Document Type : Original Article


Department of Mathematics, West Tehran Branch, Islamic Azad University, Tehran, Iran



In this paper, the Gavruta stability of two multi-quadratic functional equations is established by a known fixed point theorem. As an example, the Hyers-Ulam, and Rassias stability and hyperstability of the mentioned mappings are proved in the setting of Banach spaces.


[1] T. Aoki, On the stability of the linear transformation in Banach spaces, J. Math. Soc. Japan., 2 (1950), 64–66.
[2] M. Arunkumar, A. Bodaghi, J. M. Rassias, and E. Sathya, The general solution and approximations of a decic type functional equation in various normed spaces, J. Chung. Math. Soc., 29(1) (2016), 287–328.
[3] A. Bodaghi, Stability of a mixed type additive and quartic functional equation, Filomat, 28 (2014), 1629–1640.
[4] A. Bodaghi and I. A. Alias, Approximate ternary quadratic derivations on ternary Banach algebras and C∗-ternary rings, Adv. Diff. Equa., 2012(1) (2012), 9 pages.
[5] A. Bodaghi, I. A. Alias, and M. H. Ghahramani, Approximately cubic functional equations and cubic multipliers, J. Inequ. Appl., 2011 (2011), Paper No. 53.
[6] A. Bodaghi, C. Park, and S. Yun, Almost multi-quadratic mappings in non-Archimedean spaces, AIMS Math., 5(5) (2020), 5230–5239.
[7] J. Brzdek, J. Chudziak, and Zs. Pales, A fixed point approach to stability of functional equations, Nonlinear Anal., 74 (2011), 6728–6732.
[8] J. Brzd,ek and K. Cieplinski, Hyperstability and superstability, Abstr. Appl. Anal., 2013 (2013), Art. ID. 401756, 13 pp.
[9] K. Cieplinski, On the generalized Hyers-Ulam stability of multi-quadratic mappings, Comput. Math. Appl., 62 (2011), 3418–3426.
[10] S. Czerwik, On the stability of the quadratic mapping in normed spaces, Abh. Math. Sem. Hamburg, 62 (1992), 59–64.
[11] N. J. Daras and Th. M. Rassias, Approximation and computation in science and Eengineering, Series Springer Optimization and Its Applications (SOIA). Vol. 180, Springer 2022.
[12] P. Gavruta, A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings, J. Math. Anal. Appl., 184(3) (1994), 431–436.
[13] D. H. Hyers, On the stability of the linear functional equation, Proc. Nat. Acad. Sci. USA., 27 (1941), 222–224.
[14] P. Kannappan, Functional equations and inequalities with applications, Springer, 2009.
[15] C. Park, A. Bodaghi, and I. A. Alias, Random stability and hyperstability of multi-quadratic mappings, J. Math. Inequ., 16(3) (2022), 993–1004.  
[16] J. M. Rassias, E. Sathya, and M. Arunkumar,Generalized Ulam-Hyers stability of an alternate additive-quadratic-quartic functional equation in fuzzy Banach spaces, Math. Anal. Cont. Appl., 3(1) (2021), 13–31.
[17] J. M. Rassias, M. Arunkumar, and E. Sathya, Non-stabilities of mixed type Euler-Lagrange k-cubic-quartic functional equation in various normed spaces, Math. Anal. Cont. Appl., 1(1) (2019), 1–43.
[18] Th. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc., 72(2) (1978), 297–300.
[19] P. K. Sahoo and P. Kannappan, Introduction to functional equations, CRC Press, Boca Raton, FL, 2011.
[20] F. Skof, Proprieta locali e approssimazione di operatori , Rend. Sem. Mat. Fis. Milano, 53 (1983), 113–129.
[21] S. Salimi and A. Bodaghi, A fixed point application for the stability and hyperstability of multi-Jensen-quadratic mappings, J. Fixed Point Theory Appl., 2020 (2020).
[22] S. M. Ulam, Problems in modern mathematics, Chapter VI, Science Ed., Wiley, New York, 1940.
[23] X. Zhao, X. Yang and C.-T. Pang, Solution and stability of the multiquadratic functional equation, Abstr. Appl. Anal., (2013) Art. ID 415053, 8 pp.