Functional equations and applications

Abstract

We prove the generalized Hyers-Ulam stability of the following additive-quadratic-cubic-quartic functional equation \begin{eqnarray} && f(x+2y) + f(x-2y)=4f(x+y)+4f(x-y) \nonumber \\ && - 6 f(x) + f(2y) + f(-2y) - 4f(y) - 4 f(-y) \end{eqnarray} in Banach spaces.

Moreover, we prove the generalized Hyers-Ulam stability of the above functional equation in non-Archimedean Banach spaces.