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On Asymptotically
-Statistical Equivalent Sequences of Fuzzy Numbers
Journal of Inequalities and Applications volume 2010, Article number: 838741 (2010)
Abstract
The goal of this paper is to give the asymptotically -statistical equivalent which is a natural combination of the definition for asymptotically equivalent, invariant mean and
-statistical convergence of fuzzy numbers.
1. Introduction
The concepts of fuzzy sets and fuzzy set operation were first introduced by Zadeh [1] and subsequently several authors have discussed various aspects of the theory and applications of fuzzy sets such as fuzzy topological spaces similarity relations and fuzzy orderings, fuzzy measures of fuzzy events and fuzzy mathematical programming. Matloka [2] introduced bounded and convergent sequences of fuzzy numbers and studied their some properties. For sequences of fuzzy numbers, Nuray and Savaş [3] introduced and discussed the concepts of statistically convergent and statistically Cauchy sequences.
Quite recently, Savaş [4] introduced the idea of asymptotically -statistically equivalent sequences of fuzzy numbers. In this paper we extend his result by using invariant means.
2. Preliminaries
Before we enter the motivation for this paper and presentation of the main results we give some preliminaries.
By and
, we denote the Banach spaces of bounded and convergent sequences
normed by
, respectively. A linear functional
on
is said to be a Banach limit (see [5]) if it has the following properties:
(1) if
for all
;
(2) where
;
(3), where the shift operator
is defined by
.
Let be the set of all Banach limits on
. A sequence
is said to be almost convergent if all Banach limits of
coincide. Let
denote the space of almost convergent sequences.
Let be a one-to-one mapping from the set of natural numbers into itself. A continuous linear functional
on
is said to be an invariant mean or a
-mean if and only if
(1) when the sequence
is such that
for all
,
(2) where
, and
(3) for all
.
Throughout this paper we shall consider the mapping has having on finite orbits, that is,
for all nonnegative integers with
, where
is the
th iterate of
at
. Thus
-mean extends the limit functional on
in the sense that
for all
. Consequently,
where
is the set of bounded sequences all of whose
-mean are equal.
In the case when , the
-mean is often called the Banach limit and
is the set of almost convergent sequences.
A fuzzy real number is a fuzzy set on
, that is, a mapping
associating each real number
with its grade of membership
.
The cut of fuzzy real number
is denoted by
, where
. If
then it is the closure of the strong
. A fuzzy real number
is said to be upper semicontinuous if for each
,
, for all
is open in the usual topology of
. If there exists
such that
, then the fuzzy real number
is called normal.
A fuzzy number is said to be convex, if
where
. The class of all upper semi-continuous, normal, convex fuzzy real numbers is denoted by
and throughout the article, by a fuzzy real number we mean that the number belongs to
. Let
and the
level sets be
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F838741/MediaObjects/13660_2009_Article_2270_Equ1_HTML.gif)
Then the arithmetic operations on are defined as follows:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F838741/MediaObjects/13660_2009_Article_2270_Equ2_HTML.gif)
The above operations can be defined in terms of level sets as follows:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F838741/MediaObjects/13660_2009_Article_2270_Equ3_HTML.gif)
The additive identity and multiplicative identity in are denoted by
and
, respectively.
Let the set of all closed and bounded intervals
Then we write
, if and only if
and
, and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F838741/MediaObjects/13660_2009_Article_2270_Equ4_HTML.gif)
It is obvious that is a complete metric space. Now we define the metric
by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F838741/MediaObjects/13660_2009_Article_2270_Equ5_HTML.gif)
for
We now give the following definitions (see [6]) for fuzzy real-valued sequences.
Definition 2.1.
A fuzzy real-valued sequence is a function
from the set
of natural numbers into
). The fuzzy real-valued sequence
denotes the value of the function at
and is called the
th term of the sequence. We denote by
the set of all fuzzy real-valued sequences
.
Definition 2.2.
A fuzzy real-valued sequence is said to be convergent to a fuzzy number
written as
, if for every
there exists a positive integer
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F838741/MediaObjects/13660_2009_Article_2270_Equ6_HTML.gif)
Let denote the set of all convergent sequences of fuzzy numbers.
Definition 2.3.
A sequence of fuzzy numbers is said to be bounded if the set
of fuzzy numbers is bounded. We denote by
the set of all bounded sequences of fuzzy numbers.
It is easy to see that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F838741/MediaObjects/13660_2009_Article_2270_Equ7_HTML.gif)
It was shown that and
are complete metric spaces (see [7]).
3. Definitions and Notations
Definition 3.1.
Two fuzzy real-valued sequences and
are said to be asymptotically equivalent if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F838741/MediaObjects/13660_2009_Article_2270_Equ8_HTML.gif)
(denoted by ).
Let be a nondecreasing sequence of positive reals tending to infinity and
and
.
In [4], Savaş introduced the concept of -statistical convergence of fuzzy numbers as follows.
Definition 3.2.
A fuzzy real-valued sequences is said to be
-statistically convergent or
-convergent to L if for every
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F838741/MediaObjects/13660_2009_Article_2270_Equ9_HTML.gif)
In this case we write or
, and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F838741/MediaObjects/13660_2009_Article_2270_Equ10_HTML.gif)
The next definition is natural combination of Definitions 3.1 and 3.2., which was defined in [4].
Definition 3.3.
