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Applied Mathematics Letters 25 (2012) 466–470

Contents lists available at SciVerse ScienceDirect

Applied Mathematics Letters

journal homepage: www.elsevier.com/locate/aml

Fuzzy real valued lacunary I-convergent sequences

Bipan Hazarika∗

Department of Mathematics, Rajiv Gandhi University, Itanagar-791 112, Arunachal Pradesh, India

a r t i c l ei n f o

Article history:

Received 20 June 2011

Received in revised form 10 September

2011

Accepted 19 September 2011

Keywords:

Ideal

I-convergent

I-Cauchy

Fuzzy number

Lacunary sequence

a b s t r a c t

In this article, we introduce the concept of lacunary I-convergent sequence of fuzzy real

numbers and study some basic properties.

© 2011 Elsevier Ltd. All rights reserved.

1. Introduction

The works on I-convergence of real valued sequences was initially studied by Kostyrko et al. [1]. Later on, it was further

studied by Šalàt et al. [2,3], Tripathy et al. [4] and many others.

Let S be a non-empty set. A non-empty family of sets I ⊆ P(S) (power set of S) is called an ideal on S if (i) for each A,

B ∈ I, we have A ∪ B ∈ I; (ii) for each A ∈ I and B ⊆ A, we have B ∈ I. Let S be a non-empty set. A family F ⊆ P(S) (power

set of S) is called a filter on S if (i) φ ̸∈ F; (ii) for each A, B ∈ F, we have A ∩ B ∈ F; (iii) for each A ∈ F and B ⊇ A, we have

B ∈ F. An ideal I is called non-trivial if I ̸= φ and S ̸∈ I. It is clear that I ⊆ P(S) is a non-trivial ideal if and only if the class

F = F(I) = {S − A : A ∈ I} is a filter on S. The filter F(I) is called the filter associated with the ideal I. A non-trivial ideal

I ⊆ P(S) is called an admissible ideal on S if and only if it contains all singletons, i.e., if it contains {{x} : x ∈ S}. A non-trivial

ideal I is maximal if there cannot exist any non-trivial ideal J ̸= I containing I as a subset (for details see [1]).

The concept of fuzzy sets was initially introduced by Zadeh [5]. Later on, sequences of fuzzy real numbers have been

discussed by Nanda [6], Nuray and Savas [7] and many others.

A lacunary sequence is an increasing integer sequence θ = (kr) such that k0= 0 and hr= kr− kr−1→ ∞ as r → ∞.

The intervals are determined by θ and it will be defined by Jr= (kr−1,kr] and the ratio

Freedman et al. [8] defined the space Nθin the following way. For any lacunary sequence θ = (kr),

k∈Ir

The space Nθis a BK space with the norm

‖(xk)‖θ= sup

r

k∈Ir

N0

kr

kr−1will be defined by φr.

Nθ=

(xk) : lim

r→∞h−1

r

−

|xk− L| = 0, for some L

.

h−1

r

−

|xk|.

θdenotes the subset of these sequences in Nθfor which θ = 0, (N0

θ, ‖ · ‖θ) is also a BK space.

∗Tel.: +91 3602278512; fax: +91 360 2277881.

E-mail address: bh_rgu@yahoo.co.in.

0893-9659/$ – see front matter © 2011 Elsevier Ltd. All rights reserved.

doi:10.1016/j.aml.2011.09.037

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B. Hazarika / Applied Mathematics Letters 25 (2012) 466–470

467

Lemma 1. Every solid space is monotone.

Lemma 2 (Kostyrko et al. [1, Lemma 5.1]). If I ⊂ 2Nis a maximal ideal, then for each A ⊂ N we have either A ∈ I or N − A ∈ I.

2. Definitions and notations

A fuzzy real number X is a fuzzy set on R i.e. a mapping X: R → J (= [0,1]) associating each real number t with its grade

of membership X(t).

A fuzzy real number X is called convex if X(t) ≥ X(s) ∧ X(r) = min(X(s),X(r)), where s < t < r. If there exists t0∈ R

such that X(t0) = 1, then the fuzzy real number X is called normal.

The α-level set of a fuzzy real number X,0 < α ≤ 1 denoted by Xαis defined as Xα= {t ∈ R : X(t) ≥ α}.

