Content uploaded by Damla Barlak
Author content
All content in this area was uploaded by Damla Barlak on Dec 09, 2021
Content may be subject to copyright.
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,μ−double statistically convergence of order βin fuzzy sequences
and strongly λ−double 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 f−density 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
f−statistical convergence of order αand strong
Cesaro summability of order αwith respect to a mod-
ulus function f−for 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 λ−statistical 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+1≤λn+1, λ1=1.The generalized de la
Vallee-Poussion mean is defined by
tn(x)=1
λn
k∈In
xk,
where In=[n−λn+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,μ−statistical conver-
gence for double sequences of order
α. Savas¸ [42]
introduced and discussed the concepts of λ−statis-
tically convergence of fuzzy numbers. The theory of
λ−statistical 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,μ−double statistically convergence of order β
in fuzzy sequences and strongly λ−double 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 x0∈Rsuch
that u(x0)=1;
ii)uis fuzzy convex, that is, for x, y ∈Rand 0 ≤
λ≤1,u(λx +(1 −λ)y)≥min[u(x),u(y)];
iii)uis upper semicontinuous;
iv) suppu=cl{x∈R:u(x)>0},or denoted by
[u]0,is compact, then it is called a fuzzy number.
Definition 2.2. α−level set [u]αof a fuzzy number u
is defined by
[u]α={x∈R: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
0≤α≤1
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 uα−vα,uα−vα.
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:N→L(R).
Definition 2.5. A sequence X=(Xk)of fuzzy
numbers is called bounded if and only if the set
{Xk:k∈N}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 X0∈L(R),if and
only if for every ε>0 there exists a positive integer
k0such that d(Xk,X
0)<εfor 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<p≤1),
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]:
β≺γ⇔a<cand b<d,
βγ⇔a≤cand b≤d,
βγ⇔a=cand b=d,
β∈(0,1]⇔a, b ∈(0,1],
γ∈(0,1]⇔c, d ∈(0,1],
β1⇔a=b=1
β1⇔a>1,b>1.
Let λ=(λn)and μ=(μm)be two non-decreasing
sequences of positive real numbers tending to ∞
with λn+1≤λn+1,λ
1=0; μn+1≤μn+1,μ
1=
0 and β∈(0,1]be given.
Let K⊆N×Nbe two dimensional set of positive
integers and ϕbe an unbounded modulus function.
Definition 2.8. Then, δf2
βλ,μ−double 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,μ−double
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,μ−double 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,μ−statistically convergent of order β. Where
In=[n−λn+1,n
]and Im=[m−μm+1,m
].
It is easy to see that every convergent sequence is
ϕλ,μ−double statistically convergent of order β,but
converse does not hold as following example:
Example 3.2. Take modulus function ϕ(x)=xp
for 0 <p≤1 and λn=n,μm=m. Consider the
sequence
Xk (x)=
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
x
(k)3+1,for x∈−(k)3,0
−x
(k)3+1,for x∈0,−(k)3
0,otherwise
⎫
⎪
⎬
⎪
⎭
if k=n3,=m3
x−4,for x∈[4,5]
−x+6,for x∈[5,6]
0,otherwise
⎫
⎪
⎬
⎪
⎭
otherwise
Then, we calculate the α−level set of sequences
(Xk)as follows
[Xk]α=
(k)3(α−1),−(k)3(α−1),if k=n3,=m3
[α+4,6−α],otherwise
.
Hence (Xk)is ϕλ,μ−double statistically convergent
of order β, for β∈1
3,1to fuzzy number X0,where
[X0]α=[α+4,6−α],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 ν∈Cimplies
(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,νX
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)=
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
x−k +1,if k −1≤x≤k
−x+k +1,if k ≤x≤k +1
0,otherwise ⎫
⎬
⎭,if k=n2,=m2
x−2,if 2 ≤x≤3
−x+4,if 3 ≤x≤4
0,otherwise ⎫
⎬
⎭,otherwise
and take modulus function ϕ(x)=xp,0<p≤1.
