Multiple sequences of fuzzy numbers and their statistical convergence

Abstract

Purpose

The purpose of this paper is to extend a generalized convergence method, namely, statistical convergence to sequences of fuzzy numbers of multiplicity greater than two.

Methods

We use analytic method to obtain our results.

Results

Certain theorems on statistical convergence of real double sequences obtained by Savaş et al. and Móricz are also extended to multiple sequences of fuzzy numbers. Finally, we define Cesàro summable and strongly p-Cesàro summable multiple sequences of fuzzy numbers and obtained their relations with statistical convergence.

Conclusions

Although, we prove our results only for triple sequences, but all these results remain true for d-multiple sequences as well.

Background

Statistical convergence for real number sequences was introduced by Fast [1] and Schonenberg [2] independently. Later, the idea was further investigated from sequence space point of view and linked with summability theory by Fridy [3], Šalát [4] and many others. The idea is based on the notion of natural density of subsets of $\mathbb{N}$, the set of positive integers. For any subset A of N, the natural density of A is denoted by $\delta \left(A\right)$ and is defined by

$$\begin{array}{l}\delta \left(A\right)=\underset{n\to \infty }{lim}\frac{1}{n+1}\left|\right\{k\le n\text{:}k\in A\left\}\right|,\end{array}$$

(1)

where vertical bars denote the cardinality of the enclosed set. Using this definition, the notions of statistical convergence and statistically Cauchy for a number sequence are defined (in [5]) as follows.

A sequence $x=\left(x_k\right)$ of numbers is said to be statistically convergent to some number L, in symbol: $\mathrm{St}-lim{x}_k=L$, if for each $휖>0$,

$$\begin{array}{l}\underset{n\to \infty }{lim}\frac{1}{n+1}\left|\right\{k\le n\text{:}|x_k-L|\ge 휖\left\}\right|=0.\end{array}$$

(2)

i.e., $\delta \left(\right\{k\in N$ : $|x_k-L|\ge 휖\left\}\right)=0$.

A sequence $x=\left(x_k\right)$ of numbers is said to be a statistical Cauchy if, for each $휖>0$, there is a positive integer m such that

$$\begin{array}{l}\underset{n\to \infty }{lim}\frac{1}{n+1}\left|\right\{k\le n\text{:}|x_k-x_m|\ge 휖\left\}\right|=0.\end{array}$$

(3)

Agnew [6] studied the summability theory of multiple sequences and obtained certain theorems which have already been proved for double sequences by the author himself. Móricz [7] continued with the study of multiple sequences and gave some remarks on the notion of regular convergence of multiple series. In 2003, the author extended statistical convergence from single to multiple real sequences and obtained some results for real double sequences. Savaş et al. [8] studied a similar method of convergence with the help of lacunary sequences for multiple sequences of numbers and called it lacunary statistical convergence. However, Şahiner et al. [9] and Sharma et al. [10], respectively, developed statistical convergence for triple sequences of real numbers and for sequences on probabilistic normed spaces.

On the other side, fuzzy set theory is a powerful hand set for modelling uncertainty and vagueness in various problems arising in the field of science and engineering. It has a wide range of applications in various fields: population dynamics, chaos control, computer programming, nonlinear dynamical systems, etc. Fuzzy topology is one of the most important and useful tools to deal with such situations where the use of classical theories breaks down. While studying fuzzy topological spaces, we face many situations where we need to deal with convergence of sequences of fuzzy numbers. The concept of usual convergence of fuzzy numbers sequences was introduced by Matloka [11], where he proved some basic theorems. Nanda [12] continued with this study and showed that the set of all convergent sequences of fuzzy numbers form a complete metric space. In recent years, statistical convergence has also been adapted to the sequences of fuzzy numbers. The credit goes to Nuray and Savaş [13], who first defined the concepts of statistical convergence and statistically Cauchy for sequences of fuzzy numbers. They proved that a sequence of fuzzy numbers is statistically convergent, if and only if, it is a statistically Cauchy. Nuray [14] introduced lacunary statistical convergence of fuzzy numbers sequences, whereas Kwon [15] obtained relationship between statistical convergence and strong p-Cesàro summability of fuzzy numbers sequences. For further development on statistical convergence of fuzzy number sequences, we refer Savaş [16], and Savaş et al. [17]. Let $C\left(R^n\right)$ = {$A\subset R^n$: A is compact and convex}. The space $C\left(R^n\right)$ has a linear structure induced by the operations

$$\begin{array}{cc}A+B& =\{a+b:a\in A,b\in B\}\text{and}\\ \lambda A& =\{\lambda a:a\in A\}\end{array}$$

(4)

for $A,B\in C\left(R^n\right)$; $\lambda \in R$. The Hausdroff distance between A and B is defined by

$$\begin{array}{l}{\delta }_{\infty }(A,B)=max\{\underset{a\in A}{sup}\underset{b\in B}{inf}\Vert a-b\Vert ,\underset{b\in B}{sup}\underset{a\in A}{inf}\Vert a-b\Vert \}\end{array}$$

(5)

It is well known that $\left(C\right(R^n),{\delta }_{\infty })$ is a complete (not separable) metric space.

