On generalized double statistical convergence in a random 2-normed space

Abstract

Recently, the concept of statistical convergence has been studied in 2-normed and random 2-normed spaces by various authors. In this paper, we shall introduce the concept of λ-double statistical convergence and λ-double statistical Cauchy in a random 2-normed space. We also shall prove some new results.

MSC:40A05, 40B50, 46A19, 46A45.

1 Introduction

The probabilistic metric space was introduced by Menger [1] which is an interesting and an important generalization of the notion of a metric space. The theory of probabilistic normed (or metric) space was initiated and developed in [2–6]; further it was extended to random/probabilistic 2-normed spaces by Goleţ [7] using the concept of 2-norm which is defined by Gähler (see [8, 9]); and Gürdal and Pehlivan [10] studied statistical convergence in 2-normed spaces. Also statistical convergence in 2-Banach spaces was studied by Gürdal and Pehlivan in [11]. Moreover, recently some new sequence spaces have been studied by Savas [12–14] by using 2-normed spaces.

In order to extend the notion of convergence of sequences, statistical convergence of sequences was introduced by Fast [15] and Schoenberg [16] independently. A lot of developments have been made in this areas after the works of S̆alát [17] and Fridy [18]. Over the years and under different names, statistical convergence has been discussed in the theory of Fourier analysis, ergodic theory and number theory. Recently, Mursaleen [19] studied λ-statistical convergence as a generalization of the statistical convergence, and in [20] he considered the concept of statistical convergence of sequences in random 2-normed spaces. Quite recently, Bipan and Savas [21] defined lacunary statistical convergence in a random 2-normed space, and also Savas [22] studied λ-statistical convergence in a random 2-normed space.

The notion of statistical convergence depends on the density of subsets of N, the set of natural numbers. Let K be a subset of N. Then the asymptotic density of K denoted by $\delta (K)$ is defined as

$$\delta (K)=\underset{n\to \mathrm{\infty }}{lim}\frac{1}{n}|\{k\le n:k\in K\}|,$$

where the vertical bars denote the cardinality of the enclosed set.

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

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

In this case we write $S-limx=\ell$ or $x_k\to \ell (S)$ (see [15, 18]).

2 Definitions and preliminaries

We begin by recalling some notations and definitions which will be used in this paper.

Definition 1 A function $f:\mathbf{R}\to {\mathbf{R}}_0^{+}$ is called a distribution function if it is a non-decreasing and left continuous with ${inf}_{t\in \mathbf{R}}f(t)=0$ and ${sup}_{t\in \mathbf{R}}f(t)=1$. By $D^{+}$, we denote the set of all distribution functions such that $f(0)=0$. If $a\in {\mathbf{R}}_0^{+}$, then $H_a\in D^{+}$, where

$$H_a(t)=\{\begin{array}{cc}1,\hfill & \text{if }t>a;\hfill \\ 0,\hfill & \text{if }t\le a.\hfill \end{array}$$

It is obvious that $H_0\ge f$ for all $f\in D^{+}$.

A t-norm is a continuous mapping $\ast :[0,1]\times [0,1]\to [0,1]$ such that $([0,1],\ast )$ is an Abelian monoid with unit one and $c\ast d\ge a\ast b$ if $c\ge a$ and $d\ge b$ for all $a,b,c,d\in [0,1]$. A triangle function τ is a binary operation on $D^{+}$, which is commutative, associative and $\tau (f,H_0)=f$ for every $f\in D^{+}$.

In [8], Gähler introduced the following concept of a 2-normed space.

Definition 2 Let X be a real vector space of dimension $d>1$ (d may be infinite). A real-valued function $\parallel \cdot ,\cdot \parallel$ from $X^2$ into R satisfying the following conditions:

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

• (2) $\parallel x_1,x_2\parallel$ is invariant under permutation,

• (3) $\parallel \alpha x_1,x_2\parallel =|\alpha |\parallel x_1,x_2\parallel$, for any $\alpha \in \mathbf{R}$,

• (4) $\parallel x+\overline{x},x_2\parallel \le \parallel x,x_2\parallel +\parallel \overline{x},x_2\parallel$

is called a 2-norm on X and the pair $(X,\parallel \cdot ,\cdot \parallel )$ is called a 2-normed space.

