On $\nabla $-statistical convergence in random 2-normed space

Ayhan Esi$^\ast $

April 10, 2014.
Published online: January 23, 2015.

$^\ast $Adiyaman University, Science and Art Faculty, Department of Mathematics, 02040, Adiyaman, Turkey, e-mail: aesi23@hotmail.com.

Recently in [ 19 ] , Mursaleen introduced the concepts of statistical convergence in random 2-normed spaces. In this paper, we define and study the notion of $\nabla $-statistical convergence and $\nabla $-statistical Cauchy sequences using by $\lambda $-sequences in random 2-normed spaces and prove some theorems.

MSC. Primary 40A05; Secondary 46A70, 40A99, 46A99.

Keywords. Statistical convergence; $\lambda $-statistical convergence; $t$-norm; $2$-norm; random 2-normed space.

1 Introduction

The concept of statistical convergence play a vital role not only in pure mathematics but also in other branches of science involving mathematics, especially in information theory, computer science, biological science, dynamical systems, geographic information systems, population modelling, and motion planning in robotics.

The notion of statistical convergence was introduced by Fast [ 5 ] and Schoenberg [ 30 ] independently. Over the years and under different names statistical convergence has been discussed in the theory of Fourier analysis, ergodic theory and number theory. Later on it was further investigated by Fridy [ 6 ] , S̆alát [ 29 ] , Çakalli [ 2 ] , Maio and Kocinac [ 16 ] , Miller [ 18 ] , Maddox [ 15 ] , Leindler [ 14 ] , Mursaleen and Alotaibi [ 22 ] , Mursaleen and Edely [ 24 ] , Mursaleen and Edely [ 26 ] , and many others. In the recent years, generalization of statistical convergence have appeared in the study of strong integral summability and the structure of ideals of bounded continuous functions on Stone-C̆ech compactification of the natural numbers. Moreover statistical convergence is closely related to the concept of convergence in probability, (see [ 3 ] ).

The notion of statistical convergence depends on the density of subsets of $\mathbb {N} .$ A subset of $ \mathbb {N} $ is said to have density $\delta \left( E\right) $ if

\[ \delta \left( E\right) =\lim _{n\rightarrow \infty }\tfrac {1}{n}\textstyle \sum \limits _{k=1}^{n}\chi _{E}\left( k\right) \quad {\rm exists.} \]

Definition 1

A sequence $x=\left( x_{k}\right) $ is said to be statistically convergent to $\ell $ if for every $\varepsilon {\gt}0$

\[ \delta \big(\left\{ k\in \mathbb {N} :\left\vert x_{k}-\ell \right\vert \geq \varepsilon \right\} \big) =0. \]

In this case, we write $S-\lim x=\ell $ or $x_{k}\rightarrow \ell \big(S\big)$ and $S$ denotes the set of all statistically convergent sequences.

The probabilistic metric space was introduced by Menger [ 17 ] which is an interesting and important generalization of the notion of a metric space. Karakus [ 11 ] studied the concept of statistical convergence in probabilistic normed spaces. The theory of probabilistic normed spaces was initiated and developed in [ 1 ] , [ 31 ] , [ 32 ] , [ 33 ] , [ 35 ] and further it was extended to random/probabilistic 2-normed spaces by Goleţ [ 8 ] using the concept of 2-norm which is defined by Gähler [ 7 ] , and Gürdal and Pehlivan [ 10 ] studied statistical convergence in 2-Banach spaces. Recently, Savas [ 36 ] defined and studied generalized statistical convergence in random 2-normed space.

Let $\lambda =\left( \lambda _{n}\right) $ be non-decreasing sequence of positive numbers tending to infinity such that

\[ \lambda _{n+1}\leq \lambda _{n}+1,\text{ }\lambda _{1}=1. \]

The generalized de la Vallee-Pousin mean is defined by

\[ t_{n}\left( x\right) =\tfrac {1}{\lambda _{n}}\textstyle \sum \limits _{k\in I_{n}}x_{k} \]

where $I_{n}=\left[ n-\lambda _{n}+1,n\right] .$ The collection of such sequence $\lambda =\left( \lambda _{n}\right) $ will be denoted by $\nabla .$ Let $K\subseteq \mathbb {N} $ be a set of positive integers. Then

\[ \delta _{\lambda }\left( K\right) =\lim _{n}\tfrac {1}{\lambda _{n}}\big\vert \left\{ n-\lambda _{n}+1\leq k\leq n:\text{ }k\in K\right\} \big\vert \]

is said to be $\lambda $-density of K. In case $\lambda _{n}=n,$ the $\lambda $-density reduces to natural density.

[ 20 ] Mursaleen introduced the $\lambda $-statistical convergence as follows: A sequence $x=\left( x_{k}\right) $ is said to be $\lambda $-statistically convergent or $S_{\lambda }$-convergent to $\ell $ if for every $\varepsilon {\gt}0$

\[ \lim _{n}\tfrac {1}{\lambda _{n}}\big\vert \left\{ k\in I_{n}:\text{ }\left\vert x_{k}-\ell \right\vert \geq \varepsilon \right\} \big\vert . \]

The existing literature on statistical convergence and its generalizations appears to have been restricted to real or complex sequences, but in recent years these ideas have been also extended to the sequences in fuzzy normed [ 34 ] and intutionistic fuzzy normed spaces [ 12 ] , [ 27 ] , [ 28 ] and [ 13 ] . Further details on generalization of statistical convergence can be found in [ 23 ] , [ 24 ] , [ 25 ] and [ 26 ] .

