Ideal Statistically Pre-Cauchy Triple Sequences of Fuzzy Number and Orlicz Functions
Abstract: In this paper, we extend the notions of ideal statistically convergence for sequence of fuzzy number. We introduce the notions ideal statistically pre-Cauchy triple sequences of fuzzy number about Orlicz function, and give some correlation theorem. It is shown that x = {xijk} is ideal statistically pre-Cauchy if and only if D (xijk, xpqr) ≥ ε, i ≤ m, j ≤ n, t ≤ k}| ≥ δ} ∈ I. At the same time, we have proved x = {xijk} is ideal statistically convergent to x0 if and only if . Also, some properties of these new sequence spaces are investigated.

1. Introduction

The notion of statistical convergence was introduced by Fast  and also independently by Buck  and Schoenberg  for real and complex sequences.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 from the sequence spaces point of view and linked with summability theory by Altinok and Et , Connor , Et et al. (    ), Fridy , Fridy and Orhan , Mursaleen  and many others.

Matloka  defined the notion of fuzzy sequence and introduced bounded and convergent sequences of fuzzy real numbers and studied their some properties. After then, Nuray and Savas  defined the notion of statistical convergence for sequences of fuzzy numbers. Since then, there has been increasing interest in the study of statistical convergence of fuzzy sequences (see  -  ).

Lindesstrauss and Tzafriri  used the idea of Orlicz sequence space,

${l}_{M}:=\left\{x\in \omega :\underset{k=1}{\overset{\infty }{\sum }}\text{ }\text{ }M\left(\frac{|x|}{\rho }\right)<\infty ,\text{\hspace{0.17em}}\text{for}\text{\hspace{0.17em}}\text{some}\text{\hspace{0.17em}}\rho >0\right\}$, which is Banachi space with the norm: ${‖x‖}_{M}=\mathrm{inf}\left\{\rho >0:\underset{k=1}{\overset{\infty }{\sum }}\text{ }\text{ }M\left(\frac{|x|}{\rho }\right)\le 1\right\}$. The space ${l}_{M}$ is closely related to

the space ${l}_{p}$, which is an Orlicz sequence space with $M\left(x\right)={x}^{p}$ for $1\le p<\infty$.

Connor, Fridy and Kline  proved that statistical convergent sequences are statistically pre-Cauchy and any bounded statistically pre-Cauchy sequence with nowhere dense set of limit points is statistically convergent. They also gave an example showing statistically pre-Cauchy sequences are not necessarily statistically convergent.

In this paper, we extend the notions of ideal statistically convergence for sequence of fuzzy number. We introduce the notions ideal statistically pre-Cauchy triple sequences of fuzzy number about Orlicz function, and give some correlation theorem. Also, some properties of these new sequence spaces are investigated. It popularized the work of predecessors.

2. Definitions and Preliminaries

In this section, we give some basic notions which will be used throughout the paper.

Let $\stackrel{˜}{A}\in \stackrel{˜}{F}\left(R\right)$ be a fuzzy subset on R. If $\stackrel{˜}{A}$ is convex, normal, upper semi-continuous and has compact support, we say that $\stackrel{˜}{A}$ is a fuzzy number. Let ${\stackrel{˜}{R}}^{c}$ denote the set of all fuzzy numbers.

For $\stackrel{˜}{A}\in {\stackrel{˜}{R}}^{c}$, we write the level set of $\stackrel{˜}{A}$ as ${A}_{\lambda }=\left\{x:A\left(x\right)\ge \lambda \right\}$ and ${A}_{\lambda }=\left[{A}_{\lambda }^{-},{A}_{\lambda }^{+}\right]$. Let $\stackrel{˜}{A},\stackrel{˜}{B}\in {\stackrel{˜}{R}}^{c}$, we define $\stackrel{˜}{A}+\stackrel{˜}{B}=\stackrel{˜}{C}$ iff ${A}_{\lambda }+{B}_{\lambda }={C}_{\lambda }$, $\lambda \in \left[0,1\right]$ iff ${A}_{\lambda }^{-}+{B}_{\lambda }^{-}={C}_{\lambda }^{-}$ and ${A}_{\lambda }^{+}+{B}_{\lambda }^{+}={C}_{\lambda }^{+}$ for any $\lambda \in \left[0,1\right]$. ${A}_{\lambda }\cdot {B}_{\lambda }={C}_{\lambda }$, where

