Deferred pointwise $f$-statistical convergence of sequences of fuzzy mappings of order $\alpha$

Document Type : Research Paper

Authors

1 Tekirdağ Namık Kemal Üniversitesi, Fen Edebiyat Fakültesi,Matematik Bölümü

2 Fırat University, Faculty of Science, Department of Mathematics

3 Muş Alparslan University, Faculty of Science and Arts, Department of Mathematics

Abstract

This research paper introduces and investigates two new concepts, namely pointwise deferred $f$-statistical convergence of order $\alpha$ and strong pointwise deferred $f$-summability of order $\alpha$, within the context of sequences of fuzzy mappings. The study explores the relationships between these newly proposed concepts and establishes several inclusion theorems. These findings contribute valuable insights into the properties of the introduced concepts, shedding light on their characteristics and applications in the realm of fuzzy mappings.

Keywords

Main Subjects


[1] R. P. Agnew, On deferred Ces`aro mean, Annals of Mathematics, 33 (1932), 413-421. https://doi.org/10.2307/
1968524
[2] A. Aizpuru, M. C. Listan-Garcia, M. Rambla-Barreno, Density by moduli and statistical convergence, Quaestiones
Mathematicae, 37(4) (2014), 525-530. https://doi.org/10.2989/16073606.2014.981683
[3] K. E. Akba¸s, M. I¸sık, On asymptotically λ-statistical equivalent sequences of order α in probability, Filomat, 34(13)
(2020), 4359-4365. https://doi.org/10.2298/FIL2013359A
[4] Y. Altın, M. Et, B. C. Tripathy, On pointwise statistical convergence of sequences of fuzzy mappings, The Journal
of Fuzzy Mathematics, 15(2) (2007), 425-433.
[5] H. Altınok, M. Et, R. C¸ olak, Some remarks on generalized sequence space of bounded variation of sequences
of fuzzy numbers, Iranian Journal of Fuzzy Systems, 11(5) (2014), 39-46. https://ijfs.usb.ac.ir/
article1700270f02138ecd150c48eccb6291138b41.pdf
[6] H. Altınok, M. Kasap, f-statistical convergence of order β for sequences of fuzzy numbers, Journal of Intelligent and
Fuzzy Systems, 33 (2017), 705-712. https://doi.org/10.3233/JIFS-161654
[7] V. K. Bhardwaj, S. Dhawan, f-statistical convergence of order α and strong Ces`aro summability of order α
with respect to a modulus, Journal of Inequalities and Applications, 332 (2015). https://doi.org/10.1186/
s13660-015-0850-x
[8] N. L. Braha, H. M. Srivastava, M. Et, Some weighted statistical convergence and associated Korovkin and
Voronovskaya type theorems, Journal of Applied Mathematics Computing, 65(1-2) (2021), 429-450. https://doi.
org/10.1007/s12190-020-01398-5
[9] M. Burgin, Theory of fuzzy limits, Fuzzy Sets and Systems, 115 (2000), 433-443. https://doi.org/10.1016/
S0165-0114(98)00338-8
[10] U. C¸ akan, Y. Altın, Some classes of statistically convergent sequences of fuzzy numbers generated by a modulus
function, Iranian Journal of Fuzzy Systems, 12(3) (2015), 47-55. https://doi.org/10.22111/ijfs.2015.2019
[11] R. C¸ olak, Statistical convergence of order α, Modern Methods in Analysis and Its Applications, Anamaya Pub.,
New Delhi, (2010), 121-129.
[12] J. S. Connor, The statistical and strong p-Ces`aro convergence of sequences, Analysis, 8 (1988), 47-63. https:
//doi.org/10.1524/anly.1988.8.12.47
[13] P. Diamond, P. Kloeden, Metric spaces of fuzzy sets, Fuzzy Sets and Systems, 35(2) (1990), 241-249. https:
//doi.org/10.1016/0165-0114(90)90197-E
[14] M. Et, On pointwise λ-statistical convergence of order α of sequences of fuzzy mappings, Filomat, 28(6) (2014),
1271-1279. https://doi.org/10.2298/FIL1406271E
[15] M. Et, Y. Altın, R. C¸ olak, On pointwise λ-statistical convergence of order α and strong pointwise λ-summability
of order α of sequences of fuzzy mappings, Facta Universitatis, Series: Mathematics and Informatics, 38(2) (2023),
