General solution and generalized Ulam-Hyers stability of a generalized 3-Dimensional AQCQ functional equation

Abstract

In this paper, we achieve the general solution and generalized Ulam-Hyers stability of a generalized 3dimensional AQCQ functional equation
$$
\begin{aligned}
&f(x+r(y+z))+f(x-  r(y+z))=r^2[f(x+y+z)+f(x-y-z)]\\&+2\left(1-r^2\right) f(x) + \frac{\left(r^4-r^2\right)}{12}[f(2(y+z))+f(-2(y+z))\\&-4 f(y+z)-4 f(-(y+z))]
\end{aligned}
$$
for all positive integers \(r\) with \(r \geq 2\) in Banach Space using two different methods.