We can find the α−level set of sequence (Xk)as
follows:
[Xk]α=[k +α−1,k+1−α],if k=n2,=m2
[α+2,4−α],otherwise
.
Then, the fuzzy double sequence (Xk)is ϕ−double
statistically convergent of order γfor 1
2<γ≤1,but
not ϕ−double statistically convergent of order βfor
β∈0,1
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
k∈In
∈Im
ϕ(d(Xk,X
0)) =0,
a double sequence X=(Xk)of fuzzy numbers is
said to be strongly λ−double 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 λ−double Cesaro
summable of order β.
Theorem 3.8. Let X=(Xk)be a double sequence
of fuzzy numbers, ϕbe an unbounded modulus
function and 0<βγ≤1. 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
0≤x≤2
2≤x≤4
otherwise
⎫
⎪
⎬
⎪
⎭
,if k=n3,=m3
¯
0,otherwise
We can write
1
λc
nμd
m
k∈In
∈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
k∈In
∈Im
ϕdXk,¯
0≥43
√λn3
√μm−4
λc
nμd
m
ϕ(4)
for every n∈N.Hence we have (Xk)/∈
wβ
2(F, λ, μ, ϕ )for 0 <β≤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 0<βγ≤1. Also, ϕbe
an unbounded modulus function such that ϕ(xy)≥
cϕ (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
k∈In
∈Im
ϕ(d(Xk,X
0))
=
k∈Hnm
∈Hnm
ϕ(d(Xk,X
0))
+
k/∈Hnm
/∈Hnm
ϕ(d(Xk,X
0))
≥
k∈Hnm
∈Hnm
ϕ(d(Xk,X
0))
≥ϕ
k∈Hnm
∈Hnm
ϕ(d(Xk,X
0))
≥ϕ(|{(k, ),k ≤n, ≤m:d(Xk,X
0)≥ε}|ε)
≥cϕ (|{(k, ),k ≤n, ≤m:d(Xk,X
0)≥ε}|)ϕ(ε)
and
1
λa
nμb
m
k∈In
∈Im
ϕ(d(Xk,X
0)) ≥1
λa
nμb
m
cϕ
(|{(k, ),k ≤n, ≤m:d(Xk,X
0)≥ε}|)ϕ(ε)
≥1
λc
nμd
mϕλc
nμd
mcϕ (|{(k, ),k ≤n, ≤m:d(Xk,X
0)≥ε}|)
ϕ(ε)ϕλc
nμd
m.
Hence, we have X∈Sγ
2(F, λ, μ, ϕ )using the fact
that lim
t→∞
ϕ(t)
t>0 and X∈wβ
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,μ−statistical 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,μ−statistically convergent of order βand strongly
λ−double 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.
References
[1] A. Aizpuru, M.C. Listan-Garcia and F. Rambla-Barreno,
Density by moduli and statistical convergence, Quaest Math
37 (2014), 525–530.
[2] Y. Altin, M. Et and B.C. Tripathy, On pointwise statistical
convergence of sequences of fuzzy mappings, J Fuzzy Math
15(2) (2007), 425–433.
[3] H. Altinok, Y. Altın and M. Is¸ık, Statistical convergence
and strong p–Ces`
aro summability of order βin sequences
of fuzzy numbers, Iranian J of Fuzzy Systems 9(2) (2012),
65–75.
[4] H. Altinok, Statistical convergence of order βfor general-
ized difference sequences of fuzzy numbers, J Intell Fuzzy
Systems 26 (2014), 847–856.
[5] H. Altinok, On λ–statistical convergence of order βof
sequences of fuzzy numbers, Internat J Uncertain Fuzziness
Knowledge-Based Systems 20(2) (2012), 303–314.
[6] H. Altinok, R. C¸ olak and M. Et, λ–Difference sequence
spaces of fuzzy numbers, Fuzzy Sets and Systems 160(21)
(2009), 3128–3139.