Definition 2.1

A fuzzy number is a function X from $R^n$ to [0,1], which satisfies the following conditions:

• (i) X is normal, i.e., there exists $x_0\in R^n$ such that $X\left(x_0\right)=1$.

• (ii) X is a fuzzy convex, i.e., for any $x,y\in R^n$ and,

  $$\lambda \in \left[0,1\right],X(\lambda x+(1-\lambda \left)y\right)\ge min\left\{X\right(x),X(y\left)\right\}.$$

  (6)

• (iii) X is upper semi-continuous.

• (iv) The closure of the set $\{x\in R^n$ : $X\left(x\right)>0\}$, denoted by $X^0$, is compact.

The properties (i)-(iv) imply for each $\alpha \in (\left.0,1\right]$, the $\alpha$-level set,

$$X^{\alpha }=\{x\in R^n\text{:}X(x)\ge \alpha \}=\left[{\underset{―}{X}}^{\alpha },{\bar{X}}^{\alpha }\right]$$

(7)

is a non-empty compact convex subset of $R^n$. Let $L\left(R^n\right)$ denote the set of all fuzzy numbers. The linear structure of $L\left(R^n\right)$ induces an addition $X+Y$ and a scalar multiplication $\lambda X$ in terms of $\alpha$-level sets by

$$\begin{array}{cc}{\left[X+Y\right]}^{\alpha }& ={\left[X\right]}^{\alpha }+{\left[Y\right]}^{\alpha }\text{and}\\ {\left[\lambda X\right]}^{\alpha }& =\lambda {\left[X\right]}^{\alpha }(X,Y\in L(R^n),\lambda \in R)\end{array}$$

(8)

for each $\alpha \in \left[0,1\right]$. Define, for each $1\le q<\infty$,

$$\begin{array}{l}{d}_q(X,Y)={\left({\int }_0^1{\delta }_{\infty }{(X^{\alpha },Y^{\alpha })}^q{d}_{\alpha }\right)}^{\frac{1}{q}}\end{array}$$

(9)

and $d_{\infty }=\underset{0\le \alpha \le 1}{sup}{\delta }_{\infty }(X^{\alpha },Y^{\alpha })$. Clearly, $d_{\infty }(X,Y)=\underset{q\to \infty }{lim}{d}_q(X,Y)$ with $d_q\le d_r$ if $q\le r$. Moreover, $d_q$ is a complete, separable and locally compact metric space.

Throughout the paper, d will denote $d_q$ with $1\le q<\infty$, and $N^3$ will denote the usual product set $N\times N\times N$. We now quote the following definitions which will be needed in the sequel.

Definition 2.2

A triple sequence $X=\left(X_{\mathrm{ijk}}\right)$ of fuzzy numbers is said to be convergent to a fuzzy number $X_0$ if for each $휖>0$, there exist a positive integer m such that

$$d(X_{\mathrm{ijk}},X_0)<휖\text{for every}i,j,k\ge \mathrm{m.}$$

(10)

The fuzzy number $X_0$ is called the limit of the sequence $X_{\mathrm{ijk}})$ and we write $\underset{i,j,k\to \infty }{lim}{X}_{\mathrm{nkl}}=X_0$.

Definition 2.3

A triple sequence $X=\left(X_{\mathrm{nkl}}\right)$ of fuzzy numbers is said to be a Cauchy sequence if, for each $휖>0$, there exists a positive integer $n_0$ such that

$$d(X_{\mathrm{ijk}},X_{\mathrm{NKL}})<휖$$

(11)

for every $i\ge N\ge n_0,j\ge K\ge n_0,k\ge L\ge n_0$.

Definition 2.4

A triple sequence $X=\left(X_{\mathrm{ijk}}\right)$ of fuzzy numbers is said to be bounded if there exists a positive number M such that

$$d(X_{\mathrm{ijk}},\tilde{0})<M\text{for all}i,j,\mathrm{k.}$$

(12)

Let $l_{\infty }^3$ denote the set of all bounded triple sequences of fuzzy numbers.

Results and Discussion

In present paper, we introduce statistical convergence of sequences of fuzzy numbers having multiplicity greater than two. Certain Theorems regarding uniqueness of limit, algebraic characterization and closedness of the subspace $S{t}_3\cap l_{\infty }^3$ are obtained. We also give the following important characterization of statistical convergence for sequences of fuzzy numbers having multiplicity greater than two. "A triple sequence $X=\left(X_{\mathrm{ijk}}\right)$ of fuzzy numbers is statistical convergent to a fuzzy number $X_0$, if and only if, there exists a subset $K=\left\{\left(i_n,j_n,k_n\right)\right\}\subset N^3,n=1,2,3\cdots$ such that ${\delta }_3\left(K\right)=1$ and $\underset{n\to \infty }{lim}{X}_{i_n{j}_n{k}_n}=X_0$". Finally, we define the notions of statistically Cauchy, Cesàro summable, $p-$Cesàro summable for these kinds of sequences and establish the Cauchy convergence criterion.