2 Preliminaries

Definition 2

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

\[ H_{a}(t)=\left\{ \begin{array}[c]{cc}1 , & \mbox{ if } t{\gt}a ;\\ 0, & \mbox{if } t\leq a \end{array} \right. \]

It is obvious that $H_{0} \geq f$ for all $f\in D^{+}.$

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

In [ 7 ] , Gähler introduced the following concept of 2-normed space.

Definition 3

Let $X$ be a real vector space of dimension $d{\gt} 1$ ($d$ may be infinite). A real-valued function $\| .,.\| $ from $X^{2}$ into $\mathbb {R}$ satisfying the following conditions:

$~ \| x_{1},x_{2}\| =0$ if and only if $x_{1},x_{2}$ are linearly dependent,
$~ \| x_{1},x_{2}\| $ is invariant under permutation,
$~ \| \alpha x_{1},x_{2}\| = |\alpha |\cdot \| x_{1},x_{2}\| ,$ for any $\alpha \in \mathbb {R},$
$~ \| x+\overline{x},x_{2}\| \leq \| x,x_{2}\| + \| \overline{x},x_{2}\| $

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

A trivial example of an $2$-normed space is $X=\mathbb {R}^{2},$ equipped with the Euclidean $2$-norm $\| x_{1},x_{2}\| _{E}=$ the volume of the parallellogram spanned by the vectors $x_{1},x_{2}$ which may be given expicitly by the formula

\[ \| x_{1},x_{2}\| _{E}= |{\rm det}(x_{ij})|={\rm abs}\left( \det (\langle x_{i}, x_{j} \rangle )\right) \]

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

Recently, Goleţ [ 8 ] used the idea of 2-normed space to define the random 2-normed space.

Definition 4

Let $X$ be a linear space of dimension $d{\gt} 1$ ($d$ may be infinite), $\tau $ a triangle, and $\mathcal{F}: X\times X\rightarrow D^{+}.$ 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:

$~ \mathcal{F}(x,y;t)=H_{0}(t)$ if $x$ and $y$ are linearly dependent, where $\mathcal{F}(x,y;t)$ denotes the value of $\mathcal{F}(x,y)$ at $t\in \mathbb {R},$
$~ \mathcal{F}(x,y;t)\neq H_{0}(t)$ if $x$ and $y$ are linearly independent,
$~ \mathcal{F}(x,y;t)= \mathcal{F}(y,x;t),$ for all $x, y \in X,$
$~ \mathcal{F}(\alpha x,y;t)= \mathcal{F}(x,y; \tfrac {t}{|\alpha |}),$ for every $t{\gt}0, \alpha \neq 0$ and $x,y \in X,$
$~ \mathcal{F}( x+y,z;t)\geq \tau \left( \mathcal{F}(x,z;t),\mathcal{F}(y,z;t)\right) ,$ whenever $x,y,z \in X.$

If $(P2N_{5})$ is replaced by

$~ \mathcal{F}( x+y,z;t_{1}+t_{2})\geq \mathcal{F}(x,z;t_{1})*\mathcal{F}(y,z;t_{2}),$ for all $x,y,z \in X$ and $t_{1},t_{2} \in \mathbb {R}_{0}^{+};$

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

Remark 5

Every 2-normed space $(X, \| .,.\| )$ can be made a random 2-normed space in a natural way, by setting

$\mathcal{F}(x,y;t)=H_{0}(t-\| x,y\| ),$ for every $x,y \in X, t{\gt}0$ and $a*b= \min \{ a,b\} ,\ a,b\in [0,1];$
$\mathcal{F}(x,y;t)= \tfrac {t}{t+\| x,y\| },$ for every $x,y \in X, t{\gt}0$ and $a*b= ab, a,b\in [0,1].$ â–¡

In [ 9 ] , Gürdal and Pehlivan studied statistical convergence in 2-normed spaces and in 2-Banach spaces in [ 10 ] . In fact, Mursaleen [ 19 ] studied the concept of statistical convergence of sequences in random 2-normed space. Recently in [ 4 ] , Esi and Özdemir introduced and studied the concept of generalized $\Delta ^{m}$-statistical convergence of sequences in probabilistic normed space.

Definition 6

[ 19 ] A sequence $x=(x_{k})$ in a random $2$-normed space $(X,\mathcal{F},\ast )$ is said to be statistical-convergent or $S^{R2N}$-convergent to some $\ell \in X$ with respect to $\mathcal{F}$ if for each $\varepsilon {\gt}0,$ $\theta \in (0,1)$ and for non zero $z\in X$ such that

\[ \delta \big( \{ k\in \mathbb {N}:\mathcal{F}(x_{k}-\ell ,z;\varepsilon )\leq 1-\theta \} \big) =0, \]

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

\[ \lim _{m\rightarrow \infty } \tfrac {1}{m}\big|\{ k\leq m : \mathcal{F}(x_{k} - \ell , z;\varepsilon )\leq 1-\theta \} \big|=0. \]

or equivalently

\[ \delta \big( \{ k\in \mathbb {N}:\mathcal{F}(x_{k}-\ell ,z;\varepsilon ){\gt}1-\theta \} \big) =1, \]

i.e.

\[ S-\lim _{k\rightarrow \infty }\mathcal{F}(x_{k}-\ell ,z;\varepsilon )=1. \]

In this case we write $S^{R2N}-\lim x =\ell $ and $\ell $ is called the $S^{R2N}-limit$ of $x.$ Let $S^{R2N}(X)$ denotes the set of all statistical convergent sequences in random 2-normed space $(X, \mathcal{F},*).$

In this paper we define and study $\nabla $-statistical convergence in random 2-normed space using by $\lambda $ sequences which is quite a new and interesting idea to work with. We show that some properties of $\nabla $-statistical convergence of real numbers also hold for sequences in random 2-normed spaces. We find some relations related to $\lambda $-statistical convergent sequences in random 2-normed spaces. Also we find out the relation between $\nabla $-statistical convergent and $\nabla $-statistical Cauchy sequences in this spaces.

