On the general solution of a quartic functional equation

Abstract

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