A trivial example of a 2-normed space is $X={\mathbf{R}}^2$, equipped with the Euclidean 2-norm ${\parallel x_1,x_2\parallel }_E$ = the area of the parallelogram spanned by the vectors $x_1$, $x_2$ which may be given explicitly by the formula

$${\parallel x_1,x_2\parallel }_E=|det(x_{ij})|=abs(det(〈x_i,x_j〉)),$$

where $x_i=(x_{i1},x_{i2})\in {\mathbf{R}}^2$ for each $i=1,2$.

Recently, Goleţ [7] used the idea of a 2-normed space to define a random 2-normed space.

Definition 3 Let X be a linear space of dimension $d>1$ (d may be infinite), τ a triangle, and $\mathcal{F}:X\times X\to D^{+}$. Then $\mathcal{F}$ is called a probabilistic 2-norm and $(X,\mathcal{F},\tau )$ a probabilistic 2-normed space if the following conditions are satisfied:

($P2N_1$) $\mathcal{F}(x,y;t)=H_0(t)$ if x and y are linearly dependent, where $\mathcal{F}(x,y;t)$ denotes the value of $\mathcal{F}(x,y)$ at $t\in \mathbf{R}$,

($P2N_2$) $\mathcal{F}(x,y;t)\ne H_0(t)$ if x and y are linearly independent,

($P2N_3$) $\mathcal{F}(x,y;t)=\mathcal{F}(y,x;t)$, for all $x,y\in X$,

($P2N_4$) $\mathcal{F}(\alpha x,y;t)=\mathcal{F}(x,y;\frac{t}{|\alpha |})$, for every $t>0$, $\alpha \ne 0$ and $x,y\in X$,

($P2N_5$) $\mathcal{F}(x+y,z;t)\ge \tau (\mathcal{F}(x,z;t),\mathcal{F}(y,z;t))$, whenever $x,y,z\in X$.

If ($P2N_5$) is replaced by

($P2N_6$) $\mathcal{F}(x+y,z;t_1+t_2)\ge \mathcal{F}(x,z;t_1)\ast \mathcal{F}(y,z;t_2)$, for all $x,y,z\in X$ and $t_1,t_2\in {\mathbf{R}}_0^{+}$;

then $(X,\mathcal{F},\ast )$ is called a random 2-normed space (for short, R2NS).

Remark 1 Every 2-normed space $(X,\parallel \cdot ,\cdot \parallel )$ can be made a random 2-normed space in a natural way by setting $\mathcal{F}(x,y;t)=H_0(t-\parallel x,y\parallel )$ for every $x,y\in X$, $t>0$ and $a\ast b=min\{a,b\}$, $a,b\in [0,1]$.

Example 1 Let $(X,\parallel \cdot ,\cdot \parallel )$ be a 2-normed space with $\parallel x,z\parallel =\parallel x_1{z}_2-x_2{z}_1\parallel$, $x=(x_1,x_2)$, $z=(z_1,z_2)$ and $a\ast b=ab$, $a,b\in [0,1]$. For all $x\in X$, $t>0$ and nonzero $z\in X$, consider

$$\mathcal{F}(x,z;t)=\{\begin{array}{cc}\frac{t}{t+\parallel x,z\parallel },\hfill & \text{if }t>0;\hfill \\ 0,\hfill & \text{if }t\le 0.\hfill \end{array}$$

Then $(X,F,\ast )$ is a random 2-normed space.