3 $\nabla $-statistical convergence in random 2-normed space

In this section we define $\nabla $-statistical convergent sequence in random 2-normed $(X,\mathcal{F},\ast ).$ Also we obtained some basic properties of this notion in random 2-normed space.

Definition 7

A sequence $x=(x_{k})$ in a random $2$-normed space $(X,\mathcal{F},\ast )$ is said to be $\nabla $-convergent to $\ell \in X$ with respect to $\mathcal{F}$ if for each $\varepsilon {\gt}0,$ $\beta \in (0,1)$ there exists an positive integer $n_{0}$ such that $\mathcal{F}\Big(\tfrac {1}{\lambda _{n}}\textstyle \sum \limits _{k\in I_{n}}x_{k}-\ell ,z;\varepsilon \Big){\gt}1-\beta ,$ whenever $k\geq n_{0}$ and for non-zero $z\in X.$ In this case we write $\mathcal{F}-\lim _{k}x_{k}=\ell ,$ and $\ell $ is called the $\mathcal{F}_{\nabla }$-limit of $x=(x_{k}).$

Definition 8

A sequence $x=(x_{k})$ in a random $2$-normed space $(X,\mathcal{F},\ast )$ is said to be $\nabla $-Cauchy with respect to $\mathcal{F}$ if for each $\varepsilon {\gt}0,$ $\beta \in (0,1)$ there exists a positive integer $n_{0}=n_{0}(\varepsilon )$ such that $\mathcal{F}(\tfrac {1}{\lambda _{n}}\textstyle \sum \limits _{k\in I_{n}}\left( x_{k}-x_{s}\right) ,z;\varepsilon ){\lt}1-\theta ,$ whenever $k,s\geq n_{0}$ and for non-zero $z\in X.$

Definition 9

A sequence $x=(x_{k})$ in a random 2-normed space $(X,\mathcal{F},\ast )$ is said to be $\nabla $-satistically convergent or $S_{\nabla }$-convergent to $\ell \in X$ with respect to $\mathcal{F}$ if for every $\varepsilon {\gt}0,$ $\beta \in (0,1)$ and for non zero $z\in X$ such that

\[ \delta _{\nabla }\bigg(\bigg\{ k\in I_{n}:\mathcal{F}\bigg(\tfrac {1}{\lambda _{n}}\textstyle \sum \limits _{k\in I_{n}}x_{k}-\ell ,z;\varepsilon \bigg)\leq 1-\beta \bigg\} \bigg)=0. \]

In other ways we can write

\[ \bigg|\bigg\{ k\in I_{n}:\mathcal{F}\bigg(\lim _{n\rightarrow \infty }\tfrac {1}{\lambda _{n}}\textstyle \sum \limits _{k\in I_{n}}x_{k}-\ell ,z;\varepsilon \bigg)\leq 1-\beta \bigg\} \bigg|=0, \]

or, equivalently,

\[ \delta _{\nabla }\bigg(\bigg\{ k\in I_{n}:\mathcal{F}\bigg(\tfrac {1}{\lambda _{n}}\textstyle \sum \limits _{k\in I_{n}}x_{k}-\ell ,z;\varepsilon \bigg){\gt}1-\beta \bigg\} \bigg) =1, \]

i.e.,

\[ S_{\nabla }-\lim _{n\rightarrow \infty }\mathcal{F}\bigg(\tfrac {1}{\lambda _{n}}\textstyle \sum \limits _{k\in I_{n}}x_{k}-\ell ,z;\varepsilon \bigg)=1. \]

In this case we write $S_{\nabla }^{R2N}-\lim x=\ell $ or $x_{k}\rightarrow \ell (S_{\nabla }^{R2N})$ and

\[ S_{\nabla }^{R2N}(X)=\big\{ x=(x_{k}):\exists ~ \ell \in \mathbb {R},S_{\nabla }^{R2N}-\lim x=\ell \big\} . \]

In this case we write $S_{\nabla }^{R2N}-\lim x=\ell $ and $\ell $ is called the $S_{\nabla }^{R2N}-limit$ of $x.$ Let $S_{\nabla }^{R2N}(X)$ denotes the set of all statistical convergent sequences in random 2-normed space $(X,\mathcal{F},\ast ).$

Definition 10

A sequence $x=(x_{k})$ in a random 2-normed space $(X,\mathcal{F},\ast )$ is said to be $\nabla $-statistical Cauchy with respect to $\mathcal{F}$ if for every $\varepsilon {\gt}0,$ $\beta \in (0,1)$ and for non-zero $z\in X$ there exists a positive integer $n=n(\varepsilon )$ such that for all $k,s\geq n$

\[ \delta _{\nabla }\bigg(\bigg\{ k\in I_{n}:\mathcal{F}\bigg(\tfrac {1}{\lambda _{n}}\textstyle \sum \limits _{k\in I_{n}}\left( x_{k}-x_{s}\right) ,z;\varepsilon \bigg)\leq 1-\beta \bigg\} \bigg)=0, \]

or, equivalently,

Definition 3.3, immediately implies the following Lemma.

Lemma 11

Let $(X,\mathcal{F},\ast )$ be a random $2$-normed space. If $x=(x_{k})$ is a sequence in X, then for every $\varepsilon {\gt}0,$ $\mathit{\beta }$$\in (0,1)$ and for non zero $z\in X,$ then the following statements are equivalent.

