Ulam stability of hom-ders in fuzzy Banach algebras

Araya Kheawborisut; Siriluk Paokanta; Jedsada Senasukh; Choonkil Park; Araya Kheawborisut; Siriluk Paokanta; Jedsada Senasukh; Choonkil Park

doi:10.3934/math.2022907

AIMS Mathematics

2022, Volume 7, Issue 9: 16556-16568. doi: 10.3934/math.2022907

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Ulam stability of hom-ders in fuzzy Banach algebras

1.
Department of Mathematics, School of Science, King Mongkut's Institute of Technology Ladkrabang, Bangkok 10520, Thailand
2.
Department of Mathematics, Research Institute for Natural Sciences, Hanyang University, Seoul 04763, Korea
3.
Department of Mathematics, Faculty of Science, Khon Kaen University, Khon Kaen 40002, Thailand

Received: 11 May 2022 Revised: 28 June 2022 Accepted: 05 July 2022 Published: 08 July 2022
MSC : 47H10, 39B82

This paper aims to investigate a new type of derivations in a fuzzy Banach algebra. Moreover, by using the fixed point method, we obtain some stability results of the hom-der in fuzzy Banach algebras associated with the functional equation

$ f(x+{\textbf{k}}y) = f(x)+{\textbf{k}}f(y) $

where $ {\textbf{k}} $ is a fixed positive integer greater than $ 1 $.
- Ulam stability result,
- hom-der,
- fuzzy Banach algebra,
- fixed point method
Citation: Araya Kheawborisut, Siriluk Paokanta, Jedsada Senasukh, Choonkil Park. Ulam stability of hom-ders in fuzzy Banach algebras[J]. AIMS Mathematics, 2022, 7(9): 16556-16568. doi: 10.3934/math.2022907

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Abstract

This paper aims to investigate a new type of derivations in a fuzzy Banach algebra. Moreover, by using the fixed point method, we obtain some stability results of the hom-der in fuzzy Banach algebras associated with the functional equation

$ f(x+{\textbf{k}}y) = f(x)+{\textbf{k}}f(y) $

where $ {\textbf{k}} $ is a fixed positive integer greater than $ 1 $.

