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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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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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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
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[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.
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