Hyers-Ulam stability of an additive-quadratic functional equation

  • Vediyappan Govindan Department of Mathematics, Sri Vidya Mandir Arts & Science College, Katteri, India.
  • Choonkil Park Research Institute of Natural Sciences, Hanyang University, Seoul-04763, Korea.
  • Sandra Pinelas Departamento de Ciências Exatas e Engenharia, Academia Militar, Portugal.
  • Themistocles M. Rassias Department of Mathematics, National Technical University of Athens, Greece.
Keywords: Hyers-Ulam stability, mixed type functional equation, Banach space, fixed point


In this paper, we introduce the following \((a,b,c)\)-mixed type functional equation of the form

\(g(ax_1+bx_2+cx_3 )-g(-ax_1+bx_2+cx_3 ) + g(ax_1-bx_2+cx_3 )\)\(-g(ax_1+bx_2-cx_3 ) + 2a^2 [g(x_1 ) + g(-x_1)] + 2b^2 [g(x_2 ) + g(-x_2)] + \)\(2c^2 [g(x_3 ) + g(-x_3)]+a[g(x_1 ) - g(-x_1)]+ b[g(x_2 )-g(-x_2)] + \)  \(c[g(x_3 )-g(-x_3)]=4g(ax_1+cx_3 )+2g(-bx_2)+\)  \(2g(bx_2)\)

where \(a,b,c\) are positive integers with \(a>1\), and investigate the solution and the Hyers-Ulam stability of the above functional equation in Banach spaces by using two different methods.


J. Aczél and J. Dhombres, Functional equations in several variables, Encyclopedia of Mathematics and its Applications, 31, Cambridge University Press, Cambridge, 1989.

L. Aiemsomboon and W. Sintunavarat, Stability of the generalized logarithmic functional equations arising from fixed point theory, Rev. R. Acad. Cienc. Exactas F ́ıs. Nat. Ser. A Mat. RACSAM 112 (2018), no. 1, 229–238.

T. Aoki, On the stability of the linear transformation in Banach spaces, J. Math. Soc. Japan 2 (1950), 64–66.

I. Chang, E. Lee and H. Kim, On the Hyers-Ulam-Rassias stability of a quadratic functional equations, Math. Inequal. Appl. 6 (2003), 87–95.

K. Ciepliński, Applications of fixed point theorems to the Hyers-Ulam stability of functional equations - a survey, Ann. Funct. Anal. 3 (2012), 151–164.

St. Czerwik, On the stability of the quadratic mapping in normed spaces, Abh. Math. Sem. Univ. Hamburg 62 (1992), 59–64.

S. Czerwik, Functional equations and inequalities in several variables, World Scientific Publishing Co., Inc., River Edge, NJ, 2002.

A. Ebadian, N. Ghobadipour and M. Eshaghi Gordji, A fixed point method for perturbation of bimultipliers and Jordan bimultipliers in C∗-ternary algebras, J. Math. Phys. 51 (2010), no. 10, 103508, 10 pp.

V. Govindan, S. Murthy and M. Saravanan, Solution and stability of a cubic type functional equation: Using direct and fixed point methods, Kraguj. J. Math. 44 (2020), 7–26.

P. Gavruta, A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings, J. Math. Anal. Appl. 184 (1994), no. 3, 431–436.

O. Hadzić and E. Pap, Fixed point theory in probabilistic metric spaces, Mathematics and its Applications, 536, Kluwer Academic Publishers, Dordrecht, 2001.

D. H. Hyers, G. Isac and T. M. Rassias, Stability of functional equations in several variables, Progress in Nonlinear Differential Equations and their Applications, 34, Birkäuser Boston, Inc., Boston, MA, 1998.

K. Jun and H. Kim, On the Hyers-Ulam-Rassias stability of a generalized quadratic and additive type functional equation, Bull. Korean Math. Soc. 42 (2005), 133–148.

S.-M. Jung, Hyers-Ulam-Rassias stability of functional equations in mathematical analysis, Hadronic Press, Inc., Palm Harbor, FL, 2001.

Y. Jung, The Ulam-Gavruta-Rassias stability of module left derivations, J. Math. Anal. Appl. 339 (2008), 108–114.

Pl. Kannappan, Quadratic functional equation and inner product spaces, Results Math. 27 (1995), no. 3-4, 368–372.

Y. Lee, S.-M. Jung and M. T. Rassias. Uniqueness theorems on functional inequalities concerning cubic-quadratic-additive equation. J. Math. Inequal. 12 (2018), no. 1, 43–61.

Y. Lee and G. Kim, Generalized Hyers-Ulam stability of the pexider functional equation, Math. 7 (2019), no. 3, Article No. 280.

M. Mirzavaziri and M. S. Moslehian, A fixed point approach to stability of a quadratic equation, Bull. Braz. Math. Soc. (N.S.) 37 (2006), no. 3, 361–376.

M. S. Moslehian and Th. M. Rassias, Stability of functional equations in non-Archimedean spaces, Appl. Anal. Discrete Math. 1 (2007), 325–334.

A. Najati and M. B. Moghini, On the stability of a quadratic and additive functional equation, J. Math. Anal. Appl. 337 (2008), 399–415.

C. Park, Additive s-functional inequalities and partial multipliers in Banach algebras, J. Math. Inequal. 13 (2019), no. 3, 867–877.

C. Park, J. R. Lee and X. Zhang, Additive s-functional inequality and hom-derivations in Banach algebras, J. Fixed Point Theory Appl. 21 (2019), no. 1, Paper No. 18, 14 pp.

C. Park and T. M. Rassias, Fixed points and stability of the Cauchy functional equation, Aust. J. Math. Anal. Appl. 6 (2009), no. 1, Art. 14, 9 pp.

F. Skof, Proprieta locali e approssimazione di operatori, Rend. Semin. Mat. Fis. Milano 53 (1983), 113–129.

T. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), no. 2, 297–300.

N. Sirovnik, On certain functional equation in semiprime rings and standard operator algebras, CUBO 16 (2014), no. 1, 73–80.

S. M. Ulam, Problems in modern mathematics, Science Editions John Wiley & Sons, Inc., New York, 1964.

How to Cite
Govindan, V., Park, C., Pinelas, S., & Rassias, T. M. (2020). Hyers-Ulam stability of an additive-quadratic functional equation. CUBO, A Mathematical Journal, 22(2), 233–255. https://doi.org/10.4067/S0719-06462020000200233