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Statistical Convergence of order β for (λ, μ) double sequences of fuzzy numbers

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Abstract

In this study, we introduce the concepts of φλ,μ-double statistically convergence of order β in fuzzy sequences and strongly λ- double Cesaro summable of order β for sequences of fuzzy numbers. Also we give some inclusion theorems.
AUTHOR COPY
Journal of Intelligent & Fuzzy Systems 39 (2020) 6949–6954
DOI:10.3233/JIFS-200039
IOS Press
6949
Statistical Convergence of order βfor (λ, μ)
double sequences of fuzzy numbers
Damla Barlak
Department of Statistics, Dicle University, Diyarbakır, Turkey
Abstract. In this study, we introduce the concepts of ϕλ,μdouble statistically convergence of order βin fuzzy sequences
and strongly λdouble Cesaro summable of order βfor sequences of fuzzy numbers. Also we give some inclusion theorems.
Keywords and phrases: Statistical convergence, Ces`
aro summability, Modulus function
Mathematics Subject Classification: 40A05; 40A25; 40A30; 40C05; 03E72
1. Introduction
In order to generalize the concept of convergence
of real sequences, the notion of statistical conver-
gence was introduced by Fast [18]. Schoenberg [44]
gave some basic properties of statistical convergence.
For more details about statistical convergence one
can refer to Connor [15], Fridy [19], ˇ
Sal´
at [38], Gad-
jiev and Orhan [20] introduced the definition of order
of statistical convergence for positive linear operator
and after that generalization of statistical convergence
was introduced by C¸ olak [12] under name of statisti-
cal convergence of order α. Over the years and under
different names statistical convergence has been dis-
cussed in the theory of Fourier analysis, Ergodic
theory and Number theory. In recent years, gener-
alizations of statistical convergence have appeared
in the study of strong integral summability and the
structure of ideals of bounded continuous functions
on locally compact spaces. Statistical convergence
and its generalizations are also connected with sub-
sets of the Stone-Cech compactification of the natural
numbers. Moreover, statistical convergence is closely
related to the concept of convergence in probability.
Aizpuru et al. [1] defined the fdensity of the sub-
Corresponding author. Damla Barlak, Department of
Statistics , Dicle University, Diyarbakır, Turkey. E-mail:
damla.barlak@dicle.edu.tr.
set Aof Nby using an unbounded modulus function.
After that, Bhardwaj and Dhawan [10] introduced
fstatistical convergence of order αand strong
Cesaro summability of order αwith respect to a mod-
ulus function ffor real sequences. For a detailed
account of many more interesting investigations con-
cerning statistical convergence of order β, one may
refer to ([4, 14, 17, 28, 50]).
Mursaleen and Edely [32] introduced the defi-
nition of double statistical convergence of number
sequences using double natural density of positive
integers. Besides this topic was studied by many
authors ( [11, 13, 30, 32, 33, 47, 48]).
Recently, the fuzzy theory has emerged as the
most active area of research in many branches of
mathematics and engineering. Zadeh [51] in 1965
introduced the concept of fuzzy set, which is defined
with the help of grades of membership. Zadeh [51]
put forward the concept of fuzzy sets as a for-
mal mathematical system to model human reasoning
and decision making processes in uncertain environ-
ments. Zadeh’s [51] study attracted many researchers
in different fields of science and found numerous
applications ranging from engineering to mathemat-
ics.This theory has been developed and influenced in
many areas of application.
The idea of bounded and convergent sequences of
fuzzy numbers initially discussed by Matloka [25]
ISSN 1064-1246/20/$35.00 © 2020 – IOS Press and the authors. All rights reserved
AUTHOR COPY
6950 D. Barlak / Statistical convergence of order βfor (λ, μ)Double Sequences of Fuzzy Numbers
where it was shown that every convergent sequence
is bounded. Nanda [34] proved that the spaces of
bounded and convergent sequences of fuzzy num-
bers are complete metric spaces. After that, Nuray
and Savas¸ [36] defined the concept of statistical con-
vergence for sequences of fuzzy numbers. For some
further works in this direction we refer to ([2, 3, 9,
16, 27, 29, 37, 49]) Savas¸ [41] introduced the double
sequences of fuzzy numbers. Savas¸ and Mursaleen
[43] studied the statistical convergence for double
sequences of fuzzy numbers and others ([8, 22, 40,
46]).