Definition 4 A sequence $x=(x_{k,l})$ in a random 2-normed space $(X,\mathcal{F},\ast )$ is said to be double convergent (or $\mathcal{F}$-convergent) to $\ell \in X$ with respect to $\mathcal{F}$ if for each $\epsilon >0$, $\eta \in (0,1)$, there exists a positive integer $n_0$ such that $\mathcal{F}(x_{k,l}-\ell ,z;\epsilon )>1-\eta$, whenever $k,l\ge n_0$ and for nonzero $z\in X$. In this case we write $\mathcal{F}-{lim}_{k,l}{x}_{k,l}=\ell$, and ℓ is called the $\mathcal{F}$-limit of $x=(x_{k,l})$.

Definition 5 A sequence $x=(x_{k,l})$ in a random 2-normed space $(X,\mathcal{F},\ast )$ is said to be double Cauchy with respect to $\mathcal{F}$ if for each $\epsilon >0$, $\eta \in (0,1)$ there exist $N=N(\epsilon )$ and $M=M(\epsilon )$ such that $\mathcal{F}(x_{k,l}-x_{p,q},z;\epsilon )>1-\eta$, whenever $k,p\ge N$ and $l,q\ge M$ and for nonzero $z\in X$.

Definition 6 A sequence $x=(x_{k,l})$ in a random 2-normed space $(X,\mathcal{F},\ast )$ is said to be double statistically convergent or $S^{2R2N}$-convergent to some $\ell \in X$ with respect to $\mathcal{F}$ if for each $\epsilon >0$, $\eta \in (0,1)$ and for nonzero $z\in X$ such that

$$\delta \left(\{(k,l)\in \mathbf{N}\times \mathbf{N}:\mathcal{F}(x_{k,l}-\ell ,z;\epsilon )\le 1-\eta \}\right)=0.$$

In other words, we can write the sequence $(x_{k,l})$double statistically converges to ℓ in random 2-normed space $(X,\mathcal{F},\ast )$ if

$$\underset{m,n\to \mathrm{\infty }}{lim}\frac{1}{mn}\left|\{k\le m,l\le n:\mathcal{F}(x_{k,l}-\ell ,z;\epsilon )\le 1-\eta \}\right|=0$$

or equivalently,

$$\delta \left(\{k,l\in \mathbf{N}:\mathcal{F}(x_{k,l}-\ell ,z;\epsilon )>1-\eta \}\right)=1,$$

i.e.,

$$S^2-\underset{k,l\to \mathrm{\infty }}{lim}\mathcal{F}(x_{k,l}-\ell ,z;\epsilon )=1.$$

In this case we write $S^{2R2N}-limx=\ell$, and ℓ is called the $S^{2R2N}$-limit of x. Let $S^{2R2N}(X)$ denote the set of all double statistically convergent sequences in a random 2-normed space $(X,\mathcal{F},\ast )$.

In this article, we study λ-double statistical convergence in a random 2-normed space which is a new and interesting idea. We show that some properties of λ-double statistical convergence of real numbers also hold for sequences in random 2-normed spaces. We establish some relations related to double statistically convergent and λ-double statistically convergent sequences in random 2-normed spaces.

3 λ-double statistical convergence in a random 2-normed space

Recently, the concept of λ-double statistical convergence has been introduced and studied in [23] and [24]. In this section, we define λ-double statistically convergent sequence in a random 2-normed space $(X,\mathcal{F},\ast )$. Also we get some basic properties of this notion in a random 2-normed space.

Definition 7 Let $\lambda =({\lambda }_n)$ and $\mu =({\mu }_n)$ be two non-decreasing sequences of positive real numbers such that each is tending to ∞ and

$${\lambda }_{n+1}\le {\lambda }_n+1,{\lambda }_1=1$$

and

$${\mu }_{n+1}\le {\mu }_n+1,{\mu }_1=1.$$

Let $K\subseteq \mathbb{N}\times \mathbb{N}$. The number

$${\delta }_{\overline{\lambda }}(K)=\underset{mn}{lim}\frac{1}{{\overline{\lambda }}_{mn}}\left|\{k\in I_n,l\in J_m:(k,l)\in K\}\right|,$$

where $I_n=[n-{\lambda }_n+1,n]$, $J_m=[m-{\mu }_m+1,m]$ and ${\overline{\lambda }}_{nm}={\lambda }_n{\mu }_m$, is said to be the λ-double density of K, provided the limit exists.