Two fuzzy real-valued sequences and
are said to be asymptotically
-statistical equivalent of multiple
provided that for every
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F838741/MediaObjects/13660_2009_Article_2270_Equ11_HTML.gif)
(denoted by ) and simply asymptotically
-statistical equivalent if
.
If we take , the above definition reduces to the following definition.
Definition 3.4.
Two fuzzy real-valued sequences and
are said to be asymptotically statistical equivalent of multiple
provided that for every
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F838741/MediaObjects/13660_2009_Article_2270_Equ12_HTML.gif)
(denoted by ) and simply asymptotically statistical equivalent if
.
It is quite naturel to expect the following definition.
Definition 3.5.
Fuzzy real-valued sequences is said to be
-statistically convergent to
provided that for every
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F838741/MediaObjects/13660_2009_Article_2270_Equ13_HTML.gif)
uniformly in .
In this case we write or
, and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F838741/MediaObjects/13660_2009_Article_2270_Equ14_HTML.gif)
Following this result we introduce two new notions asymptotically -statistical equivalent of multiple
and strong
-asymptotically equivalent of multiple
.
Definition 3.6.
Two fuzzy real-valued sequences and
are said to be asymptotically
-statistical equivalent of multiple
provided that for every
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F838741/MediaObjects/13660_2009_Article_2270_Equ15_HTML.gif)
uniformly in , (denoted by
) and simply asymptotically
-statistical equivalent if
.
In case , the above definition reduces to the following definition.
Definition 3.7.
Two fuzzy real-valued sequences and
are said to be asymptotically
-statistical equivalent of multiple
provided that for every
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F838741/MediaObjects/13660_2009_Article_2270_Equ16_HTML.gif)
uniformly in , (denoted by
) and simply asymptotically
-statistical equivalent if
.
We now define the following.
Definition 3.8.
Let be a sequence of positive real numbers; two fuzzy real-valued sequences
and
are strongly asymptotically
-equivalent of multiple
, provided that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F838741/MediaObjects/13660_2009_Article_2270_Equ17_HTML.gif)
(denoted by ) and simply strongly asymptotically
-equivalent if
.
If we take for all
we write
instead of (
).
In case in above definition we get following.
Definition 3.9.
Let be a sequence of positive numbers and let us consider two fuzzy real-valued sequences
and
. Two fuzzy real-valued sequences
and
are said to be strongly asymptotically Cesáro equivalent of multiple
provided that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F838741/MediaObjects/13660_2009_Article_2270_Equ18_HTML.gif)
(denoted by ), and simply strong Cesáro asymptotically equivalent if
.
4. Main Results
Theorem 4.1.
Let . Then
-
(1)
if
then
;
-
(2)
if
and
and
then
(3)
Proof.
Part (1): if and
then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F838741/MediaObjects/13660_2009_Article_2270_Equ19_HTML.gif)
Therefore . Part (2): suppose that fuzzy real-valued sequences
and
are in
and
. Then we can assume that
. Let
be given and
be such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F838741/MediaObjects/13660_2009_Article_2270_Equ20_HTML.gif)
for all and let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F838741/MediaObjects/13660_2009_Article_2270_Equ21_HTML.gif)
Now for all we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F838741/MediaObjects/13660_2009_Article_2270_Equ22_HTML.gif)
Hence . This completes the proof.Part (3): this immediately follows from (1) and (2).
In the next theorem we prove the following relation.
Theorem 4.2.
Let . Then
implies
Proof.
Let and
be given. Then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F838741/MediaObjects/13660_2009_Article_2270_Equ23_HTML.gif)
Hence .
Theorem 4.3.
Let fuzzy real-valued sequences and
be bounded and
. Then
implies
.
Proof.
Suppose that fuzzy real-valued sequences and
be bounded and
is given. Since
and
are bounded there exists an integer
such that
for all
and
; then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F838741/MediaObjects/13660_2009_Article_2270_Equ24_HTML.gif)
Hence .
Remark 4.4.
If we take in our results, all results reduce to the results of almost convergence which have not proved so far.
References
Zadeh LA: Fuzzy sets. Information and Computation 1965, 8: 338–353.
Matloka M: Sequences of fuzzy numbers. BUSEFAL 1986, 28: 28–37.
Nuray F, Savaş E: Statistical convergence of sequences of fuzzy numbers. Mathematica Slovaca 1995, 45(3):269–273.
Savaş E: On asymptotically -statistical equivalent sequences of fuzzy numbers. New Mathematics & Natural Computation 2007, 3(3):301–306. 10.1142/S1793005707000781
Banach S: Theorie des Operations Linearies. Subwncji Funduszu Narodowej, Warszawa, Poland; 1932.
Savas E: On asymptotically lacunary -statistical equivalent sequences of Fuzzy numbers,. New Mathematics & Natural Computation 2009, 5(3):1–10.
Nanda S: On sequences of fuzzy numbers. Fuzzy Sets and Systems 1989, 33(1):123–126. 10.1016/0165-0114(89)90222-4
Acknowledgment
This work was supported by Grant (2008-FED-B162) of Yüzüncü Yil university.
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Savaş, E., Şevli, H. & Cancan, M. On Asymptotically -Statistical Equivalent Sequences of Fuzzy Numbers.
J Inequal Appl 2010, 838741 (2010). https://doi.org/10.1155/2010/838741
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DOI: https://doi.org/10.1155/2010/838741