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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 e i 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 = (k

r

) such that k

0

= 0 and h

r

= k

r

− k

r−1

→ ∞ as r → ∞.

The intervals are determined by θ and it will be defined by J

r

= (k

r−1

, k

r

] and the ratio

k

r

k

r−1

will be defined by φ

r

.

Freedman et al. [8] defined the space N

θ

in the following way. For any lacunary sequence θ = (k

r

),

N

θ

=

(x

k

) : lim

r→∞

h

−1

r

−

k∈I

r

|x

k

− L| = 0, for some L

.

The space N

θ

is a BK space with the norm

‖ (x

k

) ‖

θ

= sup

r

h

−1

r

−

k∈I

r

|x

k

|.

N

0

θ

denotes the subset of these sequences in N

θ

for which θ = 0, (N

0

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

Author's personal copy

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 ⊂ 2

N

is 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 t

0

∈ R

such that X(t

0

) = 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 X

α

is defined as X

α

= {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 = [x

1

, x

2

] on the real line R. For X = [x

1

, x

2

] and Y = [y

1

, y

2

]

in D, we define X ≤ Y if and only if x

1

≤ y

1

and x

2

≤ y

2

. Define a metric d on D by

d(X, Y ) = max{|x

1

− y

1

|, |x

2

− y

2

|}.

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) =

max{X(t), X(−t)}, if t > 0;

0, if t < 0.

Let

¯

d : R(J) × R(J) → R be defined by

¯

d (X, Y ) = sup

0≤α≤1

d(X

α

, Y

α

).

Then

¯

d defines a metric on R(J).

We define X ≤ Y if and only if X

α

≤ Y

α

, for all α ∈ J. The additive identity and multiplicative identity in R(J) are denoted

by

¯

0 and

¯

1, respectively.

A sequence (X

k

) of fuzzy real numbers is said to be convergent to a fuzzy real number X

0

if for every ε > 0, there exists

n

0

∈ N such that

¯

d(X

k

, X

0

) < ε, for all k ≥ n

0

.

A sequence (X

k

) of fuzzy real numbers is said to be I-convergent if there exists a fuzzy real number X

0

such that for each

ε > 0, the set

{k ∈ N :

¯

d(X

k

, X

0

) ≥ ε} ∈ I.

We write I-lim X

k

= X

0

.

If I = I

f

(class of all finite subsets of N), then I

f

- convergence coincides with the usual convergence.

Let E

F

denote the sequence space of fuzzy numbers. Then E

F

is said to be solid (or normal) if (Y

k

) ∈ E

F

, whenever (X

k

) ∈ E

F

and |Y

k

| ≤ |X

k

|, for all k ∈ N.

A sequence space E

F

is said to be symmetric if (X

k

) ∈ E implies (X

π(k)

) ∈ E, where π is a permutation of N.

A sequence space E

F

is 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.

3. Lacunary I -convergent sequence of fuzzy real numbers

Definition 1. Let θ = (k

r

) be a lacunary sequence. Then a sequence (X

k

) of fuzzy real numbers is said to be lacunary I-

convergent if for every ε > 0 such that

r ∈ N : h

−1

r

−

k∈I

r

¯

d(X

k

, X) ≥ ε

∈ I.

We write I

θ

− lim X

k

= X.

Definition 2. Let θ = (k

r

) be a lacunary sequence. Then a sequence (X

k

) of fuzzy real numbers is said to be lacunary I-null

if for every ε > 0 such that

r ∈ N : h

−1

r

−

k∈I

r

¯

d(X

k

,

¯

0) ≥ ε

∈ I.

We write I

θ

− lim X

k

=

¯

0.

Author's personal copy

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

Definition 3. Let θ = (k

r

) be a lacunary sequence. Then a sequence (X

k

) of fuzzy real numbers is said to be lacunary I-Cauchy

if there exists a subsequence (X

/

r

(r)) of (X

k

) such that k

′

(r) ∈ J

r

, for each r, lim

r→∞

X

/

k

(r) = X

′

and for every ε > 0 such that

r ∈ N : h

−1

r

−

k∈J

r

¯

d(X

k

, X

k

′

(r)

) ≥ ε

∈ I.

Definition 4. A lacunary sequence θ

′

= (k

′

(r)) is said to be a lacunary refinement of the lacunary sequence θ = (k

r

) if

(k

r

) ⊂ (k

′

(r)).

Throughout w

F

, ℓ

F

∞

, c

IF

θ

and (c

IF

0

)

θ

denotes all, bounded, lacunary I-convergent, and lacunary I-null class of sequences of

fuzzy real numbers, respectively.

Theorem 1. A sequence (X

k

) of fuzzy real numbers is I

θ

-convergent if and only if it is an I

θ

-Cauchy sequence.

Proof. Let (X

k

) be a sequence of fuzzy real numbers with I

θ

− lim X

k

= X.

Write H

(i)

= {r ∈ N : h

−1

r

∑

k∈J

r

¯

d(X

k

, X ) <

1

i

}, for each i ∈ N.