Definition 8 A sequence $x=(x_{k,l})$ is said to be λ-double statistically convergent or $S_{\overline{\lambda }}^2$-convergent to the number ℓ if for every $\epsilon >0$, the set $N(\epsilon )$ has λ-double density zero, where

$$N(\epsilon )=\{k\in I_n,l\in J_m:|x_{k,l}-\ell |\ge \epsilon \}.$$

In this case, we write $S_{\overline{\lambda }}^2-limx=L$.

Now we define λ-double statistical convergence in a random 2-normed space (see [25]).

Definition 9 A sequence $x=(x_{k,l})$ in a random 2-normed space $(X,\mathcal{F},\ast )$ is said to be λ-double statistically convergent or $S_{\overline{\lambda }}^2$-convergent to $\ell \in X$ with respect to $\mathcal{F}$ if for every $\epsilon >0$, $\eta \in (0,1)$ and for nonzero $z\in X$ such that

$${\delta }_{\overline{\lambda }}\left(\{k\in I_n,l\in J_m:\mathcal{F}(x_{k,l}-\ell ,z;\epsilon )\le 1-\eta \}\right)=0$$

or equivalently,

$${\delta }_{\overline{\lambda }}\left(\{k\in I_n,l\in J_m:\mathcal{F}(x_{k,l}-\ell ,z;\epsilon )>1-\eta \}\right)=1,$$

i.e.,

$$S_{\overline{\lambda }}^2-\underset{k,l\to \mathrm{\infty }}{lim}\mathcal{F}(x_{k,l}-\ell ,z;\epsilon )=1.$$

In this case we write $S_{\overline{\lambda }}^{2R2N}-limx=\ell$ or $x_{k,l}\to \ell (S_{\overline{\lambda }}^{2R2N})$ and

$$S_{\overline{\lambda }}^{2R2N}(X)=\{x=(x_{k,l}):\mathrm{\exists }\ell \in \mathbf{R},S_{\overline{\lambda }}^{2R2N}-limx=\ell \}.$$

Let $S_{\overline{\lambda }}^{2R2N}(X)$ denote the set of all λ-double statistically convergent sequences in a random 2-normed space $(X,\mathcal{F},\ast )$.

If ${\overline{\lambda }}_{mn}=mn$ for every n, m then λ-double statistically convergent sequences in a random 2-normed space $(X,\mathcal{F},\ast )$ reduce to double statistically convergent sequences in a random 2-normed space $(X,\mathcal{F},\ast )$.

Definition 9 immediately implies the following lemma.

Lemma 1 Let$(X,\mathcal{F},\ast )$be a random 2-normed space. If$x=(x_{k,l})$is a sequence in X, then for every$\epsilon >0$, $\eta \in (0,1)$and for nonzero$z\in X$, the following statements are equivalent:

• (i) $S_{\overline{\lambda }}^{R2N}-{lim}_{k,l\to \mathrm{\infty }}{x}_{k,l}=\ell$;

• (ii) ${\delta }_{\overline{\lambda }}(\{k\in I_n,l\in J_m:\mathcal{F}(x_{k,l}-\ell ,z;\epsilon )\le 1-\eta \})=0$;

• (iii) ${\delta }_{\overline{\lambda }}(\{k\in I_n,l\in J_m:\mathcal{F}(x_{k,l}-\ell ,z;\epsilon )>1-\eta \})=1$;

• (iv) $S_{\overline{\lambda }}-{lim}_{k,l\to \mathrm{\infty }}\mathcal{F}(x_{k,l}-\ell ,z;\epsilon )=1$.

Theorem 1 Let$(X,\mathcal{F},\ast )$be a random 2-normed space. If$x=(x_{k,l})$is a sequence in X such that$S_{\overline{\lambda }}^{2R2N}-lim{x}_{k,l}=\ell$exists, then it is unique.

