. Then $({\mathbb{R}}^n,\mu ,\upsilon ,\ast ,\circ )$ is an IFnNLS. Let $I_r$ and $h_r$ be as defined in Definition 2.5. Define a sequence $x=\{x_k\}$ whose terms are given by

$$x_k=\{\begin{array}{cc}(\frac{(n-[\sqrt{h_r}]+1)(-n+[\sqrt{h_r}])}{2},0,\dots ,0)\in {\mathbb{R}}^n\hfill & \text{if }1\le k\le n-[\sqrt{h_r}],\hfill \\ (-\frac{1}{2}{k}^2+\frac{1}{2}k,0,\dots ,0)\in {\mathbb{R}}^n\hfill & \text{if }n-[\sqrt{h_r}]+1\le k\le n,\hfill \\ (-\frac{1}{2}{n}^2-\frac{1}{2}n,0,\dots ,0)\in {\mathbb{R}}^n\hfill & \text{if }k>n\hfill \end{array}$$

such that

$$\mathrm{\Delta }{x}_k=\{\begin{array}{cc}(k,0,\dots ,0)\in \mathbb{N}\hfill & \text{if }n-[\sqrt{h_r}]+1\le k\le n,\hfill \\ (0,0,\dots ,0)\in \mathbb{N}\hfill & \text{otherwise}.\hfill \end{array}$$

For every $0<\epsilon <1$ and for any $y_1,y_2,\dots ,y_{n-1}\in X$, $t>0$, let

$$\begin{array}{c}K(\epsilon ,t)=\{k\in I_r:\mu (y_1,y_2,\dots ,y_{n-1},\mathrm{\Delta }{x}_k-L,t)\le 1-\epsilon \hfill \\ \text{or }\upsilon (y_1,y_2,\dots ,y_{n-1},\mathrm{\Delta }{x}_k-L,t)\ge \epsilon \}.\hfill \end{array}$$

Now,

$$\begin{array}{rl}K(\epsilon ,t)& =\{k\in I_r:\parallel y_1,y_2,\dots ,y_{n-1},\mathrm{\Delta }{x}_k\parallel \ge \frac{\epsilon t}{1-\epsilon }>0\}\\ \subseteq \{k\in I_r:\mathrm{\Delta }{x}_k=(k,0,\dots ,0)\in {\mathbb{R}}^n\}.\end{array}$$

Thus we have $\frac{1}{h_r}|\{k\in I_r:k\in K(\epsilon ,t)\}|\le \frac{[\sqrt{h_r}]}{h_r}\to 0$ as $r\to \mathrm{\infty }$. Hence ${S_{\theta }}^{{(\mu ,\upsilon )}^n}(\mathrm{\Delta })\text{-}limx=0$.

On the other hand, $x=\{x_k\}$ in X is not Δ-convergent to 0 with respect to the intuitionistic fuzzy n-norm since

$$\begin{array}{rl}\mu (y_1,y_2,\dots ,y_{n-1},\mathrm{\Delta }{x}_k,t)& =\frac{t}{t+\parallel y_1,y_2,\dots ,y_{n-1},\mathrm{\Delta }{x}_k\parallel }\\ =\{\begin{array}{cc}\frac{t}{t+\parallel y_1,y_2,\dots ,y_{n-1},\mathrm{\Delta }{x}_k\parallel }\hfill & \text{if }n-[\sqrt{h_r}]+1\le k\le n,\hfill \\ 1,\hfill & \text{otherwise},\hfill \end{array}\\ \le 1\end{array}$$

and

$$\begin{array}{rl}\upsilon (y_1,y_2,\dots ,y_{n-1},\mathrm{\Delta }{x}_k,t)& =\frac{\parallel y_1,y_2,\dots ,y_{n-1},\mathrm{\Delta }{x}_k\parallel }{t+\parallel y_1,y_2,\dots ,y_{n-1},\mathrm{\Delta }{x}_k\parallel }\\ =\{\begin{array}{cc}\frac{\parallel y_1,y_2,\dots ,y_{n-1},\mathrm{\Delta }{x}_k\parallel }{t+\parallel y_1,y_2,\dots ,y_{n-1},\mathrm{\Delta }{x}_k\parallel }\hfill & \text{if }n-[\sqrt{h_r}]+1\le k\le n,\hfill \\ 0,\hfill & \text{otherwise}\hfill \end{array}\\ \ge 0.\end{array}$$