${C}_{\lambda }^{-}=\mathrm{min}\left\{{A}_{\lambda }^{-}\cdot {B}_{\lambda }^{-},{A}_{\lambda }^{-}\cdot {B}_{\lambda }^{+},{A}_{\lambda }^{+}\cdot {B}_{\lambda }^{-},{A}_{\lambda }^{+}\cdot {B}_{\lambda }^{+}\right\},$

${C}_{\lambda }^{+}=\mathrm{max}\left\{{A}_{\lambda }^{-}\cdot {B}_{\lambda }^{-},{A}_{\lambda }^{-}\cdot {B}_{\lambda }^{+},{A}_{\lambda }^{+}\cdot {B}_{\lambda }^{-},{A}_{\lambda }^{+}\cdot {B}_{\lambda }^{+}\right\}.$

Define

$D\left(\stackrel{˜}{A},\stackrel{˜}{B}\right)=\underset{\lambda \in \left[0,1\right]}{\mathrm{sup}}d\left({A}_{\lambda },{B}_{\lambda }\right)=\underset{\lambda \in \left[0,1\right]}{\mathrm{sup}}\mathrm{max}\left\{|{A}_{\lambda }^{-}-{B}_{\lambda }^{-}|,|{A}_{\lambda }^{+}-{B}_{\lambda }^{+}|\right\},$

where d is the Hausdorff metric. $D\left(\stackrel{˜}{A},\stackrel{˜}{B}\right)$ is called the distance between $\stackrel{˜}{A}$ and $\stackrel{˜}{B}$.

Using the results of  , we see that

1) $\left({\stackrel{˜}{R}}^{c},D\right)$ is a complete metric space,

2) $D\left(u+w,v+w\right)=D\left(u,v\right)$,

3) $D\left(ku,kv\right)=|k|D\left(u,v\right),\text{\hspace{0.17em}}k\in R$,

4) $D\left(u+v,w+e\right)\le D\left(u,w\right)+D\left(v,e\right)$,

5) $D\left(u+v,\stackrel{¯}{0}\right)\le D\left(u,\stackrel{¯}{0}\right)+D\left(v,\stackrel{¯}{0}\right)$,

6) $D\left(u+v,w\right)\le D\left(u,w\right)+D\left(v+\stackrel{¯}{0}\right)$,

Where $u,v,w,e\in {\stackrel{˜}{R}}^{c}$, $\stackrel{˜}{0}$ represents zero fuzzy number.

A sequence $\left\{{x}_{n}\right\}$ of fuzzy numbers is said to be statistically convergent to a fuzzy number ${x}_{0}$ if for each $\epsilon >0$ the set $A\left(\epsilon \right)=\left\{n\in N:D\left({x}_{n},{x}_{0}\right)\ge \epsilon \right\}$ has natural density zero. The fuzzy number ${x}_{0}$ is called the statistical limit of the sequence $\left\{{x}_{n}\right\}$ and we write st- $\underset{n\to \infty }{lim}{x}_{n}={x}_{0}$ .

The concept of Orlicz function was introduced by Parashar and Choudhary , A mapping $M:\left[0,\infty \right)\to \left[0,\infty \right)$ is said to be an Orlicz funtion 

1) $M\left(0\right)=0$ iff $x=0$,

2) $M\left(x\right)>0$ for $x>0$,

3) $M\left(x\right)\to \infty$ as $x\to \infty$,

4) M is continuous, nondecreasing and convex.

An Orlicz function may be bounded or unbounded. For example, $M\left(x\right)={x}^{p}\left(0 is bounded.

A triple sequence can be defined as a function $X:N×N×N\to R\left(C\right)$ where N, R nad C denote the set of natural numbers, real numbers and complec numbers respectively. A triple sequence $\left\{{x}_{ijk}\right\}$ is said to be Cauchy sequence if for every $\epsilon >0$, there exist $N\left(\epsilon \right)\in N$ such that $|{x}_{ijk}-{x}_{pqr}|<\epsilon$ whenever $i,p\ge N$, $j,q\ge N$, $k,r\ge N$ .

A triple sequence $\left\{{x}_{ijk}\right\}$ is called statistically pre-Cauchy if for every $\epsilon >0$ there exist $p=p\left(\epsilon \right),q\left(\epsilon \right)$ and $r\left(\epsilon \right)$ such that

$\underset{m,n,t\to \infty }{lim}|\frac{1}{{m}^{2}{n}^{2}{t}^{2}}|{x}_{ijk}-{x}_{pqr}|\ge \epsilon ,i\le m,j\le n,t\le k|=0.$

where the vertical bars indicate the number of elements in the set .

