Tauberian theorems and Cesàro summability in $ \mathscr{L}- $fuzzy normed spaces

Reha Yapalı; Utku Gürdal; Reha Yapalı; Utku Gürdal

doi:10.3934/math.2026872

AIMS Mathematics

2026, Volume 11, Issue 7: 21519-21540. doi: 10.3934/math.2026872

Previous Article Next Article

Research article

Tauberian theorems and Cesàro summability in $ \mathscr{L}- $fuzzy normed spaces

Reha Yapalı ^{1
,
,},
Utku Gürdal ²

1.
Department of Mathematics, Muş Alparslan University, Muş, Türkiye
2.
Department of Mathematics, Burdur Mehmet Âkif University, Burdur, Türkiye

Received: 23 February 2026 Revised: 24 June 2026 Accepted: 03 July 2026 Published: 20 July 2026
MSC : 40A35, 40E05, 40G05

This paper investigates the Cesàro summability method in the setting of $ \mathscr{L}- $fuzzy normed spaces, extending the classical theory of summability to the framework of $ \mathscr{L}- $fuzzy norms. We also establish a Tauberian theorem characterizing the conditions under which Cesàro summability implies ordinary convergence in this setting. Furthermore, we derive an analogue of Hardy's classical two-sided Tauberian theorem by employing the notion of q-boundedness in $ \mathscr{L}- $fuzzy normed spaces, thereby providing new criteria for analyzing the convergence behavior of sequences and series in this context.
- $ \mathscr L- $fuzzy normed spaces,
- Cesàro summability,
- Tauberian theorems,
- slowly oscillating sequences,
- fuzzy functional analysis
Citation: Reha Yapalı, Utku Gürdal. Tauberian theorems and Cesàro summability in $ \mathscr{L}- $fuzzy normed spaces[J]. AIMS Mathematics, 2026, 11(7): 21519-21540. doi: 10.3934/math.2026872

Related Papers:

Abstract

This paper investigates the Cesàro summability method in the setting of $ \mathscr{L}- $fuzzy normed spaces, extending the classical theory of summability to the framework of $ \mathscr{L}- $fuzzy norms. We also establish a Tauberian theorem characterizing the conditions under which Cesàro summability implies ordinary convergence in this setting. Furthermore, we derive an analogue of Hardy's classical two-sided Tauberian theorem by employing the notion of q-boundedness in $ \mathscr{L}- $fuzzy normed spaces, thereby providing new criteria for analyzing the convergence behavior of sequences and series in this context.

