On the stability of a mixed $n$-dimensional quadratic functional equation

Abstract

In this paper, we investigate the modified Hyers-Ulam stability of a mixed $n$-dimensional quadratic functional equation in Banach spaces and also Banach modules over a Banach algebra and a $C^*-$algebra. Finally, we study the stability using the alternative fixed point of the functional equation in Banach spaces: \begin{equation*} _{n-2}C_{m-2} f(\sum^n_{j=1} x_j) + _{n-2}C_{m-1}\sum^n_{i=1} f(x_i) =\sum_{1\leq i_1 < \cdots < i_m \leq n} f(x_{i_1}+\cdots+x_{i_m}), \end{equation*} for all $x_j (j=1,\cdots,n)$ where $n\geq 3$ is an integer number and $2\leq m \leq n-1.$