Mursaleen [31] introduced λstatistical conver-
gence as an extension of (V, λ)- summability of
Leindler [24] with the help of a non-decreasing
sequence λ=(λn)of positive numbers tending to
with λn+1λn+1, λ1=1.The generalized de la
Vallee-Poussion mean is defined by
tn(x)=1
λn
kIn
xk,
where In=[nλn+1,n
].Meenaskshi et.al. also
studied these concepts (λ, μ)statistical convergence
for double sequences [26]. Later Is¸ık and Altın [21]
introduced the concepts of fλ,μstatistical conver-
gence for double sequences of order
α. Savas¸ [42]
introduced and discussed the concepts of λstatis-
tically convergence of fuzzy numbers. The theory of
λstatistical convergence for fuzzy sequences and
their properties has been studied extensively by var-
ious authors (see [5–7]).
This paper organizes as follows: In Section 2, we
give the basic notions which will be used through-
out the paper. In Section 3, we define the concepts
of ϕλ,μdouble statistically convergence of order β
in fuzzy sequences and strongly λdouble Cesaro
summable of order βfor sequences of fuzzy numbers.
Also we give some inclusion theorems.Finally in the
Section 4, we mention content of paper and studies
future time.
2. Definitions and preliminaries
In this section, we recall some basic definitions and
notations that we are going to use in this paper.
Definition 2.1. A fuzzy number is fuzzy set u:R
[0,1] with the following properties:
i)uis normal, that is, there exists an x0Rsuch
that u(x0)=1;
ii)uis fuzzy convex, that is, for x, y Rand 0
λ1,u(λx +(1 λ)y)min[u(x),u(y)];
iii)uis upper semicontinuous;
iv) suppu=cl{xR:u(x)>0},or denoted by
[u]0,is compact, then it is called a fuzzy number.
Definition 2.2. αlevel set [u]αof a fuzzy number u
is defined by
[u]α={xR:u(x)α},if α(0,1]
suppu, if α=0.
It is clear that uis a fuzzy number if and only if [u]α
is a closed interval for each α[0,1]and [u]1/=∅.
We denote space of all fuzzy numbers by L(R).
Definition 2.3. The distance between two fuzzy num-
bers uand v, we use the metric
d(u, v)=sup
0α1
dH[u]α,[v]α
Definition 2.4. Let u=uα, uαand v=vα, vα
be two fuzzy numbers. Then, the Hausdorff metric is
defined by
dH[u]α,[v]α=max uαvα,uαvα.
It is known that dis a metric on L(R),and (L(R),d)
is a complete metric space.
A sequence X=(Xk) of fuzzy numbers is a func-
tion X:NL(R).
Definition 2.5. A sequence X=(Xk)of fuzzy
numbers is called bounded if and only if the set
{Xk:kN}of fuzzy numbers consisting of the
terms the sequence Xkis a bounded set.
Definition 2.6. A sequence X=(Xk)of fuzzy num-
bers is called convergent with limit X0L(R),if and
only if for every ε>0 there exists a positive integer
k0such that d(Xk,X
0)for all k>k
0.
Let s(F),
(F)and c(F)denote the set of all
sequences, all bounded sequences and all convergent
sequences of fuzzy numbers, respectively [25].
Definition 2.7. Let ϕbe real-valued function defined
on [0,) satisfying the following:
i)ϕ(t)=0ifft=0,
ii)ϕ(t+u)ϕ(t)+ϕ(u)for t, u 0,
iii)ϕis increasing,
iv)ϕis right-continuous at 0,see [35].