Main results

In this section, we shall, for brevity, state and prove our results only for triple sequences. The reader will see that our methods can readily be applied also to double sequences of fuzzy numbers and to sequences of fuzzy numbers of any multiplicity greater than three. For $K\subset N^3$, the natural density of K is defined by

$$\begin{array}{l}\begin{array}{cc}{\delta }_3\left(K\right)=\underset{l,m,n\to \infty }{lim}\frac{1}{(l+1)(m+1)(n+1)}& \left|\right\{i\le l,j\le m,k\le n\text{:}\\ (i,j,k)\in K\left\}\right|,\end{array}\end{array}$$

(13)

provided that the limit exists. Here, vertical bars denote the cardinality of the enclosed set.

Definition 3.1

A triple sequence $X=\left(X_{\mathrm{ijk}}\right)$ of fuzzy numbers is said to be statistically convergent to some fuzzy number $X_0$, in symbol: $\mathrm{St}-lim{X}_{\mathrm{ijk}}=X_0$, if for each $휖>0$,

$$\begin{array}{l}\begin{array}{cc}\underset{l,m,n\to \infty }{lim}& \frac{1}{(l+1)(m+1)(n+1)}\left|\right\{i\le l,j\le m,k\le n\text{:}\\ d(X_{\mathrm{ijk}},X_0)\ge 휖\left\}\right|=0.\end{array}\end{array}$$

(14)

Here and in the sequel, l, m and n tend to infinity independently of one another. We shall denote the set of all statistically convergent triple sequences of fuzzy numbers by $S{t}_3$. Since the asymptotic density of finite subsets of $N^3$ is zero, it follows that every convergent triple sequence of fuzzy number is statistically convergent, although the converse is not necessarily true, as seen from the following example.

Example 3.1

For every $x\in R$, define a sequence $X=\left(X_{\mathrm{ijk}}\right)$ of fuzzy numbers as follows.

If i, j and k are all squares, define

$$X_{\mathrm{ijk}}\left(x\right)=\left\{\begin{array}{ll}0,& \text{if}x\in (-\infty ,\mathrm{ijk}-1)\cup (\mathrm{ijk}+1,\infty ),\\ x-(\mathrm{ijk}-1),& \text{if}x\in \left[\mathrm{ijk}-1,\mathrm{ijk}\right],\\ -x+(\mathrm{ijk}+1),& \text{if}x\in \left[\mathrm{ijk},\mathrm{ijk}+1\right].\end{array}\right.$$

(15)

Otherwise, $X_{\mathrm{ijk}}=X_0$ where $X_0$ is given by

$$X_0\left(x\right)=\left\{\begin{array}{ll}0,& \text{if}x\in (-\infty ,0)\cup (2,\infty ),\\ x& \text{if}x\in \left[0,1\right],\\ -x+2& \text{if}x\in (1,2].\end{array}\right.$$

(16)

Now, for $0<휖<1$, we have

$$\begin{array}{l}\begin{array}{cc}K\left(휖\right)& =\left\{\right(i,j,k)\in N^3\text{:}d(X_{\mathrm{ijk}},X_0)\ge 휖\}\\ \subset \left\{\right(i,j,k)\in N^3\text{:}i,j,\text{and}k\text{are squares}\}.\end{array}\end{array}$$

(17)

Since, the later set has triple density zero, it follows that ${\delta }_3\left(K\right(휖\left)\right)=0$, and consequently $X=\left(X_{\mathrm{ijk}}\right)$ is statistically convergent to $X_0$. But, the sequence $X=\left(X_{\mathrm{ijk}}\right)$ is not ordinarily convergent to $X_0$.

In the following theorems, we give the uniqueness and algebraic characterization of statistical limit for triple sequences of fuzzy numbers. However, the proofs are straightforward and therefore omitted.

Theorem 3.1

If a triple sequence $X=\left(X_{\mathrm{ijk}}\right)$ of fuzzy numbers is statistically convergent to some limit, then it must be unique.

Theorem 3.2

Let $X=\left(X_{\mathrm{ijk}}\right),Y=\left(Y_{\mathrm{ijk}}\right)$ be two triple sequences of fuzzy numbers.

• (i) If $X=\left(X_{\mathrm{ijk}}\right)$ is statistically convergent to $X_0$ and $c\in R$, then $\left(c{X}_{\mathrm{ijk}}\right)$ is statistically convergent to $c{X}_0$.

• (ii) If $X=\left(X_{\mathrm{ijk}}\right)$ and $Y=\left(Y_{\mathrm{ijk}}\right)$ are statistically convergent to fuzzy numbers $X_0$ and $Y_0$, respectively, then $(X_{\mathrm{ijk}}+Y_{\mathrm{ijk}})$ is statistically convergent to$X_0+Y_0$.

Theorem 3.3

A triple sequence $X=\left(X_{\mathrm{ijk}}\right)$ of fuzzy numbers is statistically convergent to a fuzzy number $X_0$, if and only if, there exists a subset $K=\left\{\right(i_n,j_n,k_n\left)\right\}\subset N^3,n=1,2,...$ such that ${\delta }_3\left(K\right)=1$ and $\underset{n\to \infty }{lim}{X}_{i_n{j}_n{k}_n}=X_0$.