Proof Suppose that $S_{\overline{\lambda }}^{2R2N}-{lim}_{k,l\to \mathrm{\infty }}{x}_{k,l}={\ell }_1;S_{\overline{\lambda }}^{2R2N}-{lim}_{k,l\to \mathrm{\infty }}{x}_{k,l}={\ell }_2$, where $({\ell }_1\ne {\ell }_2)$.

Let $\epsilon >0$ be given. Choose $a>0$ such that $(1-a)\ast (1-a)>1-\epsilon$.

Then, for any $t>0$ and for nonzero $z\in X$, we define

Since $S_{\overline{\lambda }}^{2R2N}-{lim}_{k,l\to \mathrm{\infty }}{x}_{k,l}={\ell }_1$ and $S_{\overline{\lambda }}^{2R2N}-{lim}_{k,l\to \mathrm{\infty }}{x}_{k,l}={\ell }_2$, we have Lemma 1 ${\delta }_{\overline{\lambda }}(K_1(a,t))=0$ and ${\delta }_{\overline{\lambda }}(K_2(a,t))=0$ for all $t>0$.

Now, let $K(a,t)=K_1(a,t)\cup K_2(a,t)$, then it is easy to observe that ${\delta }_{\overline{\lambda }}(K(a,t))=0$. But we have ${\delta }_{\overline{\lambda }}(K^c(r,t))=1$.

Now, if $(k,l)\in K^c(a,t)$, then we have

$$\mathcal{F}({\ell }_1-{\ell }_2,z;t)\ge \mathcal{F}(x_{k,l}-{\ell }_1,z;\frac{t}{2})\ast \mathcal{F}(x_{k,l}-{\ell }_2,z;\frac{t}{2})>(1-a)\ast (1-a).$$

It follows that

$$\mathcal{F}({\ell }_1-{\ell }_2,z;t)>(1-\epsilon ).$$

Since $\epsilon >0$ was arbitrary, we get $\mathcal{F}({\ell }_1-{\ell }_2,z;t)=0$ for all $t>0$ and nonzero $z\in X$. Hence ${\ell }_1={\ell }_2$.

This completes the proof. □

Next theorem gives the algebraic characterization of λ-statistical convergence on random 2-normed spaces. We give it without proof.

Theorem 2 Let$(X,\mathcal{F},\ast )$be a random 2-normed space, and$x=(x_{k,l})$and$y=(y_{k,l})$be two sequences in X.

• (a) If $S_{\overline{\lambda }}^{2R2N}-lim{x}_{k,l}=\ell$ and $c(\ne 0)\in \mathbf{R}$, then $S_{\overline{\lambda }}^{2R2N}-limc{x}_{k,l}=c\ell$.

• (b) If $S_{\overline{\lambda }}^{2R2N}-lim{x}_{k,l}={\ell }_1$ and $S_{\overline{\lambda }}^{R2N}-lim{y}_{k,l}={\ell }_2$, then $S_{\overline{\lambda }}^{2R2N}-lim(x_{k,l}+y_{k,l})={\ell }_1+{\ell }_2$.

Theorem 3 Let$(X,\mathcal{F},\ast )$be a random 2-normed space. If$x=(x_{k,l})$is a sequence in X such that$\mathcal{F}-lim{x}_{k,l}=\ell$, then$S_{\overline{\lambda }}^{2R2N}-lim{x}_{k,l}=\ell$.

Proof Let $\mathcal{F}-lim{x}_{k,l}=\ell$. Then for every $\epsilon >0$, $t>0$ and nonzero $z\in X$, there is a positive integer $n_0$ and $m_0$ such that

$$\mathcal{F}(x_k-\ell ,z;t)>1-\epsilon$$

for all $k\ge n_0$. Since the set

$$K(\epsilon ,t)=\{k\in I_n,l\in J_m:\mathcal{F}(x_{k,l}-\ell ,z;t)\le 1-\epsilon \}$$

has at most finitely many terms. Since every finite subset of $\mathbf{N}\times \mathbf{N}$ has ${\delta }_{\overline{\lambda }}$-density zero, finally we have ${\delta }_{\overline{\lambda }}(K(\epsilon ,t))=0$. This shows that $S_{\overline{\lambda }}^{2R2N}-lim{x}_{k,l}=\ell$. □

Remark 2 The converse of the above theorem is not true in general. It follows from the following example.

