On lacunary statistically convergent double sequences of fuzzy numbers

Applied Mathematics Letters 21 (2008) 134–141 www.elsevier.com/locate/aml

On lacunary statistically convergent double sequences of fuzzy numbers E. Savas¸ Istanbul Ticaret University, Department of Mathematics, Uskudar, Istanbul, Turkey Received 17 January 2007; accepted 17 January 2007

Abstract In this work, the concepts of double lacunary strongly p-Cesaro summability and double lacunary statistical convergence of a sequence of fuzzy numbers are introduced. The relationship between double lacunary statistical convergence and double lacunary strongly p-Cesaro summability is studied. c 2007 Elsevier Ltd. All rights reserved.

Keywords: Double sequence; Double lacunary sequence; Fuzzy numbers

1. Introduction In [3], Nanda studied sequences of fuzzy numbers and showed that the set of all convergent sequences of fuzzy numbers forms a complete metric space. Nuray [4] proved the inclusion relations between the set of statistically convergent and lacunary statistically convergent sequences of fuzzy numbers. Recently, Savas¸ [6] introduced and discussed double convergent sequences of fuzzy numbers and showed that the set of all double convergent sequences of fuzzy numbers is complete. In [7], Savas¸ generalized the statistical convergence by using de la Vallee–Poussin mean. Quite recently, Savas¸ and Mursaleen [5] introduced statistically convergent and statistically Cauchy double sequences of fuzzy numbers. In this work, we continue to study of the concepts of double lacunary statistical convergence and double lacunary strongly p-Cesaro summability for sequences of fuzzy numbers. We begin by introducing some notation and definitions which will be used throughout and we refer the readers to [2,4,5] for more details. A fuzzy number is a function X from R n to [0, 1], which is normal, fuzzy convex and upper semi-continuous, and where the closure of {x ∈ R n : X (x) > 0} is compact. These properties imply that, for each 0 < α ≤ 1, the α-level set  X α = x ∈ R n : X (x) ≥ α is a nonempty compact convex subset of R n , as is the support X 0 . Let L (R n ) denote the set of all fuzzy numbers.

E-mail address: [email protected]. c 2007 Elsevier Ltd. All rights reserved. 0893-9659/$ - see front matter doi:10.1016/j.aml.2007.01.008

E. Savas¸ / Applied Mathematics Letters 21 (2008) 134–141

135

Define for each 1 ≤ q < ∞ dq (X, Y ) =

(Z

1

δ∞ X α , Y


α q

)1 q

dα

0

and d∞ = sup0≤α≤1 δ∞ (X α , Y α ) where d∞ is the Hausdorff metric. 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 [1]. Throughout the work, d will denote d q with 1 ≤ q ≤ ∞. We will need the following definitions, (see, [5,8]). Definition 1. A double sequence X = (X kl ) of fuzzy numbers is said to be convergent in the Pringsheim’s sense or P-convergent to a fuzzy number X 0 if for every ε > 0 there exists N ∈ N such that d (X kl , X 0 ) < 

for k, l > N ,

and we define P − lim X = X 0 . The number X 0 is called the Pringsheim limit of X kl . More exactly we say that a double sequence (X kl ) converges to a finite number X 0 if X kl tends to X 0 as both k and l tend to ∞ independently of one another. Let c2 (F) denote the set of all double convergent sequences of fuzzy numbers. Definition 2. A double sequence X = (X kl ) of fuzzy numbers is bounded if there exists a positive number M such 2 (F). that d (X kl , X 0 ) < M for all k and l. We will denote the set of all bounded double sequences by l∞ Let K ⊆ N × N be a two-dimensional set of positive integers and let K m,n be the numbers of (k, l) in K such that k ≤ n and l ≤ m. Then the lower asymptotic density of K is defined as P − lim inf m,n

K m,n = δ2 (K ). mn

In the case when the sequence ( P − lim m,n

K m,n ∞,∞ mn )m,n=1,1

has a limit then we say that K has a natural density and is defined as

K m,n = δ2 (K ). mn

For example, let K = {(k 2 , l 2 ) : k, l ∈ N }, where N is the set of natural numbers. Then √ √ m n K m,n ≤ P − lim =0 δ2 (K ) = P − lim m,n m,n mn mn (i.e. the set K has double natural density zero). Double statistical convergence of the sequences of fuzzy numbers was first deduced by Savas¸ and Mursaleen [5]. They defined the statistical analogue for double sequences X = (X k,l ) of fuzzy numbers as follows. Definition 3. A double sequence X = (X kl ) of fuzzy numbers is said to be statistically convergent to X 0 provided that for each  > 0 P − lim m,n

