Modular stability results of generalized quartic functional equations by using Fatou property with $\Delta_{2}$-condition
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Abstract
In this research work, we introduce a new kind of finite variable quartic functional equation and examine the Hyers-Ulam stability of this quartic functional equation in modular spaces via fixed point technique with the help of Fatou property with $\Delta_{2}$-condition. A counterexample is also provided to prove the non-stability for singular case.
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References
[1] I. Amemiya, On the representation of complemented modular lattices, J. Math. Soc. Japan, 9 (1957), 263–279. Google Scholar
[2] T. Aoki, On the stability of the linear transformation in Banach spaces, J. Math. Soc. Japan, 2 (1950), 64–66. Google Scholar
[3] J. A. Baker, The stability of certain functional equations, Proc. Amer. Math. Soc., 112 (1991), 729–732. Google Scholar
[4] A. Baza, M. Rossafi, Generalized Hyers-Ulam stability of quadratic functional inequality in modular spaces and β-homogeneous Banach spaces, Nonlinear Funct. Anal. Appl., 29 (2024), no. 1, 295–306. https://doi.org/10.22771/nfaa.2024.29.01.18 Google Scholar
[5] L. Cadariu, V. Radu, On the stability of the Cauchy functional equation: a fixed point approach, Graz. Math., 346 (2004), 43–52. Google Scholar
[6] L. Cadariu, V. Radu, Fixed point methods for the generalized stability of functional equations in a single variable, Fixed Point Theory Appl., 2008 (2008), Art. ID. 749392. Google Scholar
[7] I. Chang, H. Kim, Almost quadratic Lie ∗-derivations on convex modular ∗-algebras, Nonlinear Funct. Anal. Appl., 28 (2023), no. 4, 887–902. https://doi.org/10.22771/nfaa.2023.28.04.03 Google Scholar
[8] Z. Gajda, On stability of additive mappings, Int. J. Math. Math. Sci., 14 (1991), 431–434. https://doi.org/10.1155/S016117129100056X Google Scholar
[9] V. Govindan, J. R. Lee, S. Pinelas, P. Muniyappan, General solution and Ulam stability of generalized CQ functional equation, Korean J. Math., 30 (2022), no. 2, 403–412. https://doi.org/10.11568/kjm.2022.30.2.403 Google Scholar
[10] D. H. Hyers, On the stability of the linear functional equation, Proc. Natl. Acad. Sci. U.S.A., 27 (1941), 222–224. Google Scholar
[11] S. Jung, On the Hyers-Ulam stability of functional equation that have the quartic property, J. Math. Anal. Appl., 222 (1998), 126–137. https://doi.org/10.1006/jmaa.1998.5916 Google Scholar
[12] H. Kim, Y. Hong, Approximate quadratic mappings in modular spaces, Int. J. Pure Appl. Math., 116 (2017), 31–43. https://doi.org/10.12732/ijpam.v116i1.3 Google Scholar
[13] S. Koshi, T. Shimogaki, On F-norms of quasi-modular spaces, J. Fac. Sci., Hokkaido Univ., Ser., 1 (1961), no. 15, 202–218. Google Scholar
[14] M. Krbec, Modular interpolation spaces, I. Z. Anal. Anwend., 1 (1982), 25–40. Google Scholar
[15] S. Lee, S. Im, I. Hwang, Quadratic functional equations, J. Math. Anal. Appl., 307 (2005), 387–394. https://doi.org/10.1016/j.jmaa.2004.12.062 Google Scholar
[16] W. A. Luxemburg, Banach function spaces, Ph.D. Thesis, Delft University of Technology, Delft, The Netherlands, 1959. Google Scholar
[17] L. Maligranda, Orlicz Spaces and Interpolation, Seminários de Mathemática, vol. 5. Universidade Estadual de Campinas, Departamento de Matemática, Campinas, 1989. Google Scholar
[18] B. Mazur, Modular curves and the Eisenstein ideal, Publ. Math. IHÉS, 47 (1978), 33–186. Google Scholar
[19] J. Musielak, Orlicz Spaces and Modular Spaces, Lecture Notes in Mathematics, 1034. Springer, Berlin, 1983. Google Scholar
[20] H. Nakano, Modulared Semi-Ordered Linear Spaces, Maruzen, Tokyo, 1950. Google Scholar
[21] W. Orlicz, Collected Papers, vols. I, II, PWN, Warszawa, 1988. Google Scholar
[22] C. Park, Fixed points and Hyers-Ulam-Rassias stability of Cauchy-Jensen functional equations in Banach algebras, Fixed Point Theory Appl., 2007 (2007), Art. ID 50175. https://doi.org/10.1155/2007/50175 Google Scholar
[23] Th. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc., 72 (1978), 297–300. Google Scholar
[24] G. Sadeghi, A fixed point approach to stability of functional equations in modular spaces, Bull. Malays. Math. Sci. Soc., 37 (2014), 333–344. Google Scholar
[25] M. Sarfraz, Y. Li, Minimum functional equation and some Pexider-type functional equation on any group, AIMS Math., 6 (2021), 11305–11317. Google Scholar
[26] K. Y. N. Sayar, A. Bergam, Approximate solutions of a quadratic functional equation in 2 Banach spaces using fixed point theorem, J. Fixed Point Theory Appl., 22 (2020), Paper No.3. https://doi.org/10.1007/s11784-019-0739-2 Google Scholar
[27] Y. Sayyari, M. Dehghanian, C. Park, Stability and solution of two functional equations in unital algebras, Korean J. Math., 31 (2023), no. 3, 363–372. https://doi.org/10.11568/kjm.2023.31.3.363 Google Scholar
[28] P. Turpin, Fubini inequalities and bounded multiplier property in generalized modular spaces, Comment. Math. Spec. Issue 1 (1978), 331–353. Google Scholar
[29] S. M. Ulam, Problems in Modern Mathematics, Wiley, New York, 1960. Google Scholar
[30] Z. Wang, Approximate mixed type quadratic-cubic functional equation, AIMS Math., 6 (2021), 3546–3561. https://doi.org/10.13934/math.2021211 Google Scholar
[31] K. Wongkum, P. Chaipunya, P. Kumam, On the generalized Ulam-Hyers-Rassias stability of quadratic mappings in modular spaces without 2-conditions, J. Funct. Spaces, 2015 (2015), Art. ID 461719. Google Scholar
[32] S. Yamamuro, On conjugate spaces of Nakano spaces, Trans. Amer. Math. Soc., 90 (1959), 291–311. Google Scholar
[33] G. Zamani Eskandani, J. M. Rassias, Stability of general A-cubic functional equations in modular spaces, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM, 112 (2018), 425–435. Google Scholar
[34] D. Zhang, J. M. Rassias, Y. Li, On the Hyers-Ulam solution and stability problem for general set-valued Euler-Lagrange quadratic functional equations, Korean J. Math., 30 (2022), no. 4, 571–592. https://doi.org/10.11568/kjm.2022.30.4.571 Google Scholar