- Research Article
- Open Access

# On Double Statistical -Convergence of Fuzzy Numbers

- Ekrem Savas
^{1}Email author and - Richard F. Patterson
^{2}

**2009**: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.

## Keywords

- Fuzzy Setting
- Fuzzy Number
- Linear Structure
- Compact Convex
- Statistical Convergence

## 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.

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 ;

(3) is upper semicontinuous;

(4)the closure of ; denoted by , is compact.

for each .

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.

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.

Let denote the set of all double Cauchy sequences of fuzzy numbers.

Definition 2.3.

We will denote the set of all bounded double sequences by .

(i.e., the set has double natural density zero).

Definition 2.4.

*statistically convergent*to provided that for each ,

In this case we write and we denote the set of all double statistically convergent sequences of fuzzy numbers by .

Definition 2.5.

That is, , a.a.( ).

Let denote the set of all double Cauchy sequences of fuzzy numbers.

Definition 2.6.

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.

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.

We now show that . Let be given, pick be given, and pick and such that , thus for , since if and if we have .

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

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

## 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.