1 |{(k, l); k ≤ m and l ≤ n : d(X kl , X 0 ) ≥ }| = 0. nm

In this case we write st2 − limk,l X k,l = X 0 and we denote the set of all double statistically convergent sequences of fuzzy numbers by st 2 (F). Definition 4. Let X = (X kl ) be a double sequence of fuzzy numbers and let p be a positive real number. The double sequence X is said to be strongly double p-Cesaro summable to X 0 such that m X n 1 X d (X kl , X 0 ) p = 0. m,n→∞ mn k=1 l=1

P − lim

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E. Savas¸ / Applied Mathematics Letters 21 (2008) 134–141

That is, ( |σ1,1 | p (F) =

) m X n 1 X p X = (X k,l ) : for some fuzzy number X 0 , P − lim d (X kl , X 0 ) = 0 . m,n→∞ mn k=1 l=1

In this case, we may say that X is strongly double p-Cesaro summable to X 0 . The double sequence θr,s = {(kr , ls )} is called double lacunary if there exist two increasing integers such that k0 = 0,

h r = kr − kk−1 → ∞ as r → ∞

l0 = 0,

h¯ s = ls − ls−1 → ∞ as s → ∞.

and

Notation: kr,s = kr ls , h r,s = h r h¯ s , θr,s is determined by Ir,s = {(k, l) : kr −1 < k ≤ kr &ls−1 < l ≤ ls }, qr = q¯s =

ls ls−1 ,

kr kr −1 ,

and qr,s = qr q¯s .

Definition 5. Let θr,s be a double lacunary sequence; the double sequence X = (X k,l ) is said to be double lacunary strongly p-Cesaro summable if there is a fuzzy number X 0 such that     X 1 Nθr,s (F) = X = (X kl ) : for some fuzzy number X 0 , P − lim d(X k,l , X 0 ) p = 0 . r,s h r,s   (k,l)∈I r,s

We now consider the double lacunary statistical convergence Definition 6. Let θr,s be a double lacunary sequence; the double fuzzy sequence X is said to be double lacunary θr,s -statistically convergent to a fuzzy number X 0 provided that for every  > 0, P − lim r,s

1 {(k, l) ∈ Ir,s : d(X k,l , X 0 ) ≥ } = 0. h r,s P

In this case, we write Sθr,s − lim X = X 0 or X kl → X 0 (Sθr,s ) and we denote the set of all double Sθr,s -statistically convergent sequences of fuzzy numbers by Sθr,s (F). There is a strong connection, which we will study in this work, between |σ1,1 | p (F) and the sequence space Nθr,s (F). 2. Main results In the following theorem we show some relations between Nθr,s (F) and Sθr,s (F)-convergence and show that Nθr,s (F) and Sθr,s (F)-convergence are equivalent for bounded double sequences. Theorem 1. Let θr,s be a double lacunary sequence and let X = {X kl } a double sequence of fuzzy numbers. Then, A. Nθr,s (F) is a subset of Sθr,s (F), 2 (F) ∩ S (F) ⊆ N (F), B. l∞ θr,s θr,s 2 2 (F). C. Sθr,s (F) ∩ l∞ = Nθr,s (F) ∩ l∞ Proof. (A) We have X d(X k,l , X 0 ) ≥ (k,l)∈Ir,s

X

d(X k,l , X 0 )

(k,l)∈Ir,s &d(X k,l ,X 0 )≥

≥ ε {(k, l) ∈ Ir,s : d(X k,l , X 0 ) ≥ } , and P − lim r,s

1 h r,s

X (k,l)∈Ir,s

d(X k,l , X 0 ) = 0.

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This implies that P − lim r,s

1 {(k, l) ∈ Ir,s : d(X k,l , X 0 ) ≥ } = 0. h r,s

This completes the proof of (A). P

(B) Assume that X k,l → X 0 (Sθr,s )(F). Let M be such that d(X k,l , X 0 ) ≤ M for all k and l. Also for given  > 0 we obtain the following: 1 h r,s

X

d(X k,l , X 0 ) =

(k,l)∈Ir,s

≤

1 h r,s

X

d(X k,l , X 0 ) +

(k,l)∈Ir,s &d(X k,l ,X 0 )≥

1 h r,s

X

d(X k,l , X 0 )

(k,l)∈Ir,s &d(X k,l ,X 0 )<

M {(k, l) ∈ Ir,s : d(X k,l , X 0 ) ≥ } + . h r,s P

P

2 (F) and X Therefore X ∈ l∞ k,l → X 0 (Sθr,s )(F) implies X k,l → X 0 (Nθr,s )(F). This completes the proof of (B). 2 (F) follows directly from (A), (B). 2  (C) Nθr,s (F) ∩ l∞ (F) = Sθr,s ∩ l∞

