© E. Savas and R. F. Patterson 2009
Received: 1 September 2009
Accepted: 2 October 2009
Published: 15 October 2009
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.
For sequences of fuzzy numbers, Nanda  studied the sequences of fuzzy numbers and showed that the set of all convergent sequences of fuzzy numbers forms a complete metric space. Kwon  introduced the definition of strongly p-Cesàro summability of sequences of fuzzy numbers. Savas  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  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  introduced and studied some new sequence spaces of fuzzy numbers generated by nonnegative regular matrix. Quite recently, Savas and Mursaleen  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.
and clearly with if . Moreover is a complete, separable, and locally compact metric space .
Throughout this paper, will denote with . We will need the following definitions (see ).
It is quite clear that if a sequence is statistically -convergent, then it is a statistically -Cauchy sequence . It is also easy to see that if there is a convergent sequence such that a.a.( ), then is statistically convergent.
3. Main Result
this completes the proof.
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.
- Nanda S: On sequences of fuzzy numbers. Fuzzy Sets and Systems 1989,33(1):123–126. 10.1016/0165-0114(89)90222-4MathSciNetView ArticleMATHGoogle Scholar
- Kwon J-S: On statistical and -Cesaro convergence of fuzzy numbers. Journal of Applied Mathematics and Computing 2000,7(1):195–203.MathSciNetMATHGoogle Scholar
- Savas E: A note on double sequences of fuzzy numbers. Turkish Journal of Mathematics 1996,20(2):175–178.MathSciNetMATHGoogle Scholar
- Savas E: On statistically convergent sequences of fuzzy numbers. Information Sciences 2001,137(1–4):277–282.MathSciNetView ArticleMATHGoogle Scholar
- Savas E: A note on sequence of fuzzy numbers. Information Sciences 2000,124(1–4):297–300.MathSciNetView ArticleMATHGoogle Scholar
- Savas E: On lacunary statistically convergent double sequences of fuzzy numbers. Applied Mathematics Letters 2008,21(2):134–141. 10.1016/j.aml.2007.01.008MathSciNetView ArticleMATHGoogle Scholar
- Mursaleen M, Basarir M: On some new sequence spaces of fuzzy numbers. Indian Journal of Pure and Applied Mathematics 2003,34(9):1351–1357.MathSciNetMATHGoogle Scholar
- Savas E, Mursaleen : On statistically convergent double sequences of fuzzy numbers. Information Sciences 2004,162(3–4):183–192. 10.1016/j.ins.2003.09.005MathSciNetView ArticleMATHGoogle Scholar
- Diamond P, Kloeden P: Metric spaces of fuzzy sets. Fuzzy Sets and Systems 1990,35(2):241–249. 10.1016/0165-0114(90)90197-EMathSciNetView ArticleMATHGoogle Scholar
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