On the Stability of Quadratic Functional Equations

Abstract

Let $X,Y$ be vector spaces and $k$ a fixed positive integer. It is shown that a mapping $f(kx+y)+f\left(kx-y\right)=2k^{2}f\left(x\right)+2f\left(y\right)$ for all $x,y\in X$ if and only if the mapping $f:X\to Y$ satisfies $f(x+y)+f\left(x-y\right)=2f\left(x\right)+2f\left(y\right)$ for all $x,y\in X$. Furthermore, the Hyers-Ulam-Rassias stability of the above functional equation in Banach spaces is proven.