Theorem 2. Let θr,s = {kr , ls } be double lacunary sequences. In order to have σ1,1 p (F) ⊂ Nθr,s (F) it is necessary and sufficient that lim infr qr > 1 and lim infs q¯s > 1. Proof. Suppose that lim infr qr > 1 and lim infs q¯s > 1; then there exist δ > 0 and δ1 > 0 such that δ + 1 < qr and δ 1 δ1 + 1 < q¯s for all r, s ≥ 1. This implies that hkrr > δ+1 and hlss > δ1δ+1 . Suppose that X = (X kl ) ∈ |σ1,1 | p (F); then we can write 1 XX Tr,s = d (X kl , X 0 ) p h r s k∈I l∈I " r s # ls−1 kX kX ls−1 ls ls kr X kr X r −1 X r −1 X X 1 X p p p p d (X kl , X 0 ) − d (X kl , X 0 ) − d (X kl , X 0 ) + = (X kl , X 0 ) . h r s k=1 l=1 k=1 l=1 k=1 l=1 k=1 l=1 ! ! ls−1 ls kr X kr X 1 X 1 X kr l s kr ls−1 p p d (X kl , X 0 ) Tr,s = d (X kl , X 0 ) − h r s kr ls k=1 l=1 hr s kr ls−1 k=1 l=1 ! ! ls−1 kr −1 X kX ls r −1 X 1 X kr −1ls−1 1 kr −1ls p p d (X kl , X 0 ) + d (X kl , X 0 ) . − hr s kr −1ls k=1 l=1 hr s kr −1ls−1 k=1 l=1 Since X ∈ σ1,1 p (F), each of the four double sums above converges to zero in the Pringsheim sense. The terms ls kr X 1 X d(X kl , X 0 ) p , kr ls k=1 l=1

ls−1 kr X 1 X d(X kl , X 0 ) p kr ls−1 k=1 l=1

and 1

kX ls r −1 X

kr −1ls

k=1 l=1

kX ls−1 r −1 X 1 (X kl , X 0 ) p kr −1ls−1 k=1 l=1

d(X kl , X 0 ) p ,

are all Pringsheim null sequences. Thus Tr,s is a null Pringsheim sequence. Therefore X is in Nθr,s (F). suppose that lim infr qr = 1 and lim infs q¯s = 1. Since θr,s = {kr , ls }, we can choose sequences  Conversely, kr j , lsi satisfying kr j −1 kr −1

> j,

kr j kr j −1

<1+

1 j

and

lsi −1 > i, ls−1

lsi 1 <1+ lsi −1 i

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E. Savas¸ / Applied Mathematics Letters 21 (2008) 134–141

where r j ≥ r j−1 + 2 and si ≥ si−1 + 2. Let A and B denote two distinct fuzzy numbers. Define X = (X kl ) by X kl = A if (k, l) ∈ (Ir j , Isi ) for some j = i = 1, 2, 3, . . . , X kl = B otherwise. Then for any fuzzy numbers F, 1 h r j ,si

X X

d (X kl , F) = d (A, F) ,

k∈Ir j l∈Isi

and 1 XX d (X kl , F) = d (B, F) , h r s k∈I l∈I r

s

for r 6= r j and s 6= si . That is X 6∈ Nθr,s . If m and n are any sufficiently large integers, we can find the unique j and i for which kr j −1 < m ≤ kr j and lsi −1 < n ≤ lsi   m X n kr j−1 lsi−1 + h r j si 2 1 X < d (B, X kl ) ≤ mn k=1 l=1 kr j −1lsi −1 ji as m, n → ∞; it follows that also j, i → ∞. Hence X ∈ |σ1,1 | p (F). This is a contradiction and so completes the proof.  Theorem 3. Let θr,s = {kr , ls } be double lacunary sequences. Nθr,s (F) ⊂ σ1,1 p (F) if and only if lim supr qr < ∞ and lim sups q¯s < ∞. Proof. Suppose that lim supr qr < ∞ and lim sups q¯s < ∞; there exists an H > 0 such that qr < H and q¯s < H for all r and s. Let X ∈ Nθr,s (F) and  > 0. Also there exist r0 > 0 and s0 > 0 such that for every i ≥ r0 and j ≥ s0 Tr,s =

