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 ℕ , the set of positive integers. For any subset A of N , the natural density of A is denoted by δ(A) and is defined by

δ(A)=limn→∞1n+1|{k≤n:k∈A}|,

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=(xk) of numbers is said to be statistically convergent to some number L , in symbol: St−limxk=L , if for each 휖>0 ,

limn→∞1n+1|{k≤n:|xk−L|≥휖}|=0.

i.e., δ({k∈N : |xk−L|≥휖})=0 .

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

limn→∞1n+1|{k≤n:|xk−xm|≥휖}|=0.

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(Rn) = { A⊂Rn : A is compact and convex}. The space C(Rn) has a linear structure induced by the operations

A+B={a+b:a∈A,b∈B}andλA={λa:a∈A}

for A,B∈C(Rn) ; λ∈R . The Hausdroff distance between A and B is defined by

δ∞(A,B)=max{supa∈Ainfb∈B‖a−b‖,supb∈Binfa∈A‖a−b‖}

It is well known that (C(Rn),δ∞) is a complete (not separable) metric space.

Definition 2.1

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

X is normal, i.e., there exists x0∈Rn such that X(x0)=1 .
X is a fuzzy convex, i.e., for any x,y∈Rn and,
λ∈0,1,X(λx+(1−λ)y)≥min{X(x),X(y)}.
X is upper semi-continuous.
The closure of the set {x∈Rn : X(x)>0} , denoted by X0 , is compact.

The properties (i)-(iv) imply for each α∈(0,1 , the α -level set,

Xα={x∈Rn:X(x)≥α}=X―α,X―α

is a non-empty compact convex subset of Rn . Let L(Rn) denote the set of all fuzzy numbers. The linear structure of L(Rn) induces an addition X+Y and a scalar multiplication λX in terms of α -level sets by

X+Yα=Xα+[Y]αandλXα=λ[X]α(X,Y∈L(Rn),λ∈R)

for each α∈0,1 . Define, for each 1≤q<∞ ,

dq(X,Y)=(∫01δ∞(Xα,Yα)qdα)1q

and d∞=sup0≤α≤1δ∞(Xα,Yα) . Clearly, d∞(X,Y)=limq→∞dq(X,Y) with dq≤dr if q≤r . Moreover, dq is a complete, separable and locally compact metric space.

Throughout the paper, d will denote dq with 1≤q<∞ , and N3 will denote the usual product set N×N×N . We now quote the following definitions which will be needed in the sequel.

Definition 2.2

A triple sequence X=(Xijk) of fuzzy numbers is said to be convergent to a fuzzy number X0 if for each 휖>0 , there exist a positive integer m such that

d(Xijk,X0)<휖for everyi,j,k≥m.

The fuzzy number X0 is called the limit of the sequence Xijk) and we write limi,j,k→∞Xnkl=X0 .

Definition 2.3

A triple sequence X=(Xnkl) of fuzzy numbers is said to be a Cauchy sequence if, for each 휖>0 , there exists a positive integer n0 such that

d(Xijk,XNKL)<휖

for every i≥N≥n0,j≥K≥n0,k≥L≥n0 .

Definition 2.4

A triple sequence X=(Xijk) of fuzzy numbers is said to be bounded if there exists a positive number M such that

d(Xijk,0~)<Mfor alli,j,k.

Let l∞3 denote the set of all bounded triple sequences of fuzzy numbers.

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⊂N3 , the natural density of K is defined by

δ3(K)=liml,m,n→∞1(l+1)(m+1)(n+1)|{i≤l,j≤m,k≤n:(i,j,k)∈K}|,

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

Definition 3.1

A triple sequence X=(Xijk) of fuzzy numbers is said to be statistically convergent to some fuzzy number X0 , in symbol: St−limXijk=X0 , if for each 휖>0 ,

liml,m,n→∞1(l+1)(m+1)(n+1)|{i≤l,j≤m,k≤n:d(Xijk,X0)≥휖}|=0.

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 St3 . Since the asymptotic density of finite subsets of N3 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∈R , define a sequence X=(Xijk) of fuzzy numbers as follows.

If i , j and k are all squares, define

Xijk(x)=0,ifx∈(−∞,ijk−1)∪(ijk+1,∞),x−(ijk−1),ifx∈ijk−1,ijk,−x+(ijk+1),ifx∈ijk,ijk+1.

Otherwise, Xijk=X0 where X0 is given by

X0(x)=0,ifx∈(−∞,0)∪(2,∞),xifx∈0,1,−x+2ifx∈(1,2].

Now, for 0<휖<1 , we have

K(휖)={(i,j,k)∈N3:d(Xijk,X0)≥휖}⊂{(i,j,k)∈N3:i,j,andkare squares}.