3. Main Results

Definition 3.1. A triple sequence of fuzzy numbers is said to be ideal statistically pre-Cauchy if for every $\epsilon >0,\delta >0$ there exist $p=p\left(\epsilon \right),q\left(\epsilon \right)$ and $r\left(\epsilon \right)$ such that

$\left\{\left(m,n,t\right)\in N×N×N:\frac{1}{{m}^{2}{n}^{2}{t}^{2}}|\left\{\left(i,j,k\right):D\left({x}_{ijk},{x}_{pqr}\right)\ge \epsilon ,i\le m,j\le n,t\le k\right\}|\ge \delta \right\}\in I.$

where the I denote the nontrivival ideal of N.

Theorem 3.1. Let $x=\left\{{x}_{ijk}\right\}$ be a triple sequence of fuzzy number and let M be a bounded Oricz function. Then x is ideal statistically pre-Cauchy if and only if

$\left\{\left(m,n,t\right)\in N×N×N:\frac{1}{{m}^{2}{n}^{2}{t}^{2}}\underset{i,p\le m}{\sum }\text{\hspace{0.17em}}\underset{j,q\le n}{\sum }\text{\hspace{0.17em}}\underset{k,r\le t}{\sum }\text{ }\text{ }M\left(\frac{D\left({x}_{ijk},{x}_{pqr}\right)}{\rho }\right)\ge \delta \right\}\in I.$

Proof. Suppose that

$\left\{\left(m,n,t\right)\in N×N×N:\frac{1}{{m}^{2}{n}^{2}{t}^{2}}\underset{i,p\le m}{\sum }\text{\hspace{0.17em}}\underset{j,q\le n}{\sum }\text{\hspace{0.17em}}\underset{k,r\le t}{\sum }\text{ }\text{ }M\left(\frac{D\left({x}_{ijk},{x}_{pqr}\right)}{\rho }\right)\ge \delta \right\}\in I.$

For each $\epsilon >0,\delta >0$ and $\rho >0$, $m,n,t\in N$, we have

$\begin{array}{l}\frac{1}{{m}^{2}{n}^{2}{t}^{2}}\underset{i,p\le m}{\sum }\text{\hspace{0.17em}}\underset{j,q\le n}{\sum }\text{\hspace{0.17em}}\underset{k,r\le t}{\sum }\text{ }\text{ }M\left(\frac{D\left({x}_{ijk},{x}_{pqr}\right)}{\rho }\right)\\ =\frac{1}{{m}^{2}{n}^{2}{t}^{2}}\underset{i,p\le m}{\sum }\text{\hspace{0.17em}}\underset{j,q\le n}{\sum }\text{\hspace{0.17em}}\underset{k,r\le tD\left({x}_{ijk},{x}_{pqr}\right)\le \epsilon }{\sum }\text{ }\text{ }M\left(\frac{D\left({x}_{ijk},{x}_{pqr}\right)}{\rho }\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }+\frac{1}{{m}^{2}{n}^{2}{t}^{2}}\underset{i,p\le m}{\sum }\text{\hspace{0.17em}}\underset{j,q\le n}{\sum }\text{\hspace{0.17em}}\underset{k,r\le tD\left({x}_{ijk},{x}_{pqr}\right)\ge \epsilon }{\sum }\text{ }\text{ }M\left(\frac{D\left({x}_{ijk},{x}_{pqr}\right)}{\rho }\right)\\ \ge \frac{1}{{m}^{2}{n}^{2}{t}^{2}}\underset{i,p\le m}{\sum }\text{\hspace{0.17em}}\underset{j,q\le n}{\sum }\text{\hspace{0.17em}}\underset{k,r\le tD\left({x}_{ijk},{x}_{pqr}\right)\le \epsilon }{\sum }\text{ }\text{ }M\left(\frac{D\left({x}_{ijk},{x}_{pqr}\right)}{\rho }\right)\\ \ge M\left(\epsilon \right)\left\{\frac{1}{{m}^{2}{n}^{2}{t}^{2}}|\left\{\left(i,j,k\right):D\left({x}_{ijk},{x}_{pqr}\right)\ge \epsilon ,i\le m,j\le n,k\le t\right\}|\ge \delta \right\}\in I.\end{array}$

Now suppose that x is ideal statistically pre-Cauchy and that $\epsilon$ has been given.

Let $\epsilon >0,\delta >0$ be such that $M\left(\xi \right)<\frac{\epsilon }{2}$.

