Gen. Math. Notes, Vol. 25, No. 2, December 2014, pp.6-18
c
ISSN 2219-7184; Copyright ICSRS
Publication, 2014
www.i-csrs.org
Available free online at http://www.geman.in
On λ-Statistical Limit Inferior and Limit Superior
of Order β for Sequences of Fuzzy Numbers
Pankaj Kumar1 , Vijay Kumar2 and S.S. Bhatia3
1
SMCA, Thapar University, Patiala-147001, Punjab, India
E-mail: pankaj.lankesh@yahoo.com
2
Department of Mathematics, HCTM Technical Campus
Kaithal-136027, Haryana, India
E-mail: vjy kaushik@yahoo.com
3
SMCA, Thapar University, Patiala-147001, Punjab, India
E-mail: ssbhatia63@yahoo.com
(Received: 3-3-14 / Accepted: 5-7-14)
Abstract
The aim of present work is to introduce and study the concepts of Sλβ limit points and cluster points and Sλβ -limit inferior and limit superior of fuzzy
number sequences.
Keywords: Fuzzy number sequences, limit inferior and limit superior,
statistical convergence.
1
Introduction
The notion of statistical convergence of sequences of number was introduced
by Fast [8] and Schoenberg [26] independently and latter dicussed in [9-12] and
[17] etc. In past years, statistical convergence has also become an interesting
area of research for sequences of fuzzy numbers. The credit goes to Nuray and
Savaş [21], who first introduced statistical convergence to sequences of fuzzy
numbers. After their pioneer work, many authors have made their contribution
to study different generalizations of statistical convergence for sequences of
fuzzy numbers(see [1-5],[14-16],[18], [20], [22], [25] etc.).
Mursaleen [19] generalized the notion of statistical convergence with the
On λ-Statistical Limit Inferior and Limit Superior...
7
help of a non-decreasing sequence λ = (λn ) of positive numbers tending to ∞
with λn+1 ≤ λn + 1, λ1 = 1 and called respectively λ-statistical convergence.
Savaş[24] extented the notion to sequences of fuzzy numbers. Tuncer and Benli
[27-28] has introduced λ-statistically Cauchy sequences and λ-statistical limit
and cluster point for the sequences of fuzzy numbers. Gadjiev and Orhan
[13] introduced the notion of statistical convergence with degree 0 < β < 1
for a number sequence. Then, Çolak [6] studied the notions of statistical
convergence and p-Cesàro summability with order α and the notion was further
generalized by Çolak and Bektaş in [7].
In the present work, we aim to introduce the concepts of λ-statistical convergence, λ-statistical limit and cluster points and λ-statistical limit inferior
and limit superior of order β for sequences of fuzzy numbers.
2
Background and Preliminaries
Given any interval A, we shall denote its end points by A1 , A2 and by D
the set of all closed bounded intervals on real line R, i.e., D = {A ⊂ R:
A = [A1 , A2 ]}. For A, B ∈ D we define A ≤ B if and only if A1 ≤ B1
and A2 ≤ B2 . Furthermore, the distance function d defined by d(A, B) =
max{|A1 − B1 |, |A2 − B2 |}, is a Housdroff metric on D and (D, d) is a complete
metric space. Also ≤ is a partial order on D.
Definition 2.1 A fuzzy number is a function X from R to [0,1] which satisfying the following conditions: (i) X is normal, i.e., there exists an x0 ∈ R such
that X(x0 ) = 1; (ii) X is fuzzy convex, i.e., for any x, y ∈ R and λ ∈ [0, 1],
X(λx + (1 − λ)y) ≥ min{X(x), X(y)}; (iii) X is upper semi-continuous; (iv)
The closure of the set {x ∈ R : X(x) > 0} denoted by X 0 is compact.
