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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 onetoone 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 semicontinuous, 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
Then the arithmetic operations on are defined as follows:
The above operations can be defined in terms of level sets as follows:
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
It is obvious that is a complete metric space. Now we define the metric by
for
We now give the following definitions (see [6]) for fuzzy realvalued sequences.
Definition 2.1.
A fuzzy realvalued sequence is a function from the set of natural numbers into ). The fuzzy realvalued 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 realvalued sequences .
Definition 2.2.
A fuzzy realvalued sequence is said to be convergent to a fuzzy number written as , if for every there exists a positive integer such that
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
It was shown that and are complete metric spaces (see [7]).
3. Definitions and Notations
Definition 3.1.
Two fuzzy realvalued sequences and are said to be asymptotically equivalent if
(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 realvalued sequences is said to be statistically convergent or convergent to L if for every ,
In this case we write or , and
The next definition is natural combination of Definitions 3.1 and 3.2., which was defined in [4].
Definition 3.3.
Two fuzzy realvalued sequences and are said to be asymptotically statistical equivalent of multiple provided that for every
(denoted by ) and simply asymptotically statistical equivalent if .
If we take , the above definition reduces to the following definition.
Definition 3.4.
Two fuzzy realvalued sequences and are said to be asymptotically statistical equivalent of multiple provided that for every ,
(denoted by ) and simply asymptotically statistical equivalent if .
It is quite naturel to expect the following definition.
Definition 3.5.
Fuzzy realvalued sequences is said to be statistically convergent to provided that for every
uniformly in .
In this case we write or , and
Following this result we introduce two new notions asymptotically statistical equivalent of multiple and strong asymptotically equivalent of multiple .
Definition 3.6.
Two fuzzy realvalued sequences and are said to be asymptotically statistical equivalent of multiple provided that for every
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 realvalued sequences and are said to be asymptotically statistical equivalent of multiple provided that for every
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 realvalued sequences and are strongly asymptotically equivalent of multiple , provided that
(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 realvalued sequences and . Two fuzzy realvalued sequences and are said to be strongly asymptotically Cesáro equivalent of multiple provided that
(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
Therefore . Part (2): suppose that fuzzy realvalued sequences and are in and . Then we can assume that . Let be given and be such that
for all and let
Now for all we have
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
Hence .
Theorem 4.3.
Let fuzzy realvalued sequences and be bounded and . Then implies .
Proof.
Suppose that fuzzy realvalued sequences and be bounded and is given. Since and are bounded there exists an integer such that for all and ; then
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
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Acknowledgment
This work was supported by Grant (2008FEDB162) 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