Since M is bounded Orlicz function, there exist an integer G such that $M\left(x\right)<\frac{G}{2}$ for all $x\ge 0$.

Not that, for each $m,n,t\in N$

$\begin{array}{l}\frac{1}{{m}^{2}{n}^{2}{t}^{2}}\underset{i,p\le m}{\sum }\text{\hspace{0.17em}}\underset{j,q\le n}{\sum }\text{\hspace{0.17em}}\underset{k,r\le t}{\sum }\text{ }\text{ }M\left(\frac{D\left({x}_{ijk},{x}_{pqr}\right)}{\rho }\right)\\ =\frac{1}{{m}^{2}{n}^{2}{t}^{2}}\underset{i,p\le m}{\sum }\text{\hspace{0.17em}}\underset{j,q\le n}{\sum }\text{\hspace{0.17em}}\underset{k,r\le tD\left({x}_{ijk},{x}_{pqr}\right)<\xi }{\sum }\text{ }\text{ }M\left(\frac{D\left({x}_{ijk},{x}_{pqr}\right)}{\rho }\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}+\frac{1}{{m}^{2}{n}^{2}{t}^{2}}\underset{i,p\le m}{\sum }\text{\hspace{0.17em}}\underset{j,q\le n}{\sum }\text{\hspace{0.17em}}\underset{k,r\le tD\left({x}_{ijk},{x}_{pqr}\right)\ge \xi }{\sum }\text{ }\text{ }M\left(\frac{D\left({x}_{ijk},{x}_{pqr}\right)}{\rho }\right)\\ \le M\left(\xi \right)+\frac{1}{{m}^{2}{n}^{2}{t}^{2}}\underset{i,p\le m}{\sum }\text{\hspace{0.17em}}\underset{j,q\le n}{\sum }\text{\hspace{0.17em}}\underset{k,r\le tD\left({x}_{ijk},{x}_{pqr}\right)\ge \xi }{\sum }\text{ }\text{ }M\left(\frac{D\left({x}_{ijk},{x}_{pqr}\right)}{\rho }\right)\\ \le \frac{\epsilon }{2}+\frac{G}{2}\left\{\frac{1}{{m}^{2}{n}^{2}{t}^{2}}|\left\{\left(i,j,k\right):D\left({x}_{ijk},{x}_{pqr}\right)\ge \xi ,i\le m,j\le n,k\le t\right\}|\ge \delta \right\}\\ \le \epsilon +G\left\{\frac{1}{{m}^{2}{n}^{2}{t}^{2}}|\left\{\left(i,j,k\right):D\left({x}_{ijk},{x}_{pqr}\right)\ge \xi ,i\le m,j\le n,k\le t\right\}|\ge \delta \right\}\in I.\end{array}$

Hence

$\left\{\left(m,n,t\right)\in N×N×N:\frac{1}{{m}^{2}{n}^{2}{t}^{2}}\underset{i,p\le m}{\sum }\text{\hspace{0.17em}}\underset{j,q\le n}{\sum }\text{\hspace{0.17em}}\underset{k,r\le t}{\sum }\text{ }\text{ }M\left(\frac{D\left({x}_{ijk},{x}_{pqr}\right)}{\rho }\right)\ge \delta \right\}\in I.$

Theorem 3.2. Let $x=\left\{{x}_{ijk}\right\}$ be a triple sequence of fuzzy numbers and let M be a bounded Orlicz function. Then x is ideal statistically convergent to ${x}_{0}$ if and only if

$\left\{\left(m,n,t\right)\in N×N×N:\frac{1}{mnt}\underset{i=1}{\overset{m}{\sum }}\text{\hspace{0.17em}}\underset{j=1}{\overset{n}{\sum }}\text{\hspace{0.17em}}\underset{k=1}{\overset{t}{\sum }}\text{ }\text{ }M\left(\frac{D\left({x}_{ijk},{x}_{0}\right)}{\rho }\right)\ge \delta \right\}\in I.$

Proof. Suppose that

$\left\{\left(m,n,t\right)\in N×N×N:\frac{1}{mnt}\underset{i=1}{\overset{m}{\sum }}\text{\hspace{0.17em}}\underset{j=1}{\overset{n}{\sum }}\text{\hspace{0.17em}}\underset{k=1}{\overset{t}{\sum }}\text{ }\text{ }M\left(\frac{D\left({x}_{ijk},{x}_{0}\right)}{\rho }\right)\ge \delta \right\}\in I.$