Such a function is called a modulus function; some
examples of modulus functions are tp,(0<p1),
AUTHOR COPY
D. Barlak / Statistical convergence of order βfor (λ, μ)Double Sequences of Fuzzy Numbers 6951
log(1 +t).A modulus function can be bounded or
unbounded.
For an extensive view on this subject we refer ([39,
45]).
Throughout the paper, we will take βinstead of
(a, b)and γinstead of (c, d)for a, b, c, d (0,1]as
follows [13]:
βγa<cand b<d,
βγacand bd,
βγa=cand b=d,
β(0,1]a, b (0,1],
γ(0,1]c, d (0,1],
β1a=b=1
β1a>1,b>1.
Let λ=(λn)and μ=(μm)be two non-decreasing
sequences of positive real numbers tending to
with λn+1λn+1
1=0; μn+1μn+1
1=
0 and β(0,1]be given.
Let KN×Nbe two dimensional set of positive
integers and ϕbe an unbounded modulus function.
Definition 2.8. Then, δf2
βλ,μdouble density of
Kis defined as
δf2
β(K)=lim
n,m→∞
1
ϕλs
nμt
mϕ
(|{(k, )In×Im:(i, j)K}|),
if the limit exists, see [21].
3. Main results
In this section we give the concept of ϕλ,μdouble
statistically convergence of order βin fuzzy
sequences.
Definition 3.1. Let λ=(λn)and μ=(μm)be two
non-decreasing sequences of positive real numbers
as above and β(0,1]be given and X=(Xk)be a
double sequence of fuzzy numbers. If for every ε>0,
lim
n,m→∞
1
ϕλa
nμb
mϕ(|{(j, k)In×Im:d(Xk ,X
0)ε}|)=0,
a double sequence X=(Xk)of fuzzy numbers
is said to be ϕλ,μdouble statistically convergent
of order βto the fuzzy number X0.In this case
we write Sβ
2(F, λ, μ, ϕ )lim Xk =X0or Xk
X0Sβ
2(F, λ, μ, ϕ )where ϕis an unbounded mod-
ulus function.By Sβ
2(F, λ, μ, ϕ ),we shall denote the
set of all double sequences of fuzzy numbers which
are ϕλ,μstatistically convergent of order β. Where
In=[nλn+1,n
]and Im=[mμm+1,m
].
It is easy to see that every convergent sequence is
ϕλ,μdouble statistically convergent of order β,but
converse does not hold as following example:
Example 3.2. Take modulus function ϕ(x)=xp
for 0 <p1 and λn=n,μm=m. Consider the
sequence
Xk (x)=
x
(k)3+1,for x(k)3,0
x
(k)3+1,for x0,(k)3
0,otherwise
if k=n3,=m3
x4,for x[4,5]
x+6,for x[5,6]
0,otherwise
otherwise
Then, we calculate the αlevel set of sequences
(Xk)as follows
[Xk]α=
(k)3(α1),(k)3(α1),if k=n3,=m3
[α+4,6α],otherwise
.
Hence (Xk)is ϕλ,μdouble statistically convergent
of order β, for β1
3,1to fuzzy number X0,where
[X0]α=[α+4,6α],but not convergent.
Theorem 3.3. Let β(0,1]and X=(Xk),Y=
(Yk)be two double sequences of fuzzy numbers. Then
(i) Xk X0Sβ
2(F, λ, μ, ϕ )and νCimplies
(νXk)νX0Sβ
2(F, λ, μ, ϕ ),
(ii) Xk X0Sβ
2(F, λ, μ, ϕ )and
Yk Y0Sβ
2(F, λ, μ, ϕ )implies (Xk +Yk )
(X0+Y0)Sβ
2(F, λ, μ, ϕ ).