Further, every real number r can be expressed as a fuzzy number r as
follows:
1, for t = r,
r(t) =
0, otherwise.
The α-cut of fuzzy real number X is denoted by [X]α , 0 < α ≤ 1, where
[X]α = {t ∈ R : Xt ≥ α}. If α = 0, then it is the closure of the strong
0-cut. A fuzzy real number X is said to be upper semi-continuous if for each
> 0, X −1 ([0, a + )) for all a ∈ [0, 1] is open in the usual topology of R. If
there exists t ∈ R such that X(t) = 1, then the fuzzy real number X is called
normal.
A fuzzy number X is said to be convex, if X(t) = X(s)∧X(r) = min(X(s), X(r))
where s < t < r. The class of all upper semi-continuous, normal, convex fuzzy
real numbers is denoted by L(R) and throughout the article, by a fuzzy real
8
Pankaj Kumar et al.
number we mean that the number belongs L(R). Let X, Y ∈ L(R) and the
α-level sets be
[X]α = [X1α , X2α ] and [Y ]α = [Y1α , Y2α ] for α ∈ [0, 1].
Then the arithmetic operations L(R) are defined as follows:
[X ⊕ Y ]α = [X1α + Y1α , X2α + Y2α ],
[X Y ]α = [X1α − Y2α , X2α − Y1α ],
[X ⊗ Y ]α = mini,j∈{1,2} {Xiα .Yjα }, maxi,j∈{1,2} {Xiα .Yjα } ,
/X
[X −1 ]α = [(X2α )−1 , (X1α )−1 ], 0 ∈
for each α ∈ [0, 1]. The additive identity and multiplicative identity in L(R)
are denoted by 0 and 1, respectively. Define a map d : L(R) × L(R) → R by
d(X, Y ) = supα∈[0,1] d(X α , Y α ). Puri and Ralescu [23] proved that (L(R), d)
is a complete metric space. Also the ordered structure on L(R) is defined as
follows.
For X, Y ∈ L(R), we define X ≤ Y if and only if X1α ≤ Y1α and X2α ≤ Y2α
for each α ∈ [0, 1]. We say that X < Y if X ≤ Y and there exist α0 ∈ [0, 1]
such that X1α0 < Y1α0 or X2α0 < Y2α0 . The fuzzy number X and Y are said to
be incomparable if neither X ≤ Y nor Y ≤ X.
Before proceeding further, we recall some definitions and results which form
the background of the present work.
Let λ = (λn ) be a non-decreasing sequence of positive numbers tending to
∞ such that λn+1 ≤ λn + 1, λ1 = 1 and the interval In = [n − λn + 1, n]. The
set of all such sequences will be denoted by Ω.
Definition 2.2 A sequence X = (Xk ) of fuzzy numbers is said to be λstatistically convergent to a fuzzy number X0 , in symbol: Sλ - limk Xk = X0 , if
for each > 0,
1
|{k ∈ In : d(Xk , X0 ) ≥ }| = 0.
n→∞ λn
lim
Let Sλ (X) denotes the set of all statistically convergent sequences of fuzzy
numbers.
3
Sλβ -Convergence
Definition 3.1 Let β be any real number such that β ∈ (0, 1]. The λβ density of a set A ⊆ N is denoted as δλβ (A) and is defined by
δλβ (A) = lim
n→∞
if the limit exists.
1
|{k ∈ In : k ∈ A}|,
λβ
On λ-Statistical Limit Inferior and Limit Superior...
9
Definition 3.2 Let λ ∈ Ω and β ∈ (0, 1] is given. A sequence X = (Xk )
of fuzzy numbers is said to be λ-statistically convergent of order β to a fuzzy
number X0 , in symbol: Sλβ - limk Xk = X0 , if for each > 0,
1
|{k ∈ In : d(Xk , X0 ) ≥ }| = 0.
n→∞ λβ
lim
Let Sλβ (X) denotes the set of all Sλβ -convergent sequences of fuzzy numbers.
Remark 3.3 1. For the choice of β = 1, the λ-statistical convergence of
order β reduces to λ-statistical convergence.
2. The λ-statistical convergence of order β is well defined for β ∈ (0, 1] but is
not well defined for β > 1. This case can be seen in the following example.
Example 3.4 Let X = (XK ) be a sequence of fuzzy number defined as
Xk (x) =
where
ν(x), if k is an odd number,
µ(x) otherwise,