For each $\epsilon >0,\delta >0$ and $\rho >0$, $m,n,t\in N$, we have

$\begin{array}{l}\frac{1}{mnt}\underset{i=1}{\overset{m}{\sum }}\text{\hspace{0.17em}}\underset{j=1}{\overset{n}{\sum }}\text{\hspace{0.17em}}\underset{k=1}{\overset{t}{\sum }}\text{ }\text{ }M\left(\frac{D\left({x}_{ijk},{x}_{0}\right)}{\rho }\right)\\ =\frac{1}{mnt}\underset{i=1}{\overset{m}{\sum }}\text{\hspace{0.17em}}\underset{j=1}{\overset{n}{\sum }}\text{\hspace{0.17em}}\underset{k=1,D\left({x}_{ijk},{x}_{0}\right)\ge \epsilon }{\overset{t}{\sum }}\text{ }\text{ }M\left(\frac{D\left({x}_{ijk},{x}_{0}\right)}{\rho }\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\frac{1}{mnt}\underset{i=1}{\overset{m}{\sum }}\text{\hspace{0.17em}}\underset{j=1}{\overset{n}{\sum }}\text{\hspace{0.17em}}\underset{k=1,D\left({x}_{ijk},{x}_{0}\right)<\epsilon }{\overset{t}{\sum }}\text{ }\text{ }M\left(\frac{D\left({x}_{ijk},{x}_{0}\right)}{\rho }\right)\\ \ge \frac{1}{mnt}\underset{i=1}{\overset{m}{\sum }}\text{\hspace{0.17em}}\underset{j=1}{\overset{n}{\sum }}\text{\hspace{0.17em}}\underset{k=1,D\left({x}_{ijk},{x}_{0}\right)\ge \epsilon }{\overset{t}{\sum }}\text{ }\text{ }M\left(\frac{D\left({x}_{ijk},{x}_{0}\right)}{\rho }\right)\\ \ge M\left(\epsilon \right)\left\{\frac{1}{mnt}|\left\{\left(i,j,k\right):D\left({x}_{ijk},{x}_{0}\right)\ge \epsilon ,i\le m,j\le n,k\le t\right\}|\ge \delta \right\}\in I.\end{array}$

We have x is ideal statistically convergent to ${x}_{0}$.

Now suppose that x is ideal statistically convergent to ${x}_{0}$, let $\epsilon >0,\delta >0$ be such that $M\left(\xi \right)<\frac{\epsilon }{2}$.

Since M is bounded Orlicz function, there exist an integer G such that $M\left(X\right)<\frac{G}{2}$ for all $x\ge 0$.

Note that, for each $m,n,t\in N$

$\begin{array}{l}\frac{1}{mnt}\underset{i=1}{\overset{m}{\sum }}\text{\hspace{0.17em}}\underset{j=1}{\overset{n}{\sum }}\text{\hspace{0.17em}}\underset{k=1}{\overset{t}{\sum }}\text{ }\text{ }M\left(\frac{D\left({x}_{ijk},{x}_{0}\right)}{\rho }\right)\\ =\frac{1}{mnt}\underset{i=1}{\overset{m}{\sum }}\text{\hspace{0.17em}}\underset{j=1}{\overset{n}{\sum }}\text{\hspace{0.17em}}\underset{k=1,D\left({x}_{ijk},{x}_{0}\right)\ge \epsilon }{\overset{t}{\sum }}\text{ }\text{ }M\left(\frac{D\left({x}_{ijk},{x}_{0}\right)}{\rho }\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }\text{ }+\frac{1}{mnt}\underset{i=1}{\overset{m}{\sum }}\text{\hspace{0.17em}}\underset{j=1}{\overset{n}{\sum }}\text{\hspace{0.17em}}\underset{k=1,D\left({x}_{ijk},{x}_{0}\right)<\epsilon }{\overset{t}{\sum }}\text{ }\text{ }M\left(\frac{D\left({x}_{ijk},{x}_{0}\right)}{\rho }\right)\\ \le M\left(\xi \right)+\frac{1}{mnt}\underset{i=1}{\overset{m}{\sum }}\text{\hspace{0.17em}}\underset{j=1}{\overset{n}{\sum }}\text{\hspace{0.17em}}\underset{k=1,D\left({x}_{ijk},{x}_{0}\right)\ge \epsilon }{\overset{t}{\sum }}\text{ }\text{ }M\left(\frac{D\left({x}_{ijk},{x}_{0}\right)}{\rho }\right)\\ \le \frac{\epsilon }{2}+\frac{G}{2}\left\{\frac{1}{mnt}|\left\{\left(i,j,k\right):D\left({x}_{ijk},{x}_{pqr}\right)\ge \xi ,i\le m,j\le n,k\le t\right\}|\ge \delta \right\}\\ \le \epsilon +G\left\{\frac{1}{mnt}|\left\{\left(i,j,k\right):D\left({x}_{ijk},{x}_{pqr}\right)\ge \xi ,i\le m,j\le n,k\le t\right\}|\ge \delta \right\}\in I.\end{array}$