Proof. (i) Proof follows from the Minkoski inequality
in [23],
lim
n,m→∞
1
ϕλa
nμb
mϕ(|{(k, )In×Im:d(νXkX
0)ε}|)
1
ϕλa
nμb
mϕ(k, )In×Im:d(Xk,X
0)ε
|ν|
AUTHOR COPY
6952 D. Barlak / Statistical convergence of order βfor (λ, μ)Double Sequences of Fuzzy Numbers
(ii) It follows from the inequality
1
ϕλa
nμb
mϕ(|{(k, )In×Im:d(Xk +Yk,X
0+Y0)ε}|)
1
ϕλa
nμb
mϕ(k, )In×Im:d(Xk,X
0)ε
2
+1
ϕλa
nμb
mϕ(k, )In×Im:d(Yk,Y
0)ε
2
Theorem 3.4. Let ϕbe unbounded modulus func-
tion, X=(Xk)be a double sequences of fuzzy
numbers and β, γ (0,1]. Then Sβ
2(F, λ, μ, ϕ )
Sγ
2(F, λ, μ, ϕ )and the inclusion is strict.
Proof. It can be easily shown the inclusion by using
the fact that ϕis increasing for βγ.Now,weshow
that the inclusion is strict. For this, consider fuzzy
sequence X=(Xk)defined by
Xk (x)=
xk +1,if k 1xk
x+k +1,if k xk +1
0,otherwise
,if k=n2,=m2
x2,if 2 x3
x+4,if 3 x4
0,otherwise
,otherwise
and take modulus function ϕ(x)=xp,0<p1.
We can find the αlevel set of sequence (Xk)as
follows:
[Xk]α=[k +α1,k+1α],if k=n2,=m2
[α+2,4α],otherwise
.
Then, the fuzzy double sequence (Xk)is ϕdouble
statistically convergent of order γfor 1
21,but
not ϕdouble statistically convergent of order βfor
β0,1
2.
Corollary 3.5. Let X=(Xk)be a fuzzy double
sequence, ϕbe an unbounded modulus function and
β(0,1]. Then Sβ
2(F, λ, μ, ϕ )S2(F, λ , μ, ϕ)
and the inclusion is strict, also the limits of sequence
X=(Xk)of fuzzy numbers are same.
Corollary 3.6. Let X=(Xk)be a fuzzy double
sequence, ϕbe an unbounded modulus function and
β, γ (0,1].Then
i) Sβ
2(F, λ, μ, ϕ )=Sγ
2(F, λ, μ, ϕ )if and only if
βγ,
ii) Sβ
2(F, λ, μ, ϕ )=S2(F, λ , μ, ϕ)if and only if
γ1,
iii) Sβ
2(F, λ, μ, ϕ )Sγ
2(F, λ, μ, ϕ )S2(F, λ ,
μ, ϕ)for βγ.
Definition 3.7. Let X=(Xk)be a double sequence
of fuzzy numbers, ϕbe an unbounded modulus func-
tion and β(0,1]. If there is a fuzzy number X0such
that
lim
n,m→∞
1
λa
nμb
m
kIn
Im
ϕ(d(Xk,X
0)) =0,
a double sequence X=(Xk)of fuzzy numbers is
said to be strongly λdouble Cesaro summable of
order βto a fuzzy number X0.By wβ
2(F, λ, μ, ϕ ),
we shall denote the set of all double sequences of
fuzzy numbers which are strongly λdouble Cesaro
summable of order β.
Theorem 3.8. Let X=(Xk)be a double sequence
of fuzzy numbers, ϕbe an unbounded modulus
function and 0γ1. Then wβ
2(F, λ, μ, ϕ )
wγ
2(F, λ, μ, ϕ )and also the limits are same.
