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ON ρ-STATISTICAL CONVERGENCE OF SEQUENCES OF FUZZY NUMBERS

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Abstract

In this study, we present the concepts of ρ-statistically convergence for sequences of fuzzy numbers as well as strong (w_ρ (F)) summability and ρ-Cauchy statistically convergence for sequences of fuzzy numbers. We also provide several results concerning these concepts.
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
16
󰇛󰇜 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
19
References
[1] Fast, H., Sur la convergence statistique”, Colloquium Math., 2, 241-244, 1951.
[2] Schoenberg, I.J., “The Integrability of Certain Functions and Related Summability Methods”,
Amer. Math. Monthly, 66, 361-375, 1959.
[3] Connor, J., “A topological and functional analytic approach to statistical convergence”, Analysis
of Divergence , Birkhauser, Boston, 403-413, 1999.
[4] Fridy, J., “On statistical convergence, Analysis 5, 301-313, 1985.
[5] Šalát, T., “On statistically convergent sequences of real numbers, Math. Slovaca, 30, 139-150,
1980.
[6] Ercan, S., Altin, Y., Bektaş, Ç., “On lacunary weak statistical convergence of order ”,
Communications in Statistics-Theory and Methods, 49(7), 1653-1664, 2020
https://doi.org/10.1080/03610926.2018.1563185
[7] Zadeh, L. A., “Fuzzy sets”, Inform and Control, 8, 338-353, 1965.
[8] Matloka, M., “Sequences of fuzzy numbers”, BUSEFAL, 28, 28-37, 1986.
[9] Altinok, H., Çolak, R., Altin, Y., “On the class of statistically convergent difference
sequences of fuzzy numvers”, Soft Computing, 16(6), 1029-1034, 2012 DOI: 10.1007/s00500-
011-0800-6
[10] Aytar, S., Pehlivan, S., “Statistical convergence of sequences of fuzzy numbers and sequences of
cuts”, International Journal of General Systems 37(2),231-237, 2008 DOI:
10.1080/03081070701251075
[11] UCakan, U., Altin, Y., “Some clases of statistically convergent sequences of fuzzy numbers
generated by modulus function”, Iranian Journal of Fuzzy Systems, 12(3), 47-55 , 2015.
[12] Çanak, İ., “On Tauberian theorems for Cesaro summability of sequences fuzzy numbers”, J.Intell.
Fuzzy Syst. 30, 2657-2661, 2016 DOI: 10.3233/IFS-131053
[13] Sezer, S.A., “Statistical harmonic summability of sequences of fuzzy numbers”, Soft Computing,
27, 1933-1940, 2023 DOI: 10.1007/s00500-020-05151-9
[14] Tripathy, B.C., Baruah, A., “Lacunary statistically convergent and lacunary strongly convergent
generalized difference sequences of fuzzy real numbers”, Kyungpook Math. Jour. 50, 565-574,
2010.
[15] Nuray, F., Savaş, E., “Statistical convergence of sequences of fuzzy real numbers”, Math. Slovaca
45(3), 269-273, 1995.
[16] Çakallı, H., “A variation on statistical ward continuity”, Bull. Malays. Math. Sci. Soc. 40, 1701-
1710, 2017. DOI: 10.1007/s40840-015-0195-0
[17] Kandemir, H.Ş., “On ρ-statistical convergence in topological groups”, Maltepe Journal of
Mathematics, 4(1), 9-14, 2022. doi:10.47087/mjm.1092559
[18] Aral, N.D., Kandemir, H.Ş., Et. M., “On Statistical convergence of sequences of Sets,
Conference Proceeding Science and Tecnology, 3(1),156-159, 2020.
[19] Gumus, H., Rho-statistical convergence of interval numbers, International Conference on
Mathematics and Its Applications in Science and Engineering. 2022.
Middle East Journal of Science (2024) 10(1):14-20 https://doi.org/10.51477/mejs.1496008
20
[20] Aral, N.D., Kandemir, H., & Et, M., "On ρ−statistical convergence of order α of sequences of
function", e-Journal of Analysis and Applied Mathematics, 2022(1), 45-55, 2022.
[21] Aral, N.D., Kandemir, H., & Et, M., "Onstatistical convergence of double sequences of order
in topological groups", The Journal of Analysis, 31, 30693078, 2023.
[22] Aral, N.D., Kandemir, H., & Et, M., "On statistical convergence of order of sequences of
sets", Miskolc Mathematical Notes, 24(2), 569578, 2023.
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