Hence

$\left\{\left(m,n,t\right)\in N×N×N:\frac{1}{mnt}\underset{i=1}{\overset{m}{\sum }}\text{\hspace{0.17em}}\underset{j=1}{\overset{n}{\sum }}\text{\hspace{0.17em}}\underset{k=1}{\overset{t}{\sum }}\text{ }\text{ }M\left(\frac{D\left({x}_{ijk},{x}_{0}\right)}{\rho }\right)\ge \delta \right\}\in I.$

Corollary 3.3. Let $x=\left\{{x}_{ijk}\right\}$ be a bound triple sequence of fuzzy number. Then x is ideal statistically pre-Cauchy if and only if

$\left\{\left(m,n,t\right)\in N×N×N:\frac{1}{{m}^{2}{n}^{2}{t}^{2}}\underset{i=1}{\overset{m}{\sum }}\text{\hspace{0.17em}}\underset{j=1}{\overset{n}{\sum }}\text{\hspace{0.17em}}\underset{k=1}{\overset{t}{\sum }}\text{ }\text{ }D\left({x}_{ijk},{x}_{pqr}\right)\ge \delta \right\}\in I.$

Proof. Let $K=\underset{i,j,k}{\mathrm{sup}}D\left({x}_{ijk},\stackrel{¯}{0}\right)$ and defined $M\left(x\right)=\frac{\left(1+2K\right)x}{1+x}$, then $M\left(\frac{D\left({x}_{ijk},{x}_{pqr}\right)}{\rho }\right)\le \left(1+2K\right)D\left({x}_{ijk},{x}_{pqr}\right)$, and

$\begin{array}{c}M\left(\frac{D\left({x}_{ijk},{x}_{pqr}\right)}{\rho }\right)=\left(1+2K\right)\frac{D\left({x}_{ijk},{x}_{pqr}\right)}{1+D\left({x}_{ijk},{x}_{pqr}\right)}\\ \ge \frac{\frac{\left(1+2K\right)D\left({x}_{ijk},{x}_{pqr}\right)}{\rho }}{1+D\left({x}_{ijk},{x}_{pqr}\right)}\\ \ge \frac{\left(1+2K\right)D\left({x}_{ijk},{x}_{pqr}\right)}{1+2A}=D\left({x}_{ijk},{x}_{pqr}\right).\end{array}$

Hence $\left\{\left(m,n,t\right)\in N×N×N:\frac{1}{{m}^{2}{n}^{2}{t}^{2}}\underset{i=1}{\overset{m}{\sum }}\text{\hspace{0.17em}}\underset{j=1}{\overset{n}{\sum }}\text{\hspace{0.17em}}\underset{k=1}{\overset{t}{\sum }}\text{ }\text{ }D\left({x}_{ijk},{x}_{pqr}\right)\ge \delta \right\}\in I$, if and only if $\left\{\left(m,n,t\right)\in N×N×N:\frac{1}{{m}^{2}{n}^{2}{t}^{2}}\underset{i,p\le m}{\sum }\text{\hspace{0.17em}}\underset{j,k\le n}{\sum }\text{\hspace{0.17em}}\underset{t,r\le t}{\sum }\text{ }\text{ }M\left(\frac{D\left({x}_{ijk},{x}_{pqr}\right)}{\rho }\right)\ge \delta \right\}\in I$, and an immediate application of Theorem 3.1 completes the proof.

