Refined stability of the additive, quartic and sextic functional equations with counter-examples

Hasanen A. Hammad; Hassen Aydi; Manuel De la Sen; Hasanen A. Hammad; Hassen Aydi; Manuel De la Sen

doi:10.3934/math.2023736

AIMS Mathematics

2023, Volume 8, Issue 6: 14399-14425. doi: 10.3934/math.2023736

Previous Article Next Article

Research article

Refined stability of the additive, quartic and sextic functional equations with counter-examples

1.
Department of Mathematics, Unaizah College of Sciences and Arts, Qassim University, Buraydah 52571, Saudi Arabia
2.
Department of Mathematics, Faculty of Science, Sohag University, Sohag 82524, Egypt
3.
Université de Sousse, Institut Supérieur d'Informatique et des Techniques de Communication, H. Sousse 4000, Tunisia
4.
China Medical University Hospital, China Medical University, Taichung 40402, Taiwan
5.
Department of Mathematics and Applied Mathematics, Sefako Makgatho Health Sciences University, Ga-Rankuwa, South Africa
6.
Institute of Research and Development of Processes, Department of Electricity and Electronics, Faculty of Science and Technology, University of the Basque Country, 48940-Leioa (Bizkaia), Spain

Received: 04 February 2023 Revised: 10 April 2023 Accepted: 12 April 2023 Published: 19 April 2023
MSC : 39B52, 39B72, 39B82

In this study, we utilize the direct method (Hyers approach) to examine the refined stability of the additive, quartic, and sextic functional equations in modular spaces with and without the $ \Delta _{2} $-condition. We also use the direct approach to discuss the Ulam stability in $ 2 $-Banach spaces. Ultimately, we ensure that stability of above equations does not hold in a particular scenario by utilizing appropriate counter-examples.
- refined stability,
- modular space,
- $ 2 $-Banach space,
- additive,
- quartic and sextic functional equations
Citation: Hasanen A. Hammad, Hassen Aydi, Manuel De la Sen. Refined stability of the additive, quartic and sextic functional equations with counter-examples[J]. AIMS Mathematics, 2023, 8(6): 14399-14425. doi: 10.3934/math.2023736

Related Papers:

Abstract

In this study, we utilize the direct method (Hyers approach) to examine the refined stability of the additive, quartic, and sextic functional equations in modular spaces with and without the $ \Delta _{2} $-condition. We also use the direct approach to discuss the Ulam stability in $ 2 $-Banach spaces. Ultimately, we ensure that stability of above equations does not hold in a particular scenario by utilizing appropriate counter-examples.