This completes the proof of the theorem. □

Theorem 3.3 Let $(X,\mu ,\upsilon ,\ast ,\circ )$ be an IFnNLS. Then ${S_{\theta }}^{{(\mu ,\upsilon )}^n}(\mathrm{\Delta })\text{-}limx=L$ if and only if there exists an increasing sequence $K=\{k_n\}$ of the natural numbers such that ${\delta }_{\theta }(\mathrm{\Delta })(K)=1$ and ${(\mu ,\upsilon )}^n\text{-}{lim}_{k\in K}\mathrm{\Delta }{x}_k=L$.

Proof Necessity. Suppose that ${S_{\theta }}^{{(\mu ,\upsilon )}^n}(\mathrm{\Delta })\text{-}limx=L$. Then, for every $y_1,y_2,\dots ,y_{n-1}\in \mathbb{X}$, $t>0$ and $j=1,2,\dots$ ,

$$\begin{array}{c}\begin{array}{rl}K(j,t)=& \{k\in \mathbb{N}:\mu (y_1,y_2,\dots ,y_{n-1},\mathrm{\Delta }{x}_k-L,t)>1-\frac{1}{j}\\ \text{and }\upsilon (y_1,y_2,\dots ,y_{n-1},\mathrm{\Delta }{x}_k-L,t)<\frac{1}{j}\}\text{and}\end{array}\hfill \\ \begin{array}{rl}M(j,t)=& \{k\in \mathbb{N}:\mu (y_1,y_2,\dots ,y_{n-1},\mathrm{\Delta }{x}_k-L,t)\le 1-\frac{1}{j}\\ \text{or }\upsilon (y_1,y_2,\dots ,y_{n-1},\mathrm{\Delta }{x}_k-L,t)\ge \frac{1}{j}\}.\end{array}\hfill \end{array}$$

Then ${\delta }_{\theta }(\mathrm{\Delta })(M(j,t))=0$ since

$$K(j,t)\supset K(j+1,t)$$

(3.1)

and

$${\delta }_{\theta }(\mathrm{\Delta })(K(j,t))=1$$

(3.2)

for $t>0$ and $j=1,2,\dots$ . Now we have to show that for $k\in K(j,t)$ suppose that for some $k\in K(j,t)$, $x=\{x_k\}$ not Δ-convergent to L with respect to the intuitionistic fuzzy n-norm ${(\mu ,\upsilon )}^n$. Therefore there is $\alpha >0$ and a positive integer $k_0$ such that $\mu (y_1,y_2,\dots ,y_{n-1},\mathrm{\Delta }{x}_k-L,t)\le 1-\alpha$ or $\upsilon (y_1,y_2,\dots ,y_{n-1},\mathrm{\Delta }{x}_k-L,t)\ge \alpha$ for all $k\ge k_0$. Let $\alpha >\frac{1}{j}$ and

$$\begin{array}{rl}K(\alpha ,t)=& \{k\in \mathbb{N}:\mu (y_1,y_2,\dots ,y_{n-1},\mathrm{\Delta }{x}_k-L,t)>1-\alpha \\ \text{and }\upsilon (y_1,y_2,\dots ,y_{n-1},\mathrm{\Delta }{x}_k-L,t)<\alpha \}.\end{array}$$

Then ${\delta }_{\theta }(\mathrm{\Delta })(K(\alpha ,t))=0$. Since $\alpha >\frac{1}{j}$, by (3.1) we have ${\delta }_{\theta }(\mathrm{\Delta })(K(j,t))=0$, which contradicts by equation (3.2).