$S_{\nabla }-\lim _{k\rightarrow \infty }x_{k}=\ell .$
$\delta _{\nabla }\bigg(\bigg\{ k\in I_{n}:\mathcal{F}\bigg(\tfrac {1}{\lambda _{n}}\textstyle \sum \limits _{k\in I_{n}}x_{k}-\ell ,z;\varepsilon \bigg)\leq 1-\beta \bigg\} \bigg)=0.$
$\delta _{\nabla }\bigg(\bigg\{ k\in I_{n}:\mathcal{F}\bigg(\tfrac {1}{\lambda _{n}}\textstyle \sum \limits _{k\in I_{n}}x_{k}-\ell ,z;\varepsilon \bigg){\gt}1-\beta \bigg\} \bigg) =1.$
$S_{\nabla }-\lim _{k\rightarrow \infty }\mathcal{F}\bigg(\tfrac {1}{\lambda _{n}}\textstyle \sum \limits _{k\in I_{n}}x_{k}-\ell ,z;\varepsilon \bigg)=1.$

Theorem 12

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

Proof â–¼

Suppose that there exist elements $\ell _{1},\ell _{2}$ $(\ell _{1}\neq \ell _{2})$ in $X$ such that

\[ S_{\nabla }^{R2N}-\lim _{k\rightarrow \infty }x_{k}=\ell _{1};\qquad S_{\nabla }^{R2N}-\lim _{k\rightarrow \infty }x_{k}=\ell _{2}. \]

Let $\varepsilon {\gt}0$ be given. Choose $s{\gt}0$ such that

\begin{equation} (1-s)\ast (1-s)>1-\varepsilon . \end{equation}

3.1

Then, for any $t {\gt} 0$ and for non zero $z\in X$ we define

\[ K_{1}(s,t)=\bigg\{ k\in I_{n}:\mathcal{F}\bigg( \tfrac {1}{\lambda _{n}}\textstyle \sum \limits _{k\in I_{n}}x_{k}-\ell _{1},z;\tfrac {t}{2}\bigg) \leq 1-s\bigg\} ; \]

\[ K_{2}(s,t)=\bigg\{ k\in I_{n}:\mathcal{F}\bigg( \tfrac {1}{\lambda _{n}}\textstyle \sum \limits _{k\in I_{n}}x_{k}-\ell _{2},z;\tfrac {t}{2}\bigg) \leq 1-s\bigg\} . \]

Since

\[ S_{\nabla }^{R2N}-\lim _{k\rightarrow \infty }x_{k}=\ell _{1} \ {\rm and}\ S_{\nabla }^{R2N}-\lim _{k\rightarrow \infty }x_{k}=\ell _{2}, \]

we have

\[ \delta _{\nabla }(K_{1}(s,t))=0 \ {\rm and}\ \delta _{\nabla }(K_{2}(s,t))=0\ {\rm for\ all} t{\gt}0. \]

Now let $K(s,t)=K_{1}(s,t)\cup K_{2}(s,t),$ then it is easy to observe that
$\delta _{\nabla }(K(s,t))=0.$ But we have $\delta _{\nabla }(K^{c}(s,t))=1.$

Now if $k\in K^{c}(s,t)$ then we have

\begin{equation*} \begin{split} & \mathcal{F}(\ell _{1}-\ell _{2},z;t)\geq \\ & \geq \mathcal{F}\bigg( \tfrac {1}{\lambda _{n}}\textstyle \sum \limits _{k\in I_{n}}x_{k}-\ell _{1},z;\tfrac {t}{2}\bigg) \ast \mathcal{F}\bigg( \tfrac {1}{\lambda _{n}}\textstyle \sum \limits _{k\in I_{n}}x_{k}-\ell _{2},z;\tfrac {t}{2}\bigg) {\gt}(1-s)\ast (1-s). \end{split}\end{equation*}

It follows by (3.1) that

\[ \mathcal{F}(\ell _{1}-\ell _{2},z;t){\gt}(1-\varepsilon ). \]

Since $\varepsilon {\gt}0$ was arbitrary, we get $\mathcal{F}(\ell _{1}-\ell _{2},z;t)=1$ for all $t{\gt}0$ and non zero $z\in X.$ Hence $\ell _{1}=\ell _{2}.$

Next theorem gives the algebraic characterization of $\lambda $-statistical convergence on random $2$-normed spaces.

Theorem 13

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

If $S_{\nabla }^{R2N}-\lim x_{k}=\ell $ and $c(\neq 0)\in \mathbb {R},$ then $S_{\nabla }^{R2N}-\lim cx_{k}=c\ell .$
If $S_{\nabla }-\lim x_{k}=\ell _{1}$ and $S_{\nabla }^{R2N}-\lim y_{k}=\ell _{2},$ then $S_{\nabla }^{R2N}-\lim (x_{k}+y_{k})=\ell _{1}+\ell _{2}.$

The proof of this theorem is straightforward, and thus will be omitted.

Theorem 14

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

Proof â–¼

Let $\mathcal{F}_{\nabla }-\lim x_{k}=\ell .$ Then for every $\varepsilon {\gt}0,t{\gt}0$ and non zero $z\in X,$ there is a positive integer $n_{0}$ such that

\[ \mathcal{F}\bigg(\tfrac {1}{\lambda _{n}}\textstyle \sum \limits _{k\in I_{n}}x_{k}-\ell ,z;t\bigg){\gt}1-\varepsilon \]

for all $k\geq n_{0}.$ Since the set

\[ K(\varepsilon ,t)=\bigg\{ k\in I_{n}:\mathcal{F}\bigg(\tfrac {1}{\lambda _{n}}\textstyle \sum \limits _{k\in I_{n}}x_{k}-\ell ,z;t\bigg)\leq 1-\varepsilon \bigg\} \]

has at most finitely many terms. Also, since every finite subset of $\mathbb {N}$ has $\delta _{\nabla }$-density zero, and consequently we have $\delta _{\nabla }(K(\varepsilon ,t))=0.$ This shows that $S_{\nabla }^{R2N}-\lim x_{k}=\ell .$