Since, the later set has triple density zero, it follows that δ3(K(휖))=0 , and consequently X=(Xijk) is statistically convergent to X0 . But, the sequence X=(Xijk) is not ordinarily convergent to X0 .

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=(Xijk) of fuzzy numbers is statistically convergent to some limit, then it must be unique.

Theorem 3.2

Let X=(Xijk),Y=(Yijk) be two triple sequences of fuzzy numbers.

If X=(Xijk) is statistically convergent to X0 and c∈R , then (cXijk) is statistically convergent to cX0 .
If X=(Xijk) and Y=(Yijk) are statistically convergent to fuzzy numbers X0 and Y0 , respectively, then (Xijk+Yijk) is statistically convergent to X0+Y0 .

Theorem 3.3

A triple sequence X=(Xijk) of fuzzy numbers is statistically convergent to a fuzzy number X0 , if and only if, there exists a subset K={(in,jn,kn)}⊂N3,n=1,2,... such that δ3(K)=1 and limn→∞Xinjnkn=X0 .

Proof

Let X=(Xijk) be statistically convergent to X0 . For each 휖>0 , if we denote

M={(i,j,k)∈N3d(Xijk,X0)≥휖}andK={(i,j,k)∈N3:d(Xijk,X0)<휖},

then δ3(M)=0 , and therefore, δ3(K)=1 . Furthermore, K∩M=∅ . Since δ3(K)=1 , it follows that K is an infinite set as otherwise δ3(K)=0 . Let K={(in,jn,kn)}⊂N3,n=1,2,... . Now, to prove the result, it is sufficient to prove that (Xinjnkn) is convergent to X0 . Suppose that (Xinjnkn) is not convergent to X0 . By definition, there exists 휖1>0 such that d(Xinjnkn,X0)≥휖1 for infinitely many terms. Let

K1={(in,jn,kn)∈K:d(Xinjnkn,X0)≥휖1}.

Clearly, ∅≠K1⊆K . Also, for all i,j,k and 휖1 , we have

M1={(i,j,k)∈N3:d(Xijk,X0)≥휖1}⊇{(in,jn,kn):d(Xinjnkn,X0)≥휖1}.

Thus, δ3(K1)=0 i.e. K1⊆M1 . Furthermore, for 휖<휖1,M1⊆M , which is impossible as K∩M=∅ . Hence, (Xinjnkn) is convergent to X0 .

Conversely, suppose that there exists a subset K={(in,jn,kn)}⊂N3,n=1,2,... such that δ3(K)=1 and limn→∞Xinjnkn=X0 . By definition, there exists a positive integer p such that d(Xinjnkn,X0)<휖 for all n≥p . Since

{(i,j,k)∈N3:d(Xijk,X0)≥휖}⊆N3−{(ip+1,jp+1,kp+1),×(ip+2,jp+2,kp+2),...},

it follows that

δ3({(i,j,k)∈N3:d(Xijk,X0)≥휖1})≤1−1=0.

Hence, X is statistically convergent to X0 .

Theorem 3.4

The set St3∩l∞3 is a closed linear subspace of the normed linear space l∞3 .

Proof

Let X(lmn)=(Xijk(lmn))∈St3∩l∞3 and X(lmn)→X∈l∞3 . Since X(lmn)∈St3∩l∞3 , therefore, there exists fuzzy number Ylmn such that

St−limi,j,kXijk(lmn)=Ylmn(l,m,n=1,2,...).

Furthermore, X(lmn)→X implies that there exists a positive integer M such that for every p≥l≥M,q≥m≥M and r≥n≥M ,

d(X(pqr),X(lmn))<휖3

Also, by Theorem 3.3, there exists subsets Kpqr,Klmn⊂N3 such that δ3(Kpqr)=δ3(Klmn)=1 and

lim(i,j,k)∈Kpqr;i,j,k→∞Xijk(pqr)=Ypqr.

lim(i,j,k)∈Klmn;i,j,k→∞Xijk(lmn)=Ylmn.

Now, the set Kpqr∩Klmn is infinite as δ3(Kpqr∩Klmn)=1 . Choose (k1,k2,k3)∈Kpqr∩Klmn , then we have, from Equations ( ² ) and ( ³ ),

d(Xk1k2k3(pqr),Ypqr)<휖3andd(Xk1k2k3(lmn),Ylmn)<휖3.

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

d(Ypqr,Ylmn)≤d(Ypqr,Xk1k2k3(pqr))+d(Xk1k2k3(pqr),Xk1k2k3(lmn))+d(Xk1k2k3(lmn),Ylmn)<휖3+휖3+휖3=휖.

This shows that (Ylmn) is a Cauchy sequence and, hence, convergent. Let

liml,m,n→∞Ylmn=Y.

