Fuzzy real valued lacunary Iconvergent sequences.
ABSTRACT In this article, we introduce the concept of lacunary IIconvergent sequence of fuzzy real numbers and study some basic properties.
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ABSTRACT: In this article we introduce the notion of $I$ convergent and $I$ Cauchy sequence in a fuzzy normed linear space and establish some basic results related to these notions. Further, we define $I$ limit points and $I$ cluster points of a sequence in a fuzzy normed linear space and investigate the relations between these concepts.Afrika Matematika.
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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 Iconvergent sequences
Bipan Hazarika∗
Department of Mathematics, Rajiv Gandhi University, Itanagar791 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
Iconvergent
ICauchy
Fuzzy number
Lacunary sequence
a b s t r a c t
In this article, we introduce the concept of lacunary Iconvergent sequence of fuzzy real
numbers and study some basic properties.
© 2011 Elsevier Ltd. All rights reserved.
1. Introduction
The works on Iconvergence 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 nonempty set. A nonempty 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 nonempty 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 nontrivial if I ̸= φ and S ̸∈ I. It is clear that I ⊆ P(S) is a nontrivial 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 nontrivial 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 nontrivial
ideal I is maximal if there cannot exist any nontrivial 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.
Email address: bh_rgu@yahoo.co.in.
08939659/$ – 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
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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 αlevel 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 semicontinuous 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 semicontinuous, 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 Iconvergent if there exists a fuzzy real number X0such that for each
ε > 0, the set
{k ∈ N :¯d(Xk,X0) ≥ ε} ∈ I.
We write Ilim 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 Iconvergent 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 Inull
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)offuzzyrealnumbersissaidtobelacunaryICauchy
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 Iconvergent, and lacunary Inull class of sequences of
Theorem 1. A sequence (Xk) of fuzzy real numbers is Iθconvergent if and only if it is an IθCauchy 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 IθCauchy sequence.
Conversely, suppose (Xk) is an IθCauchy 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 Iθconvergent 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 Iconvergence of sequences point of view.
Still there are a lot to be investigated on sequence spaces applying the notion of Iconvergence. The workers will apply the
techniques used in this article for further investigations on Iconvergence.
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, Iconvergence, Real. Anal. Exchange 26 (2) (2000–2001) 669–686.
[2] T. Šalàt, B.C. Tripathy, M. Ziman, On some properties of Iconvergence, Tatra Mt. Math. Publ. 28 (2004) 279–286.
[3] T. Šalàt, B.C. Tripathy, M. Ziman, On Iconvergence field, Indian J. Pure Appl. Math. 17 (2005) 45–54.
[4] B.C. Tripathy, B. Hazarika, B. Choudhary, Lacunary Iconvergent 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 Cesarotype summability spaces, Proc. Lond. Math. Soc. 37 (1978) 508–520.
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AML3812