Statistical cluster and extreme limit points of sequences of fuzzy numbers
Section snippets
Introduction and background
Statistical convergence is a generalization of the usual notion of convergence that parallels the theory of ordinary convergence. Some of the results obtained in the theory of ordinary convergence have been extended to the theory of statistical convergence by using the concept of density. For instance, Fridy and Orhan [11] introduced the statistical analogues of the concepts of limit inferior and limit superior of a sequence of real numbers. Moreover, they showed that both statistical limit
Main results
Fridy and Orhan [11] showed that the sli (or sls) of a sequence of real numbers is a statistical cluster point of the sequence. But this result may not be valid for a sequence of fuzzy numbers in general, as can be seen in the following example. Example 1 Define the sequence bywhereandIt follows that
Further results
The results given in this paper are still valid if we replace the δ-natural density with any density which satisfies the axiomatic properties of the density function given by Freedman and Sember [9]. For instance, we say that a set has A-density if exists where is a nonnegative regular matrix. A sequence of fuzzy numbers is said to be A-statistically convergent to X0 provided that the set has A-density zero for every . The case
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2014, Information SciencesCitation Excerpt :The importance of the introduced notion of fuzzy set was realized and has successfully been applied in almost all the branches of science and technology. Recently fuzzy set theory has been applied in studying sequence spaces by Aytar et al. [1,2], Aytar and Pehlivan [3], Dutta [4–6], Fang and Hung [7], Hong et al. [9], Nanda [11], Wu and Wu [19,20], Talo and Başar [12–14], Tripathy and Baruah [15], Tripathy and Borgohain [16], Tripathy et al. [17], Tripathy and Debnath [18] and many others. According to the representation theorem obtained by Goetschel and Voxman [8], a fuzzy real number is completely determined by the endpoints of the level sets.
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