General solution and two methods of generalized Ulam - Hyers stability of n-dimensional AQCQ functional equation

Abstract

In this paper, we achieve the general solution and generalized Ulam - Hyers stability of a $n$-dimensional additive-quadratic-cubic-quartic (AQCQ) functional equation$$\begin{aligned}f\left(\sum_{i=1}^{n-1} v_i+2 v_n\right)+f\left(\sum_{i=1}^{n-1} v_i-2 v_n\right)= & 4 f\left(\sum_{i=1}^n v_i\right)+4 f\left(\sum_{i=1}^{n-1} v_i-v_n\right)-6 f\left(\sum_{i=1}^{n-1} v_i\right) \\& +f\left(2 v_n\right)+f\left(-2 v_n\right)-4 f\left(v_n\right)-4 f\left(-v_n\right)\end{aligned}$$where $n$ is a positive integer with $n \geq 3$ in Banach Space (BS) via direct and fixed point methods. The stability results are discussed in two different ways by assuming $n$ is an odd positive integer and $n$ is an even positive integer.