 x − 7 if x ∈ [7, 8],
8 − x if x ∈ (8, 9],
ν(x) =

0,
otherwise.
and

 x − 4 if x ∈ [4, 5],
6 − x if x ∈ (5, 6]
µ(x) =

0
otherwise.
Then, for α ∈ (0, 1], the α-level sets of Xk are
[7 + α, 9 − α], if k is odd
α
[Xk ] =
[4 + α, 6 − α] otherwise.
Hence, for β > 1 the sequence (Xk ) is λ-statistical convergence of order β to
η1 and η2 as well, since
λn
1 | k ∈ In : d(Xk , η1 ) ≥ | = lim
=0
β
n→∞ 2λβ
n→∞ λ
n
lim
1 λn
| k ∈ In : d(Xk , η2 ) ≥ | = lim
=0
β
n→∞ λ
n→∞ 2λβ
n
lim
where [η1 ]α = [7 + α, 9 − α] and [η2 ]α = [4 + α, 6 − α]. Hence, Sλβ - lim Xk is
not unique. 10
Pankaj Kumar et al.
Theorem 3.5 For 0 < β ≤ γ ≤ 1,
Sλβ (X) ⊆ Sλγ (X)
and the inclusion is strict.
Proof. Let 0 < β ≤ γ ≤ 1, then obliviously Sλβ (X) ⊆ Sλγ (X) as
1
1
|{k ∈ In : d(Xk , X0 ) ≥ }| ≤ β |{k ∈ In : d(Xk , X0 ) ≥ }|.
γ
λ
λ
To prove the strictness of the inclusion, let us consider the following example.


x
−
2
if
x
∈
[2,
3],





4
−
x
if
x
∈
(3,
4],
if k = n3




0,
otherwise 
Xk (x) =
x
−
6
if
x ∈ [6, 7], 




8 − x if x ∈ (7, 8],
if k 6= n3 .




