Stability of a Functional Equation Deriving from Quadratic and Additive Functions in Non-Archimedean Normed Spaces

Abstract

We obtain the general solution of the generalized mixed additive and quadratic functional equation $f\left(x+my\right)+f\left(x-my\right)=2f\left(x\right)-2m^{2}f\left(y\right)+m^{2}f\left(2y\right)$, $m$ is even; $f\left(x+y\right)+f\left(x-y\right)-2\left(m^2-1\right)f\left(y\right)+\left(m^2-1\right)f\left(2y\right)$, $m$ is odd, for a positive integer $m$. We establish the Hyers-Ulam stability for these functional equations in non-Archimedean normed spaces when $m$ is an even positive integer or $m=3$.