The generalized Hyers-Ulam-Rassias stability of a cubic functional equation.

*(English)*Zbl 1140.39014*K. W. Jun* and *H. M. Kim* [J. Math. Anal. Appl. 274, No. 2, 867–878 (2002; Zbl 1021.39014)] considered the following functional equation

$$f(2x+y)+f(2x-y)=2f(x+y)+2f(x-y)+12f\left(x\right)\xb7$$

Every solution of this equation is said to be a cubic function. In this paper the author considers the following functional equation

$$f(mx+y)+f(mx-y)=mf(x+y)+mf(x-y)+2({m}^{3}-m)f\left(x\right),$$

where $m\ge 2$. He shows that every solution of the functional equation above is a cubic function and he also establishes the generalized Hyers-Ulam-Rassias stability for this generalized cubic functional equation.

Reviewer: Maryam Amyari (Mashhad)

##### MSC:

39B82 | Stability, separation, extension, and related topics |

39B22 | Functional equations for real functions |

39B52 | Functional equations for functions with more general domains and/or ranges |