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(3s.) v. 30 2 (2012): 57–62.
ISSN-00378712 in press
doi:10.5269/bspm.v30i2.14007
Asymptotically Lacunary Statistical Equivalent Sequences of Fuzzy
Numbers
Ayhan Esi and Necdet Çatalbaş
abstract:
In this article we present the following definition which is natural
combination of the definition for asymptotically equivalent and lacunary statistical
convergence of fuzzy numbers. Let θ = (kr ) be a lacunary sequence. The two
sequnces X = (Xk ) and Y = (Yk ) of fuzzy numbers are said to be asymptotically
lacunary statistical equivalent to multiple L provided that for every ε > 0
1 Xk
k ∈ Ir : d
, L ≥ ε = 0.
lim
r hr
Yk
Key Words: Asymptotically equivalent, lacunary sequence, fuzzy numbers.
Contents
1 Introduction
57
2 Definitions and Notations
58
3 Main Results
59
1. Introduction
In 1993, Marouf [2] presented definitions for asymptotically equivalent sequences
of real numbers. In 2003, Patterson [4] extended these definitions by presenting
on asymptotically statistical equivalent analog of these definitions. For sequences
of fuzzy numbers Savaş [5] introduced and studied asymptotically λ−statistical
equivalent sequences. Later Esi and Esi [1] studied ∆−asymptotically equivalent
sequences of fuzzy numbers. The goal of this paper is to extended the idea to apply
to asymptotically equivalent and lacunary statistical convergence of fuzzy numbers.
By a lacunary sequence θ = (kr ) ; r = 0, 1, 2, 3, ... where k0 = 0 , we shall
mean an increasing sequence of nonnegative integers with hr = kr − kr−1 → ∞ as
r → ∞. The intervals determined by θ will be denoted by Ir = (kr−1 , kr ] . the ratio
kr
kr−1 will be denoted by qr .
Let D denote the set of all closed and bounded intervals on R , the real line.
For X, Y ∈ D we define
d (X, Y ) = max (|a1 − b1 | , |a2 − b2 |)
where X = [a1 , a2 ] and Y = [b1 , b2 ] . It is known that (D, d) is a complete metric
space. A fuzzy real number X is a fuzzy set on R , i.e. a mapping X :R → I
(= [0, 1]) associating each real number t with its grade of membership X (t) .
2000 Mathematics Subject Classification: 40A05, 40C05, 46D25
57
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P style.
c Soc. Paran. de Mat.
58
Ayhan Esi and Necdet Çatalbaş
The set of all upper - semi continous, normal,convex fuzzy real numbers is
denoted by R (I) . Throughout the paper, by a fuzzy real number X ,we mean
that X ∈ R (I) .
α
The α− cut or α− level set [X] of the fuzzy real number X , for 0 < α ≤ 1
α
defined by [X] = {t ∈ R : X (t) ≥ α} ; for α = 0 , it is the closure of the strong
0−cut , i.e. closure of the set {t ∈ R : X (t) > 0} . The linear structure of R (I)
induces addition X + Y and scalar multipliction µX , µ ∈ R, in terms of α− level
α
α
α
α
α
set , by [X + Y ] = [X] + [Y ] , [µX] = µ [X] for each α ∈ [0, 1].
Let
d : R (I) × R (I) → R
be defined by
α
α
d (X, Y ) = sup d ([X] , [Y ] ) .
α∈(0,1]
Then d defines a metric on R (I) . It is well known that R (I) is complete with
respect to d.
A sequence (Xk ) of fuzzy real numbers is said to be convergent to the fuzzy
real numbers X0 ,if for every ε > 0 , there exists n0 ∈ N such that d (Xk , X0 ) < ε
for all k ≥ n0 .Nuray and Savaş [3] defined the notion of statistical convergence for
sequences of fuzzy numbers as follows :A sequence of fuzzy numbers is said to be
statistically convergent to the fuzzy real number X0 , if for evey ε > 0
lim
n→∞
1 k ≤ n : d (Xk , X0 ) ≥ ε = 0.
n
2. Definitions and Notations
Definition 2.1 Let θ = (kr ) be lacunary sequence. The two sequences X = (Xk )
and Y = (Yk ) of fuzzy numbers are said to be asymptotically lacunary statistical
equivalent of multiple L provided that for every ε ≻ 0,
1 Xk
lim
k ∈ Ir : d
, L ≥ ε = 0.
r hr Yk
S L (F )
denoted by X θ∼ Y
and simply asymptotically lacunary statistical equivalent if L = 1 (where 1 is unity
element for addition in R (I) ). Furthermore, let SθL (F ) denotes the set of X =
(Xk ) and Y = (Yk ) of fuzzy numbers such that X
SθL (F )
∼
Y.
Definition 2.2 Let θ = (kr ) be lacunary sequence. The two sequences X = (Xk )
and Y = (Yk ) of fuzzy numbers are strong asymptotically lacunary statistical equivalent of multiple L provided that
1 X
Xk
lim
, L = 0,
d
r hr
Yk
k∈Ir
59
Fuzzy Numbers
denoted by X
NθL (F )
∼
Y
and simply strong asymptotically lacunary statistical equiv-
alent if L = 1. In addition, let NθL (F ) denotes the set of X = (Xk ) and Y = (Yk )
of fuzzy numbers such that X
NθL (F )
∼
Y.
3. Main Results
Theorem 3.1 Let θ = (kr ) be lacunary sequence. Then
(a) If X
NθL (F )
∼
SθL (F )
Y then X
∼
Y,
N L (F )
SθL (F )
(b) if X ∈ ℓ∞ (F ) and X ∼ Y then X θ∼ Y,
(c) SθL (F ) ∩ ℓ∞ (F ) = NθL (F ) ∩ ℓ∞ (F )
where ℓ∞ (F ) the set of all bounded sequences of fuzzy numbers.