Proof

Let $X=\left(X_{\mathrm{ijk}}\right)$ be statistically convergent to $X_0$. For each $휖>0$, if we denote

$$\begin{array}{cc}M& =\left\{\right(i,j,k)\in N^{3}d(X_{\mathrm{ijk}},X_0)\ge 휖\}\text{and}\\ K& =\left\{\right(i,j,k)\in N^3\text{:}d(X_{\mathrm{ijk}},X_0)<휖\},\end{array}$$

(18)

then ${\delta }_3\left(M\right)=0$, and therefore, ${\delta }_3\left(K\right)=1$. Furthermore, $K\cap M=\varnothing$. Since ${\delta }_3\left(K\right)=1$, it follows that K is an infinite set as otherwise ${\delta }_3\left(K\right)=0$. Let $K=\left\{\right(i_n,j_n,k_n\left)\right\}\subset N^3,n=1,2,\mathrm{...}$. Now, to prove the result, it is sufficient to prove that $\left(X_{i_n{j}_n{k}_n}\right)$ is convergent to $X_0$. Suppose that $\left(X_{i_n{j}_n{k}_n}\right)$ is not convergent to $X_0$. By definition, there exists ${휖}_1>0$ such that $d(X_{i_n{j}_n{k}_n},X_0)\ge {휖}_1$ for infinitely many terms. Let

$$\begin{array}{l}{K}_1=\left\{\right(i_n,j_n,k_n)\in K\text{:}d(X_{i_n{j}_n{k}_n},X_0)\ge {휖}_1\}.\end{array}$$

(19)

Clearly, $\varnothing \ne K_1\subseteq K$. Also, for all $i,j,k$ and ${휖}_1$, we have

$$\begin{array}{cc}{M}_1& =\left\{\right(i,j,k)\in N^3\text{:}d(X_{\mathrm{ijk}},X_0)\ge {휖}_1\}\\ \supseteq \left\{\right(i_n,j_n,k_n\left)\text{:}d\right(X_{i_n{j}_n{k}_n},X_0)\ge {휖}_1\}.\end{array}$$

(20)

Thus, ${\delta }_3\left(K_1\right)=0$ i.e. $K_1\subseteq M_1$. Furthermore, for $휖<{휖}_1,M_1\subseteq M$, which is impossible as $K\cap M=\varnothing$. Hence, $\left(X_{i_n{j}_n{k}_n}\right)$ is convergent to $X_0$.

Conversely, suppose that there exists a subset $K=\left\{\right(i_n,j_n,k_n\left)\right\}\subset N^3,n=1,2,...$ such that ${\delta }_3\left(K\right)=1$ and $\underset{n\to \infty }{lim}{X}_{i_n{j}_n{k}_n}=X_0$. By definition, there exists a positive integer p such that $d(X_{i_n{j}_n{k}_n},X_0)<휖$ for all $n\ge p$. Since

$$\begin{array}{cc}\left\{\right(i,j,k)\in N^3\text{:}d(X_{\mathrm{ijk}},X_0)& \ge 휖\}\\ \subseteq N^3-\left\{\right(i_{p+1},j_{p+1},k_{p+1}),\\ \times (i_{p+2},j_{p+2},k_{p+2}),...\},\end{array}$$

(21)

it follows that

$${\delta }_3\left(\right\{(i,j,k)\in N^3\text{:}d(X_{\mathrm{ijk}},X_0)\ge {휖}_1\left\}\right)\le 1-1=0.$$

(22)

Hence, X is statistically convergent to $X_0$.

Theorem 3.4

The set $S{t}_3\cap l_{\infty }^3$ is a closed linear subspace of the normed linear space $l_{\infty }^3$.

Proof

Let $X^{\left(\mathrm{lmn}\right)}=\left(X_{\mathrm{ijk}}^{\left(\mathrm{lmn}\right)}\right)\in S{t}_3\cap l_{\infty }^3$ and $X^{\left(\mathrm{lmn}\right)}\to X\in l_{\infty }^3$. Since $X^{\left(\mathrm{lmn}\right)}\in S{t}_3\cap l_{\infty }^3$, therefore, there exists fuzzy number $Y_{\mathrm{lmn}}$ such that

$$\mathrm{St}-\underset{i,j,k}{lim}{X}_{\mathrm{ijk}}^{\left(\mathrm{lmn}\right)}=Y_{\mathrm{lmn}}(l,m,n=1,2,...).$$

(23)

Furthermore, $X^{\left(\mathrm{lmn}\right)}\to X$ implies that there exists a positive integer M such that for every $p\ge l\ge M,q\ge m\ge M$ and $r\ge n\ge M$,

$$d(X^{\left(\mathrm{pqr}\right)},X^{\left(\mathrm{lmn}\right)})<\frac{휖}{3}$$

(24)

Also, by Theorem 3.3, there exists subsets $K_{\mathrm{pqr}},K_{\mathrm{lmn}}\subset N^3$ such that ${\delta }_3\left(K_{\mathrm{pqr}}\right)={\delta }_3\left(K_{\mathrm{lmn}}\right)=1$ and

$$\underset{(i,j,k)\in K_{\mathrm{pqr}};i,j,k\to \infty }{lim}{X}_{\mathrm{ijk}}^{\left(\mathrm{pqr}\right)}=Y_{\mathrm{pqr}}.$$