References

[1]	A. Aasma, H. Dutta, P. N. Natarajan, An introductory course in summability theory, John Wiley & Sons, Inc., 2017. http://dx.doi.org/10.1002/9781119397786
[2]	K. T. Atanassov, Intuitionistic fuzzy sets, Fuzzy Sets Syst., 20 (1986), 87–96. http://dx.doi.org/10.1016/S0165-0114(86)80034-3 doi: 10.1016/S0165-0114(86)80034-3
[3]	İ. Çanak, Ü. Totur, Z. Önder, A Tauberian theorem for $(C, 1, 1)$ summable double sequences of fuzzy numbers, Iran. J. Fuzzy Syst., 14 (2017), 61–75.
[4]	S. Çetin, Unidefiners, J. Nat. Appl. Sci., 29 (2025), 220–227. http://dx.doi.org/10.19113/sdufenbed.1666339 doi: 10.19113/sdufenbed.1666339
[5]	A. A. Das, S. K. Paikray, T. Pradhan, H. Dutta, Statistical $(C, 1)(E, \mu)$-summability and associated fuzzy approximation theorems with statistical fuzzy rates, Soft Comput., 24 (2020), 10883–10892. http://dx.doi.org/10.1007/s00500-019-04591-2 doi: 10.1007/s00500-019-04591-2
[6]	P. Debnath, Statistical Cesàro summability in intuitionistic fuzzy $n$-normed linear spaces leading towards Tauberian theorems, Axioms, 13 (2024), 557. http://dx.doi.org/10.3390/axioms13080557 doi: 10.3390/axioms13080557
[7]	H. Dutta, J. Gogoi, Weighted $\lambda$-statistical convergence connecting a statistical summability of sequences of fuzzy numbers and Korovkin-type approximation theorems, Soft Comput., 23 (2019), 12883–12895. http://dx.doi.org/10.1007/s00500-019-03846-2 doi: 10.1007/s00500-019-03846-2
[8]	H. Dutta, B. E. Rhoades, Current topics in summability theory and applications, Singapore: Springer, 2016. http://dx.doi.org/10.1007/978-981-10-0913-6
[9]	U. Gürdal, Ö. Oğur, S. Çetin, Unima on $[0, 1]$, Fuzzy Sets Syst., 516 (2025), 109439. http://dx.doi.org/10.1016/j.fss.2025.109439 doi: 10.1016/j.fss.2025.109439
[10]	U. Gürdal, E. U. Ekici, S. Çetin, Modulative categories and fixed point theorems on modulative metric spaces, Hacet. J. Math. Stat., 55 (2026), 525–545. http://dx.doi.org/10.15672/hujms.1643663 doi: 10.15672/hujms.1643663
[11]	H. Efe, C. Alaca, Compact and bounded sets in intuitionistic fuzzy metric spaces, Demonstr. Math., 40 (2007), 449–456. http://dx.doi.org/10.1515/dema-2007-0216 doi: 10.1515/dema-2007-0216
[12]	Ö. Kişi, M. Gürdal, On deferred Cesàro summability and statistical convergence for the sets of triple sequences, Ann. Fuzzy Math. Inform., 24 (2022), 115–127. http://dx.doi.org/10.30948/afmi.2022.24.2.115 doi: 10.30948/afmi.2022.24.2.115
[13]	Ö. Kişi, M. Gürdal, Certain aspects of deferred statistical convergence of fuzzy variables in credibility space, J. Anal., 32 (2024), 2057–2075. http://dx.doi.org/10.1007/s41478-023-00583-6 doi: 10.1007/s41478-023-00583-6
[14]	Ö. Kişi, R. Savaş, On the structure of Zweier $(\lambda, \mu)$-statistical convergence in neutrosophic $n$-normed space, J. Nat. Appl. Sci., 29 (2025), 649–660. http://dx.doi.org/10.19113/sdufenbed.1793560 doi: 10.19113/sdufenbed.1793560
[15]	J. Korevaar, Tauberian theory: A century of developments, Heidelberg: Springer, 2004. http://dx.doi.org/10.1007/978-3-662-10225-1
[16]	F. Móricz, Tauberian conditions under which statistical convergence follows from statistical summability $(C, 1)$, J. Math. Anal. Appl., 275 (2002), 277–287. http://dx.doi.org/10.1016/S0022-247X(02)00338-4 doi: 10.1016/S0022-247X(02)00338-4
[17]	F. Móricz, Necessary and sufficient Tauberian conditions, under which convergence follows from summability $(C, 1)$, Bull. Lond. Math. Soc., 26 (1994), 288–294. http://dx.doi.org/10.1112/blms/26.3.288 doi: 10.1112/blms/26.3.288
[18]	Z. Önder, İ. Çanak, Ü. Totur, Tauberian theorems for statistically $(C, 1, 1)$ summable double sequences of fuzzy numbers, Open Math., 15 (2017), 157–178. http://dx.doi.org/10.1515/math-2017-0006 doi: 10.1515/math-2017-0006