Proof: (a) If ε > 0 and X
X
d
k∈Ir
Therefore X
NθL (F )
∼
Y, then
Xk
,L ≥
Yk
k∈Ir
SθL (F )
∼
Y.
&
X
X
d Y k ,L ≥ε
d
Xk
,L
Yk
k
Xk
, L ≥ ε .
≥ ε. k ∈ Ir : d
Yk
(b) Suppose that
= (Xk ) and Y = (Yk ) ∈ ℓ∞ (F ) and X
X
Xk
we can assume that d Yk , L ≤ T for all k.Given ε > 0
Xk
1
1 X
d
,L =
hr
Yk
hr
k∈Ir
≤
X
Xk
,L ≥ε
k∈Ir & d
Yk
d
Xk
1
,L +
Yk
hr
X
SθL (F )
∼
d
Y.Then
Xk
,L
Yk
Xk
,L <ε
k∈Ir & d
Yk
T Xk
+ε
k
∈
I
:
d
,
L
≥
ε
r
hr
Yk
N L (F )
Therefore X θ∼ Y.
(c) It follows from (a) and (b).
Theorem 3.2 Let θ = (kr ) be lacunary sequence with lim inf r qr > 1, then X
SθL (F )
2
SL (F )
∼
Y implies X ∼ Y,
where S L (F ) (see [5]) the set of X = (Xk ) and Y = (Yk ) of fuzzy numbers such
that
1
Xk
, L ≥ ε = 0.
lim k ≤ n : d
n n
Yk
60
Ayhan Esi and Necdet Çatalbaş
Proof: Suppose that lim inf r qr > 1, then there exists a δ > 0 such that qr ≥ 1 + δ
for sufficiently large r, which implies
δ
hr
≥
kr
1+δ
S L (F )
if X
∼
Y , then for every ε > 0 and for sufficiently large r, we have
Xk
Xk
1 1 d
d
≥
k
≤
k
:
k
∈
I
:
,
L
≥
ε
,
L
≥
ε
r
r
kr
Yk
kr
Yk
1 δ
Xk
. k ∈ Ir : d
, L ≥ ε .
≥
1 + δ hr
Yk
This complete the proof.
2
Theorem 3.3 Let θ = (kr ) be lacunary sequence with lim supr qr < ∞ , then
X
SθL (F )
∼
Y implies X
SL (F )
∼
Y.
Proof: If lim supr qr < ∞,then there exists B > 0 such that qr < C for all r ≥
1.Let X
SθL (F )
∼
Y and ε > 0.There exists B > 0 such that for every j ≥ B
AJ =
Xk
1 < ε.
d
,
L
≥
ε
k
∈
I
:
j
hj
Yk
we can also find K > 0 such that Aj < K for all j = 1, 2, 3, .... Now let n be
any integer with kr−1 < n < kr ,where r ≥ B. Then
Xk
1
1 , L ≥ ε ≤
k≤n:d
n
Yk
kr−1
k ≤ kr : d Xk , L ≥ ε Yk
Fuzzy Numbers
61
Xk
Xk
1 ,L ≥ ε +
, L ≥ ε =
k ∈ I1 : d
k ∈ I2 : d
kr−1
Yk
kr−1
Yk
Xk
1 k ∈ Ir : d
, L ≥ ε + ... +
kr−1 Yk
k2 − k1
k1 Xk
Xk
k ∈ I1 : d
k ∈ I2 : d
=
,L ≥ ε +
, L ≥ ε kr−1 k1 Yk
kr−1 (k2 − k1 ) Yk
kB − kB−1
Xk
+ ... +
k ∈ IB : d
, L ≥ ε kr−1 (kB − kB−1 ) Yk
kr − kr−1
Xk
k ∈ Ir : d
, L ≥ ε + ... +
kr−1 (kr − kr−1 ) Yk
k2 − k1
kB − kB−1
kr − kr−1
k1
A1 +
A2 + ... +
AB + ... +
Ar
=
kr−1
kr−1
kr−1
kr−1
kB
kr − kB
≤ sup Aj
+ sup Aj
kr−1
kr−1
j≥1
jB
kB
≤ K.
+ ε.C.
kr−1
1
This complete the proof.
2
Theorem 3.4 Let θ = (kr ) be lacunary sequence 1 < lim inf r qr ≤ lim supr qr <
∞, then X
SL (F )
∼
Y ⇔X
SθL (F )
∼
Y.
Proof: The result follow from Theorem 3.2 and Theorem 3.3.
2
References
1. Esi, A. and Esi, A., On ∆−asymptotically equivalent sequences of fuzzy numbers, International Journal of Mathematics and Computation, 1(8)(2008), 29-35.
2. Marouf , M., Asymptotic equivalence and summability,Inf. J. Math.Sci.,16 (1993),755 -762.
3. Nuray, F. and Savaş, E., Statistical convergence of sequences of fuzzy real numbers, Math.
Slovaca , 45(3) (1995), 269 -273.
4. Patterson , R.F., On asymptotically statistically equivalent sequences, Demonstratio Math.,
36 (1) (2003), 149 -153.
5. Savaş , E ., On asymptotically λ−statistical equivalent sequence of fuzzy numbers, New
Mathematics and Natural Computation, 3(3) (2007), 301 -306.
62
Ayhan Esi and Necdet Çatalbaş
Ayhan Esi
Adiyaman University, Science and Art Faculty
Department of Mathematics
02040,Adiyaman,Turkey.
E-mail address: [email protected]
and
Necdet Çatalbaş
Firat University, Science and Art Faculty
Department of Mathematics
23199, Elazığ, Turkey.
E-mail address: [email protected]