A fuzzy real number X is said to be upper semi-continuous if for each ε > 0,X−1([0,a + ε]), for all a ∈ J is open in the

usual topology of R. The set of all upper semi-continuous, normal, convex fuzzy number is denoted by R(J).

Let D denote the set of all closed and bounded intervals X = [x1,x2] on the real line R. For X = [x1,x2] and Y = [y1,y2]

in D, we define X ≤ Y if and only if x1≤ y1and x2≤ y2. Define a metric d on D by

d(X,Y) = max{|x1− y1|,|x2− y2|}.

It is known that (D, d) is a complete metric space and ‘‘≤’’ is a partial order on D.

The absolute value |X| of X ∈ R(J) is defined as

|X|(t) =

Let¯d : R(J) × R(J) → R be defined by

¯d (X,Y) = sup

0≤α≤1

Then¯d defines a metric on R(J).

We define X ≤ Y if and only if Xα≤ Yα, for allα ∈ J. The additive identity and multiplicative identity in R(J) are denoted

by¯0 and¯1, respectively.

A sequence (Xk) of fuzzy real numbers is said to be convergent to a fuzzy real number X0if for every ε > 0, there exists

n0∈ N such that¯d(Xk,X0) < ε, for all k ≥ n0.

A sequence (Xk) of fuzzy real numbers is said to be I-convergent if there exists a fuzzy real number X0such that for each

ε > 0, the set

{k ∈ N :¯d(Xk,X0) ≥ ε} ∈ I.

We write I-lim Xk= X0.

If I = If(class of all finite subsets of N), then If- convergence coincides with the usual convergence.

LetEFdenote the sequencespace of fuzzy numbers. ThenEFis said tobe solid(ornormal) if(Yk) ∈ EF, whenever(Xk) ∈ EF

and |Yk| ≤ |Xk|, for all k ∈ N.

A sequence space EFis said to be symmetric if (Xk) ∈ E implies (Xπ(k)) ∈ E, where π is a permutation of N.

A sequence space EFis said to be monotoneif it contains the canonical preimages of its step space.

Throughout the article, we assume that I is an admissible ideal of N.

max{X(t),X(−t)},

0,

if t > 0;

if t < 0.

d(Xα,Yα).

3. Lacunary I-convergent sequence of fuzzy real numbers

Definition 1. Let θ = (kr) be a lacunary sequence. Then a sequence (Xk) of fuzzy real numbers is said to be lacunary I-

convergent if for every ε > 0 such that

k∈Ir

We write Iθ− limXk= X.

Definition 2. Let θ = (kr) be a lacunary sequence. Then a sequence (Xk) of fuzzy real numbers is said to be lacunary I-null

if for every ε > 0 such that

k∈Ir

We write Iθ− limXk=¯0.

r ∈ N : h−1

r

−

¯d(Xk,X) ≥ ε

∈ I.

r ∈ N : h−1

r

−

¯d(Xk,¯0) ≥ ε

∈ I.

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B. Hazarika / Applied Mathematics Letters 25 (2012) 466–470

Definition 3. Letθ = (kr)bealacunarysequence.Thenasequence(Xk)offuzzyrealnumbersissaidtobelacunaryI-Cauchy

if there exists a subsequence (X/

k∈Jr

r(r)) of (Xk) such that k′(r) ∈ Jr, for each r, limr→∞X/

k(r) = X′and for everyε > 0 such that

r ∈ N : h−1

r

−

¯d(Xk,Xk′(r)) ≥ ε

∈ I.

Definition 4. A lacunary sequence θ′= (k′(r)) is said to be a lacunary refinement of the lacunary sequence θ = (kr) if

(kr) ⊂ (k′(r)).

Throughout wF, ℓF

fuzzy real numbers, respectively.

∞, cIF

θand (cIF

0)θdenotes all, bounded, lacunary I-convergent, and lacunary I-null class of sequences of

Theorem 1. A sequence (Xk) of fuzzy real numbers is Iθ-convergent if and only if it is an Iθ-Cauchy sequence.

Proof. Let (Xk) be a sequence of fuzzy real numbers with Iθ− limXk= X.

