Bulletin of the
Korean Mathematical Society BKMS

Let $X, Y$ be vector spaces. It is shown that if an even mapping $f : X \rightarrow Y$ satisfies $f(0)=0$ and $$\aligned &f\left(\frac{x+y}{2}+z\right) + f\left(\frac{x+y}{2}-z\right) + f\left(\frac{x-y}{2}+z\right)\\ &\ + f\left(\frac{x-y}{2}-z\right) = f(x)+f(y)+4f(z) \endaligned \tag 0.1 $$ for all $x, y, z\in X$, then the mapping $f : X \rightarrow Y$ is quadratic. Furthermore, we prove the Cauchy--Rassias stability of the functional equation {\rm (0.1)} in Banach spaces.

Keywords: Cauchy--Rassias stability, quadratic mapping, functional equation

MSC numbers: 39B52