Sufficiency. Suppose that there exists an increasing sequence $K=\{k_n\}$ of the natural numbers such that ${\delta }_{\theta }(\mathrm{\Delta })(K)=1$ and ${(\mu ,\upsilon )}^n\text{-}{lim}_{k\in K}\mathrm{\Delta }{x}_k=L$, i.e., for every $y_1,y_2,\dots ,y_{n-1}\in \mathbb{X}$, $\epsilon >0$ and $t>0$, there exists $n_0\in \mathbb{N}$ such that $\mu (y_1,y_2,\dots ,y_{n-1},\mathrm{\Delta }{x}_k-L,t)>1-\epsilon$ and $\upsilon (y_1,y_2,\dots ,y_{n-1},\mathrm{\Delta }{x}_k-L,t)<\epsilon$.

Let

$$\begin{array}{rl}M(\epsilon ,t):=& \{k\in \mathbb{N}:\mu (y_1,y_2,\dots ,y_{n-1},\mathrm{\Delta }{x}_k-L,t)\le 1-\epsilon \\ \text{or }\upsilon (y_1,y_2,\dots ,y_{n-1},\mathrm{\Delta }{x}_k-L,t)\ge \epsilon \}\\ \subseteq & -\{k_{n_0+1},k_{n_0+2},\dots \}\end{array}$$

and consequently ${\delta }_{\theta }(\mathrm{\Delta })(M(\epsilon ,t))\le 1-1=0$. Hence ${S_{\theta }}^{{(\mu ,\upsilon )}^n}(\mathrm{\Delta })\text{-}limx=L$. This completes proof of the theorem. □

Theorem 3.4 Let $(X,\mu ,\upsilon ,\ast ,\circ )$ be an IFnNLS. Then ${S_{\theta }}^{{(\mu ,\upsilon )}^n}(\mathrm{\Delta })\text{-}limx=L$ if and only if there exist a convergent sequence $y=\{y_k\}$ and a lacunary Δ-statistically null sequence $z=\{z_k\}$ with respect to the intuitionistic fuzzy n-norm ${(\mu ,\upsilon )}^n$ such that ${(\mu ,\upsilon )}^n\text{-}limy=L$, $\mathrm{\Delta }x=y+\mathrm{\Delta }z$ and ${\delta }_{\theta }(\mathrm{\Delta })(\{k\in \mathbb{N}:\mathrm{\Delta }{z}_k=0\})=1$.

Proof Necessity. Suppose that ${S_{\theta }}^{{(\mu ,\upsilon )}^n}(\mathrm{\Delta })\text{-}limx=L$ and

$$\begin{array}{rl}K(j,t)=& \{k\in \mathbb{N}:\mu (y_1,y_2,\dots ,y_{n-1},\mathrm{\Delta }{x}_k-L,t)>1-\frac{1}{j}\\ \text{and }\upsilon (y_1,y_2,\dots ,y_{n-1},\mathrm{\Delta }{x}_k-L,t)<\frac{1}{j}\}.\end{array}$$

Using Theorem 3.3 for any $y_1,y_2,\dots ,y_{n-1}\in X$, $t>0$ and $j\in \mathbb{N}$, we can construct an increasing index sequence $\{r_j\}$ of the natural numbers such that $r_j\in K(j,t)$, ${\delta }_{\theta }(\mathrm{\Delta })(K(j,t))=1$, and so we can conclude that for all $r>r_j$ ($j\in \mathbb{N}$),

$$\begin{array}{r}\frac{1}{h_r}\left|\right\{k\in I_r:\mu (y_1,y_2,\dots ,y_{n-1},\mathrm{\Delta }{x}_k-L,t)>1-\frac{1}{j}\\ \text{and }\upsilon (y_1,y_2,\dots ,y_{n-1},\mathrm{\Delta }{x}_k-L,t)<\frac{1}{j}\left\}\right|>\frac{j-1}{j}.\end{array}$$