0,
otherwise
Then, for α ∈ (0, 1],
α
[Xk ] =
[2 + α, 4 − α], if k = n3
[6 + α, 8 − α] if k 6= n3 .
For γ ∈ ( 31 , 1], we have
1
lim γ |{k ∈ In : d(Xk , ζ) ≥ }| = lim
n→∞ λ
n→∞
√
3
λn − 1
=0
λγn
where ζ = [−2 + 2α, 2 − 2α]. This means, the sequence (Xk ) is Sλβ -convergent
of order γ but is not Sλβ -convergent of order β, for β ∈ (0, 31 ]. Theorem 3.6 Let λ = (λn ) and µ = (µn ) both are in Ω such that λn ≤ µn
β
for all n ≥ n0 for some n0 ∈ N. If, for 0 < β ≤ γ ≤ 1, lim inf n→∞ µλnγn > 0,
then
(i) Sµγ (X) ⊆ Sλβ (X),
(ii) Sµβ (X) ⊆ Sλβ (X),
(iii) Sµ (X) ⊆ Sλβ (X).
β
Proof. (i) Suppose lim inf n→∞ µλnγn > 0 for the sequences λ, µ ∈ Ω such
that λn ≤ µn for all n ≥ n0 for some n0 ∈ N. Assume that Sµγ - limk Xk = X0 .
Now for > 0, we have
k ∈ In : d(Xk , X0 ) ≥ ⊂ k ∈ Jn : d(Xk , X0 ) ≥ On λ-Statistical Limit Inferior and Limit Superior...
11
where Jn = [n − µn + 1, n]. Now, we can write
1 1 k ∈ In : d(Xk , X0 ) ≥ ≤ γ k ∈ Jn : d(Xk , X0 ) ≥ γ
µn
µn
or
1 λβn 1 k ∈ In : d(Xk , X0 ) ≥ ≤ γ k ∈ Jn : d(Xk , X0 ) ≥ γ. β
µn λn
µn
Since Sµγ - limk Xk = X0 and lim inf n→∞
λβ
n
µγn
> 0, taking the limit as n → ∞
1 k ∈ In : d(Xk , X0 ) ≥ = 0
β
n→∞ λn
lim
which completes the proof.
The results (ii) and (iii) are the immediate consequences of (i). Theorem 3.7 Let λ = (λn ) and µ = (µn ) both are in Ω such that λn ≤ µn
β
for all n ≥ n0 for some n0 ∈ N. If, for 0 < β ≤ γ ≤ 1, limn→∞ µλnγn = 1 and
limn→∞ µµnγn = 1, then
(i) Sλβ (X) = Sµγ (X),
(ii) Sλβ (X) = Sµβ (X),
(iii) Sλβ (X) = Sµ (X).
β
Proof. (i) Suppose limn→∞ µλnγn = 1 and limn→∞ µµnγn = 1 for the sequences
λ, µ ∈ Ω such that λn ≤ µn for all n ≥ n0 for some n0 ∈ N. Assume that
(Xk ) ∈ Sλβ (X). Since In ⊂ Jn then, for > 0, we may write
k ∈ Jn : d(Xk , X0 ) ≥ ⊃ k ∈ In : d(Xk , X0 ) ≥ Thus, we obtain
1 1 k ∈ Jn : d(Xk , X0 ) ≥ = γ n − µn + 1 ≤ k ≤ n − λn : d(Xk , X0 ) ≥ γ
µn
µn
1 + γ k ∈ In : d(Xk , X0 ) ≥ µn
µn − λn
1 ≤
+
k ∈ In : d(Xk , X0 ) ≥ γ
γ
µn
µn
β
µn − λn
1 ≤
+ γ k ∈ In : d(Xk , X0 ) ≥ γ
µn
µn
µn λβn
1 ≤
+ β k ∈ In : d(Xk , X0 ) ≥ γ − γ
µn µn
λn
12
Pankaj Kumar et al.
for all n ≥ n0 for some n0 ∈ N. The right hand side of the above inequality
tends to 0 as n → ∞. Hence, (Xk ) ∈ Sµγ (X) and therefore Sλβ (X) ⊆ Sµγ (X).
Now, together with the (i) part of Theorem 2.1, this immediately implies that
Sλβ (X) = Sµγ (X).
It is very easy to prove (ii) and (iii), hence omitted. 4
Sλβ -Limit and Cluster Point
Let λ = (λn ) ∈ Ω and β ∈ (0, 1]. A subsequence (X)K , where K = {k(j) : j ∈
N}, of a fuzzy number sequence X = (Xk ) is a λβ -thin subsequence if
1
lim
r→∞
λβn
|{k(j) ∈ Ir : j ∈ N}| = 0,
On the other hand, (X)K is a λβ -nonthin subsequence of (Xk ) if
lim sup
r→∞
1
λβn
|{k(j) ∈ Ir : j ∈ N}| > 0.
.
Theorem 4.1 Let λ = (λn ) ∈ Ω and β ∈ (0, 1]. A fuzzy number X0 is
said to be a Sλβ -limit point of the sequence (X) of fuzzy numbers provided that
there is a λβ -nonthin subsequence of (Xk ) that is convergent to X0 .
Let ΛS β denotes the set of all Sλβ -limit point of the sequence (Xk ) of fuzzy
λ
numbers.
Theorem 4.2 Let λ = (λn ) ∈ Ω and β ∈ (0, 1]. A fuzzy number Y0 is said
to be a Sλβ -cluster point of the sequence (Xk ) of fuzzy numbers provided that
for each > 0,
lim sup
n→∞
1
λβn
|{k ∈ In : d(Xk , Y0 ) < }| > 0.
Let ΓS β denotes the set of all Sλβ -cluster point of the sequence (Xk ) of fuzzy
λ
numbers.
Theorem 4.3 Let λ = (λn ) and µ = (µn ) both are in Ω such that λn ≤ µn
β
