Elsevier

Information Sciences

Volume 177, Issue 16, 15 August 2007, Pages 3290-3296
Information Sciences

Statistical cluster and extreme limit points of sequences of fuzzy numbers

https://doi.org/10.1016/j.ins.2007.02.017Get rights and content

Abstract

In this paper, we investigate the relations between statistical limit superior (or statistical limit inferior) and statistical cluster points of a statistically bounded sequence of fuzzy numbers and show that the set of statistical cluster points of a sequence of this type might be empty.

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 X={Xk} byXk(x):=μ1(x),ifkis an odd numberμ2(x),otherwise,whereμ1(x):=0,ifx∈(-∞,1)∪(6,∞)x-14,ifx∈[1,5]6-x,otherwiseandμ2(x):=0,ifx∈(-∞,2)∪(4,∞)x-2,ifx∈[2,3]4-x,otherwise.It follows thatst-liminfX=0,ifx∈(-∞,1)∪(4,∞)x-14,ifx∈1,73x-2,ifx∈(

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 K⊂N has A-density if δA(K):=limn→∞∑k∈Kank exists where A=(ank) is a nonnegative regular matrix. A sequence X={Xk} of fuzzy numbers is said to be A-statistically convergent to X0 provided that the set K(ε):={k∈N:d¯(Xk,X0)⩾ε} has A-density zero for every ε>0. The case

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