Research Article
Open Access
On Double Statistical
-Convergence of Fuzzy Numbers
- Ekrem Savas1Email author and
- Richard F. Patterson2
Journal of Inequalities and Applications20092009:423792
https://doi.org/10.1155/2009/423792
© E. Savas and R. F. Patterson 2009
Received: 1 September 2009
Accepted: 2 October 2009
Published: 15 October 2009
Abstract
Savas and Mursaleen defined the notions of statistically convergent and statistically Cauchy for double sequences of fuzzy numbers. In this paper, we continue the study of statistical convergence by proving some theorems.
1. Introduction
For sequences of fuzzy numbers, Nanda [1] studied the sequences of fuzzy numbers and showed that the set of all convergent sequences of fuzzy numbers forms a complete metric space. Kwon [2] introduced the definition of strongly p-Cesàro summability of sequences of fuzzy numbers. Savas [3] introduced and discussed double convergent sequences of fuzzy numbers and showed that the set of all double convergent sequences of fuzzy numbers is complete. Savas [4] studied some equivalent alternative conditions for a sequence of fuzzy numbers to be statistically Cauchy and he continue to study in [5, 6]. Recently Mursaleen and Basarir [7] introduced and studied some new sequence spaces of fuzzy numbers generated by nonnegative regular matrix. Quite recently, Savas and Mursaleen [8] defined statistically convergent and statistically Cauchy for double sequences of fuzzy numbers. In this paper, we continue the study of double statistical convergence and introduce the definition of double strongly
-Cesàro summabilty of sequences of fuzzy numbers.
2. Definitions and Preliminary Results
Since the theory of fuzzy numbers has been widely studied, it is impossible to find either a commonly accepted definition or a fixed notation. We therefore being by introducing some notations and definitions which will be used throughout.
Let
. The spaces
has a linear structure induced by the operations
. The spaces
has a linear structure induced by the operations
(2.1)
for
and
. The Hausdorff distance between
and
of
is defined as
and
. The Hausdorff distance between
and
of
is defined as
(2.2)
It is well known that
is a complete (not separable) metric space.
A fuzzy number is a function
from
to
satisfying
(1)
is normal, that is, there exists an
such that
;
(2)
is fuzzy convex, that is, for any
and
,
is fuzzy convex, that is, for any
and
,
(2.3)
(3)
is upper semicontinuous;
(4)the closure of
; denoted by
, is compact.
These properties imply that for each
, the
-level set
, the
-level set
(2.4)
is a nonempty compact convex, subset of
, as is the support
. Let
denote the set of all fuzzy numbers. The linear structure of
induces the addition
and scalar multiplication
,
, in terms of
-level sets, by
, as is the support
. Let
denote the set of all fuzzy numbers. The linear structure of
induces the addition
and scalar multiplication
,
, in terms of
-level sets, by
(2.5)
for each
.
Define for each
,
,
(2.6)
and
clearly
with
if
. Moreover
is a complete, separable, and locally compact metric space [9].
Throughout this paper,
will denote
with
. We will need the following definitions (see [8]).
Definition 2.1.
A double sequence
of fuzzy numbers is said to be convergent in Pringsheim's sense or
-convergent to a fuzzy number
, if for every
there exists
such that
of fuzzy numbers is said to be convergent in Pringsheim's sense or
-convergent to a fuzzy number
, if for every
there exists
such that
(2.7)
and we denote by
. The number
is called the Pringsheim limit of
.
More exactly we say that a double sequence
converges to a finite number
if
tend to
as both
and
tend to
independently of one another.
Let
denote the set of all double convergent sequences of fuzzy numbers.
Definition 2.2.
A double sequence
of fuzzy numbers is said to be
-Cauchy sequence if for every
there exists
such that
of fuzzy numbers is said to be
-Cauchy sequence if for every
there exists
such that
(2.8)
Let
denote the set of all double Cauchy sequences of fuzzy numbers.
Definition 2.3.
A double sequence
of fuzzy numbers is bounded if there exists a positive number
such that
for all
and
,
of fuzzy numbers is bounded if there exists a positive number
such that
for all
and
,
(2.9)
We will denote the set of all bounded double sequences by
.