Next, we show that X is statistically convergent to Y . Since X(lmn)→X , so for each 휖>0 , there exists l,m,n and N0∈N such that

d(Xijk(lmn),Xijk)<휖3fori,j,k≥N0.

Also from Equation ⁵ , we have, for every 휖>0 , N1∈N such that

d(Yijk,Y)<휖3fori,j,k≥N1.

Furthermore, by virtue of the fact that (X(lmn)) is statistically convergent to Ylmn , there is a set Kijk={(i,j,k)}⊆N3 such that δ3(Kijk)=1 , and for each 휖>0 , there exists N2∈N such that, for (i,j,k)∈Kijk , we have

d(Xijk(lmn),Ylmn)<휖3fori,j,k≥N2.

Let N3=max{N0,N1,N2} . Now, for 휖>0 and (i,j,k)∈Kijk ,

d(Xijk,Y)≤d(Xijk,Xijklmn)+d(Xijklmn,Yijk)+d(Yijk,Y)<휖3+휖3+휖3=휖.

This shows that X is statistically convergent to Y , i.e., X∈St3∩l∞3 . This shows that St3∩l∞3 is a closed linear subspace of l∞3 , and therefore, the proof of the theorem is complete.

Definition 3.2

A triple sequence X=(Xijk) of fuzzy numbers is said to be a statistically Cauchy if, for each 휖>0 , there exist integers L=L(휖),M=M(휖), and N=N(휖) such that

liml,m,n→∞1(l+1)(m+1)(n+1)|{i≤l,j≤m,k≤n:d(Xijk,XLMN)≥휖}|=0.

Theorem 3.5

A triple sequence X=(Xijk) of fuzzy numbers is statistically convergent, if and only if, it is a statistical Cauchy.

Proof

Let X=(Xijk) be statistically convergent to X0 . By definition, for each 휖≥0 we have

δ3({(i,j,k)∈N3:d(Xijk,X0)≥휖})=0.

We can choose numbers L , M and N such that d(XLMN,X0)≥휖 . If we denote

A={(i,j,k)∈N3,i≤l,j≤m,k≤n:d(Xijk,XLMN)≥휖};B={(i,j,k)∈N3,i≤l,j≤m,k≤n:d(Xijk,X0)≥휖};C={(L,M,N):d(XLMN,X0)≥휖},

then it is clear that A⊆B∪C and consequently δ3(A)≤δ3(B)+δ3(C) . Hence X=(Xijk) is statistically Cauchy.

Conversely, suppose that X=(Xijk) is a statistically Cauchy. We shall prove that (Xijk) is statistically convergent. To this effect, let (휖p:p=1,2,...) be a strictly decreasing sequence of numbers converging to zero. Since X=(Xijk) is a statistically Cauchy, therefore, there exists three strictly increasing sequences ( Lp,Mp and Np ) of positive integers such that

liml,m,n→∞1(l+1)(m+1)(n+1)|{i≤l,j≤m,k≤n:d(Xijk,XLpMpNp)≥휖p}|=0.

Clearly, for each p and q pair (p≠q) of positive integers, we can select (ipq,jpq,kpq)∈N3 such that

d(Xipqjpqkpq,XLpMpNp)<휖pandd(Xipqjpqkpq,XLqMqNq)<휖q.

It follows that

d(XLpMpNp,XLqMqNq)≤d(Xipqjpqkpq,XLpMpNp)+d(Xipqjpqkpq,XLqMqNq)<휖p+휖q→0asp,q→∞.

Thus, (XLpMpNp:p=1,2,...) is a Cauchy sequence and satisfies the Cauchy convergence criterion. Let (XLpMpNp) converge to X0 . Since (휖p:p=1,2,...)→0 , so for 휖>0 , there exists p0∈N such that

휖p0<휖2andd(XLpMpNp,X0)<휖2,p≥p0.

Now, consider (i,j,k)∈N3 arbitrary. By Equation ( ⁷ ),

d(Xijk,X0)≤d(Xijk,XLp0Mp0Np0)+d(XLp0Mp0Np0,X0)≤d(Xijk,XLp0Mp0Np0)+휖2

where, by Equation ( ⁶ ),

1(l+1)(m+1)(n+1)|{i≤l,j≤m,k≤n:d(Xijk,X0)≥휖}|≤1(l+1)(m+1)(n+1)|{i≤l,j≤m,k≤n:d(Xijk,XLp0Mp0Np0)≥휖2}|≤1(l+1)(m+1)(n+1)|{i≤l,j≤m,k≤n:d(Xijk,XLp0Mp0Np0)>휖p0}|→0asl,m,n→∞.

This shows that X=(Xijk) is statistically convergent to X0 , and therefore, the proof of the theorem is complete.

