- Research Article
- Open Access
© Ekrem Savaş et al. 2010
- Received: 30 December 2009
- Accepted: 13 April 2010
- Published: 20 May 2010
- Topological Space
- Fuzzy Number
- Positive Real Number
- Arithmetic Operation
- Statistical Equivalent
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 ).
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.
In the next theorem we prove the following relation.
This work was supported by Grant (2008-FED-B162) of Yüzüncü Yil university.
- Zadeh LA: Fuzzy sets. Information and Computation 1965, 8: 338–353.MATHMathSciNetGoogle Scholar
- Matloka M: Sequences of fuzzy numbers. BUSEFAL 1986, 28: 28–37.MATHGoogle Scholar
- Nuray F, Savaş E: Statistical convergence of sequences of fuzzy numbers. Mathematica Slovaca 1995, 45(3):269–273.MATHMathSciNetGoogle Scholar
- Savaş E: On asymptotically -statistical equivalent sequences of fuzzy numbers. New Mathematics & Natural Computation 2007, 3(3):301–306. 10.1142/S1793005707000781MATHView ArticleGoogle Scholar
- Banach S: Theorie des Operations Linearies. Subwncji Funduszu Narodowej, Warszawa, Poland; 1932.Google Scholar
- Savas E: On asymptotically lacunary -statistical equivalent sequences of Fuzzy numbers,. New Mathematics & Natural Computation 2009, 5(3):1–10.MathSciNetView ArticleGoogle Scholar
- Nanda S: On sequences of fuzzy numbers. Fuzzy Sets and Systems 1989, 33(1):123–126. 10.1016/0165-0114(89)90222-4MATHMathSciNetView ArticleGoogle Scholar
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