Example 2 Let $X={\mathbf{R}}^2$, with the 2-norm $\parallel x,z\parallel =|x_1{z}_2-x_2{z}_1|$, $x=(x_1,x_2)$, $z=(z_1,z_2)$ and $a\ast b=ab$ for all $a,b\in [0,1]$. Let $\mathcal{F}(x,y;t)=\frac{t}{t+\parallel x,y\parallel }$, for all $x,z\in X$, $z_2\ne 0$, and $t>0$. We define a sequence $x=(x_k)$ by

$$x_{k,l}=\{\begin{array}{cc}(kl,0),\hfill & \text{if }n-[\sqrt{{\lambda }_n}]+1\le k\le n\text{ and }m-[\sqrt{{\mu }_m}]+1\le k\le m;\hfill \\ (0,0),\hfill & \text{otherwise}.\hfill \end{array}$$

Now for every $0<\epsilon <1$ and $t>0$, we write

$$K_n(\epsilon ,t)=\{k\in I_n,l\in J_m:\mathcal{F}(x_{k,l}-\ell ,z;t)\le 1-\epsilon \}.$$

Therefore, we get

$${\delta }_{\overline{\lambda }}(K(\epsilon ,t))=\underset{nm\to \mathrm{\infty }}{lim}\frac{[\sqrt{{\overline{\lambda }}_{nm}}]}{{\overline{\lambda }}_{nm}}=0.$$

This shows that $S_{\overline{\lambda }}^{2R2N}-lim{x}_{k,l}=0$, while it is obvious that $\mathcal{F}-lim{x}_{k,l}\ne 0$.

Theorem 4 Let$(X,\mathcal{F},\ast )$be a random 2-normed space. If$x=(x_{k,l})$is a sequence in X, then$S_{\overline{\lambda }}^{2R2N}-lim{x}_{k,l}=\ell$if and only if there exists a subset$K=\{(k_n,l_n):k_1<k_2,\dots ;l_1<l_2,\dots \}\subseteq \mathbf{N}\times \mathbf{N}$such that${\delta }_{\overline{\lambda }}(K)=1$and$\mathcal{F}-{lim}_{n\to \mathrm{\infty }}{x}_{k_n,l_n}=\ell$.

Proof Suppose first that $S_{\lambda }^{2R2N}-lim{x}_{k,l}=\ell$. Then for any $t>0$, $a=1,2,3,\dots$ and nonzero $z\in X$, let

$$A(a,t)=\{k\in I_n;l\in J_m:\mathcal{F}(x_{k,l}-\ell ,z;t)>1-\frac{1}{a}\}$$

and

$$K(a,t)=\{k\in I_n;l\in J_m:\mathcal{F}(x_{k,l}-\ell ,z;t)\le 1-\frac{1}{a}\}.$$

Since $S_{\overline{\lambda }}^{2R2N}-lim{x}_{k,l}=\ell$, it follows that

$${\delta }_{\overline{\lambda }}(K(a,t))=0.$$

Now, for $t>0$ and $a=1,2,3,\dots$ , we observe that

$$A(a,t)\supset A(a+1,t)$$

and

$${\delta }_{\overline{\lambda }}(A(a,t))=1.$$

(3.1)