(25)

$$\underset{(i,j,k)\in K_{\mathrm{lmn}};i,j,k\to \infty }{lim}{X}_{\mathrm{ijk}}^{\left(\mathrm{lmn}\right)}=Y_{\mathrm{lmn}}.$$

(26)

Now, the set $K_{\mathrm{pqr}}\cap K_{\mathrm{lmn}}$ is infinite as ${\delta }_3(K_{\mathrm{pqr}}\cap K_{\mathrm{lmn}})=1$. Choose $(k_1,k_2,k_3)\in K_{\mathrm{pqr}}\cap K_{\mathrm{lmn}}$, then we have, from Equations (2) and (3),

$$d(X_{k_1{k}_2{k}_3}^{\left(\mathrm{pqr}\right)},Y_{\mathrm{pqr}})<\frac{휖}{3}\text{and}d(X_{k_1{k}_2{k}_3}^{\left(\mathrm{lmn}\right)},Y_{\mathrm{lmn}})<\frac{휖}{3}.$$

(27)

Hence, for every $p\ge l\ge M,q\ge m\ge M$ and $r\ge n\ge M$, we have, from Equations (1) to (4),

$$\begin{array}{cc}d(Y_{\mathrm{pqr}},Y_{\mathrm{lmn}})& \le d(Y_{\mathrm{pqr}},X_{k_1{k}_2{k}_3}^{\left(\mathrm{pqr}\right)})+d(X_{k_1{k}_2{k}_3}^{\left(\mathrm{pqr}\right)},X_{k_1{k}_2{k}_3}^{\left(\mathrm{lmn}\right)})\\ +d(X_{k_1{k}_2{k}_3}^{\left(\mathrm{lmn}\right)},Y_{\mathrm{lmn}})<\frac{휖}{3}+\frac{휖}{3}+\frac{휖}{3}=\mathrm{휖.}\end{array}$$

(28)

This shows that $\left(Y_{\mathrm{lmn}}\right)$ is a Cauchy sequence and, hence, convergent. Let

$$\underset{l,m,n\to \infty }{lim}{Y}_{\mathrm{lmn}}=\mathrm{Y.}$$

(29)

Next, we show that X is statistically convergent to Y . Since $X^{\left(\mathrm{lmn}\right)}\to X$, so for each $휖>0$, there exists $l,m,n$ and $N_0\in N$ such that

$$\begin{array}{l}d(X_{\mathrm{ijk}}^{\left(\mathrm{lmn}\right)},X_{\mathrm{ijk}})<\frac{휖}{3}\mathrm{for}i,j,k\ge N_0.\end{array}$$

(30)

Also from Equation 5, we have, for every $휖>0$, $N_1\in N$ such that

$$\begin{array}{l}d(Y_{\mathrm{ijk}},Y)<\frac{휖}{3}\mathrm{for}i,j,k\ge N_1.\end{array}$$

(31)

Furthermore, by virtue of the fact that $\left(X^{\left(\mathrm{lmn}\right)}\right)$ is statistically convergent to $Y_{\mathrm{lmn}}$, there is a set $K_{\mathrm{ijk}}=\left\{\right(i,j,k\left)\right\}\subseteq N^3$ such that ${\delta }_3\left(K_{\mathrm{ijk}}\right)=1$, and for each $휖>0$, there exists $N_2\in N$ such that, for $(i,j,k)\in K_{\mathrm{ijk}}$, we have

$$\begin{array}{l}d(X_{\mathrm{ijk}}^{\left(\mathrm{lmn}\right)},Y_{\mathrm{lmn}})<\frac{휖}{3}\mathrm{for}i,j,k\ge N_2.\end{array}$$

(32)

Let $N_3=max\{N_0,N_1,N_2\}$. Now, for $휖>0$ and $(i,j,k)\in K_{\mathrm{ijk}}$,

$$\begin{array}{cc}d(X_{\mathrm{ijk}},Y)& \le d(X_{\mathrm{ijk}},X_{\mathrm{ijk}}^{\mathrm{lmn}})+d(X_{\mathrm{ijk}}^{\mathrm{lmn}},Y_{\mathrm{ijk}})+d(Y_{\mathrm{ijk}},Y)\\ <\frac{휖}{3}+\frac{휖}{3}+\frac{휖}{3}=\mathrm{휖.}\end{array}$$

(33)

This shows that X is statistically convergent to Y , i.e., $X\in S{t}_3\cap l_{\infty }^3$. This shows that $S{t}_3\cap l_{\infty }^3$ is a closed linear subspace of $l_{\infty }^3$, and therefore, the proof of the theorem is complete.

Definition 3.2

A triple sequence $X=\left(X_{\mathrm{ijk}}\right)$ of fuzzy numbers is said to be a statistically Cauchy if, for each $휖>0$, there exist integers $L=L\left(휖\right),M=M\left(휖\right),$ and $N=N\left(휖\right)$ such that

$$\begin{array}{cc}\underset{l,m,n\to \infty }{lim}& \frac{1}{(l+1)(m+1)(n+1)}\left|\right\{i\le l,j\le m,k\le n\text{:}\\ d(X_{\mathrm{ijk}},X_{\mathrm{LMN}})\ge 휖\left\}\right|=0.\end{array}$$

(34)

Theorem 3.5

A triple sequence $X=\left(X_{\mathrm{ijk}}\right)$ of fuzzy numbers is statistically convergent, if and only if, it is a statistical Cauchy.