Remark 15

The converse of the above theorem is not true in general. It follows from the following example. â–¡

Example 16

Let $X=\mathbb {R}^{2},$ with the 2-norm $\| x,z\| =|x_{1}z_{2}-x_{2}z_{1}|,$ $x=(x_{1},x_{2}),z=(z_{1},z_{2})$ and $a\ast b=ab$ for all $a,b\in \lbrack 0,1].$ Let $\mathcal{F}(x,z;t)=\tfrac {t}{t+\| x,z\| },$ for all $x,z\in X,z_{2}\neq 0,$ and $t{\gt}0.$ Now we define a sequence $x=(x_{k})$ by

\[ \tfrac {1}{\lambda _{n}}\textstyle \sum \limits _{k\in I_{n}}x_{k}=\\ \begin{cases} (k,0), & \mbox{ if }n-[\sqrt{\lambda _{n}}]+1\leq k\leq n;\ n\in \mathbb {N}\\ (0,0), & \mbox{otherwise. } \end{cases} \]

Now for every $0{\lt}\varepsilon {\lt}1$ and $t{\gt}0,$ write

\[ K(\varepsilon ,t)=\bigg\{ k\in I_{n}:\mathcal{F}\bigg(\tfrac {1}{\lambda _{n}}\textstyle \sum \limits _{k\in I_{n}}x_{k}-\ell ,z;t\bigg)\leq 1-\varepsilon \bigg\} , \]

where $\ell =(0,0).$ Then

\begin{align*} K(\varepsilon ,t)& =\bigg\{ k\in I_{n}:\tfrac {t}{t+|\frac{1}{\lambda _{n}}\sum _{k\in I_{n}}x_{k}|}\leq 1-\varepsilon \bigg\} \\ & =\bigg\{ k\in I_{n}:|\tfrac {1}{\lambda _{n}}\textstyle \sum \limits _{k\in I_{n}}x_{k}|\geq \tfrac {t\varepsilon }{1-\varepsilon }{\gt}0\bigg\} \\ & =\{ k\in I_{n}:x_{k}=\left( k,0\right) \} \\ & =\{ k\in I_{n}:n-[\sqrt{\lambda _{n}}]+1\leq k\leq n;n\in \mathbb {N} \} , \end{align*}

so we get

\[ \tfrac {1}{\lambda _{n}}|K(\varepsilon ,t)|\text{ }\leq \tfrac {1}{\lambda _{n}}|\{ k\in I_{n}:n-[\sqrt{\lambda _{n}}]+1\leq k\leq n;n\in \mathbb {N} \} |\text{ }\leq \tfrac {\lbrack \sqrt{\lambda _{n}}]}{\lambda _{n}}. \]

Taking limit $n$ approaches to $\infty $, we get

\[ \delta _{\theta }(K(\varepsilon ,t))=\lim _{n\rightarrow \infty }\tfrac {1}{\lambda _{n}}|K(\varepsilon ,t)|\leq \lim _{n\rightarrow \infty }\tfrac {[\sqrt{\lambda _{n}}]}{\lambda _{n}}=0. \]

This shows that $x_{k}\rightarrow 0$ $(S_{\nabla }^{R2N}(X)).$

On the other hand the sequence is not $\mathcal{F}_{\nabla }$-convergent to zero as

\begin{align*} \mathcal{F}\Big(\tfrac {1}{\lambda _{n}}\textstyle \sum \limits _{k\in I_{n}}x_{k}-\ell ,z;t\Big)& =\tfrac {t}{t+\bigg|\tfrac {1}{\lambda _{n}}\textstyle \sum \limits _{k\in I_{n}}x_{k}\bigg|}\\ & =\begin{cases} \tfrac {t}{t+kz_{2}}, & \mbox{ if }n-[\sqrt{\lambda _{n}}]+1\leq k\leq n;n\in \mathbb {N}\\ 1, & \mbox{otherwise. } \end{cases}\\ & \leq 1. \hfil \qed \end{align*}

Theorem 17

Let $(X,\mathcal{F},\ast )$ be a random $2$-normed space. If $x=(x_{k})$ be a sequence in X, then $S_{\nabla }^{R2N}-\lim x_{k}=\ell $ if and only if there exists a subset $K\subseteq \mathbb {N} $ such that $\delta _{\nabla }(K)=1$ and $\mathcal{F}_{\nabla }-\lim x_{k}=\ell .$

Proof â–¼

Suppose first that $S_{\nabla }^{R2N}-\lim x_{k}=\ell .$ Then for any $t{\gt}0,$ $s=1,2,3,\ldots $ and non zero $z\in X,$ let

\[ A(s,t)=\Big\{ k\in I_{n}:\mathcal{F}\Big(\tfrac {1}{\lambda _{n}}\textstyle \sum \limits _{k\in I_{n}}x_{k}-\ell ,z;t\Big){\gt}1-\tfrac {1}{s}\Big\} \]

and

\[ K(s,t)=\Big\{ k\in I_{n}:\mathcal{F}\Big(\tfrac {1}{\lambda _{n}}\textstyle \sum \limits _{k\in I_{n}}x_{k}-\ell ,z;t\Big)\leq 1-\tfrac {1}{s}\Big\} . \]

Since $S_{\nabla }^{R2N}-\lim x_{k}=\ell $ it follows that

\[ \delta _{\nabla }(K(s,t))=0. \]

