ON STABILITY OF A GENERALIZED QUADRATIC FUNCTIONAL EQUATION WITH n-VARIABLES AND m-COMBINATIONS IN QUASI-β-NORMED SPACES

In this paper, we establish a general solution of the following functional equation$$\begin{aligned}&m f\left( \sum_{k=1}^{n}{x_k} \right) +\sum_{t=1}^{m}{ f\left( \sum_{k=1}^{n-i_{t}}{x_k} -\sum_{k=n-i_{t}+1}^{n}{x_k} \right) }\\=&2 \sum_{t=1}^{m}{\left( f\left( \sum_{k=1}^{n-i_{t}}{x_k} \right) +f\left( \sum_{k=n-i_{t}+1}^{n}{x_k} \right) \right)}\end{aligned}$$where $m, n, t, i_t \in \mathbb{N}$ such that $1 \leq t \leq m < n$. Also, we study Hyers-Ulam-Rassias stability for the generalized quadratic functional equation with $n$-variables and $m$-combinations form in quasi-$\beta$-normed spaces and then we investigate its application.