1 XX d (X kl , X 0 ) p < ε. h r s k∈I l∈I r

s

Let M = max Tr,s : 1 ≤ r ≤ r0 and 1 ≤ s ≤ s0 , and m and n be such that kr −1 < m ≤ kr and ls−1 < n ≤ ls . Thus we obtain the following: 

ls kr X m X n X 1 1 X d (X kl , X 0 ) p ≤ d (X kl , X 0 ) p mn k=1 l=1 kr −1ls−1 k=1 l=1   r,s X X 1 p  ≤ d (X kl , X 0 )  kr −1ls−1 p,u=1,1 (k,l)∈( I p ,Iu ) rX 0 ,s0 X 1 1 h p,u A p,u + = kr −1ls−1 p,u=1,1 kr −1ls−1 (r < p≤r )∪(s 0

≤

M kr −1ls−1

rX 0 ,s0 p,u=1,1

Mkr0 ls0 r0 s0 + ≤ kr −1ls−1 ≤ ≤

h p,u +

1 kr −1ls−1

X

( p≥r0 )∪(u≥s0 )

Mkr0 ls0 r0 s0 1 +ε kr −1ls−1 kr −1ls−1 Mkr0 ls0 r0 s0 + ε H 2. kr −1ls−1

h p,u A p,u

(r0 < p≤r )∪(s0
! sup

h p,u A p,u 0
A p,u

1 kr −1ls−1

X (r0 < p≤r )∪(s0
X (r0 < p≤r )∪(s0
h p,u

h p,u

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E. Savas¸ / Applied Mathematics Letters 21 (2008) 134–141

Now kr and ls both approach infinity as both m and n approach infinity. Thus m X n 1 X d (X kl , X 0 ) p → 0. mn k=1 l=1

Therefore X ∈ |σ1,1 | p (F). Conversely, we shall assume that lim supr qr = ∞ or lim sups q¯s = ∞. We shall prove that there is a bounded Nθr,s (F)-convergent sequence that is not |σ1,1 | p (F). Now θr,s is double lacunary; we could construct two subsequences kr j and q¯si of θr,s satisfying qr j > j, q¯si > i and let A and B be distinct fuzzy numbers. Define X = (X kl ) by if kr j −1 < k ≤ 2kr j −1 and lsi −1 < l ≤ 2lsi −1 ;

X kl = A,

and

X kl = B,

otherwise.

Let us write Tr j ,si =

1

X X

h r j ,si

d (X kl , B) <

k∈Ir j l∈Isi



1 j −1



 1 . i −1

This implies that lim j,i Tr j ,si = 0 if r 6= r j and s 6= s j . Therefore X ∈ Nθr,s (F). On the other hand, for the double sequence {X k,l } above and for a fuzzy number F, we can write   2kr j−1 2lsi−1 kr j l s kr j 2lsi−1 i X X X X 1 X 1  X d (A, F) + d (X kl , F) ≥ d (B, F) kr j lsi k=1 l=1 kr j lsi k=k k=2kr j −1 +1 l=lsi −1 r j −1 l=lsi −1 ! !   kr j − 2kr j −1 kr j −1 lsi −1 lsi − 2lsi −1 + d(B, F) = d(A, F) kr j lsi kr j lsi ! !   kr j −1 lsi −1 2 2 = d(A, F) + d(B, F) 1 − 1− . kr j lsi qr j q¯si Since

1 qr j

<

1 j

and

1 qsi

< 1i , we obtain

k

r j lsi X kr j −1 1 X d (X kl , F) ≥ d(A, F) kr j lsi k=1 l=1 kr j

Therefore P − lim j,m l = 1, . . . , 2lsi −1

1 kr j lsi

Pkr j Plsi k=1

l=1 d(X kl ,

!

lsi −1 lsi



2 + d(B, F) 1 − j

  2 1− → d(B, F). i

F) ≥ d(B, F). On the other hand, for k = 1, . . . , 2kr j −1 and

2k

r j −1 2lsi −1 X X 1 1 d(X kl , F) ≥ 2kr j −1 2lsi −1 k=1 l=1 2kr j −1 2lsi −1

≥



2kr j −1

2lsi −1

X

X

d (X kl , F)

k=1+kr j −1 l=1+lsi −1

d(B, F) 4

which implies that X 6∈ σ1,1 p (F). This completes the proof.



Theorem 4. Let θr,s = {kr , ls } be a double lacunary sequence. Nθr,s (F) = σ1,1 p (F) if and only if 1 < lim infr qr ≤ lim supr qr < ∞ and 1 < lim infs q¯s ≤ lim sups q¯s < ∞. Proof. Theorem 4 follows from Theorems 2 and 3.