Now for $t{\gt}0$ and $s=1,2,3,\ldots ,$ we observe that

\[ A(s,t)\supset A(s+1,t) \]

and

\begin{equation} \delta _{\theta }(A(s,t))=1. \end{equation}

3.8

Now we have to show that, for $k\in A(s,t),\mathcal{F}_{\nabla }-\lim x_{k}=\ell .$ Suppose that for $k\in A(s,t),x=(x_{k})$ not convergent to $\ell $ with respect to $\mathcal{F}_{\nabla }.$ Then there exists some $u{\gt}0$ such that

\[ \Big\{ k\in I_{n}:\mathcal{F}\Big(\tfrac {1}{\lambda _{n}}\textstyle \sum \limits _{k\in I_{n}}x_{k}-\ell ,z;t\Big)\leq 1-u\Big\} \]

for infinitely many terms $x_{k}.$ Let

\[ A(u,t)=\Big\{ k\in I_{n}:\mathcal{F}\Big(\tfrac {1}{\lambda _{n}}\textstyle \sum \limits _{k\in I_{n}}x_{k}-\ell ,z;t\Big){\gt}1-u\Big\} \]

and

\[ u{\gt}\tfrac {1}{s},\qquad s=1,2,3,\ldots \]

Then we have

\[ \delta _{\lambda }(A(u,t))=0. \]

Furthermore, $A(s,t)\subset A(u,t)$ implies that $\delta _{\nabla }(A(s,t))=0,$ which contradicts (3.2) as $\delta _{\nabla }(A(s,t))=1.$ Hence $\mathcal{F}_{\nabla }-\lim x_{k}=\ell .$

Conversely, suppose that there exists a subset $K\subseteq \mathbb {N} $ such that $\delta _{\nabla }(K)=1$ and $\mathcal{F}_{\nabla }-\lim x_{k}=\ell .$

Then for every $\varepsilon {\gt} 0,$ $t {\gt} 0$ and non zero $z\in X,$ we can find out a positive integer $n$ such that

\[ \mathcal{F}\Big(\tfrac {1}{\lambda _{n}}\textstyle \sum \limits _{k\in I_{n}}x_{k}-\ell ,z;t\Big){\gt}1-\varepsilon \]

for all $k\geq n.$ If we take

\[ K(\varepsilon ,t)=\Big\{ k\in I_{n}:\mathcal{F}\Big(\tfrac {1}{\lambda _{n}}\textstyle \sum \limits _{k\in I_{n}}x_{k}-\ell ,z;t\Big)\leq 1-\varepsilon \Big\} \]

then it is easy to see that

\[ K(\varepsilon ,t)\subset \mathbb {N}-\{ n_{k+1},n_{k+2},\ldots \} \]

and consequently

\[ \delta _{\nabla }(K(\varepsilon ,t))\leq 1-1. \]

Hence $S_{\nabla }^{R2N}-\lim x_{k}=\ell .$

Finally, we establish the Cauchy convergence criteria in random 2-normed spaces.

Theorem 18

Let $(X,\mathcal{F},\ast )$ be a random 2-normed space. Then a sequence $x=(x_{k})$ in $\mathit{X}$ is $\nabla $-statistically convergent if and only if it is $\nabla $-statistically Cauchy.

Proof â–¼

Let $x=(x_{k})$ be a $\nabla $-statistically convergent sequence in $X$. We assume that $S_{\nabla }^{R2N}-\lim x_{k}=\ell .$ Let $\varepsilon {\gt}0$ be given. Choose $s{\gt}0$ such that (3.1) is satisfied. For $t{\gt}0$ and for non zero $z\in X$ define

\[ A(s,t)=\Big\{ k\in I_{n}:\mathcal{F}\Big(\tfrac {1}{\lambda _{n}}\textstyle \sum \limits _{k\in I_{n}}x_{k}-\ell ,z;\tfrac {t}{2}\Big)\leq 1-s\Big\} \]

and

\[ A^{c}(s,t)=\Big\{ k\in I_{n}:\mathcal{F}\Big(\tfrac {1}{\lambda _{n}}\textstyle \sum \limits _{k\in I_{n}}x_{k}-\ell ,z;\tfrac {t}{2}\Big){\gt}1-s\Big\} . \]

Since $S_{\nabla }^{R2N}-\lim x_{k}=\ell $ it follows that $\delta _{\nabla }(A(s,t))=0$ and consequently $\delta _{\nabla }(A^{c}(s,t))=1.$ Let $p\in A^{c}(s,t).$ Then

\begin{equation} \mathcal{F}\Big(\tfrac {1}{\lambda _{n}}\textstyle \sum \limits _{k\in I_{n}}x_{k}-\ell ,z;\tfrac {t}{2}\Big)\leq 1-s. \end{equation}

3.9

If we take

\[ B(\varepsilon ,t)=\bigg\{ k\in I_{n}:\mathcal{F}\Big(\tfrac {1}{\lambda _{n}}\textstyle \sum \limits _{k\in I_{n}}x_{k}-\ell ,z;t\Big)\leq 1-\varepsilon \bigg\} \]

then to prove the result it is sufficient to prove that $B(\varepsilon ,t)\subseteq A(s,t).$ Let $n\in B(\varepsilon ,t),$ then for non zero $z\in X$

\begin{equation} \mathcal{F}\Big(\tfrac {1}{\lambda _{n}}\textstyle \sum \limits _{n\in I_{n}}( x_{n}-x_{p}) ,z;t\Big)\leq 1-\varepsilon . \end{equation}