Proof. It is easy to show the inclusion relation. For
strictness of inclusion, let ϕbe a modulus function
and consider the fuzzy double sequence X=(Xk)
defined by
Xk (x)=
x
2,
x
2+2,
0
0x2
2x4
otherwise
,if k=n3,=m3
¯
0,otherwise
We can write
1
λc
nμd
m
kIn
Im
ϕdXk,¯
043
λn3
μm
λc
nμd
m
ϕ(4)
for every n, m N.So, we have (Xk )
wγ
2(F, λ, μ, ϕ )since the right side tends to zero for
γ> 1
3as m, n →∞.On the other hand, we obtain
1
λa
nμb
m
kIn
Im
ϕdXk,¯
043
λn3
μm4
λc
nμd
m
ϕ(4)
for every nN.Hence we have (Xk)/
wβ
2(F, λ, μ, ϕ )for 0 1
3as n, m →∞.
Corollary 3.9. Let ϕbe an unbounded modulus func-
tion and β, γ (0,1],then
i) wβ
2(F, λ, μ, ϕ )=wγ
2(F, λ, μ, ϕ )if and only if
βγ,
AUTHOR COPY
D. Barlak / Statistical convergence of order βfor (λ, μ)Double Sequences of Fuzzy Numbers 6953
ii) wβ
2(F, λ, μ, ϕ )=w2(F, λ , μ, ϕ)for β(0,1]
and γ1.
Theorem 3.10. Let X=(Xk)be a double sequence
of fuzzy numbers and 0γ1. Also, ϕbe
an unbounded modulus function such that ϕ(xy)
(x)ϕ(y)for some positive constant cand for all
x, y 0and lim
t→∞
ϕ(t)
t>0.Then wβ
2(F, λ, μ, ϕ )
Sγ
2(F, λ, μ, ϕ ).
Proof. Take any double sequence X=(Xk )
of fuzzy numbers and ε>0.Let Hnm =
{(k, ),k n,  m:d(Xk,X
0)ε}.From
the definition of modulus function, we can write
kIn
Im
ϕ(d(Xk,X
0))
=
kHnm
Hnm
ϕ(d(Xk,X
0))
+
k/Hnm
/Hnm
ϕ(d(Xk,X
0))
kHnm
Hnm
ϕ(d(Xk,X
0))
ϕ
kHnm
Hnm
ϕ(d(Xk,X
0))
ϕ(|{(k, ),k n,  m:d(Xk,X
0)ε}|ε)
(|{(k, ),k n,  m:d(Xk,X
0)ε}|)ϕ(ε)
and
1
λa
nμb
m
kIn
Im
ϕ(d(Xk,X
0)) 1
λa
nμb
m
(|{(k, ),k n,  m:d(Xk,X
0)ε}|)ϕ(ε)
1
λc
nμd
mϕλc
nμd
m(|{(k, ),k n,  m:d(Xk,X
0)ε}|)
ϕ(ε)ϕλc
nμd
m.
Hence, we have XSγ
2(F, λ, μ, ϕ )using the fact
that lim
t→∞
ϕ(t)
t>0 and Xwβ
2(F, λ, μ, ϕ ).So, the
proof is completed.
Corollary 3.11. Let ϕbe an unbounded modulus
function and β, γ (0,1],then wβ
2(F, λ, μ, ϕ )=
Sγ
2(F, λ, μ, ϕ )if and only if βγ.
4. Conclusion
Meenaskshi et al. [26] studied the concept of
(λ, μ)statistical convergence for double sequences.
Later Is¸ık and Altın [21] introduced the concepts of
fλ,μstatistical convergence for double sequences
of order
α. Now in this paper we generalized the
study of Meenaskshi et.al. [26] grading the interval
[0,1]by helping a value β, defined the sequence
classes Sβ
2(F, λ, μ, ϕ )and wβ
2(F, λ, μ, ϕ )which are
ϕλ,μstatistically convergent of order βand strongly
λdouble Cesaro summable of order β, and gave
some inclusion relations between them. Statistical
convergence has several generalizations and applica-
tions in different fields of mathematics such as: rough
convergence, rough continuity, rough statistical con-
vergence. Regarding this topic, the concept of rough
convergence of order βfor sequences of fuzzy or real
numbers can be studied future time.
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