References

[1]	S. M. Ulam, A collection of mathematical problems, New York: Interscience, 1960.
[2]	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
[3]	T. Aoki, On the stability of the linear transformation in Banach spaces, J. Math. Soc. Japan, 2 (1950), 64–66. https://doi.org/10.2969/jmsj/00210064 doi: 10.2969/jmsj/00210064
[4]	T. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc., 72 (1978), 297–300.
[5]	P. Gavruta, 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
[6]	A. Charifi, R. Lukasik, D. Zeglami, A special class of functional equations, Math. Slovaca, 68 (2018), 397–404. https://doi.org/10.1515/ms-2017-0110 doi: 10.1515/ms-2017-0110
[7]	H. A. Hammad, R. A. Rashwan, A. Nafea, M. E. Samei, S. Noeiaghdam, Stability analysis for a tripled system of fractional pantograph differential equations with nonlocal conditions, J. Vib. Control, 2023. https://doi.org/10.1177/10775463221149232 doi: 10.1177/10775463221149232
[8]	H. A. Hammad, H. Aydi, H. Isik, M. De la Sen, Existence and stability results for a coupled system of impulsive fractional differential equations with Hadamard fractional derivatives, AIMS Math., 8 (2023), 6913–6941. https://doi.org/10.3934/math.2023350 doi: 10.3934/math.2023350
[9]	H. A. Hammad, R. A. Rashwan, A. Nafea, M. E. Samei, M. De la Sen, Stability and existence of solutions for a tripled problem of fractional hybrid delay differential equations, Symmetry, 14 (2022), 2579. https://doi.org/10.3390/sym14122579 doi: 10.3390/sym14122579
[10]	H. A. Hammad, M. Zayed, Solving a system of differential equations with infinite delay by using tripled fixed point techniques on graphs, Symmetry, 14 (2022), 1388. https://doi.org/10.3390/sym14071388 doi: 10.3390/sym14071388
[11]	H. A. Hammad, P. Agarwal, S. Momani, F. Alsharari, Solving a fractional-order differential equation using rational symmetric contraction mappings, Fractal Fract., 5 (2021), 159. https://doi.org/10.3390/fractalfract5040159 doi: 10.3390/fractalfract5040159
[12]	R. A. Rashwan, H. A. Hammad, M. G. Mahmoud, Common fixed point results for weakly compatible mappings under implicit relations in complex valued $G$-metric spaces, Inform. Sci. Lett., 8 (2019), 111–119. https://doi.org/10.18576/isl/080305 doi: 10.18576/isl/080305
[13]	H. A. Hammad, M. De la Sen, Fixed-point results for a generalized almost $(s, q)$-Jaggi $F$-contraction-type on $b$-metric-like spaces, Mathematics, 8 (2020), 63. https://doi.org/10.3390/math8010063 doi: 10.3390/math8010063
[14]	H. Nakano, Modulared semi-ordered linear spaces, Tokyo: Maruzen Company, Ltd., 1950.
[15]	W. A. J. Luxemburg, Banach function spaces, Ph.D. Thesis, Delft, the Netherlands: Technische Hogeschool Delft, 1955.
[16]	I. Amemiya, On the representation of complemented modular lattices, J. Math. Soc. Japan, 9 (1957), 263–279. https://doi.org/10.2969/jmsj/00920263 doi: 10.2969/jmsj/00920263
[17]	J. Musielak, Orlicz spaces and nodular spaces, Berlin, Heidelberg: Springer, 1983.
[18]	S. Koshi, T. Shimogaki, On $F$-norms of quasi-modular spaces, J. Fac. Sci. Hokkaido Univ. Ser. 1 Math., 15 (1961), 202–218.
[19]	B. Mazur, Modular curves and the Eisenstein ideal, Publ. Math. l'IHÉS, 47 (1977), 33–186. https://doi.org/10.1007/BF02684339 doi: 10.1007/BF02684339
[20]	P. Turpin, Fubini inequalities and bounded multiplier property in generalized modular spaces, Comment. Math., 1 (1978), 331–353.
[21]	W. Orlicz, Collected papers. Part I, II, Warsaw, Poland: PWN-Polish Scientific Publishers, 1988.
[22]	L. Maligranda, Orlicz spaces and interpolation, Campinas: Universidade Estadual de Campinas, 1989.
[23]	M. A. Khamsi, Quasicontraction mappings in modular spaces without $\Delta _{2}$-condition, Fixed Point Theory Appl., 2008 (2008), 916187. https://doi.org/10.1155/2008/916187 doi: 10.1155/2008/916187
[24]	G. Sadeghi, A fixed point approach to stability of functional equations in modular spaces, Bull. Malays. Math. Sci. Soc., 37 (2014), 333–344.
[25]	H. M. Kim, H. Y. Shin, Refined stability of additive and quadratic functional equations in modular spaces, J. Inequal. Appl., 2017 (2017), 146. https://doi.org/10.1186/s13660-017-1422-z doi: 10.1186/s13660-017-1422-z
[26]	S. Gähler, $2$-metrische Räume und ihre topologische struktur, Math. Nachr., 26 (1963), 115–148. https://doi.org/10.1002/mana.19630260109 doi: 10.1002/mana.19630260109
[27]	W. G. Park, Approximate additive mappings in 2-Banach spaces and related topics, J. Math. Anal. Appl., 376 (2011), 193–202. https://doi.org/10.1016/j.jmaa.2010.10.004 doi: 10.1016/j.jmaa.2010.10.004
[28]	K. Wongkum, P. Chaipunya, P. Kumam, On the generalized Ulam-Hyers-Rassias stability of quadratic mappings in modular spaces without $\Delta _{2}$-conditions, J. Funct. Spaces, 2015 (2015), 461719. https://doi.org/10.1155/2015/461719 doi: 10.1155/2015/461719
[29]	H. M. Kim, I. S. Chang, E. Son, Stability of Cauchy additive functional equation in fuzzy Banach spaces, Math. Inequal. Appl., 16 (2013), 1123–1136.
[30]	Z. Gajda, On stability of additive mappings, Int. J. Math. Math. Sci., 14 (1991), 431–434.
[31]	N. Uthirasamy, K. Tamilvanan, H. K. Nashine, R. George, Solution and stability of quartic functional equations in modular spaces by using Fatou property, J. Funct. Spaces, 2022 (2022), 5965628. https://doi.org/10.1155/2022/5965628 doi: 10.1155/2022/5965628
[32]	S. A. Mohiuddine, K. Tamilvanan, M. Mursaleen, T. Alotaibi, Stability of quartic functional equation in modular spaces via Hyers and fixed-point methods, Mathematics, 10 (2022), 1938. https://doi.org/10.3390/math10111938 doi: 10.3390/math10111938

Reader Comments

Your name:*

Email:*
© 2023 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)