Proof

Let $X=\left(X_{\mathrm{ijk}}\right)$ be statistically convergent to $X_0$. By definition, for each $휖\ge 0$ we have

$$\begin{array}{l}{\delta }_3\left(\right\{(i,j,k)\in N^3\text{:}d(X_{\mathrm{ijk}},X_0)\ge 휖\left\}\right)=0.\end{array}$$

(35)

We can choose numbers L, M and N such that $d(X_{\mathrm{LMN}},X_0)\ge 휖$. If we denote

$$\begin{array}{cc}A& =\left\{\right(i,j,k)\in N^3,i\le l,j\le m,k\le n\text{:}d(X_{\mathrm{ijk}},X_{\mathrm{LMN}})\ge 휖\};\\ B& =\left\{\right(i,j,k)\in N^3,i\le l,j\le m,k\le n\text{:}d(X_{\mathrm{ijk}},X_0)\ge 휖\};\\ C& =\left\{\right(L,M,N\left)\text{:}d\right(X_{\mathrm{LMN}},X_0)\ge 휖\},\end{array}$$

(36)

then it is clear that $A\subseteq B\cup C$ and consequently ${\delta }_3\left(A\right)\le {\delta }_3\left(B\right)+{\delta }_3\left(C\right)$. Hence $X=\left(X_{\mathrm{ijk}}\right)$ is statistically Cauchy.

Conversely, suppose that $X=\left(X_{\mathrm{ijk}}\right)$ is a statistically Cauchy. We shall prove that $\left(X_{\mathrm{ijk}}\right)$ is statistically convergent. To this effect, let $({휖}_p\text{:}p=1,2,...)$ be a strictly decreasing sequence of numbers converging to zero. Since $X=\left(X_{\mathrm{ijk}}\right)$ is a statistically Cauchy, therefore, there exists three strictly increasing sequences ($L_p,M_p$ and $N_p$) of positive integers such that

$$\begin{array}{ll}\underset{l,m,n\to \infty }{lim}& \frac{1}{(l+1)(m+1)(n+1)}\left|\right\{i\le l,j\le m,k\le n\text{:}\\ d(X_{\mathrm{ijk}},X_{L_p{M}_p{N}_p})\ge {휖}_p\left\}\right|=0.\end{array}$$

(37)

Clearly, for each p and q pair $(p\ne q)$ of positive integers, we can select $(i_{\mathrm{pq}},j_{\mathrm{pq}},k_{\mathrm{pq}})\in N^3$ such that

$$d(X_{i_{\mathrm{pq}}{j}_{\mathrm{pq}}{k}_{\mathrm{pq}}},X_{L_p{M}_p{N}_p})<{휖}_p\mathrm{and}d(X_{i_{\mathrm{pq}}{j}_{\mathrm{pq}}{k}_{\mathrm{pq}}},X_{L_q{M}_q{N}_q})<{휖}_q.$$

(38)

It follows that

$$\begin{array}{cc}d(X_{L_p{M}_p{N}_p},X_{L_q{M}_q{N}_q})& \le d(X_{i_{\mathrm{pq}}{j}_{\mathrm{pq}}{k}_{\mathrm{pq}}},X_{L_p{M}_p{N}_p})\\ +d(X_{i_{\mathrm{pq}}{j}_{\mathrm{pq}}{k}_{\mathrm{pq}}},X_{L_q{M}_q{N}_q})\\ <{휖}_p+{휖}_q\to 0\text{as}p,q\to \infty .\end{array}$$

(39)

Thus, $(X_{L_p{M}_p{N}_p}\text{:}p=1,2,...)$ is a Cauchy sequence and satisfies the Cauchy convergence criterion. Let $\left(X_{L_p{M}_p{N}_p}\right)$ converge to $X_0$. Since $({휖}_p\text{:}p=1,2,...)\to 0$, so for $휖>0$, there exists $p_0\in N$ such that

$${휖}_{p_0}<\frac{휖}{2}\text{and}d(X_{L_p{M}_p{N}_p},X_0)<\frac{휖}{2},p\ge p_0.$$

(40)

Now, consider $(i,j,k)\in N^3$ arbitrary. By Equation (7),

$$\begin{array}{cc}d(X_{\mathrm{ijk}},X_0)& \le d(X_{\mathrm{ijk}},X_{L_{p_0}{M}_{p_0}{N}_{p_0}})+d(X_{L_{p_0}{M}_{p_0}{N}_{p_0}},X_0)\\ \le d(X_{\mathrm{ijk}},X_{L_{p_0}{M}_{p_0}{N}_{p_0}})+\frac{휖}{2}\end{array}$$

