Nearly Quartic Mappings in β-Homogeneous F-Spaces

Abstract

In this paper, we investigate the Hyers–Ulam stability of the following quartic equation

$$\begin{array}{ll} {\sum\limits^{n}_{k=2}}\left({\sum\limits^{k}_{i_{1}=2}}{\sum\limits^{k+1}_{i_{2}=i_{1}+1}} \ldots {\sum\limits^{n}_{i_{n-k+1}=i_{n-k}+1}}\right)\\ \quad\times f \left({\sum\limits^{n}_{i=1,i \neq i_{1},\ldots,i_{n-k+1}}} x_{i}-{\sum\limits^{n-k+1}_{r=1}}x_{i_{r}}\right) + f \left({\sum\limits^{n}_{i=1}}x_{i}\right)\\ \quad-2^{n-2}{\sum\limits^{}_{1 \leq{i} \leq{j} \leq{n}}}(f(x_{i} + x_{j}){+f(x_{i} - x_{j})){+2^{n-5}(n - 2){\sum\limits^{n}_{i=1}}f(2x_{i})}} = \theta \end{array} $$

$({n \in \mathbb{N}, n \geq 3})$ in β-homogeneous F-spaces.