We define $y=\{y_k\}$ and $z=\{z_k\}$ as follows. If $1<k<r_1$, we set $y_k=\mathrm{\Delta }{x}_k$ and $z_k=0$. Now suppose that $j\ge 1$ and $r_j<k\le r_{j+1}$. If $k\in K(j,t)$, i.e., $\mu (y_1,y_2,\dots ,y_{n-1},\mathrm{\Delta }{x}_k-L,t)>1-\frac{1}{j}$ and $\upsilon (y_1,y_2,\dots ,y_{n-1},\mathrm{\Delta }{x}_k-L,t)<\frac{1}{j}$, we set $y_k=\mathrm{\Delta }{x}_k$ and $\mathrm{\Delta }{z}_k=0$. Otherwise $y_k=L$ and $\mathrm{\Delta }{z}_k=\mathrm{\Delta }{x}_k-L$. Hence it is clear that $\mathrm{\Delta }x=y+\mathrm{\Delta }z$.

We claim that ${(\mu ,\upsilon )}^n\text{-}limy=L$. Let $\epsilon >\frac{1}{j}$. If $k\in K(j,t)$ for all $k>r_j$, $\mu (y_1,y_2,\dots ,y_{n-1},y_k-L,t)>1-\epsilon$ and $\upsilon (y_1,y_2,\dots ,y_{n-1},y_k-L,t)<\epsilon$. Since ε was arbitrary, we have proved the claim.

Next we claim that $z=\{z_k\}$ is a lacunary Δ-statistically null sequence with respect to the intuitionistic fuzzy n-norm ${(\mu ,\upsilon )}^n$, i.e., ${S_{\theta }}^{{(\mu ,\upsilon )}^n}(\mathrm{\Delta })\text{-}limz=0$. It suffices to see that ${\delta }_{\theta }(\mathrm{\Delta })(\{k\in \mathbb{N}:\mathrm{\Delta }{z}_k=0\})=1$ to prove the claim. This follows from observing that

$$\begin{array}{r}|\{k\in I_r:\mathrm{\Delta }{z}_k=0\}|\\ \le \left|\{k\in I_r:\mu (y_1,y_2,\dots ,y_{n-1},\mathrm{\Delta }{z}_k,t)>1-\epsilon \text{ and }\upsilon (y_1,y_2,\dots ,y_{n-1},\mathrm{\Delta }{z}_k,t)<\epsilon \}\right|\end{array}$$

for any $r\in \mathbb{N}$ and $\epsilon >0$.

We show that if $\delta >0$ and $j\in \mathbb{N}$ such that $\frac{1}{j}<\delta$, then

$$\frac{1}{h_r}|\{k\in I_r:\mathrm{\Delta }{z}_k=0\}|>1-\delta$$

for all $r>r_j$. Recall from the construction that if $k\in K(j,t)$, then $\mathrm{\Delta }{z}_k=0$ for $r_j<k\le r_{j+1}$.

Now, for $t>0$ and $s\in \mathbb{N}$, let

$$\begin{array}{rl}K(s,t)=& \{k\in \mathbb{N}:\mu (y_1,y_2,\dots ,y_{n-1},\mathrm{\Delta }{x}_k-L,t)>1-\frac{1}{s}\\ \text{and }\upsilon (y_1,y_2,\dots ,y_{n-1},\mathrm{\Delta }{x}_k-L,t)<\frac{1}{s}\}.\end{array}$$

For $s>j$ and $r_s<k\le r_{s+1}$ by (3.2),

$$\begin{array}{rl}K(s,t)=& \{k\in \mathbb{N}:\mu (y_1,y_2,\dots ,y_{n-1},\mathrm{\Delta }{x}_k-L,t)>1-\frac{1}{s}\\ \text{and }\upsilon (y_1,y_2,\dots ,y_{n-1},\mathrm{\Delta }{x}_k-L,t)<\frac{1}{s}\}\\ \subset & \{k\in \mathbb{N}:\mathrm{\Delta }{z}_k=0\}.\end{array}$$

