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Middle East Journal of Science (2024) 10(1):14-20 https://doi.org/10.51477/mejs.1496008
14
Middle East Journal of Science
https://dergipark.org.tr/mejs
e-ISSN:2618-6136
MEJS
Research Article
ON STATISTICAL CONVERGENCE OF SEQUENCES OF FUZZY NUMBERS
Damla BARLAK*
Dicle University, Faculty of Science, Department of Statistics, 21280, Diyarbakır, Turkiye
* Corresponding author; dyagdiran@hotmail.com
Abstract: In this study, we present the concepts of statistically convergence for sequences of fuzzy
numbers as well as strong summability and Cauchy statistically convergence for sequences
of fuzzy numbers. We also provide several results concerning these concepts.
Keywords: Cesàro summability, Statistical convergence, Strongly Cesàro summability.
Received: June 4, 2024
Accepted: June 28, 2024
1. Introduction
Fast [1] gave short description of statistical convergence 1951. Schoenberg [2] investigated
statistical convergence as a summability method and outlined several fundamental properties associated
with it. This concept has been applied by many researchers under different names to measurement
theory, locally convex spaces, summability theory, Banach spaces, trigonometric series in Fourier
analysis and theory of fuzzy set ([3],[4],[5],[6]). The concept of statistical convergence depend on the
density subsets of the set The natural density of a subset of is defined by
if the limit exists, where the vertical bars indicate number of the
elements in
If is a sequence such that satisfies feature for all apart from a set of naturally density zero,
then we say that satifies for "almost all " and we shortened this by " ."
Fuzzy set theory, which is a very valuable logic with accuracy, was first introduced by Zadeh [7]
in 1965. The applications of this theory span various fields, including fuzzy topological spaces, fuzzy
measurements, fuzzy mathematical programming, and fuzzy logic. The concept of fuzzy number
sequence is first encountered in Matloka’s paper [8].
Matloka [8] defined the concept of bounded and convergent sequences of fuzzy numbers and
studied their some properties. Since then, many studies on sequences of fuzzy numbers have been made
and studies on this subject are still ongoing ([9], [10], [11], [12], [13], [14]).
A fuzzy number is fuzzy set with the following properties:
is normal, that is, there exists an such that
is fuzzy convex, that is, for and
is upper semicontinuous;
or denoted by is compact.
The definition level set of a fuzzy number is determined by
Middle East Journal of Science (2024) 10(1):14-20 https://doi.org/10.51477/mejs.1496008
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It is evident that is a fuzzy number is necessary and sufficient for is a closed interval for
each and The set of all fuzzy number sequences will be denoted as The
distance between two fuzzy numbers and we use the metric
Let and . Then, the Hausdorff metric is characterized by
It is known that is a metric on and is a complete metric space.
Nuray and Savaş [15] defined the concept of statistical convergence for sequences of fuzzy
numbers. A sequence of fuzzy numbers is a function Let be a
sequence of fuzzy numbers. Then the sequence fuzzy numbers, is called statistically
convergent to the fuzzy number if for each
The set of all fuzzy number sequences demonstrating statistically convergent will be denoted as
Çakallı [16] defined the concept of statistically convergence. Subsequently, many authors
have done a great deal of work on statistical convergence([17],[18],[19],[20],[21],[22]). The aim of
this paper is to extend the investigation conducted by Çakallı [16].
2. Main Results
In this section, we present the concepts of statistically convergence for sequences of fuzzy
numbers, strong summability for sequences of fuzzy numbers and Cauchy statistically
convergence for sequences of fuzzy numbers. We also provide several results pertaining to these
concepts.
Definition 2.1. Let be a fuzzy number sequence, the sequence is is called
statistically convergent to the fuzzy number if
for each where is a non-decreasing sequence for each tending to such that
and for each .
In this case, either or is used as a notation. The set of all
fuzzy number sequences demonstrating -statistical convergence will be denoted as If for each
Middle East Journal of Science (2024) 10(1):14-20 https://doi.org/10.51477/mejs.1496008
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the concept of being statistically convergent is equivalent to being statistically
convergent.
Definition 2.2. Let be a fuzzy number sequence, the sequence is is called strong
convergent (or convergent) to if
In this case, either or
is used as a notation. The
set of all fuzzy number sequences demonstrating strong convergent will be denoted as
Theorem 2.1 Let and be two fuzzy numbers sequences, is a non-decreasing
sequence for each tending to such that
and
for each . Then
(i) and implies
(ii) and implies
Proof. (i) For the proof is clear. Let the inequality leads to the proof
(ii) Let and , we can write
for each and thus if and then
Definition 2.3 Let be a fuzzy number sequence, the sequence is called
Cauchy sequence if there exists a subsequence of such that for every
and for each
where is non-decreasing sequence for each tending to such that
and for each
Middle East Journal of Science (2024) 10(1):14-20 https://doi.org/10.51477/mejs.1496008
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Theorem 2.2. The subsequent statements are mutually equivalent:
(i) is a statistical convergence,
(ii) is a Cauchy statistical convergence,
(iii) is a sequence of fuzzy numbers for which there is a statistically convergent sequence
of fuzzy numbers such that
Theorem 2.3 Let be a fuzzy number sequence, the sequence is
convergent a necessary and sufficient condition is that is an Cauchy sequence.
Proof. Let's consider is an Cauchy sequence. For each we can say
Hence, we get
The proof to the contrary is obvious.
Theorem 2.4 Let be a fuzzy number sequence, is non-decreasing sequence for
each tending to such that
and for each
If for each
then
Proof. Let's consider the following inequality leads to the proof, for every
Theorem 2.5. Let be a fuzzy number sequence, and be two sequences
such that for all If
then
Proof. Suppose that the following inequality leads to the proof, for every
Corollary 2.1 Let be a fuzzy number sequence, and be two sequences
such that for all If
then
Theorem 2.6. If then
Proof. Suppose that for , we can write
Middle East Journal of Science (2024) 10(1):14-20 https://doi.org/10.51477/mejs.1496008
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If we take the limit for we have
Thus, the desired outcome is obtained.
Corollary 2.2 Let be a fuzzy number sequence. If then
The opposite of the Theorem 2.6 and Corollary 2.2 aren’t true, mostly. For example, let the
sequence be as follows:
If we choose
Moreover,
so, is not convergent
Ethical statement
The author declares that this document does not require ethics committee approval or any special
permission. Our study does not cause any harm to the environment.
Conflict of interest
The author declares no potential conflicts of interest related to this article's research, authorship,
and publication.
Middle East Journal of Science (2024) 10(1):14-20 https://doi.org/10.51477/mejs.1496008
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