References

[1]	T. Aoki, On the stability of the linear transformation in Banach spaces, J. Math. Soc. Japan, 2 (1950), 64–66. http://dx.doi.org/10.2969/jmsj/00210064 doi: 10.2969/jmsj/00210064
[2]	T. Bag, S. K. Samanta, Finite dimensional fuzzy normed linear spaces J. Fuzzy Math., 11 (2003), 687–705.
[3]	T. Bînzar, F. Pater, S. Nǎdǎban, On fuzzy normed algebras, J. Nonlinear Sci. Appl., 9 (2016), 5488–5496. http://dx.doi.org/10.22436/jnsa.009.09.16 doi: 10.22436/jnsa.009.09.16
[4]	L. Cǎdariu, V. Radu, Fixed points and the stability of Jensen's functional equation, J. Inequal. Pure Appl. Math., 4 (2003), 4.
[5]	L. Cǎdariu, V. Radu, On the stability of the Cauchy functional equation: A fixed point approach, Grazer Math. Ber., 346 (2004), 43–52.
[6]	Y. Cho, C. Park, Y. Yang, Stability of derivations in fuzzy normed algebras, J. Nonlinear Sci. Appl., 8 (2015), 1–7. http://dx.doi.org/10.22436/jnsa.008.01.01 doi: 10.22436/jnsa.008.01.01
[7]	J. Diaz, B. Margolis, A fixed point theorem of the alternative for contractions on a generalized complete metric space, Bull. Am. Math. Soc., 74 (1968), 305–309. http://dx.doi.org/10.1090/S0002-9904-1968-11933-0 doi: 10.1090/S0002-9904-1968-11933-0
[8]	P. Gǎvruta, A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings, J. Math. Anal. Appl., 184 (1994), 431–436. http://dx.doi.org/10.1006/jmaa.1994.1211 doi: 10.1006/jmaa.1994.1211
[9]	Z. Gajda, On stability of additive mappings, Int. J. Math. Math. Sci., 14 (1991), 817959. http://dx.doi.org/10.1155/S016117129100056X doi: 10.1155/S016117129100056X
[10]	D. H. Hyers, On the stability of the linear functional equation, Proc. Nat. Acad. Sci., 27 (1941), 222–224. http://dx.doi.org/10.1073/pnas.27.4.222 doi: 10.1073/pnas.27.4.222
[11]	M. Isar, G. Lu, Y. Jin, C. Park, A general additive functional inequality and derivation in Banach algebras, J. Math. Inequal., 15 (2021), 305–321. http://dx.doi.org/10.7153/jmi-2021-15-23 doi: 10.7153/jmi-2021-15-23
[12]	Y. Lee, S. Jung, M. T. Rassias, Uniqueness theorems on functional inequalities concerning cubic-quadratic-additive equation, J. Math. Inequal., 12 (2018), 43–61. http://dx.doi.org/10.7153/jmi-2018-12-04 doi: 10.7153/jmi-2018-12-04
[13]	M. Mirzavaziri, M. S. Moslehian, A fixed point approach to stability of a quadratic equation, Bull. Braz. Math. Soc., 37 (2006), 361–376. http://dx.doi.org/10.1007/s00574-006-0016-z doi: 10.1007/s00574-006-0016-z
[14]	A. K. Mirmostafaee, M. Mirzavaziri, M. S. Moslehian, Fuzzy stability of the Jensen functional equation, Fuzzy Sets Syst., 159 (2008), 730–738. http://dx.doi.org/10.1016/j.fss.2007.07.011 doi: 10.1016/j.fss.2007.07.011
[15]	A. K. Mirmostafaee, M. S. Moslehian, Fuzzy approximately cubic mappings, Inf. Sci., 178 (2008), 3791–3798. https://doi.org/10.1016/j.ins.2008.05.032 doi: 10.1016/j.ins.2008.05.032
[16]	A. K. Mirmostafaee, Perturbation of generalized derivations in fuzzy Menger normed algebras, Fuzzy Sets Syst., 195 (2012), 109–117. http://dx.doi.org/10.1016/j.fss.2011.10.015 doi: 10.1016/j.fss.2011.10.015
[17]	S. Nǎdǎban, I. Dzitac, Atomic decompositions of fuzzy normed linear spaces for wavelet applications, Informatica, 25 (2014), 643–662. https://doi.org/10.15388/Informatica.2014.33 doi: 10.15388/Informatica.2014.33
[18]	B. Noori, M. B. Moghimi, B. Khosravi, C. Park, Stability of some functional equations on bounded domains, J. Math. Inequal., 14 (2020), 455–472. http://dx.doi.org/10.7153/jmi-2020-14-29 doi: 10.7153/jmi-2020-14-29
[19]	C. Park, Fixed points and Hyers-Ulam-Rassias stability of Cauchy-Jensen functional equations in Banach algebras, Fixed Point Theory Appl., 2007 (2007), 50175. http://dx.doi.org/10.1155/2007/50175 doi: 10.1155/2007/50175
[20]	C. Park, Generalized Hyers-Ulam-Rassias stability of quadratic functional equations: A fixed point approach, Fixed Point Theory Appl., 2008 (2008), 493751. http://dx.doi.org/10.1155/2008/493751 doi: 10.1155/2008/493751
[21]	C. Park, Homomorphisms between Poisson $JC^{\ast}$-algebras, Bull. Braz. Math. Soc. (N.S.), 36 (2005), 79–97. http://dx.doi.org/10.1007/s00574-005-0029-z doi: 10.1007/s00574-005-0029-z
[22]	V. Radu, The fixed point alternative and the stability of functional equations, Fixed Point Theory, 4 (2003), 91–96.
[23]	Th. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Am. Math. Soc., 72 (1978), 297–300. http://dx.doi.org/10.1090/S0002-9939-1978-0507327-1 doi: 10.1090/S0002-9939-1978-0507327-1
[24]	R. Saadati, C. Park, Non-Archimedean $\mathcal{L}$-fuzzy normed spaces and stability of functional equations, Comput. Math. Appl., 60 (2010), 2488–2496. http://dx.doi.org/10.1016/j.camwa.2010.08.055 doi: 10.1016/j.camwa.2010.08.055
[25]	M. Sarfraz, Y. Li, Minimum functional equation and some Pexider-type functional equation on any group, AIMS Math., 6 (2021), 11305–11317. http://dx.doi.org/10.3934/math.2021656 doi: 10.3934/math.2021656
[26]	Z. Wang, Approximate mixed type quadratic-cubic functional equation, AIMS Math., 6 (2021), 3546–3561. http://dx.doi.org/10.3934/math.2021211 doi: 10.3934/math.2021211
[27]	S. M. Ulam, A collection of the mathematical problems, Interscience, New York, 1960.

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