Consequently, if $r_s<k\le r_{s+1}$ and $s>j$, then

$$\begin{array}{r}\frac{1}{h_r}|\{k\in I_r:\mathrm{\Delta }{z}_k=0\}|\\ \ge \frac{1}{h_r}\left|\right\{k\in I_r:\mu (y_1,y_2,\dots ,y_{n-1},\mathrm{\Delta }{x}_k-L,t)>1-\frac{1}{s}\\ \text{and }\upsilon (y_1,y_2,\dots ,y_{n-1},\mathrm{\Delta }{x}_k-L,t)<\frac{1}{s}\left\}\right|\\ >1-\frac{1}{s}>1-\frac{1}{j}>1-\delta .\end{array}$$

Hence we get ${\delta }_{\theta }(\mathrm{\Delta })(\{k\in \mathbb{N}:\mathrm{\Delta }{z}_k=0\})=1$, which establishes the claim.

Sufficiency. Let x, y and z be sequences such that ${(\mu ,\upsilon )}^n\text{-}limy=L$, $\mathrm{\Delta }x=y+\mathrm{\Delta }z$ and ${\delta }_{\theta }(\mathrm{\Delta })(\{k\in \mathbb{N}:\mathrm{\Delta }{z}_k=0\})=1$. Then, for any $y_1,y_2,\dots ,y_{n-1}\in X$, $\epsilon >0$ and $t>0$, we have

$$\begin{array}{r}\{k\in \mathbb{N}:\mu (y_1,y_2,\dots ,y_{n-1},\mathrm{\Delta }{x}_k-L,t)\le 1-\epsilon \text{ or }\upsilon (y_1,y_2,\dots ,y_{n-1},\mathrm{\Delta }{x}_k-L,t)\ge \epsilon \}\\ \subseteq \{k\in \mathbb{N}:\mu (y_1,y_2,\dots ,y_{n-1},y_k-L,t)\le 1-\epsilon \text{ or }\upsilon (y_1,y_2,\dots ,y_{n-1},y_k-L,t)\ge \epsilon \}\\ \cup \{k\in \mathbb{N}:\mathrm{\Delta }{z}_k\ne 0\}.\end{array}$$

Therefore

$$\begin{array}{r}{\delta }_{\theta }(\mathrm{\Delta })\left(\right\{k\in \mathbb{N}:\mu (y_1,y_2,\dots ,y_{n-1},\mathrm{\Delta }{x}_k-L,t)\le 1-\epsilon \\ \text{or }\upsilon (y_1,y_2,\dots ,y_{n-1},\mathrm{\Delta }{x}_k-L,t)\ge \epsilon \left\}\right)\\ \le {\delta }_{\theta }\left(\{k\in \mathbb{N}:\mu (y_1,y_2,\dots ,y_{n-1},y_k-L,t)\le 1-\epsilon \text{ or }\upsilon (y_1,y_2,\dots ,y_{n-1},y_k-L,t)\ge \epsilon \}\right)\\ +{\delta }_{\theta }(\mathrm{\Delta })(\{k\in \mathbb{N}:\mathrm{\Delta }{z}_k\ne 0\}).\end{array}$$

Since ${(\mu ,\upsilon )}^n\text{-}limy=L$, the set

$$\{k\in \mathbb{N}:\mu (y_1,y_2,\dots ,y_{n-1},y_k-L,t)\le 1-\epsilon \text{ or }\upsilon (y_1,y_2,\dots ,y_{n-1},y_k-L,t)\ge \epsilon \}$$

contains at most finitely many terms and thus

$${\delta }_{\theta }\left(\{k\in \mathbb{N}:\mu (y_1,y_2,\dots ,y_{n-1},y_k-L,t)\le 1-\epsilon \text{ or }\upsilon (y_1,y_2,\dots ,y_{n-1},y_k-L,t)\ge \epsilon \}\right).$$

