Bulletin of the
Korean Mathematical Society BKMS

In this paper, we determine the general solution of the quartic equation $f(x+2y) + f(x-2y) + 6 f(x) = 4 [f(x+y) + f(x-y) + 6 f(y)]$ for all $x, y \in \mathbb{R}$ without assuming any regularity conditions on the unknown function $f$. The method used for solving this quartic functional equation is elementary but exploits an important result due to M. Hossz\'u \cite{MH}. The solution of this functional equation is also determined in certain commutative groups using two important results due to L. Sz\'{e}kelyhidi \cite{LS}.

Keywords: additive function, difference operator, Frechet functional equation, $n$-additive function, quartic map, and quartic functional equation

MSC numbers: Primary 39B22