3.10

\[ \mathcal{F}\Big(\tfrac {1}{\lambda _{n}}\textstyle \sum \limits _{n\in I_{n}}( x_{n}-x_{p}) ,z;t\Big)\leq 1-\varepsilon , \]

then we have

\[ \mathcal{F}\Big(\tfrac {1}{\lambda _{n}}\textstyle \sum \limits _{n\in I_{n}}x_{n}-\ell ,z;\tfrac {t}{2}\Big)\leq 1-s \]

and therefore $n\in A(s,t).$ As otherwise i.e., if

\[ \mathcal{F}\bigg(\tfrac {1}{\lambda _{n}}\textstyle \sum \limits _{n\in I_{n}}x_{n}-\ell ,z;\tfrac {t}{2}\bigg){\gt}1-s \]

then by (3.1), (3.3) and (3.4) we get

\begin{align*} 1-\varepsilon & \geq \mathcal{F}\Big(\tfrac {1}{\lambda _{n}}\textstyle \sum \limits _{n\in I_{n}}( x_{n}-x_{p}),z;t\Big) \\ & \geq \mathcal{F}\Big(\tfrac {1}{\lambda _{n}}\textstyle \sum \limits _{n\in I_{n}}x_{n}-\ell ,z;\tfrac {t}{2}\Big)\ast \mathcal{F}\Big(\tfrac {1}{\lambda _{p}}\textstyle \sum \limits _{p\in I_{n}}x_{p}-\ell ,z;\tfrac {t}{2}\Big)\\ & {\gt}(1-s)\ast (1-s){\gt}(1-\varepsilon ) \end{align*}

which is not possible. Thus $B(\varepsilon ,t)\subset A(s,t).$ Since $\delta _{\nabla }(A(s,t))=0,$ it follows that $\delta _{\nabla }(B(\varepsilon ,t))=0.$ This shows that $(x_{k})$ is $\nabla $-statistically Cauchy.

Conversely, suppose $(x_{k})$ is $\nabla $-statistically Cauchy but not $\nabla $-statistically convergent. Then there exists positive integer $p$ and for non zero $z\in X$ such that if we take

\[ A(\varepsilon ,t)=\bigg\{ k\in I_{n}:\mathcal{F}\Big(\tfrac {1}{\lambda _{n}}\textstyle \sum \limits _{n\in I_{n}}( x_{n}-x_{p}),z;t\Big)\leq 1-\varepsilon \bigg\} \]

and

\[ B(\varepsilon ,t)=\bigg\{ k\in I_{n}:\mathcal{F}\Big(\tfrac {1}{\lambda _{n}}\textstyle \sum \limits _{k\in I_{n}}x_{k}-\ell ,z;\tfrac {t}{2}\Big){\gt}1-\varepsilon \bigg\} \]

then

\[ \delta _{\nabla }(A(\varepsilon ,t))=0=\delta _{\nabla }(B(\varepsilon ,t)) \]

and consequently

\begin{equation} \delta _{\nabla }(A^{c}(\varepsilon ,t))=1=\delta _{\nabla }(B^{c}(\varepsilon ,t)). \end{equation}

3.11

Since

\[ \mathcal{F}\Big(\tfrac {1}{\lambda _{n}}\textstyle \sum \limits _{n\in I_{n}}( x_{n}-x_{p}) ,z;t\Big)\geq 2\mathcal{F}\Big(\tfrac {1}{\lambda _{n}}\textstyle \sum \limits _{k\in I_{n}}x_{k}-\ell ,z;\tfrac {t}{2}\Big){\gt}1-\varepsilon , \]

\[ \mathcal{F\Big(}\tfrac {1}{\lambda _{n}}\textstyle \sum \limits _{k\in I_{n}}x_{k}-\ell ,z;\tfrac {t}{2}\Big){\gt}\tfrac {1-\varepsilon }{2} \]

then we have

\[ \delta _{\nabla }\Big(\Big\{ k\in I_{n}:\mathcal{F}\Big(\tfrac {1}{\lambda _{n}}\textstyle \sum \limits _{n\in I_{n}}\left( x_{n}-x_{p}\right) ,z;t\Big){\gt}1-\varepsilon \Big\} \Big)=0 \]

i.e. $\delta _{\nabla }(A^{c}(\varepsilon ,t))=0,$ which contradicts (3.5) as $\delta _{\nabla }(A^{c}(\varepsilon ,t))=1.$ Hence $x=(x_{k})$ is $\nabla $-statistically convergent.

Combining Theorem 3.7 and Theorem 3.8 we get the following corollary.

Corollary 19

Let $(X,\mathcal{F},\ast )$ be a random 2-normed space and and $x=(x_{k})$ be a sequence in X. Then the following statements are equivalent:

x is $\nabla $-statistically convergent.
x is $\nabla $-statistically Cauchy.
there exists a subset $K\subseteq $$ \mathbb {N} $ such that $\delta _{\nabla }(K)=1$ and $\mathcal{F}_{\nabla }-\lim x_{k}=\ell .$

