A generalized mixed type of quartic, cubic, quadratic and additive functional equation

Abstract

We determine the general solution of the functional equation $f(x + ky) + f(x — ky) = g(x + y) + g(x — y) + h(x) + \tilde{h}(y)$ forfixed integers $k$ with $k \neq 0, \pm 1$ without assuming any regularity condition on the unknown functions $f, g, h, \tilde{h}$. The method used for solving these functional equations is elementary but exploits an important result due to Hosszii. The solution of this functional equation can also be determined in certain type of groups using two important results due to Szekelyhidi.