[7] M. Et, H. Altinok and R. C¸ olak, On λ–statistical conver-
gence of difference sequences of fuzzy numbers, Inform Sci
176(15) (2006), 2268–2278.
[8] H. Altinok, Y. Altin and M. Isik, Statistical Convergence of
order βfor Double Sequences of Fuzzy Numbers defined by
a Modulus Function, 6th International Eurasian Conference
on Mathematical Sciences and Applications (2017).
[9] S. Aytar and S. Pehlivan, Statistical convergence of
sequences of fuzzy numbers and sequences of α–cuts,
International Journal of General Systems 37(2) (2008),
231–237.
[10] V.K. Bhardwaj and S. Dhawan, f–statistical convergence
of order αand strong Cesaro summability of order αwith
respect to a modulus, J Inequal Appl 2015 (2015), 332 DOI
10.1186/s13660-015-0850-x
[11] S. Bhunia, P. Das and S.K. Pal, Restricting statistical con-
vergence, Acta Math Hungar 134 (2012), 1-2, 153–161.
AUTHOR COPY
6954 D. Barlak / Statistical convergence of order βfor (λ, μ)Double Sequences of Fuzzy Numbers
[12] R. C¸ olak, Statistical convergence of order αModern Meth-
ods in Analysis and Its Applications, New Delhi, India:
Anamaya Pub, (2010), 121–129.
[13] R. C¸ olak and Y. Altin, Statistical convergence of double
sequences of order ˜
α,J Funct Spaces Appl (2013), Art. ID
682823, pp. 5.
[14] R. C¸ olak and C.A. Bektas¸, λ–statistical convergenceof order
α,Acta Math Sci 31(3) (2011), 953–959.
[15] J.S. Connor, The statistical and strong p–Cesaro conver-
gence of sequences, Analysis 8(1988), 47–63.
[16] S.N. Deepmala and L.N. Mishra, The Triple χof Ideal Fuzzy
Real Numbers Over p–Metric Spaces Defined by Musielak
Orlicz Function, Southeast Asian Bulletin of Mathematics
40(6) (2016), 823–836.
[17] M. Et, M. C¸ ınar and M. Karakas¸, On λ–statistical conver-
gence of order αof sequences of functions, J Inequal Appl
2013, Article ID 204 (2013).
[18] H. Fast, Sur la convergence statistique, Colloquium Math 2
(1951), 241–244.
[19] J. Fridy, On statistical convergence, Analysis 5(1985),
301–313.
[20] A.D. Gadjiev and C. Orhan, Some approximation theorems
via statistical convergence, Rocky Mountain J Math 32(1)
(2002), 129–138.
[21] M. Isik and Y. Altin, fλ,μ -statistical convergence of order
¯
αfor double sequences, J Inequal Appl (2017), Paper No.
246, pp. 8.
[22] A. Karakas¸, Y. Altın and H. Altınok, On generalized statis-
tical convergence of order βof sequences of fuzzy numbers,
J Intell Fuzzy Systems 26(4) (2014), 1909–1917.
[23] J.S. Kwon, On statistical and p-Cesaro Convergence of
fuzzy numbers, Korean J Comput 8 Appl Math 7(1) (2000),
195–203.
[24] L. Leindler, ¨
Uber die de la Vall´
ee-Pousinsche Summier-
barkeit allgemeiner Orthogonalreihen, Acta Math Acad Sci
Hungar 16 (1965), 375–387.
[25] M. Matloka, Sequences of fuzzy numbers, BUSEFAL 28
(1986), 28–37.
[26] C. Meenakshi, V. Kumar and M.S. Saroa, Some remarks on
statistical sum-mability of order ¯
αdefined by generalized De
la Vall´
ee-Poussin mean, Bol Soc Parana Mat (3) 33 (2015),
no. 1, 147–156.
[27] V.N. Mishra and L.N. Mishra, Trigonometric Approxi-
mation of Signals (Functions) in Lp-norm, International
Journal of Contemporary Mathematical Sciences 7(19)
(2012), 909–918.
