Salih Aytar

Statistical limit points of sequences of fuzzy numbers

In the current work, we extend the concepts of statistical limit and cluster points of a sequence of fuzzy numbers corresponding to the definitions in [Proc. Am. Math. Soc. 4 (1993) 1187] for the sequences of real numbers. Later we discuss the relations among sets of ordinary limit points, statistical limit points and statistical cluster points of sequences of fuzzy numbers.

Statistical limit inferior and limit superior for sequences of fuzzy numbers

In this paper, we extend the concepts of statistical limit superior and limit inferior (as introduced by Fridy and Orhan [Statistical limit superior and limit inferior, Proc. Amer. Math. Soc. 125 (12) (1997) 3625-3631. [12]]) to statistically bounded sequences of fuzzy numbers and give some fuzzy-analogues of properties of statistical limit superior and limit inferior for sequences of real numbers.

Statistically monotonic and statistically bounded sequences of fuzzy numbers

We introduce the statistical monotonicity and boundedness of a sequence of fuzzy numbers. We also derive the analogue of monotone convergence theorem and prove the decomposition theorems for this type of sequences.

Statistical cluster and extreme limit points of sequences of fuzzy numbers

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.

Rough Statistical Convergence

In this work, using the concept of natural density, we introduce the notion of rough statistical convergence. We define the set of rough statistical limit points of a sequence and obtain two statistical convergence criteria associated with this set. Later, we prove that this set is closed and convex. Finally, we examine the relations between the set of statistical cluster points and the set of rough statistical limit points of a sequence.

The Rough Limit Set and the Core of a Real Sequence

In this paper, we prove that the ordinary core of a sequence x = (x i ) of real numbers is equal to its 2 -limit set, where : = inf {r ≥ 0:LIM r x ≠ Ø}. Defining the sets r-limit inferior and r-limit superior of a sequence, we show that the r-limit set of the sequence is equal to the intersection of these sets and that r-core of the sequence is equal to the union of these sets. Finally, we prove an ordinary convergence criterion that says a sequence is convergent iff its rough core is equal to its rough limit set for the same roughness degree.

Statistical convergence of sequences of fuzzy numbers and sequences of α−cuts

In this short paper, we show that the statistical convergence of a sequence of fuzzy numbers with respect to the supremum metric is equivalent to the uniform statistical convergence of the sequences of functions which are defined via the endpoints of α-cuts of the same fuzzy numbers sequence. We introduce the concept of levelwise statistical convergence and present the analogous relation between levelwise statistical convergence of a sequence of fuzzy numbers and pointwise statistical convergence of the sequences of endpoint functions.

The core of a sequence of fuzzy numbers

In this paper, based on level sets we define the limit inferior and limit superior of a bounded sequence of fuzzy numbers and prove some properties. We extend the concept of the core of a sequence of complex numbers, first introduced by Knopp in 1930, to a bounded sequence of fuzzy numbers and prove that the core of a sequence of fuzzy numbers is the interval [@n,@m] where @n and @m are extreme limit points of the sequence.

Levelwise statistical cluster and limit points of a sequence of fuzzy numbers

In the current work we introduce the notions of levelwise statistical limit point and levelwise statistical cluster point of a sequence of fuzzy numbers and discuss the relations between the sets of ordinary limit points, levelwise limit points, levelwise statistical limit points, statistical cluster points and levelwise statistical cluster points of a sequence of fuzzy numbers. Finally, we show that the Bolzano–Weierstrass Theorem is not valid for levelwise statistical convergence.

Addendum to “Statistically monotonic and statistically bounded sequences of fuzzy numbers” [Informat. Sci. 176 (2006) 734–744]

Abstract In this short note, we state that one side of implication of the statistical monotone convergence theorem given in [S. Aytar, S. Pehlivan, Statistically monotonic and statistically bounded sequences of fuzzy numbers, Informat. Sci. 176 (2006) 734–744] is incorrect and elaborate on this shortcoming.