Hyers-Ulam Stability for a Class of Quadratic Functional Equations via a Typical Form

Abstract

We investigate the following typical form of a certain class of quadratic functional equations: $f\left(ax+by\right)+f\left(ax-by\right)+c\left[f\left(x+y\right)+f\left(x-y\right)-2f\left(x\right)-2f\left(y\right)\right]=2a^{2}f\left(x\right)+2b^{2}f\left(y\right)$. Furthermore, we provide a systematic program to prove the generalized Hyers-Ulam stability for the class of functional equations via the stability for the typical form.