Corollary 3.4. Let $x=\left\{{x}_{ijk}\right\}$ be a bound triple sequence of fuzzy number. Then x is ideal statistically convergent ${x}_{0}$ if and only if

$\left\{\left(m,n,t\right)\in N×N×N:\frac{1}{mnt}\underset{i=1}{\overset{m}{\sum }}\text{\hspace{0.17em}}\underset{j=1}{\overset{n}{\sum }}\text{\hspace{0.17em}}\underset{k=1}{\overset{t}{\sum }}\text{ }\text{ }D\left({x}_{ijk},{x}_{0}\right)\ge \delta \right\}\in I.$

Proof. Let $K=\underset{i,j,k}{\mathrm{sup}}D\left({x}_{ijk},\stackrel{¯}{0}\right)$ and defined $M\left(x\right)=\frac{\left(1+K+{x}_{0}\right)x}{1+x}$, then $M\left(\frac{D\left({x}_{ijk},{x}_{0}\right)}{\rho }\right)\le \left(1+K+{x}_{0}\right)D\left({x}_{ijk},{x}_{0}\right)$, and

$\begin{array}{c}M\left(\frac{D\left({x}_{ijk},{x}_{0}\right)}{\rho }\right)=\left(1+K+{x}_{0}\right)\frac{D\left({x}_{ijk},{x}_{0}\right)}{1+D\left({x}_{ijk},{x}_{0}\right)}\\ \ge \frac{\frac{\left(1+K+{x}_{0}\right)D\left({x}_{ijk},{x}_{0}\right)}{\rho }}{1+D\left({x}_{ijk},{x}_{0}\right)}\\ \ge \frac{\left(1+K+{x}_{0}\right)D\left({x}_{ijk},{x}_{0}\right)}{1+K+{x}_{0}}=D\left({x}_{ijk},{x}_{0}\right).\end{array}$

Hence $\left\{\left(m,n,t\right)\in N×N×N:\frac{1}{mnt}\underset{i=1}{\overset{m}{\sum }}\text{\hspace{0.17em}}\underset{j=1}{\overset{n}{\sum }}\text{\hspace{0.17em}}\underset{k=1}{\overset{t}{\sum }}\text{ }D\left({x}_{ijk},{x}_{0}\right)\ge \delta \right\}\in I$, if and only if $\left\{\left(m,n,t\right)\in N×N×N:\frac{1}{{m}^{2}{n}^{2}{t}^{2}}\underset{i=1}{\overset{m}{\sum }}\text{\hspace{0.17em}}\underset{j=1}{\overset{n}{\sum }}\text{\hspace{0.17em}}\underset{k=1}{\overset{t}{\sum }}\text{ }\text{ }M\left(\frac{D\left({x}_{ijk},{x}_{0}\right)}{\rho }\right)\ge \delta \right\}\in I$, and an immediate application of Theorem 3.1 completes the proof.

4. Conclusion

In this article, we introduced ideal statistically pre-Cauchy triple sequences of fuzzy numbers about Orlicz function. At the same time, we have proved some properties and relationships.

Fund

This work is supported by National Natural Science Fund of China (11761056); the Natural Science Foundation of Qinghai Province (2020-ZJ-920); University level planning project of Qinghai Minzu University (2021XJGH24).

Cite this paper: Feng, X. (2021) Ideal Statistically Pre-Cauchy Triple Sequences of Fuzzy Number and Orlicz Functions. Applied Mathematics, 12, 767-774. doi: 10.4236/am.2020.129051.
References

   Fast, H. (1951) Sur la convergence statistique. Colloquium Mathematicum, 2, 241-244.
https://doi.org/10.4064/cm-2-3-4-241-244

   Buck, R.C. (1953) Generalized Asymptote Density. American Journal of Mathematics, 75, 335-346.
https://doi.org/10.2307/2372456

   Schoenberg, I.J. (1959) The Integrability of Certain Functions and Related Summability Methods. The American Mathematical Monthly, 66, 361-375.
https://doi.org/10.2307/2308747

   Altinok, H. and Et, M. (2015) On λ-Statistical Boundedness of Order β of Sequences of Fuzzy Numbers. Soft Computing, 19, 2095-2100.
https://doi.org/10.1007/s00500-015-1660-2

   Connor, J.S. (1988) The Statistical and Strong p-Cesàro Convergence of Sequences. Analysis, 8, 47-63.
https://doi.org/10.1524/anly.1988.8.12.47

   Et, M. (2013) Generalized Cesàro Difference Sequence Spaces of Non-Absolute Type Involving Lacunary Sequences. Applied Mathematics and Computation, 219, 9372-9376.
https://doi.org/10.1016/j.amc.2013.03.039

   Et, M., Altinok, H. and Altin, Y. (2004) On Some Generalized Sequence Spaces. Applied Mathematics and Computation, 154, 167-173.
https://doi.org/10.1016/S0096-3003(03)00700-8

