© Ekrem Savaş et al. 2010
Received: 30 December 2009
Accepted: 13 April 2010
Published: 20 May 2010
The concepts of fuzzy sets and fuzzy set operation were first introduced by Zadeh  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  introduced bounded and convergent sequences of fuzzy numbers and studied their some properties. For sequences of fuzzy numbers, Nuray and Savaş  introduced and discussed the concepts of statistically convergent and statistically Cauchy sequences.
Quite recently, Savaş  introduced the idea of asymptotically -statistically equivalent sequences of fuzzy numbers. In this paper we extend his result by using invariant means.
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 ) if it has the following properties:
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.
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.
We now give the following definitions (see ) for fuzzy real-valued sequences.
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 .
It was shown that and are complete metric spaces (see ).
3. Definitions and Notations
In , Savaş introduced the concept of -statistical convergence of fuzzy numbers as follows.
The next definition is natural combination of Definitions 3.1 and 3.2., which was defined in .
It is quite naturel to expect the following definition.
We now define the following.
4. Main Results
In the next theorem we prove the following relation.
This work was supported by Grant (2008-FED-B162) of Yüzüncü Yil university.
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