Open Access

On Asymptotically -Statistical Equivalent Sequences of Fuzzy Numbers

Journal of Inequalities and Applications20102010:838741

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

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.

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
(2.1)
Then the arithmetic operations on are defined as follows:
(2.2)
The above operations can be defined in terms of level sets as follows:
(2.3)

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
(2.4)
It is obvious that is a complete metric space. Now we define the metric by
(2.5)

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
(2.6)

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
(2.7)

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
(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.

A fuzzy real-valued sequences is said to be -statistically convergent or -convergent to L if for every ,
(3.2)
In this case we write or , and
(3.3)

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
(3.4)

(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 ,
(3.5)

(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
(3.6)

uniformly in .

In this case we write or , and
(3.7)

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
(3.8)

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
(3.9)

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
(3.10)

(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
(3.11)

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

4. Main Results

Theorem 4.1.

Let . Then
  1. (1)

    if then ;

     
  2. (2)

    if and and then

     

(3)

Proof.

Part (1): if and then
(4.1)
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
(4.2)
for all and let
(4.3)
Now for all we have
(4.4)

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
(4.5)

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
(4.6)

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

(1)
Department of Mathematics, Istanbul Commerce University
(2)
Department of Mathematics, Yüzüncü Yil University

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Copyright

© Ekrem Savaş et al. 2010

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.