Now we are going to ask whether both Nθr,s (F) and |σ1,1 | p (F) limits of fuzzy numbers are the same. We will give the answer in the following theorem. Theorem 5. If X ∈ Nθr,s (F) ∩ |σ1,1 | p (F), and p ≤ 1, then Nθr,s (F) − lim X kl = |σ1,1 | p (F) − lim X kl .

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E. Savas¸ / Applied Mathematics Letters 21 (2008) 134–141

Proof. Let Nθr,s (F) − lim X kl = X 0 and |σ1,1 | p (F) − X kl = X 00 and suppose that X 0 6= X 00 . Then write
p 1 XX 1 XX d (X kl , X 0 ) p + d X kl , X 00 Tr s + Tr0s = h r s k∈I l∈I h r s k∈I l∈I r s r s X X
1 p ≥ d X 0 , X 00 h r s k∈I l∈I r s
p ≥ d X 0 , X 00 where we used the condition p ≤ 1. Now since X ∈ Nθr,s , P − limr,s→∞ Tr0s = 0. Thus, for min(r, p) > N , we have Tr p >


1 d X 0 , X 00 . 2

Now ls kr X
p 1 X 1 d X k,l , X 0 = kr ls k=1 l=1 kr l s

!

r,s X

XX

(u,v)=1,1

k∈Iu l∈Iv

d (X kl , X 0 )

p

1 XX d (X kl , X 0 ) p kr ls k∈I l∈l r s    1 1 1− Tr s = 1− qr qs   
p 1 1 1 > 1− d X 0 , X 00 . 1− 2 qr qs

≥

Since X ∈ |σ1,1 | p (F), the left side converges to zero in Pringsheim’s sense, i.e., ls kr X 1 X d (X kl , X 0 ) p → 0 kr ls k=1 l=1

as r → ∞ and s → ∞. So we must have qr → 1 and q¯s → 1. But these statements imply, by the proof of Theorem 2, Nθr,s (F) ⊂ σ1,1 p (F). Thus, since Nθr,s (F) − lim X kl = X 00 , it follows that |σ1,1 | p (F) − X kl = X 00 . Therefore m X n 1 X d (X kl , X 0 ) p = 0. mn→∞ mn k=1 l=1

P − lim But,

m X n m X n  
p 1 X 1 X 0p d (X kl , X 0 ) p + d X kl , X 0 ≥ d X 0 , X 00 ≥ 0 mn k=1 l=1 mn k=1 l=1

which yields a contradiction since both terms on the left converge to 0. This completes the proof.



Acknowledgement This research was completed while the author was a TUBITAK (the Scientific and Technological Research Council of Turkey) scholar at Indiana University, Bloomington, IN, USA, during the summer of 2006. References [1] P. Diamond, P. Kloeden, Metric spaces of fuzzy sets, Fuzzy Sets and Systems 35 (1990) 241–249. [2] Mursaleen, M. Bas¸arir, On some new sequence spaces of fuzzy numbers, Indian J. Pure Appl. Math. 34 (9) (2003) 1351–1357.

E. Savas¸ / Applied Mathematics Letters 21 (2008) 134–141 [3] [4] [5] [6] [7] [8]

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S. Nanda, On sequence of fuzzy numbers, Fuzzy Sets and System 33 (1989) 123–126. F. Nuray, Lacunary statistical convergence of sequences of fuzzy numbers, Fuzzy Sets and Systems 99 (3) (1998) 353–355. E. Savas¸, Mursaleen, On statistically convergent double sequences of fuzzy numbers, Inform. Sci. 162 (3–4) (2004) 183–192. E. Savas¸, A note on double sequence of fuzzy numbers, Turkish J. Math. 20 (1996) 175–178. E. Savas¸, On strongly λ-summable sequences of fuzzy numbers, Inform. Sci. 125 (2000) 181–186. E. Savas¸, R.F. Patterson, Lacunary statistical convergence of multiple sequences, Appl. Math. Lett. 19 (6) (2006) 527–534.

Further reading [1] [2] [3] [4]

J.S. Kwon, S.H. Sung, On lacunary statistical and p-Cesaro summability of fuzzy numbers, J. Fuzzy Math. 9 (3) (2001) 603–610. E. Savas¸, A note on sequence of fuzzy numbers, Inform. Sci. 124 (2000) 297–300. E. Savas¸, On statistically convergent sequence of fuzzy numbers, Inform. Sci. 137 (2001) 272–282. E. Savas¸, V. Karakaya, R.F. Patterson, Inclusion theorems for double lacunary sequence spaces, Acta Sci. Math. (Szeged) 71 (1–2) (2005) 147–157.