Solution and stability of system of quartic functional equations

Abstract

In this paper, the authors introduced and investigated the general solution of system of quartic functional equations
$$
\begin{aligned}
& f(x+y+z)+f(x+y-z)+f(x-y+z)+f(x-y-z) \\
& =2[f(x+y)+f(x-y)+f(x+z)+f(x-z)+f(y+z)+f(y-z)] \\
& -4[f(x)+f(y)+f(z)] \\
& f(3 x+2 y+z)+f(3 x+2 y-z)+f(3 x-2 y+z)+f(3 x-2 y-z) \\
& =72[f(x+y)+f(x-y)]+18[f(x+z)+f(x-z)]\\&+8[f(y+z)+f(y-z)]+144 f(x)-96 f(y)-48 f(z) \\
& f(x+2 y+3 z)+f(x+2 y-3 z)+f(x-2 y+3 z)+f(x-2 y-3 z) \\
& =8[f(x+y)+f(x-y)]+18[f(x+z)+f(x-z)]\\&+72[f(y+z)+f(y-z)]-48 f(x)-96 f(y)+144 f(z)
\end{aligned}
$$
Its generalized Hyers-Ulam stability using Hyers direct method and fixed point method are discussed. Counter examples for non stable cases are also given.