1. Introduction
In 1965, Zadeh [1], an expert in cybernetics at University of California, first proposed the concept of fuzzy set theory. Since its inception, fuzzy set theory and its applications have been attracting the attention of researchers from various areas of science, engineering and technology. In daily life, the practical problems we have to solve often involve uncertainty, which can be expressed by fuzzy number [2]. Therefore, in the following research work, the convergence problem of sequences of fuzzy numbers is particularly important. The concept of statistical convergence of fuzzy sequence is defined by Savas [3], at the same time, statistical convergence of sequences of fuzzy numbers is expressed by the sequences of fuzzy numbers with zero natural density and the general convergent sequences of fuzzy numbers. In 1986, Matloka [4] introduced the concepts of bounded and convergent sequences of fuzzy numbers and studied their properties. In 1989, Nanda [5] studied the bounded and convergent spaces of fuzzy numbers and established that they are complete metric spaces. In 1995, Naray and Savas [6] extended the concept of statistical convergence to sequences of fuzzy numbers and showed that a sequence of fuzzy numbers is statistically convergent if and only if it is statistically Cauchy. In recent years, the problem of statistical convergence of sequences of fuzzy numbers has been studied extensively by Talo [7], Balen [8], Cinar [9] and Dutta [10], some interesting results related to statistical convergence of sequences of fuzzy numbers and related notions can also be found.
In this paper, we give the concept of weighted
-ideal statistical convergence and strongly weighted
-ideal convergence of double sequences of fuzzy numbers. And we have examined relevant inclusion relations concerning different types of weight ideal statistical convergence and strongly weight ideal convergence of double sequences of fuzzy numbers.
2. Definitions and Preliminaries
In this section, we give some basic notions which will be used throughout the paper.
Let
be a fuzzy subset on R. If
is convex, normal, upper semi-continuous and has compact support, we say that
is a fuzzy number [11] [12] [13]. Let
denote the set of all fuzzy numbers.
For
, we write the level set of
as
and
. Let
, we define
iff
,
iff
and
for any
.
, where
(2-1)
(2-2)
Define
(2-3)
where d is the Hausdorff metric.
is called the distance between
and
.
Using the results of [11] [12] [13], we see that
1)
is a complete metric space,
2)
,
3)
,
4)
,
5)
,
6)
,
Where
,
represents zero fuzzy number.
Let X is a nonempty set,
is said to be ideal on X [14] [15], if:
1)
;
2) if
, then
;
3) For
, if
, then
.
Especially, if
and
, then I is said to be a nontrivial ideal on X.
A sequence
of fuzzy numbers is said to be statistically convergent to a fuzzy number
if for each
the set
has natural density zero. The fuzzy number
is called the statistical limit of the sequence
and we write
. A sequence
of fuzzy numbers is said to be ideal statistically convergent to a fuzzy number
if for each
the set
, where I ia a nontrivial ideal on X [16] [17].
A double sequence of fuzzy numbers
is said to be bounded if there exists a positive number M such that
for all
, i.e. if
, where
[14].
Let
and
. The number
is called the double natural density of K, provided the limit exists [18] [19].
A double sequence of fuzzy numbers
is said to be statistically convergent to
if for every
,
, where
, i.e.,
.
In this case, we write,
. The set of all double statistically convergent sequences of fuzzy numbers is denoted by
[20] [21].
3. Main Results
Definition 3.1. Let
and
be sequences of nonnegative numbers such that
,
,
and
,
,
with
, as
.
, as
.
The weighted mean
is defined as
where
and
.
Definition 3.2. A double sequence of fuzzy numbers
is weighted ideal statistically convergent to
if for every
,
, we have
In this case, we write,
.
Where let
. We define the double weighted density of K by
where
,
,
.
Definition 3.3. A double sequence of fuzzy numbers
is strongly weight ideal convergent to
if
and we write
.
Definition 3.4. Let
and
be two nondecreasing sequence of positive real numbers such that each tending to
and
,
;
,
.
Let
and
be two sequence of nonnegative real numbers such that
,
,
and
,
,
with
, as
.
, as
.
where
,
.
We define generalized weighted mean as follows:
Definition 3.5. A double sequence of fuzzy numbers
is said to be weighted
-ideal statistically convergent to
if for every
,
, we have
In this case, we write
. We denote the set of all weight
-ideal statistically convergent double sequences of fuzzy numbers by
.
Definition 3.6. A double sequence of fuzzy numbers
is said to be strongly weight
-ideal convergent to
if
In this case, we write
.
Remark 3.7. When we take
for all
, weighted
-ideal statistically convergence reduces to weighted ideal statistically convergence; strongly weight
-ideal convergence reduces to strongly weight ideal convergence.
Remark 3.8. When we take
for all
and
for all
, weighted
-ideal statistically convergence reduces to ideal statistically convergence; strongly weight
-ideal convergence reduces to strongly ideal convergence.
Theorem 3.9. Let
are the sequence of fuzzy numbers:
1) If
and
, then
;
2) If
then
.
Proof. 1) When
, the conclusion is clearly established.
Let
, we have
So
We have
.
2) Let
, then
On the other hand,
for
, we have
So
We can get
.
In case
for all
,
-ideal statistical convergence reduces to
-ideal statistical convergence and then we have the following corollary.
Corollary 3.10. Let
are the sequence of fuzzy numbers:
1) If
and
, then
;
2) If
then
.
Theorem 3.11. Let
is the sequence of fuzzy number, there is a
-ideal statistically convergent sequence of fuzzy number
, such that
for almost all
, then
also
-ideal statistical convergence.
Proof. For almost all
, we have
, and
. Let
,
, then
Let
is the number of elements in the set of
, then
So
.
The theorem proved.
In case
for all
,
-ideal statistical convergence reduces to
-ideal statistical convergence and then we have the following corollary.
Corollary 3.12. Let
is the sequence of fuzzy number, there is a
-ideal statistically convergent sequence of fuzzy number
, such that
for almost all
, then
also
-ideal statistical convergence.
Theorem 3.13. Let
for all
. If a double sequence of fuzzy numbers
is weight
-ideal statistically convergent to
then it is strongly weight
-ideal convergent to
.
Proof. Suppose
for all
and the double sequence of fuzzy numbers
is weight
-ideal statistically convergent to
. We note
which implies that
i.e.
.
Theorem 3.14. Let a double sequence of fuzzy numbers
is strongly weighted
-ideal convergent to
, then
is weighted
-ideal statistically convergent to
.
Proof. Let
, then
where
.
We have
We get
is weighted
-ideal statistically convergent to
.
4. Conclusion
In this article, we aim to investigate different types of weighted ideal statistical convergence and strongly weighted ideal convergence of double sequences of fuzzy numbers. Relations connecting ideal statistical convergence and strongly ideal convergence have been investigated in the environment of the newly defined classes of double sequences of fuzzy numbers. At the same time, we have examined relevant inclusion relations concerning weighted
-ideal statistical convergence and strongly weighted
-ideal convergence of double sequences of fuzzy numbers.
Supported
This work is supported by National Natural Science Fund of China (11761056); the Natural Science Foundation of Qinghai Province (2020-ZJ-920).