[28] V.N. Mishra, S.N. Deepmala and L.N. Mishra, The Gen-
eralized semi normed Difference of χ3Sequence Spaces
defined by Orlicz function, J Appl Computat Math 5 316
(2016).
[29] V.N. Mishra, K. Khatri and L.N. Mishra, Statistical approx-
imation by Kantorovich-type Discrete q–Beta operators,
Advances in Difference Equations 2013 (2013), 345.
[30] F. M´
oricz, Statistical convergence of multiple sequences,
Arch Math 81 (2003), 82–89.
[31] M. Mursaleen, λ–statistical convergence, Math Slovaca
50(1) (2000), 111–115.
[32] M.O. Mursaleen and H.H. Edely, Statistical convergence of
double sequences, J Math Anal Appl 288 (2003), 223–231.
[33] M. Mursaleen and S.A. Mohiuddine, Convergence Methods
for Double Sequences and Applications, Springer, (2014),
(ISBN 978-81-322-1610-0).
[34] S. Nanda, On sequences of fuzzy numbers, Fuzzy Sets Syst
33 (1989), 123–126.
[35] H. Nakano, Concave modulars, J Math Soc Japan 5(1953),
29–49.
[36] F. Nuray and E. Savas¸, Statistical convergence of sequences
of fuzzy real numbers, Math Slovaca 45(3) (1995), 269–273.
[37] D. Rai, N. Subramanian and V.N. Mishra, The General-
ized difference of fχ2/of fuzzy real numbers over p–metric
spaces defined by Musielak Orlicz function, New Trends in
Math 4(3) (2016), 296–306.
[38] T. ˇ
Sal´
at, On statistically convergent sequences of real num-
bers, Math Slovaca 30 (1980), 139–150.
[39] B. Sarma, On a class of sequences of fuzzy numbers defined
by modulus function, International Journal of Science &
Technology 2(1) (2007), 25–28.
[40] E. Savas¸, A note on sequence of fuzzy numbers, Inform Sci
124(1-4) (2000), 297–300.
[41] E. Savas¸, A note on double sequences of fuzzy numbers, Tr
J of Mathematics 20 (1996), 175–178.
[42] E. Savas, On strongly λ–summable sequences of fuzzy num-
bers, Inform Sci 125 (2000), 181–186.
[43] E. Savas¸ and M. Mursaleen, On statistically convergent dou-
ble sequences of fuzzy numbers, Inform Sci 162 (2004),
183–192.
[44] I.J. Schoenberg, The integrability of certain functions and
related summability methods, Amer Math Monthly 66
(1959), 361–375.
[45] ¨
O. Talo and F. Bas¸ar, Certain spaces of sequences of fuzzy
numbers defined by a modulus function, Demonstratio Math
43(1) (2010), 139–149.
[46] B.C. Tripathy and A. Baruah, Lacunary statistically con-
vergent and lacunary strongly convergent generalized
difference sequences of fuzzy real numbers, Kyungpook
Math Jour 50 (2010), 565–574.
[47] B.C. Tripathy and B. Sarma, Statistically convergent differ-
ence double sequence spaces, Acta Math Sin (Engl. Ser.)
24(5) (2008), 737–742.
[48] B.C. Tripathy, Statistically convergent double sequences,
Tamkang J Math 34(3) (2003), 231–237.
[49] G. Wang and X. Xi, Convergence of sequences on the fuzzy
real line, Fuzzy Sets and Systems 127(3) (2002), 323–331.
[50] D. Vandana, N. Subramanian and V.N. Mishra, Riesz Triple
Probabilisitic of Almost Lacunary Cesaro C111 Statistical
Convergence of χ3Defined by Musielak Orlicz Function,
Boletim da Sociedade Paranaense de Matematica 36(4)
(2018), 23–32.
[51] L.A. Zadeh, Fuzzy sets, Inform and Control 8(1965),
338–353.