Let
be a two-dimensional set of positive integers and let
be the numbers of
in
such that
and
. Then the lower asymptotic density of
is defined as
be a two-dimensional set of positive integers and let
be the numbers of
in
such that
and
. Then the lower asymptotic density of
is defined as
(2.10)
In the case when the sequence
has a limit, then we say that
has a natural density and is defined as
has a limit, then we say that
has a natural density and is defined as
(2.11)
For example, let
, where
is the set of natural numbers. Then
, where
is the set of natural numbers. Then
(2.12)
(i.e., the set
has double natural density zero).
Definition 2.4.
A double sequence
of fuzzy numbers is said to be statistically convergent to
provided that for each
,
of fuzzy numbers is said to be statistically convergent to
provided that for each
,
(2.13)
In this case we write
and we denote the set of all double statistically convergent sequences of fuzzy numbers by
.
Definition 2.5.
A double sequence
of fuzzy numbers is said to be statistically
-Cauchy if for every
there exist
and
such that
of fuzzy numbers is said to be statistically
-Cauchy if for every
there exist
and
such that
(2.14)
That is,
, a.a.(
).
Let
denote the set of all double Cauchy sequences of fuzzy numbers.
Definition 2.6.
A double sequence
of fuzzy and let
be a positive real numbers. The sequence
is said to be strongly double
-Cesaro summable if there is a fuzzy number
such that
of fuzzy and let
be a positive real numbers. The sequence
is said to be strongly double
-Cesaro summable if there is a fuzzy number
such that
(2.15)
In which case we say that
is strongly
-Cesaro summable to
.
It is quite clear that if a sequence
is statistically
-convergent, then it is a statistically
-Cauchy sequence [8]. It is also easy to see that if there is a convergent sequence
such that
a.a.(
), then
is statistically convergent.
3. Main Result
Theorem 3.1.
A double sequence
of fuzzy numbers is statistically
-Cauchy then there is a
-convergent double sequence
such that
a.a.(
).
Proof.
Let us begin with the assumption that
is statistically
-Cauchy this grant us a closed ball
that contains
a.a.(
) for some positive numbers
and
. Clearly we can choose
and
such that
contains
a.a.(
). It is also clear that
a.a.(
); for
is statistically
-Cauchy this grant us a closed ball
that contains
a.a.(
) for some positive numbers
and
. Clearly we can choose
and
such that
contains
a.a.(
). It is also clear that
a.a.(
); for
(3.1)
we have
(3.2)
Thus
is a closed ball of diameter less than or equal to 1 that contains
a.a.(
). Now we let us consider the second stage to this end we choose
and
such that
In a manner similar to the first stage we have
, a.a.(
). Note the diameter of
is less than or equal
. If we now consider the (
)th general stage we obtain the following. First a sequence
of closed balls such that for each (
),
, the diameter of
is not greater than
with
, a.a.(
). By the nested closed set theorem of a complete metric space we have
. So there exists a fuzzy number
. Using the fact that
, a.a.(
), we can choose an increasing sequence
of positive integers such that
is a closed ball of diameter less than or equal to 1 that contains
a.a.(
). Now we let us consider the second stage to this end we choose
and
such that
In a manner similar to the first stage we have
, a.a.(
). Note the diameter of
is less than or equal
. If we now consider the (
)th general stage we obtain the following. First a sequence
of closed balls such that for each (
),
, the diameter of
is not greater than
with
, a.a.(
). By the nested closed set theorem of a complete metric space we have
. So there exists a fuzzy number
. Using the fact that
, a.a.(
), we can choose an increasing sequence
of positive integers such that
(3.3)
Now define a double subsequence
of
consisting of all terms
such that
and if
of
consisting of all terms
such that
and if
(3.4)
Next we define the sequence (
) by
) by
(3.5)
Then
indeed if
and
then either
is a term of
. Which means
or
and
diameter of
. We will now show that
a.a.(
). Note that if
then
indeed if
and
then either
is a term of
. Which means
or
and
diameter of
. We will now show that
a.a.(
). Note that if
then
(3.6)
and by (3.3)
(3.7)
Hence the limit as
is 0 and
a.a.(
). This completes the proof.
Theorem 3.2.
