On the Generalized Hyers-Ulam-Rassias Stability of Quadratic Functional Equations

Abstract

We achieve the general solution and the generalized Hyers-Ulam-Rassias and Ulam-Gavruta-Rassias stabilities for quadratic functional equations $f\left(ax+by\right)+f\left(ax-by\right)=\left(b\left(a+b\right)/2\right)f\left(x+y\right)+$ $\left(b\left(a+b\right)/2\right)f\left(x-y\right)+$ $\left(2a^2-ab-b^2\right)f\left(x\right)+$ $\left(b^2-ab\right)f\left(y\right)$ where $\mathrm{a,\; b}$ are nonzero fixed integers with $b\ne \pm a,-3a$, and $f\left(ax+by\right)+f\left(ax-by\right)=2a^{2}f\left(x\right)+2b^{2}f\left(y\right)$ for fixed integers $\mathrm{a,\; b}$ with $a,b\ne 0$ and $a\pm b\ne 0$.