Now we have to show that for $(k,l)\in A(a,t),\mathcal{F}-lim{x}_{k,l}=\ell$. Suppose that for some $(k,l)\in A(a,t)$, $(x_{k,l})$ is not convergent to ℓ with respect to $\mathcal{F}$. Then there exist some $s>0$ and a positive integer $k_0$, $l_0$ such that

$$\{k\in I_n;l\in J_m:\mathcal{F}(x_{k,l}-\ell ,z;t)\le 1-s\}$$

for all $k\ge k_0$ and $l\ge l_0$. Let

$$A(s,t)=\{k\in I_n;l\in J_m:\mathcal{F}(x_{k,l}-\ell ,z;t)>1-s\}$$

for $k<k_0$ and $l<l_0$ and

$$s>\frac{1}{a},a=1,2,3,\dots .$$

Then we have

$${\delta }_{\overline{\lambda }}(A(s,t))=0.$$

Furthermore, $A(a,t)\subset A(s,t)$ implies that ${\delta }_{\overline{\lambda }}(A(a,t))=0$, which contradicts (3.1) as ${\delta }_{\overline{\lambda }}(A(a,t))=1$. Hence $\mathcal{F}-lim{x}_{k,l}=\ell$.

Conversely, suppose that there exists a subset $K=\{(k_n,l_n):k_1<k_2,\dots ;l_1<l_2,\dots \}\subseteq \mathbf{N}\times \mathbf{N}$ such that ${\delta }_{\overline{\lambda }}(K)=1$ and $\mathcal{F}-{lim}_{n,m\to \mathrm{\infty }}{x}_{k_n,l_n}=\ell$. Then for every $\epsilon >0$, $t>0$ and nonzero $z\in X$, we can find a positive integer $n_0$ such that

$$\mathcal{F}(x_{k,l},z;t)>1-\epsilon$$

for all $k,l\ge n_0$. If we take

$$K(\epsilon ,t)=\{k\in I_n;l\in J_m:\mathcal{F}(x_{k,l}-\ell ,z;t)\le 1-\epsilon \},$$

then it is easy to see that

$$K(\epsilon ,t)\subset \mathbf{N}\times \mathbf{N}-\{(k_{n_0+1},l_{n_0+1}),(k_{n_0+2},l_{n_0+2}),\dots \},$$

and finally,

$${\delta }_{\overline{\lambda }}(K(\epsilon ,t))\le 1-1=0.$$

Thus $S_{\overline{\lambda }}^{R2N}-lim{x}_{k,l}=\ell$. This completes the proof. □

We now have

Definition 10 A sequence $x=(x_{k,l})$ in a random 2-normed space $(X,\mathcal{F},\ast )$ is said to be λ-double statistically Cauchy with respect to $\mathcal{F}$ if for each $\epsilon >0$, $\eta \in (0,1)$ and for nonzero $z\in X$, there exist $N=N(\epsilon )$ and $M=M(\epsilon )$ such that for all $k,m>N$ and $l,n>M$,

$${\delta }_{\overline{\lambda }}\left(\{k\in I_n;l\in J_m:\mathcal{F}(x_{k,l}-x_{MN},z;\epsilon )\le 1-\eta \}\right)=0,$$

or equivalently,

$${\delta }_{\overline{\lambda }}\left(\{k\in I_n;l\in J_m:\mathcal{F}(x_{k,l}-x_{MN},z;\epsilon )>1-\eta \}\right)=1.$$

Theorem 5 Let$(X,\mathcal{F},\ast )$be a random 2-normed space. Then a sequence$(x_{k,l})$in X is λ-double statistically convergent if and only if it is λ-double statistically Cauchy in random 2-normed space X.

Proof Let $(x_{k,l})$ be a λ-double statistically convergent to ℓ with respect to random 2-normed space, i.e., $S_{\overline{\lambda }}^{2R2N}-lim{x}_k=\ell$. Let $\epsilon >0$ be given. Choose $a>0$ such that

$$(1-a)\ast (1-a)>1-\epsilon .$$

(3.2)

For $t>0$ and for nonzero $z\in X$, define

$$A(a,t)=\{k\in I_n;l\in J_m:\mathcal{F}(x_{k,l}-\ell ,z;\frac{t}{2})\le 1-a\}.$$

Then

$$A^c(a,t)=\{k\in I_n;l\in J_m:\mathcal{F}(x_{k,l}-\ell ,z;\frac{t}{2})>1-a\}.$$

Since $S_{\overline{\lambda }}^{2R2N}-lim{x}_{k,l}=\ell$, it follows that ${\delta }_{\overline{\lambda }}(A(a,t))=0$, and finally, ${\delta }_{\overline{\lambda }}(A^c(a,t))=1$.

