- Research Article
- Open Access

# On Asymptotically -Statistical Equivalent Sequences of Fuzzy Numbers

- Ekrem Savaş
^{1}Email author, - H Şevli
^{1}and - M Cancan
^{2}

**2010**:838741

https://doi.org/10.1155/2010/838741

© Ekrem Savaş et al. 2010

**Received:**30 December 2009**Accepted:**13 April 2010**Published:**20 May 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.

## Keywords

- Topological Space
- Fuzzy Number
- Positive Real Number
- Arithmetic Operation
- Statistical Equivalent

## 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.

The additive identity and multiplicative identity in are denoted by and , respectively.

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.

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 was shown that and are complete metric spaces (see [7]).

## 3. Definitions and Notations

Definition 3.1.

(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.

*L*if for every ,

The next definition is natural combination of Definitions 3.1 and 3.2., which was defined in [4].

Definition 3.3.

(denoted by ) and simply asymptotically -statistical equivalent if .

If we take , the above definition reduces to the following definition.

Definition 3.4.

(denoted by ) and simply asymptotically statistical equivalent if .

It is quite naturel to expect the following definition.

Definition 3.5.

uniformly in .

Following this result we introduce two new notions asymptotically -statistical equivalent of multiple and strong -asymptotically equivalent of multiple .

Definition 3.6.

uniformly in , (denoted by ) and simply asymptotically -statistical equivalent if .

In case , the above definition reduces to the following definition.

Definition 3.7.

uniformly in , (denoted by ) and simply asymptotically -statistical equivalent if .

We now define the following.

Definition 3.8.

(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.

(denoted by ), and simply strong Cesáro asymptotically equivalent if .

## 4. Main Results

Theorem 4.1.

- (1)
if then ;

- (2)
if and and then

(3)

Proof.

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.

Hence .

Theorem 4.3.

Let fuzzy real-valued sequences and be bounded and . Then implies .

Proof.

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.

## Declarations

### Acknowledgment

This work was supported by Grant (2008-FED-B162) of Yüzüncü Yil university.

## Authors’ Affiliations

## References

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## Copyright

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.