Journal of the
Korean Mathematical Society JKMS

Let $X, Y$ be real normed vector spaces. We exhibit all the solutions $f:X\to Y$ of the functional equation $ f(rx+sy)+rsf(x-y)=rf(x)+sf(y) $ for all $x, y\in X$, where $r, s$ are nonzero real numbers satisfying $r+s=1$. In particular, if $Y$ is a Banach space, we investigate the Hyers-Ulam stability problem of the equation. We also investigate the Hyers-Ulam stability problem on a restricted domain of the following form $\Omega\cap\{(x, y)\in X^2:\|x\|+\|y\|\ge d\}$, where $\Omega$ is a rotation of $H\times H\subset X^2$ and $H^c$ is of the first category. As a consequence, we obtain a measure zero Hyers-Ulam stability of the above equation when $f:\mathbb R\to Y$.

Keywords: Baire category theorem, first category, Lebesgue measure, quadratic functional equation, $(r, s)$-quasi-quadratic functional equation, second category, Hyers-Ulam stability

MSC numbers: 39B82