[19]	Z. Önder, S. Karakahya, İ. Çanak, Tauberian theorems for the statistical Cesàro summability method in intuitionistic fuzzy normed spaces, Soft Comput., 28 (2024), 87–104. http://dx.doi.org/10.1007/s00500-023-09335-x doi: 10.1007/s00500-023-09335-x
[20]	P. Parida, B. B. Jena, S. K. Paikray, M. Mursaleen, On fuzzy approximation theorems for functions of two variables via statistical deferred Nörlund summability, Iran. J. Fuzzy Syst., 23 (2026), 17–35.
[21]	R. Saadati, J. H. Park, On the intuitionistic fuzzy topological spaces, Chaos Solitons Fract., 27 (2006), 331–344. http://dx.doi.org/10.1016/j.chaos.2005.03.019 doi: 10.1016/j.chaos.2005.03.019
[22]	R. Saadati, A. Razani, H. Adibi, A common fixed point theorem in $L$-fuzzy metric spaces, Chaos Solitons Fract., 33 (2007), 358–363. http://dx.doi.org/10.1016/j.chaos.2006.01.023 doi: 10.1016/j.chaos.2006.01.023
[23]	E. Savaş, M. Gürdal, Certain summability methods in intuitionistic fuzzy normed spaces, J. Intell. Fuzzy Syst., 27 (2014), 1621–1629. http://dx.doi.org/10.3233/IFS-141128 doi: 10.3233/IFS-141128
[24]	S. A. Sezer, İ. Çanak, Power series methods of summability for series of fuzzy numbers and related Tauberian theorems, Soft Comput., 21 (2017), 1057–1064. http://dx.doi.org/10.1007/s00500-015-1840-0 doi: 10.1007/s00500-015-1840-0
[25]	S. A. Sezer, İ. Çanak, Deferred Cesàro means of fuzzy number-valued sequences with applications to Tauberian theorems, Filomat, 37 (2023), 5993–6004. http://dx.doi.org/10.2298/FIL2318993S doi: 10.2298/FIL2318993S
[26]	S. Shakeri, R. Saadati, C. Park, Stability of the quadratic functional equation in non-Archimedean $L$-fuzzy normed spaces, Int. J. Nonlinear Anal. Appl., 1 (2010), 72–83. http://dx.doi.org/10.22075/ijnaa.2010.77 doi: 10.22075/ijnaa.2010.77
[27]	Ö. Talo, F. Başar, Necessary and sufficient Tauberian conditions for the $A^r$ method of summability, Math. J. Okayama Univ., 60 (2018), 209–219.
[28]	Ö. Talo, E. Yavuz, Cesàro summability of sequences in intuitionistic fuzzy normed spaces and related Tauberian theorems, Soft Comput., 25 (2021), 2315–2323. http://dx.doi.org/10.1007/s00500-020-05301-z doi: 10.1007/s00500-020-05301-z
[29]	Ü. Totur, İ. Çanak, Tauberian theorems for $(\bar{N}; p;q)$ summable double sequences of fuzzy numbers, Soft Comput., 24 (2020), 2301–2310. http://dx.doi.org/10.1007/s00500-019-04060-w doi: 10.1007/s00500-019-04060-w
[30]	R. Yapalı, H. Çoşkun, Lacunary statistical convergence for double sequences on $\mathscr{L}$-fuzzy normed space, J. Math. Sci. Model., 6 (2023), 24–31. http://dx.doi.org/10.33187/jmsm.1127905 doi: 10.33187/jmsm.1127905
[31]	R. Yapalı, H. Çoşkun, U. Gürdal, Statistical convergence on $L$-fuzzy normed space, Filomat, 37 (2023), 2077–2085. http://dx.doi.org/10.2298/FIL2307077Y doi: 10.2298/FIL2307077Y
[32]	R. Yapalı, U. Gürdal, Pringsheim and statistical convergence for double sequences on $L$-fuzzy normed space, AIMS Mathematics, 6 (2021), 13726–13733. https://doi.org/10.3934/math.2021796 doi: 10.3934/math.2021796
[33]	R. Yapalı, E. Korkmaz, M. Çınar, H. Çoşkun, Lacunary statistical convergence on $L$-fuzzy normed space, J. Intell. Fuzzy Syst., 46 (2022), 1985–1993. https://doi.org/10.3233/JIFS-222695 doi: 10.3233/JIFS-222695
[34]	R. Yapalı, Ö. Talo, Tauberian conditions for double sequences which are statistically summable $(C, 1, 1)$ in fuzzy number space, J. Intell. Fuzzy Syst., 33 (2017), 947–956. http://dx.doi.org/10.3233/JIFS-162211 doi: 10.3233/JIFS-162211
[35]	L. A. Zadeh, Fuzzy sets, Inform. Control, 8 (1965), 338–353. http://dx.doi.org/10.1016/S0019-9958(65)90241-X doi: 10.1016/S0019-9958(65)90241-X

Reader Comments

Your name:*

Email:*
© 2026 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)