Write H(i)= {r ∈ N : h−1

Hence for each i,H(i)⊇ H(i+1)and {r ∈ N : h−1

We choose k1such that r ≥ k1, then {r ∈ N : h−1

Next we choose k2> k1such that r ≥ k2, then {r ∈ N : h−1

For each r satisfying k1≤ r < k2, choose k′(r) ∈ Jrsuch that

k′(r)∈Jr

In general, we choose kp+1> kp, such that r > kp+1then

kp+1∈Jr

Then for all r satisfying kp≤ r < kp+1, such that

k′(r)∈Jr

Thus we get k′(r) ∈ Jr, for each r and limr→∞Xk′(r)= X.

Therefore, for every ε > 0, we have

k,k′∈Jr

r

∑

k∈Jr¯d(Xk,X) <

1

i}, for each i ∈ N.

r

∑

∑

k∈Jr¯d(Xk,X) <

k1∈Jr¯d(Xk1,X) < 1} ̸∈ I.

r

1

i} ̸∈ I.

r

∑

k2∈Jr¯d(Xk2,X) <

1

2} ̸∈ I.

r ∈ N : h−1

r

−

¯d(Xk′(r),X) < 1

̸∈ I.

r ∈ N : h−1

r

−

¯d(Xkp+1,X) <

1

p

̸∈ I.

r ∈ N : h−1

r

−

¯d(Xk′(r),X) <

1

p

̸∈ I.

r ∈ N : h−1

r

−

¯d(Xk,Xk′(r)) ≥ ε

⊆

r ∈ N : h−1

r

−

−

k∈Jr

¯d(Xk,X) ≥ε

2

∪

r ∈ N : h−1

r

k′(r)∈Jr

¯d(Xk′(r),X) ≥ε

2

.

i.e.

r ∈ N : h−1

Then (Xk) is an Iθ-Cauchy sequence.

Conversely, suppose (Xk) is an Iθ-Cauchy sequence. Then for every ε > 0, we have

k∈Jr

r

∑

k,k′∈Jr¯d(Xk,Xk′(r)) ≥ ε∈ I.

r ∈ N : h−1

r

−

¯d(Xk,X) ≥ ε

⊆

r ∈ N : h−1

r

−

−

k,k′∈Jr

¯d(Xk,Xk′(r)) ≥ε

2

∪

r ∈ N : h−1

r

k′(r)∈Jr

¯d(Xk′(r),X) ≥ε

2

.

It follows that (Xk) is a Iθ-convergent sequence.

?

Theorem 2. If θ′is a lacunary refinement of a lacunary sequence θ and (Xk) ∈ cIF

θ′, then (Xk) ∈ cIF

θ.

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469

Proof. Suppose that for each Jrof θ contains the points (k/

r,t)η(r)

t=1of θ′such that

r,t= (k/

kr−1< k/

r,1< k/

r,2< ··· < k/

r,η(r)= kr,

where J/

r,t−1,k/

r,t].

Since kr⊆ (k′(r)), so r, η(r) ≥ 1.

Let(J∗

j)∞

j=1be the sequence of intervals (J/

r,t) ordered by increasing right end points. Since(Xk) ∈ cIF

θ′, then for eachε > 0,

j ∈ N : (h∗

For each ε > 0, we have

j)−1−

J∗

j⊂Jr

¯d(Xk,X) ≥ ε

∈ I.

Also since hr= kr− kr−1, so h/

r,t= k/

r,t− k/

r,t−1.

r ∈ N : h−1

r

−

k∈Jr

¯d(Xk,X) ≥ ε

⊆

r ∈ N : h−1

r

−

k∈Jr

j ∈ N : (h∗

j)−1−

J∗

k∈J∗

j⊂Jr

j

¯d(Xk,X)

≥ ε

.

Therefore {r ∈ N : h−1

Hence (Xk) ∈ cIF

Theorem 3. Let ψ be a set of all lacunary sequences.

(a) If ψ is closed under arbitrary union, then cIF

(b) If ψ is closed under arbitrary intersection, then cIF

(c) If ψ is closed under union and intersection, then cIF

r

∑

k∈Jr¯d(Xk,X) ≥ ε} ∈ I.

θ.

?

µ=

θ∈ψcIF

θ, where µ =

µ⊆ cIF

θ∈ψθ;

ν=

θ∈ψcIF

θ⊆ cIF

θ, where υ =

θ∈ψθ;

ν.