Also by hypothesis, ${\delta }_{\theta }(\mathrm{\Delta })(\{k\in \mathbb{N}:\mathrm{\Delta }{z}_k\ne 0\})$. Hence,

$$\begin{array}{r}{\delta }_{\theta }(\mathrm{\Delta })\left(\right\{k\in \mathbb{N}:\mu (y_1,y_2,\dots ,y_{n-1},\mathrm{\Delta }{x}_k-L,t)\le 1-\epsilon \\ \text{or }\upsilon (y_1,y_2,\dots ,y_{n-1},\mathrm{\Delta }{x}_k-L,t)\ge \epsilon \left\}\right)=0,\end{array}$$

and consequently ${S_{\theta }}^{{(\mu ,\upsilon )}^n}(\mathrm{\Delta })\text{-}limx=L$. □

4 Δ-Cauchy and lacunary Δ-statistically Cauchy sequences in IFnNLS

In this section, we introduce the notion of Cauchy sequences and lacunary statistically Cauchy sequences in IFnNLS.

Definition 4.1 Let $(X,\mu ,\upsilon ,\ast ,\circ )$ be an IFnNLS. A sequence $x=\{x_k\}$ in X is said to be Δ-Cauchy with respect to the intuitionistic fuzzy n-norm ${(\mu ,\upsilon )}^n$ if, for every $\epsilon >0$, $t>0$ and $y_1,y_2,\dots ,y_{n-1}\in \mathbb{X}$, there exists $k_0\in \mathbb{N}$ such that $\mu (y_1,y_2,\dots ,y_{n-1},\mathrm{\Delta }{x}_k-\mathrm{\Delta }{x}_m,t)>1-\epsilon$ and $\upsilon (y_1,y_2,\dots ,y_{n-1},\mathrm{\Delta }{x}_k-\mathrm{\Delta }{x}_m,t)<\epsilon$ for all $k,m\ge k_0$.

Definition 4.2 Let $(X,\mu ,\upsilon ,\ast ,\circ )$ be an IFnNLS. A sequence $x=\{x_k\}$ in X is said to be lacunary Δ-statistically Cauchy or $S_{\theta }(\mathrm{\Delta })$-Cauchy with respect to the intuitionistic fuzzy n-norm ${(\mu ,\upsilon )}^n$ if, for every $\epsilon >0$, $t>0$ and $y_1,y_2,\dots ,y_{n-1}\in \mathbb{X}$, there exists a number $m\in \mathbb{N}$ satisfying

$$\begin{array}{c}{\delta }_{\theta }(\mathrm{\Delta })\left(\right\{k\in \mathbb{N}:\mu (y_1,y_2,\dots ,y_{n-1},\mathrm{\Delta }{x}_k-\mathrm{\Delta }{x}_m,t)\le 1-\epsilon \hfill \\ \text{or }\upsilon (y_1,y_2,\dots ,y_{n-1},\mathrm{\Delta }{x}_k-\mathrm{\Delta }{x}_m,t)\ge \epsilon \left\}\right)=0.\hfill \end{array}$$

Theorem 4.1 Let $(X,\mu ,\upsilon ,\ast ,\circ )$ be an IFnNLS. If a sequence $x=\{x_k\}$ in X is lacunary Δ-statistically convergent with respect to the intuitionistic fuzzy n-norm ${(\mu ,\upsilon )}^n$ if and only if it is lacunary Δ-statistically Cauchy with respect to the intuitionistic fuzzy n-norm ${(\mu ,\upsilon )}^n$.