Bibliography

1: C. Alsina, B. Schweizer and A. Sklar, Continuity properties of probabilistic norms, J. Math. Anal. Appl., 208 (1997), pp. 446–452. $\includegraphics[scale=0.1]{ext-link.png}$
2: H. Çakalli, A study on statistical convergence, Funct. Anal. Approx. Comput., 1(2) (2009), pp. 19-24, MR2662887.
3: J.Connor and M.A. Swardson, Measures and ideals of $C^{\ast }(X) $, Ann. N. Y. Acad. Sci., 704 (1993), pp. 80–91.
4: A. Esi and M. K. Özdemir, Generalized $\Delta ^{m}$-Statistical convergence in probabilistic normed space, J. Comput. Anal. Appl., 13(5) (2011), pp. 923–932.
5: H. Fast, Sur la convergence statistique, Colloq. Math., 2 (1951), pp. 241–244.
6: J. A. Fridy, On statistical convergence, Analysis, 5 (1985) pp. 301–313.
7: S. Gähler, 2-metrische Raume and ihre topologische Struktur, Math. Nachr., 26 (1963), pp. 115–148. $\includegraphics[scale=0.1]{ext-link.png}$
8: I . Goleţ, On probabilistic 2-normed spaces, Novi Sad J. Math., 35 (2006), pp. 95–102.
9: M. Gürdal and S. Pehlivan, Statistical convergence in 2-normed spaces, South. Asian Bull. Math., 33 (2009), pp. 257-264.
10: M. Gürdal and S. Pehlivan, The statistical convergence in 2-Banach spaces, Thai J. Math., 2(1) (2004), pp. 107–113.
11: S. Karakus, Statistical convergence on probabilistic normed spaces, Math. Commun., 12 (2007), pp. 11–23.
12: S. Karakus, K. Demirci and O. Duman, Statistical convergence on intuitionistic fuzzy normed spaces, Chaos, Solitons and Fractals, 35 (2008), pp. 763–769. $\includegraphics[scale=0.1]{ext-link.png}$
13: V. Kumar and M. Mursaleen, On $(\lambda , \mu )$-statistical convergence of double sequences on intuitionistic fuzzy normed spaces, Filomat, 25(2) (2011), pp. 109–120.
14: L. Leindler, Uber die de la Vallee-Pousinsche Summierbarkeit allgenmeiner Othogonalreihen, Acta Math. Acad. Sci. Hungar, 16 (1965), pp. 375–387.
15: I. J. Maddox, Statistical convergence in a locally convex space, Math. Proc. Cambridge Philos. Soc., 104(1) (1988), pp. 141–145. $\includegraphics[scale=0.1]{ext-link.png}$
16: G. D. Maio and L. D. R. Kočinac, Statistical convergence in topology, Topology Appl., 156 (2008), pp. 28–45. $\includegraphics[scale=0.1]{ext-link.png}$
17: K. Menger, Statistical metrics, Proc. Natl. Acad. Sci. USA, 28 (1942), pp. 535–537.
18: H. I. Miller, A measure theoretical subsequence characterization of statistical convergence, Trans. Amer. Math. Soc., 347(5) (1995), pp. 1811–1819.
19: M. Mursaleen, Statistical convergence in random 2-normed spaces, Acta Sci. Math. (Szeged), 76(1–2) (2010), pp. 101–109.
20: M. Mursaleen, $\lambda $-statistical convergence, Mathematica Slovaca, 50(1) (2000), pp. 111–115.
21: M. Mursaleen and A. K. Noman, On the spaces of $\lambda $-convergent and bounded sequences, Thai J. Math., 8(2) (2010), pp. 311–329.
22: M. Mursaleen and A. Alotaibi, Statistical summability and approximation by de la Vallee-Pousin mean, Applied Math. Letters, 24 (2011), pp. 320–324. $\includegraphics[scale=0.1]{ext-link.png}$
23: M. Mursaleen, C. Çakan, S. A. Mohiuddine and E. Savaş, Generalized statistical convergence and statistical core of double sequences, Acta Math. Sinica, 26(11) (2010), pp. 2131–2144. $\includegraphics[scale=0.1]{ext-link.png}$
24: M. Mursaleen and Osama H. H. Edely, Statistical convergence of double sequences, J. Math. Anal. Appl., 288 (2003), pp. 223–231. $\includegraphics[scale=0.1]{ext-link.png}$
25: M. Mursaleen and Osama H. H. Edely, Generalized statistical convergence, Information Sciences, 162 (2004), pp. 287–294.
26: M. Mursaleen and Osama H. H. Edely, On the invariant mean and statistical convergence, Appl. Math. Letters, 22 (2009), pp. 1700–1704. $\includegraphics[scale=0.1]{ext-link.png}$
27: S. A. Mohiuddine and Q.M. Danish Lohani, On generalized statistical convergence in intuitionistic fuzzy normed space, Chaos, Solitons and Fractals, 42 (2009), pp. 1731–1737. $\includegraphics[scale=0.1]{ext-link.png}$
28: M. Mursaleen and S.A. Mohiuddine, On lacunary statistical convergence with respect to the intuitionistic fuzzy normed space, J. Comput. Appl. Math., 233(2) (2009), pp. 142–149. $\includegraphics[scale=0.1]{ext-link.png}$
29: T. S̆alát, On statistical convergence sequences of real numbers, Math. Slovaca, 30 (1980), pp. 139–150.
30: I. J. Schoenberg, The integrability of certain functions and related summability methods, Amer. Math. Monthly 66 (1959), pp. 361–375.
31: B. Schweizer and A. Sklar, Statistical metric spaces, Pacific J. Math. 10 (1960), pp. 313–334.
32: B. Schweizer and A. Sklar, Probabilistic Metric Spaces, North Holland, New York- Amsterdam-Oxford, 1983.
33: C. Sempi, A short and partial history of probabilistic normed spaces, Mediterr. J. Math., 3 (2006), pp. 283–300. $\includegraphics[scale=0.1]{ext-link.png}$
34: C. Şençimen and S. Pehlivan, Statistical convergence in fuzzy normed linear spaces, Fuzzy Sets and Systems, 159 (2008), pp. 361–370. $\includegraphics[scale=0.1]{ext-link.png}$
35: A. N. Serstnev, On the notion of a random normed space, Dokl. Akad. Nauk SSSR 149 (1963), pp. 280–283.
36: On generalized statistical convergence in random 2-normed space, Iranian Journal of Science & Technology, (2012) A4: 417-423.

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On \(\nabla \)-statistical convergence in random 2-normed space

1 Introduction

2 Preliminaries

3 \(\nabla \)-statistical convergence in random 2-normed space

Bibliography