(41)

where, by Equation (6),

$$\begin{array}{c}\frac{1}{(l+1)(m+1)(n+1)}\left|\right\{i\le l,j\le m,k\le n\text{:}\\ d(X_{\mathrm{ijk}},X_0)\ge 휖\left\}\right|\\ \le \frac{1}{(l+1)(m+1)(n+1)}\left|\right\{i\le l,j\le m,k\le n\text{:}\\ d(X_{\mathrm{ijk}},X_{L_{p_0}{M}_{p_0}{N}_{p_0}})\ge \frac{휖}{2}\left\}\right|\\ \le \frac{1}{(l+1)(m+1)(n+1)}\left|\right\{i\le l,j\le m,k\le n\text{:}\\ d(X_{\mathrm{ijk}},X_{L_{p_0}{M}_{p_0}{N}_{p_0}})>{휖}_{p_0}\left\}\right|\to 0\\ \mathrm{as}l,m,n\to \infty .\end{array}$$

(42)

This shows that $X=\left(X_{\mathrm{ijk}}\right)$ is statistically convergent to $X_0$, and therefore, the proof of the theorem is complete.

Definition 3.3

A triple sequence $X=\left(X_{\mathrm{ijk}}\right)$ of fuzzy numbers is said to be $C_{111}$-summable or Cesàro summable to $X_0$ provided that

$$\underset{l,m,n}{lim}\frac{1}{(l+1)(m+1)(n+1)}\sum _{i=0}^l\sum _{j=0}^m\sum _{k=0}^n{X}_{\mathrm{ijk}}=X_0.$$

(43)

Definition 3.4

Let p be a positive real number. A triple sequence $X=\left(X_{\mathrm{ijk}}\right)$ of fuzzy numbers is said to be strongly p-Cesàro summable to a fuzzy number $X_0$ if

$$\underset{l,m,n}{lim}\frac{1}{(l+1)(m+1)(n+1)}\sum _{i=0}^l\sum _{j=0}^m\sum _{k=0}^{n}d{(X_{\mathrm{ijk}},X_0)}^p=0.$$

(44)

We denote the space of all strongly p-Cesàro summable triple sequences of fuzzy numbers by $w_p^3$.

Remark 3.1

• (i) If $0<p<q<\infty$, $w_q^3\subseteq w_p^3$ (Holder inequality) and $w_p^3\cap l_{\infty }^3=w_1^3\cap l_{\infty }^3\subseteq C_{111}\cap l_{\infty }^3$.

• (ii) If $X=\left(X_{\mathrm{ijk}}\right)$ is convergent but unbounded, then $X=\left(X_{\mathrm{ijk}}\right)$ is statistically convergent; however, $X=\left(X_{\mathrm{ijk}}\right)$ need not to be Cesàro nor strongly Cesàro.

• (iii) If $X=\left(X_{\mathrm{ijk}}\right)$ is a bounded convergent triple sequence of fuzzy numbers, then it is also $C_{111}$, $w_p^3$ and statistically convergent.

Theorem 3.6

• (a) Let $p\in (0,\infty )$. If a triple sequence $X=\left(X_{\mathrm{ijk}}\right)$ of fuzzy numbers is strongly p-Cesàro summable to a fuzzy number $X_0$, then it is also statistically convergent to $X_0$.

(b) Let $p\in (0,\infty )$. If a triple bounded sequence $X=\left(X_{\mathrm{ijk}}\right)$ of fuzzy numbers is statistically convergent to a fuzzy number $X_0$, then it is strongly p-Cesàro summable to $X_0$.

Proof

• (a) Let $K_{\mathrm{lmn}}\left(휖\right)=\left\{\right(i,j,k),i\le l,j\le m,k\le n\text{:}d{(X_{\mathrm{ijk}},X_0)}^p\ge 휖\}$. Now, we have

  $$\begin{array}{c}\frac{1}{(l+1)(m+1)(n+1)}\sum _{i=0}^l\sum _{j=0}^m\sum _{k=0}^{n}d{(X_{\mathrm{ijk}},X_0)}^p\\ =\frac{1}{(l+1)(m+1)(n+1)}\{\sum _{(i,j,k)\in K_{\mathrm{lmn}}\left(휖\right)}d{(X_{\mathrm{ijk}},X_0)}^p\\ +\sum _{(i,j,k)\notin K_{\mathrm{lmn}}\left(휖\right)}d{(X_{\mathrm{ijk}},X_0)}^p\}\end{array}$$

  (45)
  $$\begin{array}{c}\ge \frac{1}{(l+1)(m+1)(n+1)}\left\{\sum _{(i,j,k)\in K_{\mathrm{lmn}}\left(휖\right)}d{(X_{\mathrm{ijk}},X_0)}^p\right\}\\ \ge \frac{1}{(l+1)(m+1)(n+1)}\left|\right\{(i,j,k),i\le l,j\le m,\\ k\le n\text{:}d{(X_{\mathrm{ijk}},X_0)}^p\ge 휖\left\}\right|.\end{array}$$

  (46)

Since $X=\left(X_{\mathrm{ijk}}\right)$ is strongly p-Cesàro summable to $X_0$, therefore, we have

$$\begin{array}{cc}0\ge \underset{l,m,n}{lim}\frac{1}{(l+1)(m+1)(n+1)}& \left|\right\{(i,j,k),i\le l,j\le m,\\ k\le n\text{:}d{(X_{\mathrm{ijk}},X_0)}^p\ge 휖\left\}\right|.\end{array}$$

(47)

Hence,

$$\begin{array}{cc}\underset{l,m,n}{lim}\frac{1}{(l+1)(m+1)(n+1)}& \left|\right\{(i,j,k),i\le l,j\le m,k\le n\text{:}\\ d{(X_{\mathrm{ijk}},X_0)}^p\ge 휖\left\}\right|=0\end{array}$$

(48)

as it cannot be negative. This shows that $X=\left(X_{\mathrm{ijk}}\right)$ is statistically convergent to $X_0$.