Proof Let $x=\{x_k\}$ be a lacunary Δ-statistically convergent sequence which converges to L. For a given $\epsilon >0$, choose $s>0$ such that $(1-\epsilon )\ast (1-\epsilon )>1-s$ and $\epsilon \circ \epsilon <s$. Let

$$\begin{array}{rl}A(\epsilon ,t)=& \{k\in \mathbb{N}:\mu (y_1,y_2,\dots ,y_{n-1},\mathrm{\Delta }{x}_k-L,t/2)\le 1-\epsilon \\ \text{or }\upsilon (y_1,y_2,\dots ,y_{n-1},\mathrm{\Delta }{x}_k-L,t/2)\ge \epsilon \}.\end{array}$$

Then, for any $t>0$ and $y_1,y_2,\dots ,y_{n-1}\in \mathbb{X}$,

$${\delta }_{\theta }(\mathrm{\Delta })(A(\epsilon ,t))=0,$$

(4.1)

which implies that ${\delta }_{\theta }(\mathrm{\Delta })(A^c(\epsilon ,t))=1$.

Let $q\in A^c(\epsilon ,t)$. Then

$$\mu (y_1,y_2,\dots ,y_{n-1},\mathrm{\Delta }{x}_q-L,t/2)>1-\epsilon$$

and

$$\upsilon (y_1,y_2,\dots ,y_{n-1},\mathrm{\Delta }{x}_k-L,t/2)<\epsilon .$$

Now, let

$$\begin{array}{rl}B(s,t)=& \{k\in \mathbb{N}:\mu (y_1,y_2,\dots ,y_{n-1},\mathrm{\Delta }{x}_k-\mathrm{\Delta }{x}_q,t)\le 1-s\\ \text{or }\upsilon (y_1,y_2,\dots ,y_{n-1},\mathrm{\Delta }{x}_k-\mathrm{\Delta }{x}_q,t)\ge s\}.\end{array}$$

We need to show that $B(s,t)\subset A(\epsilon ,t)$. Let $k\in B(s,t)\cap A^c(\epsilon ,t)$. Hence $\mu (y_1,y_2,\dots ,y_{n-1},\mathrm{\Delta }{x}_k-\mathrm{\Delta }{x}_q,t)\le 1-s$ and $\mu (y_1,y_2,\dots ,y_{n-1},\mathrm{\Delta }{x}_k-L,t/2)>1-\epsilon$, in particular, $\mu (y_1,y_2,\dots ,y_{n-1},\mathrm{\Delta }{x}_q-L,t/2)>1-\epsilon$. Then

$$\begin{array}{rl}1-s& \ge \mu (y_1,y_2,\dots ,y_{n-1},\mathrm{\Delta }{x}_k-\mathrm{\Delta }{x}_q,t)\\ \ge \mu (y_1,y_2,\dots ,y_{n-1},\mathrm{\Delta }{x}_k-L,t/2)\ast \mu (y_1,y_2,\dots ,y_{n-1},\mathrm{\Delta }{x}_q-L,t/2)\\ >(1-\epsilon )\ast (1-\epsilon )>1-s,\end{array}$$

which is not possible. On the other hand, $\upsilon (y_1,y_2,\dots ,y_{n-1},\mathrm{\Delta }{x}_k-\mathrm{\Delta }{x}_q,t)\ge s$ and $\upsilon (y_1,y_2,\dots ,y_{n-1},\mathrm{\Delta }{x}_k-L,t/2)<\epsilon$, in particular, $\upsilon (y_1,y_2,\dots ,y_{n-1},\mathrm{\Delta }{x}_q-L,t/2)<\epsilon$. Hence,

$$\begin{array}{rl}s& \le \upsilon (y_1,y_2,\dots ,y_{n-1},\mathrm{\Delta }{x}_k-\mathrm{\Delta }{x}_q,t)\\ \le \upsilon (y_1,y_2,\dots ,y_{n-1},\mathrm{\Delta }{x}_k-L,t/2)\circ \upsilon (y_1,y_2,\dots ,y_{n-1},\mathrm{\Delta }{x}_q-L,t/2)\\ <\epsilon \circ \epsilon <s,\end{array}$$

which is not possible. Hence $B(s,t)\subset A(\epsilon ,t)$ and by (4.1) ${\delta }_{\theta }(\mathrm{\Delta })(B(\epsilon ,t))=0$. This proves that x is lacunary Δ-statistically Cauchy with respect to the intuitionistic fuzzy n-norm ${(\mu ,\upsilon )}^n$.