219-230. https://doi.org/10.22190/FUMI200406015E
[16] M. Et, P. Baliarsingh, H. S¸. Kandemir, M. K¨u¸c¨ukaslan, On μ-deferred statistical convergence and strongly deferred
summable functions, Revista de la Real Academia de Ciencias Exactas, F´ısicas y Naturales, Series A Matem´aticas,
115(1) (2021), 34. https://doi.org/10.1007/s13398-020-00983-4
[17] M. Et, S. A. Mohiuddine, A. Alotaibi, On λ-statistical convergence and strongly λ-summable functions of order α,
Journal of Inequalities and Applications, 469 (2013). https://doi.org/10.1186/1029-242X-2013-469
[18] M. Et, H. S¸eng¨ul, On pointwise lacunary statistical convergence of order α of sequences of functions, Proceedings
of the National Academy of Sciences, India Section A: Physical Sciences, 85(2) (2015), 253-258. https://doi.org/
10.1007/s40010-015-0199-z
[19] H. Fast, Sur la convergence statistique, Colloquium Mathematicum, 2 (1951), 241-244. https://eudml.org/doc/
209960
[20] J. A. Fridy, On statistical convergence, Analysis, 5 (1985), 301-313. https://dx.doi.org/10.1524/anly.1985.
5.4.301
[21] M. I¸sık, K. E. Akba¸s, On λ-statistical convergence of order α in probability, Journal of Inequalities and Special
Functions, 8(4) (2017), 57-64.
[22] M. I¸sık, K. E. Akba¸s, On Asymptotically lacunary statistical equivalent sequences of order α in Probability, ITM
Web of Conferences, 13 (2017), Article ID 01024. https://doi.org/10.1051/itmconf/20171301024
[23] M. I¸sık, K. E. Et, On lacunary statistical convergence of order α in probability, AIP Conference Proceedings, 1676
(2015), 020045. https://doi.org/10.1063/1.4930471
[24] M. K¨u¸c¨ukaslan, Y. Yılmazt¨urk, On deferred statistical convergence of sequences, Kyungpook Mathematical Journal,
56 (2016), 357-366. https://kmj.knu.ac.kr/journal/view.html?doi=10.5666/KMJ.2016.56.2.357
[25] M. Matloka, Fuzzy mappings sequences and series, Busefal, 30 (1987), 18-25.
[26] H. Nakano, Concave modulars, Journal of the Mathematical Society of Japan, 5 (1953), 29-49. https://doi.org/
10.2969/jmsj/00510029
[27] L. Nayak, M. Mursaleen, P. Baliarsingh, On deferred statistical A-convergence of fuzzy sequence and applications,
Iranian Journal of Fuzzy Systems, 19(2) (2022), 119-131. https://doi.org/10.22111/IJFS.2022.6794
[28] S. Pehlivan, B. Fisher, On some sequence spaces, Indian Journal of Pure and Applied Mathematics, 25(10) (1994),
1067-1071.
[29] T. Salat, On statistically convergent sequences of real numbers, Mathematica Slovaca, 30 (1980), 139-150. https:
//eudml.org/doc/34081
[30] I. J. Schoenberg, The integrability of certain functions and related summability methods, The American Mathematical
Monthly, 66 (1959), 361-375. https://doi.org/10.1080/00029890.1959.11989303
[31] H. Steinhaus, Sur la convergence ordinaire et la convergence asymptotique, Colloquium Mathematicum, 2 (1951),
73-74.
[32] F. Temizsu, M. Et, On deferred f-statistical boundedness, TWMS Journal of Pure and Applied Mathematics, 14(1)
(2023), 106-119. https://doi.org/10.30546/2219-1259.14.1.2023.106
[33] B. C. Tripathy, M. Et, On generalized difference lacunary statistical convergence, Studia Universitatis Babe¸s-Bolyai
Mathematica, 50(1) (2005), 119-130.
[34] B. C. Tripathy, G. C. Ray, Fuzzy δ-continuity in mixed fuzzy ideal topological spaces, Journal of Applied Analysis,
24(2) (2018), 233-239. https://doi.org/10.1515/jaa-2018-0022
[35] B. C. Tripathy, M. Sen, On lacunary strongly almost convergent double sequences of fuzzy numbers, Annals of the
University of Craiova, Mathematics and Computer Science Series, 42(2) (2015), 254-259. https://doi.org/10.
52846/ami.v42i2.535
[36] L. A. Zadeh, Fuzzy sets, Information and Control, 8 (1965), 338-353. https://dx.doi.org/10.1016/
S0019-9958(65)90241-X