Definition 3.3

A triple sequence X=(Xijk) of fuzzy numbers is said to be C111 -summable or Cesàro summable to X0 provided that

liml,m,n1(l+1)(m+1)(n+1)∑i=0l∑j=0m∑k=0nXijk=X0.

Definition 3.4

Let p be a positive real number. A triple sequence X=(Xijk) of fuzzy numbers is said to be strongly p -Cesàro summable to a fuzzy number X0 if

liml,m,n1(l+1)(m+1)(n+1)∑i=0l∑j=0m∑k=0nd(Xijk,X0)p=0.

We denote the space of all strongly p -Cesàro summable triple sequences of fuzzy numbers by wp3 .

Remark 3.1

If 0<p<q<∞ , wq3⊆wp3 (Holder inequality) and wp3∩l∞3=w13∩l∞3⊆C111∩l∞3 .
If X=(Xijk) is convergent but unbounded, then X=(Xijk) is statistically convergent; however, X=(Xijk) need not to be Cesàro nor strongly Cesàro.
If X=(Xijk) is a bounded convergent triple sequence of fuzzy numbers, then it is also C111 , wp3 and statistically convergent.

Theorem 3.6

Let p∈(0,∞) . If a triple sequence X=(Xijk) of fuzzy numbers is strongly p-Cesàro summable to a fuzzy number X0 , then it is also statistically convergent to X0 .

(b) Let p∈(0,∞) . If a triple bounded sequence X=(Xijk) of fuzzy numbers is statistically convergent to a fuzzy number X0 , then it is strongly p-Cesàro summable to X0 .

Proof

Let Klmn(휖)={(i,j,k),i≤l,j≤m,k≤n:d(Xijk,X0)p≥휖} . Now, we have
1(l+1)(m+1)(n+1)∑i=0l∑j=0m∑k=0nd(Xijk,X0)p=1(l+1)(m+1)(n+1){∑(i,j,k)∈Klmn(휖)d(Xijk,X0)p+∑(i,j,k)∉Klmn(휖)d(Xijk,X0)p}

≥1(l+1)(m+1)(n+1){∑(i,j,k)∈Klmn(휖)d(Xijk,X0)p}≥1(l+1)(m+1)(n+1)|{(i,j,k),i≤l,j≤m,k≤n:d(Xijk,X0)p≥휖}|.

Since X=(Xijk) is strongly p -Cesàro summable to X0 , therefore, we have

0≥liml,m,n1(l+1)(m+1)(n+1)|{(i,j,k),i≤l,j≤m,k≤n:d(Xijk,X0)p≥휖}|.

Hence,

liml,m,n1(l+1)(m+1)(n+1)|{(i,j,k),i≤l,j≤m,k≤n:d(Xijk,X0)p≥휖}|=0

as it cannot be negative. This shows that X=(Xijk) is statistically convergent to X0 .

Let Klmn(휖)={(i,j,k),i≤l,j≤m,k≤n:d(Xijk,X0)p≥(휖2)p} and M=||X||(∞,3)+d(X0,0~) , where ||X||(∞,3) is the sup-norm for bounded triple sequences X=(Xijk) . Since X=(Xijk) is bounded and statistically convergent, we can choose a positive integer r=r(휖)∈N such that, for all i,j,k≥r , we have
1(l+1)(m+1)(n+1)|{(i,j,k),i≤l,j≤m,k≤n:d(Xijk,X0)p≥휖2p}|<휖2Mp.

Now, for all l,m,n≥r ,

1(l+1)(m+1)(n+1)∑i=0l∑j=0m∑k=0nd(Xijk,X0)p=1(l+1)(m+1)(n+1){∑(i,j,k)∈Klmnd(Xijk,X0)p+∑(i,j,k)∉Klmnd(Xijk,X0)p}≥1(l+1)(m+1)(n+1)×(l+1)(m+1)(n+1)휖2MpMp+1(l+1)(m+1)(n+1)(l+1)(m+1)(n+1)휖2=휖.

This shows that X=(Xijk) is strongly p -Cesàro summable to X0 .

Acknowledgments

The authors are thankful to the reviewers of the paper for careful reading and suggestions.

Multiple sequences of fuzzy numbers and their statistical convergence

Abstract

Purpose

Methods

Results

Conclusions

Background

Definition 2.1

Definition 2.2

Definition 2.3

Definition 2.4

Results and Discussion

Main results

Definition 3.1

Example 3.1

Theorem 3.1

Theorem 3.2

Theorem 3.3

Proof

Theorem 3.4

Proof

Definition 3.2

Theorem 3.5

Proof

Definition 3.3

Definition 3.4

Remark 3.1

Theorem 3.6

Proof

Multiple sequences of fuzzy numbers

Definition 4.1

Definition 4.2

Acknowledgments

Competing interests

References