If
is strongly
-Cesaro summable or statistically
-convergent to
then there is a
-convergent double sequences
and a statistically
-null sequence
such that
and
.
Proof.
Note that if
is strongly
-Cesaro summable to
then
is statistically
-convergent to
. Let
and
and select two increasing index sequences of positive integers
and
such that
and
, we have
is strongly
-Cesaro summable to
then
is statistically
-convergent to
. Let
and
and select two increasing index sequences of positive integers
and
such that
and
, we have
(3.8)
Define
and
as follows: if
and
, set
and
. Suppose that
and
,
then
and
as follows: if
and
, set
and
. Suppose that
and
,
then
(3.9)
We now show that
. Let
be given, pick
be given, and pick
and
such that
, thus for
, since
if
and
if
we have
.
Next we show that
is a statistically
-null double sequence, that is, we need to show that
. Let
if
such that
then
for all
. From the construction of
, if
and
then
only if
. It follows that if
and
then
is a statistically
-null double sequence, that is, we need to show that
. Let
if
such that
then
for all
. From the construction of
, if
and
then
only if
. It follows that if
and
then
(3.10)
Thus for
and
and
then
and
and
then
(3.11)
this completes the proof.
Corollary 3.3.
If
is a strongly
-Cesaro summable to
or statistically
-convergent to
then
has a double subsequence which is
-converges to
.
4. Conclusion
In recent years the statistical convergence has been adapted to the sequences of fuzzy numbers. Double statistical convergence of sequences of fuzzy numbers was first deduced in similar form by Savas and Mursaleen as we explain now: a double sequences
is said to be
-statistically convergent to
provided that for each
,
is said to be
-statistically convergent to
provided that for each
,
(4.1)
Since the set of real numbers can be embedded in the set of fuzzy numbers, statistical convergence in reals can be considered as a special case of those fuzzy numbers. However, since the set of fuzzy numbers is partially ordered and does not carry a group structure, most of the results known for the sequences of real numbers may not be valid in fuzzy setting. Therefore, this theory should not be considered as a trivial extension of what has been known in real case. In this paper, we continue the study of double statistical convergence and also some important theorems are proved.
Authors’ Affiliations
(1)
Department of Mathematics, Istanbul Commerce University
(2)
Department of Mathematics and Statistics, University of North Florida
References
- Nanda S: On sequences of fuzzy numbers. Fuzzy Sets and Systems 1989,33(1):123–126. 10.1016/0165-0114(89)90222-4MathSciNetView ArticleMATHGoogle Scholar
- Kwon J-S: On statistical and -Cesaro convergence of fuzzy numbers. Journal of Applied Mathematics and Computing 2000,7(1):195–203.MathSciNetMATHGoogle Scholar
- Savas E: A note on double sequences of fuzzy numbers. Turkish Journal of Mathematics 1996,20(2):175–178.MathSciNetMATHGoogle Scholar
- Savas E: On statistically convergent sequences of fuzzy numbers. Information Sciences 2001,137(1–4):277–282.MathSciNetView ArticleMATHGoogle Scholar
- Savas E: A note on sequence of fuzzy numbers. Information Sciences 2000,124(1–4):297–300.MathSciNetView ArticleMATHGoogle Scholar
- Savas E: On lacunary statistically convergent double sequences of fuzzy numbers. Applied Mathematics Letters 2008,21(2):134–141. 10.1016/j.aml.2007.01.008MathSciNetView ArticleMATHGoogle Scholar
- Mursaleen M, Basarir M: On some new sequence spaces of fuzzy numbers. Indian Journal of Pure and Applied Mathematics 2003,34(9):1351–1357.MathSciNetMATHGoogle Scholar
- Savas E, Mursaleen : On statistically convergent double sequences of fuzzy numbers. Information Sciences 2004,162(3–4):183–192. 10.1016/j.ins.2003.09.005MathSciNetView ArticleMATHGoogle Scholar
- Diamond P, Kloeden P: Metric spaces of fuzzy sets. Fuzzy Sets and Systems 1990,35(2):241–249. 10.1016/0165-0114(90)90197-EMathSciNetView ArticleMATHGoogle Scholar
Copyright
© E. Savas and R. F. Patterson 2009
This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
