From single-variable to Pexider-type: A new direct proof for Hyers-Ulam stability of functional equations in fuzzy Banach spaces

Chun Ji; Gang Lyu; Ming Fang; Qi Liu; Chun Ji; Gang Lyu; Ming Fang; Qi Liu

doi:10.3934/math.2026472

AIMS Mathematics

2026, Volume 11, Issue 4: 11473-11488. doi: 10.3934/math.2026472

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Research article

From single-variable to Pexider-type: A new direct proof for Hyers-Ulam stability of functional equations in fuzzy Banach spaces

Chun Ji ¹,
Gang Lyu ^{2
,
,},
Ming Fang ^{3
,
,},
Qi Liu ⁴

1.
Vocational Education Teaching and Research Training Center, Jilin Provincial Institute of Education, Changchun 130022, China
2.
School of General Education, Guangzhou College of Technology and Business, Guangzhou 510850, China
3.
Department of Mathematics, Yanbian University, Yanji 133001, China
4.
School of Mathematics and Physics, Anqing Normal University, Anqing 246133, China

Received: 09 March 2026 Revised: 07 April 2026 Accepted: 15 April 2026 Published: 24 April 2026
MSC : 39B52, 39B62, 46B25

We investigate the Hyers-Ulam stability of functional equations involving a single variable in fuzzy Banach spaces using a new direct method. This method imposes no restrictions on the domain or range of functions and is shown to be simpler and more effective for various functional equations. Furthermore, we establish a fuzzy version of the generalized Hyers-Ulam stability for a Pexider-type functional inequality and a linear functional equation with multiple coefficients in a fuzzy Banach linear space. For both equations, we obtain the existence and uniqueness of approximating solutions. To validate the theoretical results, numerical experiments are conducted using the Monte Carlo random sampling method. The results show that the mean ratio of the true error to the theoretical upper bound is only 0.0027, and the 95th percentile is 0.0051, indicating that the derived error bound is both reliable and tight. The proposed method enriches the proof techniques for stability problems of functional equations in fuzzy spaces, and the findings can serve as a reference for theoretical research and practical applications in related fields.
- new direct method,
- fuzzy approximation,
- Pexider functional inequality,
- fuzzy Banach space
Citation: Chun Ji, Gang Lyu, Ming Fang, Qi Liu. From single-variable to Pexider-type: A new direct proof for Hyers-Ulam stability of functional equations in fuzzy Banach spaces[J]. AIMS Mathematics, 2026, 11(4): 11473-11488. doi: 10.3934/math.2026472

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Abstract

We investigate the Hyers-Ulam stability of functional equations involving a single variable in fuzzy Banach spaces using a new direct method. This method imposes no restrictions on the domain or range of functions and is shown to be simpler and more effective for various functional equations. Furthermore, we establish a fuzzy version of the generalized Hyers-Ulam stability for a Pexider-type functional inequality and a linear functional equation with multiple coefficients in a fuzzy Banach linear space. For both equations, we obtain the existence and uniqueness of approximating solutions. To validate the theoretical results, numerical experiments are conducted using the Monte Carlo random sampling method. The results show that the mean ratio of the true error to the theoretical upper bound is only 0.0027, and the 95th percentile is 0.0051, indicating that the derived error bound is both reliable and tight. The proposed method enriches the proof techniques for stability problems of functional equations in fuzzy spaces, and the findings can serve as a reference for theoretical research and practical applications in related fields.