Hence for each i, H

(i)

⊇ H

(i+1)

and {r ∈ N : h

−1

r

∑

k∈J

r

¯

d(X

k

, X ) <

1

i

} ̸∈ I.

We choose k

1

such that r ≥ k

1

, then {r ∈ N : h

−1

r

∑

k

1

∈J

r

¯

d(X

k

1

, X ) < 1} ̸∈ I.

Next we choose k

2

> k

1

such that r ≥ k

2

, then {r ∈ N : h

−1

r

∑

k

2

∈J

r

¯

d(X

k

2

, X ) <

1

2

} ̸∈ I.

For each r satisfying k

1

≤ r < k

2

, choose k

′

(r) ∈ J

r

such that

r ∈ N : h

−1

r

−

k

′

(r)∈J

r

¯

d(X

k

′

(r)

, X ) < 1

̸∈ I.

In general, we choose k

p+1

> k

p

, such that r > k

p+1

then

r ∈ N : h

−1

r

−

k

p+1

∈J

r

¯

d(X

k

p+1

, X ) <

1

p

̸∈ I.

Then for all r satisfying k

p

≤ r < k

p+1

, such that

r ∈ N : h

−1

r

−

k

′

(r)∈J

r

¯

d(X

k

′

(r)

, X ) <

1

p

̸∈ I.

Thus we get k

′

(r) ∈ J

r

, for each r and lim

r→∞

X

k

′

(r)

= X.

Therefore, for every ε > 0, we have

r ∈ N : h

−1

r

−

k,k

′

∈J

r

¯

d(X

k

, X

k

′

(r)

) ≥ ε

⊆

r ∈ N : h

−1

r

−

k∈J

r

¯

d(X

k

, X ) ≥

ε

2

∪

r ∈ N : h

−1

r

−

k

′

(r)∈J

r

¯

d(X

k

′

(r)

, X ) ≥

ε

2

.

i.e.

r ∈ N : h

−1

r

∑

k,k

′

∈J

r

¯

d(X

k

, X

k

′

(r)

) ≥ ε

∈ I.

Then (X

k

) is an I

θ

-Cauchy sequence.

Conversely, suppose (X

k

) is an I

θ

-Cauchy sequence. Then for every ε > 0, we have

r ∈ N : h

−1

r

−

k∈J

r

¯

d(X

k

, X ) ≥ ε

⊆

r ∈ N : h

−1

r

−

k,k

′

∈J

r

¯

d(X

k

, X

k

′

(r)

) ≥

ε

2

∪

r ∈ N : h

−1

r

−

k

′

(r)∈J

r

¯

d(X

k

′

(r)

, X ) ≥

ε

2

.

It follows that (X

k

) is a I

θ

-convergent sequence.

Theorem 2. If θ

′

is a lacunary refinement of a lacunary sequence θ and (X

k

) ∈ c

IF

θ

′

, then (X

k

) ∈ c

IF

θ

.

Author's personal copy

B. Hazarika / Applied Mathematics Letters 25 (2012) 466–470 469

Proof. Suppose that for each J

r

of θ contains the points (k

/

r,t

)

η(r)

t=1

of θ

′

such that

k

r−1

< k

/

r,1

< k

/

r,2

< · · · < k

/

r,η(r)

= k

r

, where J

/

r,t

= (k

/

r,t−1

, k

/

r,t

].

Since k

r

⊆ (k

′

(r)), so r, η(r) ≥ 1.

Let (J

∗

j

)

∞

j=1

be the sequence of intervals (J

/

r,t

) ordered by increasing right end points. Since (X

k

) ∈ c

IF

θ

′

, then for each ε > 0,

j ∈ N : (h

∗

j

)

−1

−

J

∗

j

⊂J

r

¯

d(X

k

, X ) ≥ ε

∈ I.

Also since h

r

= k

r

− k

r−1

, so h

/

r,t

= k

/

r,t

− k

/

r,t−1

.

For each ε > 0, we have

r ∈ N : h

−1

r

−

k∈J

r

¯

d(X

k

, X ) ≥ ε

⊆

r ∈ N : h

−1

r

−

k∈J

r

j ∈ N : (h

∗

j

)

−1

−

J

∗

j

⊂J

r

k∈J

∗

j

¯

d(X

k

, X )

≥ ε

.

Therefore {r ∈ N : h

−1

r

∑

k∈J

r

¯

d(X

k

, X ) ≥ ε} ∈ I.

Hence (X

k

) ∈ c

IF

θ

.

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

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

IF

µ

=

θ∈ψ

c

IF

θ

, where µ =

θ∈ψ

θ;

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

IF

ν

=

θ∈ψ

c

IF

θ

, where υ =

θ∈ψ

θ;

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

IF

µ

⊆ c

IF

θ

⊆ c

IF

ν

.

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

k

) ∈ c

IF

µ

implies that (X

k

) ∈ c

IF

θ

.

Thus for each θ ∈ ψ , we get c

IF

µ

⊆ c

IF

θ

. The reverse inclusion is obvious.