Conversely, let $x=\{x_k\}$ be lacunary Δ-statistically Cauchy but not lacunary Δ-statistically convergent with respect to the intuitionistic fuzzy n-norm ${(\mu ,\upsilon )}^n$. For a given $\epsilon >0$, choose $s>0$ such that $(1-\epsilon )\ast (1-\epsilon )>1-s$ and $\epsilon \circ \epsilon <s$. Since x is not lacunary Δ-convergent

$$\begin{array}{c}\begin{array}{r}\mu (y_1,y_2,\dots ,y_{n-1},\mathrm{\Delta }{x}_k-\mathrm{\Delta }{x}_m,t)\\ \ge \mu (y_1,y_2,\dots ,y_{n-1},\mathrm{\Delta }{x}_k-L,t/2)\ast \mu (y_1,y_2,\dots ,y_{n-1},\mathrm{\Delta }{x}_q-L,t/2)\\ >(1-\epsilon )\ast (1-\epsilon )>1-s,\end{array}\hfill \\ \begin{array}{r}\upsilon (y_1,y_2,\dots ,y_{n-1},\mathrm{\Delta }{x}_k-\mathrm{\Delta }{x}_m,t)\\ \le \upsilon (y_1,y_2,\dots ,y_{n-1},\mathrm{\Delta }{x}_k-L,t/2)\circ \upsilon (y_1,y_2,\dots ,y_{n-1},\mathrm{\Delta }{x}_q-L,t/2)\\ <\epsilon \circ \epsilon <s.\end{array}\hfill \end{array}$$

Therefore ${\delta }_{\theta }(\mathrm{\Delta })(E^c(s,t))=0$, where

$$\begin{array}{rl}B(s,t)=& \{k\in \mathbb{N}:\mu (y_1,y_2,\dots ,y_{n-1},\mathrm{\Delta }{x}_k-\mathrm{\Delta }{x}_q,t)\le 1-s\\ \text{or }\upsilon (y_1,y_2,\dots ,y_{n-1},\mathrm{\Delta }{x}_k-\mathrm{\Delta }{x}_q,t)\ge s\}\end{array}$$

and so ${\delta }_{\theta }(\mathrm{\Delta })(E(s,t))=1$, which is a contradiction, since x was lacunary Δ-statistically Cauchy with respect to the intuitionistic fuzzy n-norm ${(\mu ,\upsilon )}^n$. So, x must be lacunary Δ-statistically convergent with respect to the intuitionistic fuzzy n-norm ${(\mu ,\upsilon )}^n$. □

Corollary 4.1 Let $(X,\mu ,\upsilon ,\ast ,\circ )$ be an IFnNLS and θ be a lacunary sequence. Then, for any sequence $x=\{x_k\}$ in X, the following conditions are equivalent:

• (i) x is $S_{\theta }(\mathrm{\Delta })$-convergent with respect to the intuitionistic fuzzy n-norm ${(\mu ,\upsilon )}^n$.

• (ii) x is $S_{\theta }(\mathrm{\Delta })$-Cauchy with respect to the intuitionistic fuzzy n-norm ${(\mu ,\upsilon )}^n$.

• (iii) There exists an increasing sequence $K=\{k_n\}$ of the natural numbers such that ${\delta }_{\theta }(\mathrm{\Delta })(K)=1$ and the subsequence $\{x_{k_n}\}$ is $S_{\theta }(\mathrm{\Delta })$-Cauchy with respect to the intuitionistic fuzzy n-norm ${(\mu ,\upsilon )}^n$.