Fuzzy real valued lacunary I-convergent sequences

Article (PDF Available)inApplied Mathematics Letters 25(3):466-470 · March 2012with25 Reads
DOI: 10.1016/j.aml.2011.09.037 · Source: DBLP
Abstract
In this article, we introduce the concept of lacunary II-convergent sequence of fuzzy real numbers and study some basic properties.

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Author's personal copy
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
r1
as r .
The intervals are determined by θ and it will be defined by J
r
= (k
r1
, k
r
] and the ratio
k
r
k
r1
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
kI
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
kI
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
0α1
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
kI
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
kI
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
kJ
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
,
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
kJ
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
kJ
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
kJ
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
kJ
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
r1
< k
/
r,1
< k
/
r,2
< · · · < k
/
r,η(r)
= k
r
, where J
/
r,t
= (k
/
r,t1
, 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
r1
, so h
/
r,t
= k
/
r,t
k
/
r,t1
.
For each ε > 0, we have
r N : h
1
r
kJ
r
¯
d(X
k
, X ) ε
r N : h
1
r
kJ
r
j N : (h
j
)
1
J
j
J
r
kJ
j
¯
d(X
k
, X )
ε
.
Therefore {r N : h
1
r
kJ
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
=
θψ
c
IF
θ
, where µ =
θψ
θ;
(b) If ψ is closed under arbitrary intersection, then c
IF
ν
=
θψ
c
IF
θ
, where υ =
θψ
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
=
θψ
c
IF
θ
.
(b) By part (a) and Theorem 2, we have c
IF
ν
=
θψ
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
F
is a closed subset of
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
kJ
r
¯
d(X
k
,
¯
0) ε
I.
Again the set D =
r N : h
1
r
kJ
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.
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