Hence c

IF

µ

=

θ∈ψ

c

IF

θ

.

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

IF

ν

=

θ∈ψ

c

IF

θ

.

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

IF

µ

⊆ c

IF

θ

⊆ c

IF

ν

.

Theorem 4. c

IF

θ

∩ ℓ

F

∞

is a closed subset of ℓ

F

∞

.

Proof of the theorem is easy, so omitted.

Theorem 5. Let θ = (k

r

) be a lacunary sequence. Then the spaces c

IF

θ

and (c

IF

0

)

θ

are normal and monotone, in general.

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

IF

0

)

θ

only. Let (X

k

) ∈ (c

IF

0

)

θ

and (Y

k

) be such that

¯

d(Y

k

,

¯

0) ≤

¯

d(X

k

,

¯

0), for

all k ∈ N.

Then for a given ε > 0, we have

B =

r ∈ N : h

−1

r

−

k∈J

r

¯

d(X

k

,

¯

0) ≥ ε

∈ I.

Again the set D =

r ∈ N : h

−1

r

∑

k∈J

r

¯

d(Y

k

,

¯

0) ≥ ε

⊆ B.

Hence D ∈ I and so (Y

k

) ∈ (c

IF

0

)

θ

. Thus the space (c

IF

0

)

θ

is normal. Also from Lemma 1, it follows that (c

IF

0

)

θ

is

monotone.

Theorem 6. Let θ = (k

r

) be a lacunary sequence. Then the spaces (c

IF

0

)

θ

and c

IF

θ

are symmetric, in general.

Proof. We will give the proof for c

IF

θ

only. Suppose I is not maximal and I ̸= I

f

. Let us consider a sequence X = (X

k

) of fuzzy

real numbers defined by

X

k

(t) =

1 + t − 2k, if t ∈ [2k − 1, 2k];

1 − t + 2k, if t ∈ [2k, 2k + 1];

0, otherwise.

for k ∈ A ⊂ I an infinite set.

Author's personal copy

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

Then (X

k

) ∈ c

IF

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).

Consider a sequence Y = (Y

k

), a rearrangement of the sequence (X

k

) defined as follows:

Y

k

=

X

k

, if k ∈ K ,

¯

1, otherwise.

Then (Y

k

) ̸∈ c

IF

θ

. Therefore the space c

IF

θ

is 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.

- CitationsCitations20
- ReferencesReferences13

- ", for all x, y ∈ X; (4) If (σ mn ) is a sequence of scalars with σ mn → σ as m, n → ∞ and (x mn ) is a sequence of vectors with [25,26] and also more applications of ideals can be deals with various authors by B.Hazarika [27][28][29][30][31][32][33][34][35][36][37][38][39]and B.C.Tripathy and B. Hazarika [40][41][42][43]. "

- "It is well known that the metric of any linear metric space is given by some total paranorm (see [23], Theorem 10.4.2, p. 183). The notion of deal convergence was introduced first by Kostyrko et al. [24] as a generalization of statistical convergence which was further studied in toplogical spaces by Kumar et al. [25,26] and also more applications of ideals can be deals with various authors by B.Hazarika27282930313233343536373839 and B.C.Tripathy and B. Hazarika40414243. A family I & 2 Y of subsets of a non-empty set Y is said to be an ideal in Y if "

[Show abstract] [Hide abstract]**ABSTRACT:**In this paper we introduce the I- of χ2χ2 sequence spaces over p-metric spaces defined by Musielak function. We also examine some topological properties and prove some inclusion relation between these spaces.- "Further details on ideal convergence can be found in ([6, 14, 15, 16, 26, 28, 29, 30, 31, 32, 39, 50, 55, 57, 58] ), and many others. The notion of lacunary ideal convergence of real sequences was introduced in ([9, 56]) and Hazarika ([24, 25]), was introduced the lacunary ideal convergent sequences of fuzzy real numbers and studied some properties. Debnath [13] introduced the notion lacunary ideal convergence in intuitionistic fuzzy normed linear spaces. "

[Show abstract] [Hide abstract]**ABSTRACT:**An ideal $I$ is a family of subsets of positive integers $\mathbb{N}$ which is closed under taking finite unions and subsets of its elements. A sequence $(x_k)$ of real numbers is said to be lacunary $I$-convergent to a real number $\ell$, if for each $ \varepsilon> 0$ the set $$\left\{r\in \mathbb{N}:\frac{1}{h_r}\sum_{k\in J_r} |x_{k}-\ell|\geq \varepsilon\right\}$$ belongs to $I.$ The aim of this paper is to study the notion of lacunary $I$-convergence in probabilistic normed spaces as a variant of the notion of ideal convergence. Also lacunary $I$-limit points and lacunary $I$-cluster points have been defined and the relation between them has been established. Furthermore, lacunary-Cauchy and lacunary $I$-Cauchy sequences are introduced and studied. Finally, we provided example which shows that our method of convergence in probabilistic normed spaces is more general.

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