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

$$\mathcal{F}(x_{p,q}-\ell ,z;\frac{t}{2})>1-a.$$

(3.3)

If we take

$$B(\epsilon ,t)=\{k\in I_n;l\in J_m:\mathcal{F}(x_{k,l}-x_{p,q},z;t)\le 1-\epsilon \},$$

then to prove the result it is sufficient to prove that $B(\epsilon ,t)\subseteq A(a,t)$.

Let $(k,l)\in B(\epsilon ,t)\cap A^c(a,t)$, then for nonzero $z\in X$, we have

$$\mathcal{F}(x_{k,l}-x_{p,q},z;t)\le 1-\epsilon \text{and}\mathcal{F}(x_{k,l}-\ell ,z;\frac{t}{2})>1-a.$$

(3.4)

Now, from (3.1), (3.3) and (3.4), we get

$$\begin{array}{rcl}1-\epsilon & \ge & \mathcal{F}(x_{k,l}-x_{p,q},z;t)\ge \mathcal{F}(x_{k,l}-\ell ,z;\frac{t}{2})\ast \mathcal{F}(x_p-\ell ,z;\frac{t}{2})\\ >& (1-a)\ast (1-a)>(1-\epsilon ),\end{array}$$

which is not possible. Thus $B(\epsilon ,t)\subset A(a,t)$. Since ${\delta }_{\overline{\lambda }}(A(a,t))=0$, it follows that ${\delta }_{\overline{\lambda }}(B(\epsilon ,t))=0$. This shows that $(x_{k,l})$ is λ-double statistically Cauchy.

Conversely, suppose $(x_{k,l})$ is λ-double statistically Cauchy but not λ-double statistically convergent with respect to $\mathcal{F}$. Then for each $\epsilon >0$, $t>0$ and for nonzero $z\in X$, there exist a positive integer $N=N(\epsilon )$ and $M=M(\epsilon )$ such that

$$A(\epsilon ,t)=\{k\in I_n;l\in J_m:\mathcal{F}(x_{k,l}-x_{NM},z;t)\le 1-\epsilon \}.$$

Then

$${\delta }_{\overline{\lambda }}(A(\epsilon ,t))=0$$

and

$${\delta }_{\overline{\lambda }}(A^c(\epsilon ,t))=1.$$

(3.5)

For $t>0$, choose $a>0$ such that

$$(1-a)\ast (1-a)>1-\epsilon$$

(3.6)

is satisfied, and we take

$$B(a,t)=\{k\in I_n;l\in J_m:\mathcal{F}(x_{k,l}-\ell ,z;\frac{t}{2})>1-a\}.$$

If $N,M\in B(a,t)$, then $\mathcal{F}(x_{N,M}-\ell ,z;\frac{t}{2})>1-a$.

Since

$$\mathcal{F}(x_{k,l}-x_{NM},z;t)\ge \mathcal{F}(x_{k,l}-\ell ,z;\frac{t}{2})\ast \mathcal{F}(x_{N,M}-\ell ,z;\frac{t}{2})>(1-a)\ast (1-a)>1-\epsilon ,$$

then we have

$${\delta }_{\overline{\lambda }}\left(\{x_{k,l}:\mathcal{F}(x_{k,l}-x_{NM},z;t)>1-\epsilon \}\right)=0,$$

i.e., ${\delta }_{\overline{\lambda }}(A^c(\epsilon ,t))=0$, which contradicts (3.5) as ${\delta }_{\overline{\lambda }}(A^c(\epsilon ,t))=1$. Hence $(x_{k,l})$ is λ-double statistically convergent.

This completes the proof. □