• (b) Let $K_{\mathrm{lmn}}\left(휖\right)=\left\{\right(i,j,k),i\le l,j\le m,k\le n\text{:}d{(X_{\mathrm{ijk}},X_0)}^p\ge {\left(\frac{휖}{2}\right)}^p\}$ and $M=||X{||}_{(\infty ,3)}+d(X_0,\tilde{0})$, where $||X{||}_{(\infty ,3)}$ is the sup-norm for bounded triple sequences $X=\left(X_{\mathrm{ijk}}\right)$. Since $X=\left(X_{\mathrm{ijk}}\right)$ is bounded and statistically convergent, we can choose a positive integer $r=r\left(휖\right)\in N$ such that, for all $i,j,k\ge r$, we have

  $$\begin{array}{cc}\frac{1}{(l+1)(m+1)(n+1)}& \left|\right\{(i,j,k),i\le l,j\le m,k\le n\text{:}\\ d{(X_{\mathrm{ijk}},X_0)}^p\ge {\left(\frac{휖}{2}\right)}^p\left\}\right|<\frac{휖}{2M^p}.\end{array}$$

  (49)

Now, for all $l,m,n\ge r$,

$$\begin{array}{c}\frac{1}{(l+1)(m+1)(n+1)}\sum _{i=0}^l\sum _{j=0}^m\sum _{k=0}^{n}d{(X_{\mathrm{ijk}},X_0)}^p\\ =\frac{1}{(l+1)(m+1)(n+1)}\{\sum _{(i,j,k)\in K_{\mathrm{lmn}}}d{(X_{\mathrm{ijk}},X_0)}^p\\ +\sum _{(i,j,k)\notin K_{\mathrm{lmn}}}d{(X_{\mathrm{ijk}},X_0)}^p\}\ge \frac{1}{(l+1)(m+1)(n+1)}\\ \times (l+1)(m+1)(n+1)\frac{휖}{2M^p}{M}^p\\ +\frac{1}{(l+1)(m+1)(n+1)}(l+1)(m+1)(n+1)\frac{휖}{2}\\ =\mathrm{휖.}\end{array}$$

(50)

This shows that $X=\left(X_{\mathrm{ijk}}\right)$ is strongly p-Cesàro summable to $X_0$.

Multiple sequences of fuzzy numbers

The concepts and results presented in the previous section can be extended to d-multiple sequences of fuzzy numbers where d is a fixed positive integer. Let $N^d$ = $\left\{\right(k_1,k_2,...k_d)\text{:}{k}_j\in N,\forall j\}$. The d-tuple k$\ne$n, where k = $(k_1,k_2,...k_d)$ and n = $(n_1,n_2,...n_d)$, if and only if, $n_j\ne k_j$ for at least one j. Moreover, the partial order on $N^d$ is defined as follows.

For $k,n$$\in$$N^d$, we say that k$\le$n if, and only if, $k_j\le n_j$ for each j. The natural density of a set S$\subseteq$$N^d$ can be defined as

$$\begin{array}{l}{\delta }_d\left(S\right)=\underset{min{n}_j\to \infty }{lim}\frac{1}{|n+1|}\left|\right\{k\le n\text{:}k\in S\left\}\right|,\end{array}$$

(51)

provided that this limit exists. With the help of ${\delta }_d$-density, the notions of statistical convergence and statistical Cauchy for multiple sequences of fuzzy numbers can be define as follows.

Definition 4.1

A d-tuple sequence $(X=X_k\text{:}k\in N^d)$ of fuzzy numbers is said to be statistically convergent to some fuzzy number $X_0$ if, for each $휖>0$,

$$\begin{array}{l}\underset{min{n}_j\to \infty }{lim}\frac{1}{|n+1|}\left|\right\{k\le n\text{:}d(X_{\mathbf{\text{k}}},X_0)\ge 휖\left\}\right|=0,\end{array}$$

(52)

where

$$\begin{array}{l}\frac{1}{|n+1|}=\prod _{j=1}^d(n_j+1).\end{array}$$

(53)

Definition 4.2

A d-tuple sequence $(X=X_k\text{:}k\in N^d)$ of fuzzy numbers is said to be statistically Cauchy if for each $휖>0$ and $l\ge 0$ there exist $m$ = $(m_1,m_2,...m_d)$$\in$$N^d$ such that $min{m}_j>l$ and

$$\begin{array}{l}\underset{min{n}_j\to \infty }{lim}\frac{1}{|n+1|}\left|\right\{k\le n\text{:}d(X_k,X_m)\ge 휖\left\}\right|=0.\end{array}$$

(54)

All the results presented in previous sections remain true for d-multiple sequences as well.