Proof. (a) By hypothesis, we have µ ∈ ψ which is a refinement of each θ ∈ ψ. Then from Theorem 2, we have if (Xk) ∈ cIF

implies that (Xk) ∈ cIF

Thus for each θ ∈ ψ, we get cIF

Hence cIF

(b) By part (a) and Theorem 2, we have cIF

(c) By part (a) and (b), we get cIF

?

µ

θ.

µ⊆ cIF

θ. The reverse inclusion is obvious.

µ=

θ∈ψcIF

θ.

ν=

θ∈ψcIF

θ.

µ⊆ cIF

θ⊆ cIF

ν.

Theorem 4. cIF

θ∩ ℓF

∞is a closed subset of ℓF

∞.

Proof of the theorem is easy, so omitted.

Theorem 5. Let θ = (kr) be a lacunary sequence. Then the spaces cIF

Proof. We shall give the proof of the theorem for (cIF

all k ∈ N.

Then for a given ε > 0, we have

k∈Jr

Again the set D =

monotone.

?

θand (cIF

0)θare normal and monotone, in general.

0)θand (Yk) be such that¯d(Yk,¯0) ≤¯d(Xk,¯0), for

0)θonly. Let (Xk) ∈ (cIF

B =

r ∈ N : h−1

r

−

¯d(Xk,¯0) ≥ ε

∑

∈ I.

r ∈ N : h−1

r

k∈Jr¯d(Yk,¯0) ≥ ε⊆ B.

Hence D ∈ I and so (Yk) ∈ (cIF

0)θ. Thus the space (cIF

0)θ is normal. Also from Lemma 1, it follows that (cIF

0)θ is

Theorem 6. Let θ = (kr) be a lacunary sequence. Then the spaces (cIF

Proof. We will give the proof for cIF

real numbers defined by

1 + t − 2k,

0,

otherwise.

0)θand cIF

θare symmetric, in general.

θonly. Suppose I is not maximal and I ̸= If. Let us consider a sequence X = (Xk) of fuzzy

Xk(t) =

if t ∈ [2k − 1,2k];

if t ∈ [2k,2k + 1];

1 − t + 2k,

for k ∈ A ⊂ I an infinite set.

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B. Hazarika / Applied Mathematics Letters 25 (2012) 466–470

Then (Xk) ∈ cIF

Consider a sequence Y = (Yk), a rearrangement of the sequence (Xk) defined as follows:

Xk,

Then (Yk) ̸∈ cIF

θ. Let K ⊆ N be such that K ̸∈ I and N − K ̸∈ I (the set K exists by Lemma 2, as I is not maximal).

Yk=

if k ∈ K,

otherwise.

¯1,

θ. Therefore the space cIF

θis not symmetric. This completes the proof of the theorem.

?

4. Conclusions

In this article, we have investigated the notion of lacunary convergence from I-convergence of sequences point of view.

Still there are a lot to be investigated on sequence spaces applying the notion of I-convergence. The workers will apply the

techniques used in this article for further investigations on I-convergence.

Acknowledgments

The author thanks the reviewers for the comments on the paper and several constructive comments that have improved

the presentation of the results.

References

[1] P. Kostyrko, T. Šalàt, W. Wilczyński, I-convergence, Real. Anal. Exchange 26 (2) (2000–2001) 669–686.

[2] T. Šalàt, B.C. Tripathy, M. Ziman, On some properties of I-convergence, Tatra Mt. Math. Publ. 28 (2004) 279–286.

[3] T. Šalàt, B.C. Tripathy, M. Ziman, On I-convergence field, Indian J. Pure Appl. Math. 17 (2005) 45–54.

[4] B.C. Tripathy, B. Hazarika, B. Choudhary, Lacunary I-convergent sequences, Kyungpook Math. J (in press).

[5] L.A. Zadeh, Fuzzy sets, Inf. Control 8 (1965) 338–353.

[6] S. Nanda, On sequences of fuzzy numbers, Fuzzy Sets Syst. 33 (1989) 123–126.

[7] F. Nuray, E. Savas, Statistical convergence of sequences of fuzzy real numbers, Math. Slovaca 45 (3) (1995) 269–273.

[8] A.R. Freedman, J.J. Sember, M. Raphael, Some Cesaro-type summability spaces, Proc. Lond. Math. Soc. 37 (1978) 508–520.

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