for all n ≥ n0 for some n0 ∈ N. If, for 0 < β ≤ γ ≤ 1, lim inf n→∞ µλnγn > 0,
then
(i) ΛSµγ ⊇ ΛS β ,
λ
(ii) ΛSµβ ⊇ ΛS β ,
λ
(iii) ΛSµ ⊇ ΛS β .
λ
13
On λ-Statistical Limit Inferior and Limit Superior...
β
Proof. (i) Suppose, for 0 < β ≤ γ ≤ 1, lim inf n→∞ µλnγn > 0. Assume
that X0 ∈ ΛS β , then there is λβ -nonthin subsequence (Xk(j) ) of (Xk ) that is
λ
convergent to X0 and
lim sup
n→∞
1
λβn
|{k(j) ∈ In : j ∈ N}| = d > 0
(1)
Now for > 0, we have
{k(j) ∈ Jn : j ∈ N} ⊃ {k(j) ∈ In : j ∈ N}
1
λβ 1
Since, γ |{k(j) ∈ Jn : j ∈ N}| ≥ nγ . β |{k(j) ∈ In : j ∈ N}|
µn
µn λn
it follows by (1) that lim supn→∞ µ1γn |{k(j) ∈ Jn : j ∈ N}| > 0. Since (Xk(j) ) is
already convergent to X0 , so we have X0 ∈ ΛSµγ . Hence ΛSµγ ⊇ ΛS β .
λ
The results (ii) and (iii) are the immediate consequences of (i). Theorem 4.4 Let λ = (λn ) and µ = (µn ) both are in Ω such that λn ≤ µn
β
for all n ≥ n0 for some n0 ∈ N. If, for 0 < β ≤ γ ≤ 1, lim inf n→∞ µλnγn > 0,
then
(i) ΓSµγ ⊇ ΓS β ,
λ
(ii) ΓSµβ ⊇ ΓS β ,
λ
(iii) ΓSµ ⊇ ΓS β .
λ
Proof. The proof of the theorem can be obtain on the similar lines as that
of the above theorem and therefore is omitted here.
Theorem 4.5 Let λ = (λn ) and µ = (µn ) both are in Ω such that λn ≤ µn
β
for all n ≥ n0 for some n0 ∈ N. If, for 0 < β ≤ γ ≤ 1, lim inf n→∞ µλnγn = 1 and
lim inf n→∞ µµnγn = 1, then
(i) ΛSµγ = ΛS β ,
λ
(ii) ΛSµβ = ΛS β ,
λ
(iii) ΛSµ = ΛS β .
λ
β
Proof. (i) Suppose limn→∞ µλnγn = 1 and limn→∞ µµnγn = 1 for the sequences
λ, µ ∈ Ω such that λn ≤ µn for all n ≥ n0 for some n0 ∈ N. Assume
that X0 ∈ ΛSµγ , then there is µγ -nonthin subsequence (Xk(j) ) of (Xk ) that is
convergent to X0 and
lim sup
n→∞
1
|{k(j) ∈ In : j ∈ N}| = d > 0
µγn
(2)
14
Pankaj Kumar et al.
Since In ⊂ Jn then, for > 0, we may write
{k(j) ∈ Jn : j ∈ N} ⊃ {k(j) ∈ In : j ∈ N}
Thus, as in the proof of Theorem 3.3 (i), we obtain
1
µn λβn
1
+ β |{k ∈ In : j ∈ N}|
γ |{k ∈ Jn : j ∈ N}| ≤
γ − γ
µn
µn µn
λn
for all n ≥ n0 for some n0 ∈ N. The right hand side of the above inequality
tends to 0 as n → ∞ and therefore (Xk ) ∈ Sµγ (X). Hence, ΛSµγ ⊆ ΛS β .
λ
Now, together with the (i) part of Theorem 4.1, this immediately implies that
ΛSµγ = ΛS β .
λ
It is very easy to prove (ii) and (iii), hence omitted. Theorem 4.6 Let λ = (λn ) and µ = (µn ) both are in Ω such that λn ≤ µn
β
for all n ≥ n0 for some n0 ∈ N. If, for 0 < β ≤ γ ≤ 1, lim inf n→∞ µλnγn = 1 and
lim inf n→∞ µµnγn = 1, then
(i) ΓSµγ = ΓS β ,
λ
(ii) ΓSµβ = ΓS β ,
λ
(iii) ΓSµ = ΓS β .
λ
Proof. The proof of the theorem can be obtain on the similar lines as that
of the above theorem and therefore is omitted here.
5
Sλβ -Limit Inferior and Superior
In this section, we introduce the notions of Sλβ -statistical limit inferior and
superior for sequences of fuzzy numbers.
Let λ = (λn ) ∈ Ω and β ∈ (0, 1]. For a sequence X = (Xk ) of fuzzy
numbers, let us define the sets:
M = {µ ∈ L(R) : δλβ ({k ∈ In : Xk < µ}) 6= 0};
N = {µ ∈ L(R) : δλβ ({k ∈ In : Xk > µ}) 6= 0}.
Theorem 5.1 Let λ = (λn ) ∈ Ω and β ∈ (0, 1]. The Sλβ -limit superior of
(Xk ) is defined by
sup N, if N 6= φ,
Sλβ - lim sup X =
−∞,
otherwise.
On λ-Statistical Limit Inferior and Limit Superior...
15
Similarly, the Sλβ -limit inferior is defined by
inf M, if M 6= φ,
Sλβ - lim inf X =
∞,
otherwise.
The concepts defined above can be illustrated with help of the following example.
Example 5.2 Let λ = (λn ) ∈ Ω. We define a sequence of fuzzy numbers
X = (Xk ) as follows. For x ∈ R, define