References

[1]	A. K. Katsaras, Fuzzy topological vector spaces $II$, Fuzzy Set. Syst., 12 (1984), 143–154. https://doi.org/10.1016/0165-0114(84)90034-4 doi: 10.1016/0165-0114(84)90034-4
[2]	C. Felbin, Finite dimensional fuzzy normed linear space, Fuzzy Set. Syst., 48 (1992), 239–248. https://doi.org/10.1016/0165-0114(92)90338-5 doi: 10.1016/0165-0114(92)90338-5
[3]	S. V. Krishna, K. K. M. Sarma, Separation of fuzzy normed linear spaces, Fuzzy Set. Syst., 63 (1994), 207–217. https://doi.org/10.1016/0165-0114(94)90351-4 doi: 10.1016/0165-0114(94)90351-4
[4]	J.-Z. Xiao, X.-H. Zhu, Fuzzy normed spaces of operators and its completeness, Fuzzy Set. Syst., 133 (2003), 389–399. https://doi.org/10.1016/S0165-0114(02)00274-9 doi: 10.1016/S0165-0114(02)00274-9
[5]	S. C. Cheng, J. N. Mordeson, Fuzzy linear operator and fuzzy normed linear spaces, Bulletin of the Calcutta Mathematical Society, 86 (1994), 429–436.
[6]	V. Balopoulos, A. G. Hatzimichailidis, B. K. Papadopoulos, Distance and similarity measures for fuzzy operators, Inform. Sciences, 177 (2007), 2336–2348. https://doi.org/10.1016/j.ins.2007.01.005 doi: 10.1016/j.ins.2007.01.005
[7]	T. Bag, S. K. Samanta, Finite dimensional fuzzy normed linear spaces, The Journal of Fuzzy Mathematics, 11 (2003), 687–705.
[8]	S. M. Ulam, A collection of the mathematical problems, New York: Interscience, 1960. https://doi.org/10.1126/science.132.3428.6
[9]	D. H. Hyers, On the stability of the linear functional equation, Proc. N. A. S., 27 (1941), 222–224. https://doi.org/10.1073/pnas.27.4.222 doi: 10.1073/pnas.27.4.222
[10]	K. Tamilvanan, F. Özger, S. A. Mohiuddine, N. Ahmad, M. J. Kabeto, Fixed point technique: Stability analysis of quadratic functional equation in various Quasi-Banach spaces, J. Math., 2025 (2025), 6689441. https://doi.org/10.1155/jom/6689441 doi: 10.1155/jom/6689441
[11]	G. Senthil, N. Vijaya, S. Gayathri, P. Vijayalakshmi, M. Balamurugan, S. Karthikeyan, Hyers-Ulam stability of quartic functional equation in IFN-spaces and 2-Banach spaces by classical methods, Int. J. Anal. Appl., 23 (2025), 68. https://doi.org/10.28924/2291-8639-23-2025-68 doi: 10.28924/2291-8639-23-2025-68
[12]	H. Koh, Hyers-Ulam-Rassias stability of reciprocal-type functional equations: Comparative study of direct and fixed point methods, Symmetry, 17 (2025), 1626. https://doi.org/10.3390/sym17101626 doi: 10.3390/sym17101626
[13]	T. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc., 72 (1978), 297–300. https://doi.org/10.1090/S0002-9939-1978-0507327-1 doi: 10.1090/S0002-9939-1978-0507327-1
[14]	J. M. Rassias, On approximation of approximately linear mappings by linear mappings, Bull. Sci. Math., 108 (1984), 445–446.
[15]	Z. Gajda, On stability of additive mappings, International Journal of Mathematics and Mathematical Sciences, 14 (1991), 431–434. https://doi.org/10.1155/S016117129100056X doi: 10.1155/S016117129100056X
[16]	P. Găvruța, A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings, J. Math. Anal. Appl., 184 (1994), 431–436. https://doi.org/10.1006/jmaa.1994.1211 doi: 10.1006/jmaa.1994.1211
[17]	S.-M. Jung, Hyers-Ulam-Rassias stability of functional equations in mathematical analysis, Florida: Hadronic Press, 2001.
[18]	G. Lu, C. Park, Hyers-Ulam stability of additive set-valued functional equations, Appl. Math. Lett., 24 (2011), 1312–1316. https://doi.org/10.1016/j.aml.2011.02.024 doi: 10.1016/j.aml.2011.02.024
[19]	I.-S Chang, Stability of higher ring derivations in fuzzy Banach algebras, J. Comput. Anal. Appl., 14 (2012), 1059–1066.
[20]	C.-G. Park, Homomorphisms between Poisson $JC^$-algebra, Bull. Braz. Math. Soc.*, 36 (2005), 79–97. https://doi.org/10.1007/s00574-005-0029-z doi: 10.1007/s00574-005-0029-z
[21]	C. Park, Hyers-Ulam-Rassias stability of homomorphisms in quasi-Banach algebras, B. Sci. Math., 132 (2008), 87–96. https://doi.org/10.1016/j.bulsci.2006.07.004 doi: 10.1016/j.bulsci.2006.07.004
[22]	P. Malliavin, Stochastic analysis, Berlin: Springer, 1997. https://doi.org/10.1007/978-3-642-15074-6
[23]	B. Hu, Y. K. Liao, Convergence conditions for extreme solutions of an impulsive differential system, AIMS Mathematics, 10 (2025), 10591–10604. https://doi.org/10.3934/math.2025481 doi: 10.3934/math.2025481
[24]	Y. Li, Z. H. Rui, B. Hu, Monotone iterative and quasilinearization method for a nonlinear integral impulsive differential equation, AIMS Mathematics, 10 (2025), 21–37. https://doi.org/10.3934/math.2025002 doi: 10.3934/math.2025002
[25]	C. Park, Y. S. Cho, M.-H. Han, Functional inequalities associated with Jordan-von Neumann type additive functional equations, J. Inequal. Appl., 2007 (2007), 41820. https://doi.org/10.1155/2007/41820 doi: 10.1155/2007/41820
[26]	Y. J. Cho, C. Park, R. Saadati, Functional inequalities in non-Archimedean Banach spaces, Appl. Math. Lett., 23 (2010), 1238–1242. https://doi.org/10.1016/j.aml.2010.06.005 doi: 10.1016/j.aml.2010.06.005
[27]	G. Lu, C. Park, Additive functional inequalities in Banach spaces, J. Inequal. Appl., 2012 (2012), 294. https://doi.org/10.1186/1029-242X-2012-294 doi: 10.1186/1029-242X-2012-294
[28]	G. Lu, J. C. Xin, Y. F. Jin, C. Park, Approximation of general Pexider functional inequalities in fuzzy Banach spaces, J. Nonlinear Sci. Appl., 12 (2019), 206–216. https://doi.org/10.22436/jnsa.012.04.02 doi: 10.22436/jnsa.012.04.02
[29]	M. Fang, G. Lu, D. H. Pei, Functional inequalities in generalized quasi-Banach spaces, J. Nonlinear Sci. Appl., 9 (2016), 2481–2491. http://doi.org/10.22436/jnsa.009.05.47 doi: 10.22436/jnsa.009.05.47
[30]	G. L. Forti, Comments on the core of the direct method for proving Hyers-Ulam stability of functional equations, J. Math. Anal. Appl., 295 (2004), 127–133. https://doi.org/10.1016/j.jmaa.2004.03.011 doi: 10.1016/j.jmaa.2004.03.011
[31]	J.-R. Wu, Z.-Y. Jin, A note on Ulam stability of some fuzzy number-valued functional equations, Fuzzy Set. Syst., 375 (2019), 191–195. https://doi.org/10.1016/j.fss.2018.10.018 doi: 10.1016/j.fss.2018.10.018

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