   Et, M. and Sengül, H. (2014) Some Cesàro-Type Summability Spaces of Order α and Lacunary Statistical Convergence of Order α. Filomat, 28, 1593-1602.
https://doi.org/10.2298/FIL1408593E

   Fridy, J. (1985) On Statistical Convergence. Analysis, 5, 301-313.
https://doi.org/10.1524/anly.1985.5.4.301

   Fridy, J. and Orhan, C. (1993) Lacunary Statistical Convergence. Pacific Journal of Mathematics, 160, 43-51.
https://doi.org/10.2140/pjm.1993.160.43

   Mursaleen, M. (2000) λ-Statistical Convergence. Math Slovaca, 50, 28-37.

   Matloka, M. (1986) Sequences of Fuzzy Numbers. BUSEFAL, 28, 28-37.

   Nuray, F. and Savas, E. (1995) Statistical Convergence of Fuzzy Real Numbers. Mathematica Slovaca, 45, 269-273.

   Tripathy, B.C. and Baruah, A. (2010) Lacunary Statistically Convergent and Lacunary Strongly Convergent Generalized Difference Sequences of Fuzzy Real Numbers. Kyungpook Mathematical Journal, 50, 565-574.
https://doi.org/10.5666/KMJ.2010.50.4.565

   Canak, I. (2014) Tauberian Theorems for Cesàro Summability of Sequences of Fuzzy Numbers. Journal of Intelligent & Fuzzy Systems, 27, 937-942.
https://doi.org/10.3233/IFS-131053

   Mursaleen, M. and Mohiuddine, S.A. (2009) On Lacunary Statistical Convergence with Respect to the Intuitionistic Fuzzy Normed Space. Journal of Computational and Applied Mathematics, 233, 142-149.
https://doi.org/10.1016/j.cam.2009.07.005

   Savas, E. (2000) A Note on Sequence of Fuzzy Numbers. Information Sciences, 124, 297-300.
https://doi.org/10.1016/S0020-0255(99)00073-0

   Hazarika, B. (2013) Lacunary Difference Ideal Convergent Sequence Spaces of Fuzzy Numbers. Journal of Intelligent and Fuzzy Systems, 25, 157-166.
https://doi.org/10.3233/IFS-2012-0622

   Aytar, S. and Pehlivan, S. (2008) Statistical Convergence of Sequences of Fuzzy Numbers and Sequences of α-Cuts. International Journal of General Systems, 37, 231-237.
https://doi.org/10.1080/03081070701251075

   Lindenstrauss, J. and Tzafiri, L. (1971) On Orlicz Sequence Spaces. Israel Journal of Mathematics, 10, 379-390.
https://doi.org/10.1007/BF02771656

   Connor, J.S., Fridy, J. and Kline, J. (1994) Statistically Pre-Cauchy Sequences. Analysis, 14, 311-317.
https://doi.org/10.1524/anly.1994.14.4.311

   Barlak, D. (2020) Statistical Convergence of Order for Double Sequences of Fuzzy Numbers. Journal of Intelligent and Fuzzy Systems, 39, 6949-6954.
https://doi.org/10.3233/JIFS-200039

   Saini, K., Raj, K. and Mursaleen M. (2021) Deferred Cesaro and Deferred Euler Equi-Statistical Convergence and Its Applications to Korovkin-Type Approximation Theorem. International Journal of General Systems, 50, 567-579.
https://doi.org/10.1080/03081079.2021.1942867

   Nuray, F. and Savaş, E. (1998) Lacunary Statistical Convergence of Sequences of Fuzzy Numbers. Fuzzy Sets & Systems, 99, 353-355.
https://doi.org/10.1016/S0165-0114(98)00031-1

   Parashar, S.D. and Choudhary, B. (1994) Sequence Spaces Defined by Orlicz Functions. Indian Journal of Pure and Applied Mathematics, 25, 419-428.

   Tripathy, B.C. and Mehta, S. (2003) On a Class of Sequences Related to Space Defined by Orlicz Functions. Soochow Journal of Mathematics, 29, 379-391.

   Sahiner, A., Gürdal, M. and Düden, K. (2007) Triple Sequences and Their Statistical Convergence. Selcuk Journal of Applied Mathematics, 8, 49-55.

   Sabiha, T. (2018) Statistically Pre-Cauchy Triple Sequences and Orlicz Functions. Pure Mathematical Sciences, 7, 1-9.
https://doi.org/10.12988/pms.2018.756

Top