x − k + 1,
if k − 1 ≤ x ≤ k 




, if k = n2
 −x + k + 1, if k < x ≤ k + 1

0,
otherwise
Xk (x) =


ϑ
,
if
k
is
even

1

,
if k 6= n2 .

ϑ2 , if k is odd
where the fuzzy numbers ϑ1 and ϑ2 are

 x − 5,
7−x
ϑ1 =

0
and
given by
if 5 ≤ x ≤ 6,
if 6 < x ≤ 7,
otherwise.

 x + 7, if −7 ≤ x ≤ −6,
−5 − x if −6 < x ≤ −5,
ϑ2 =

0
otherwise.
Then, we obtain

 [k − 1 + α, k + 1 − α], if k = n2
α
[5 + α, 7 − α],
if k 6= n2 and k is even,
[Xk ] =

[−7 + α, −5 − α],
if k =
6 n2 . and k is odd,
Clearly,
M = (η1 , ∞) and N = (−∞, η2 )
for the sequences (Xk ), where
[η1 ]α = [−7 + α, −5 − α] and [η2 ]α = [5 + α, 7 − α].
Therefore Sλβ - lim inf X = η1 and Sλβ - lim sup X = η2 , only for β ∈ ( 21 , 1]
Theorem 5.3 Let λ = (λn ) ∈ Ω and β ∈ (0, 1]. Then, for a sequence
X = (Xk ) of fuzzy numbers, Sλβ - lim sup X = η if and only if, for each positive
fuzzy number δλβ ({k ∈ In : Xk > η } =
6 0 and δλβ ({k ∈ In : Xk > η ⊕ }) = 0.
16
Pankaj Kumar et al.
Theorem 5.4 Let λ = (λn ) ∈ Ω and β ∈ (0, 1]. Then, for a sequence
X = (Xk ) of fuzzy numbers, Sλβ - lim inf X = ξ if and only if, for each positive
fuzzy number δλβ ({k ∈ In : Xk > ξ ⊕ } =
6 0 and δλβ ({k ∈ In : Xk > ξ }) = 0.
The proofs of the above theorems are routine work so is omitted here.
Theorem 5.5 Let λ = (λn ) ∈ Ω and β ∈ (0, 1]. Then, for a sequence
X = (Xk ) of fuzzy numbers
Sλβ - lim inf X ≤ Sλβ - lim sup X.
Proof. If Sλβ - lim sup X = +∞, then the result is obvious one but if
Sλβ - lim sup X = −∞, then N = φ. It means for every µ ∈ L(R), δλβ ({k ∈
In : Xk > µ}) = 0. Hence, for every ν ∈ L(R), δλβ ({k ∈ In : Xk < ν}) 6= 0.
By the definition Sλβ - lim inf X = −∞.
Now, let us consider the case when Sλβ - lim sup X = η is finite and Sλβ - lim inf X =
θ. Since Sλβ - lim sup X = η, therefore δλβ ({k ∈ In : Xk > η ⊕ 2 }) = 0. This
implies that δλβ ({k ∈ In : Xk ≤ η ⊕ 2 }) 6= 0, which gives in return that
δλβ ({k ∈ In : Xk < η ⊕ }) 6= 0. Hence η ⊕ ∈ M . But, by the definition of
Sλβ − lim inf, inf M = θ so θ ≤ β ⊕ . Since is arbitrary, hence completes the
result. Corollary 5.6 Let λ = (λn ) ∈ Ω and β ∈ (0, 1]. Then, for a sequence
X = (Xk ) of fuzzy numbers
lim inf X ≤ Sλβ - lim inf X ≤ Sλβ - lim sup X ≤ lim sup X.
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