International Journal of Nonlinear Analysis and Applications

Stability of orthogonally quintic functional equation on C*-algebras with the type fixed point alternative

Document Type : Research Paper

Authors

1 Department of Mathematics, Shiraz Branch, Islamic Azad University, Shiraz, Iran

2 Department of Mathematics, Semnan University, P.O. Box 35195-363, Semnan, Iran

Abstract

Using fixed point methods, we prove the stability of orthogonally quintic functional equation on C ∗ -algebras for the functional equation
$$Df(x, y) = f(3x + y) − 5f(2x + y) + f(2x − y) + 10f(x + y) − 5f(x − y) − 10f(y) − f(3x) + 3f(2x) + 27f(x).$$

Keywords

[1] N. Ansari, M.H. Hooshmand, M. Eshaghi Gordji, and K. Jahedi, Stability of fuzzy orthogonally ∗-n-derivation in orthogonally fuzzy C∗-algebras, Int. J. Nonlinear Anal. Appl. 12 (2021), no. 1, 533–540.
[2] A. Bahraini, G. Askari, M. Eshaghi Gordji, and R. Gholami, Stability and hyperstability of orthogonally ∗-m-homomorphisms in orthogonally Lie C∗-algebras: A fixed point approach, J. Fixed Point Theory Appl. 20 (2018), 2, 1–12.
[3] A. Bahraini, G. Askari, and M. Eshaghi Gordji, On approximate orthogonally ring homomorphisms and orthogonally ring derivations in Banach algebras with the new type fixed point, J. Mahani Math. Res. Center 10 (2021), no. 2, 115–124.
[4] A. Bahyrycz, J. Brzdek, E.S. El-Hady, and Z. Lesniak, On Ulam stability of functional equations in 2-normed spaces: A survey, Symmetry 13 (2021), 2200.
[5] A. Bodaghi, Quintic functional equations in non-Archimedean normed spaces, J. Math. Exten. 9 (2015), 51–63.
[6] A. Bodaghi and A. Fosner, Characterization, stability and hyperstability of multi-quadratic–cubic mappings, J. Ineq. Appl. 2021 (2021), no. 1, 1–12.
[7] L. Cadariu and V. Radu, Fixed points and the stability of Jensen’s functional equation, J. Inequal. Pure Appl. Math. 4 (2003), no. 1, 4.
[8] L. Cadariu, L. Gavruta, and P. Gavruta, Fixed points and generalized Hyers-Ulam stability, Abstr. Appl. Anal. 2012 (2012).
[9] Y.J. Cho, C. Park, and M. Eshaghi Gordji, Approximate additive and quadratic mappings in 2-Banach spaces and related topics, Int. J. Nonlinear Anal. Appl. 3 (2012), no. 2, 75–81.
[10] A. Dinmohammadi, Orthogonally fixed points and (m, m)-hom-derivation equations, J. Math. 2022 (2022).
[11] Z. Eivazi Damirchi, M. Eshaghi Gordji, and D. Ebrahimi Bagha, Banach fixed point theorem on orthogonal cone metric spaces, Facta Univer. (NIS) Ser. Math. Inf. 35 (2020), 1239–1250.
[12] E.S. El-Hady and S. Ogrekci, On Hyers-Ulam-Rassias stability of fractional differential equations with Caputo derivative, J. Math. Comput. Sci. 22 (2021), 325–332.
[13] M. Eshaghi Gordji, M. Ramezani, M. De La Sen, and Y.J. Cho, On orthogonal sets and Banach fixed point theorem, Fixed Point Theory 18 (2017), no. 2, 569–578.
[14] M. Eshaghi Gordji, G. Askari, N. Ansari, G.A. Anastassiou, and Choonkil Park, Stability and hyperstability of generalized orthogonally quadratic ternary homomorphisms in non-Archimedean ternary Banach algebras: a fixed point approach, J. Comput. Anal. Appl. 21 (2016), no. 3, 1–7.
[15] P. Gavruta, A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings, J. Math. Anal. Appl. 184, (1994), no. 3, 431–436.
[16] M.B. Ghaemi, M. Choubin, G. Sadeghi, and M. Eshaghi Gordji, A fixed point approach to stability of quintic functional equations in modular spaces, Kyungpook Math. J. 55 (2015), no. 2, 313–326.
[17] R. Gholami, G. Askari, and M. Eshaghi Gordji, Stability and hyperstability of orthogonally ring ∗-n-derivations and orthogonally ring ∗-n-homomorphisms on C∗-algebras, J. Linear Topol. Algebra 7 (2018), no. 2, 109–119.
[18] H.D. Hyers, On the stability of the linear functional equation, Proc. Nat. Acad. Sci. United States Amer. 27 (1941), no. 4, 222–224.
[19] S.M. Jung, and J.M. Rassias, A fixed point approach to the stability of a functional equation of the spiral of Theodorus, Fixed Point Theory Appl. 2008 (2008), 1–7.
[20] V. Keshavarz, S. Jahedi, and M. Eshaghi Gordji, Ulam-Hyers stability of C*-ternary 3-Jordan derivations, Southeast Asian Bull. Math. 41 (2021), no. 1, 55–64.
[21] V. Keshavarz, S. Jahedi, M. Eshaghi Gordji, and S. Bazeghi, Hyperstability of orthogonally Pexider Lie functional equation: An orthogonally fixed point approach, Thai J. Math. 20 (2022), no. 1, 235–243.
[22] M.B. Moghimi and A. Najati, Some hyperstability and stability results for the Cauchy and Jensen equations, J. Inequal. Appl. 2022 (2022), no. 1, 1–12.
[23] I. Nikoufar, Lipschitz stability of n-cubic functional equations and applications, Miskolc Math. Notes 23 (2022), no. 1, 389–399.
[24] C. Park, J.L. Cui, and M. Eshaghi Gordji, Orthogonality and quintic functional equations, Acta Math. Sinica English Ser. 29 (2013), no. 7, 1381.
[25] T.M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), no. 2, 297–300.
[26] S.M. Ulam, Problems in Modern Mathematics New York, Wiley, 1940.
[27] S. Zolfaghari, A. Ebadian, S. Ostadbashi, M. De La Sen, and M. Eshaghi Gordji, A fixed point approach to the Hyers-Ulam stability of an AQ functional equation in probabilistic modular spaces, Int. J. Nonlinear Anal. Appl. 4 (2013), no. 2, 89-101.
International Journal of Nonlinear Analysis and Applications
Volume 16, Issue 3
March 2025
Pages 379-384
  • Receive Date: 01 July 2022
  • Revise Date: 10 September 2022
  • Accept Date: 17 September 2022