\(a_i \) Type \( n\) - variable multi \(n\)-dimensional additive functional equation

Abstract

In this paper, the authors investigated the general solution and generalized Ulam - Hyers stability of \(a_i\) type \(n\)- variable multi \(n\) - dimensional additive functional equation
$$
\begin{aligned}
&2 h\left(\sum_{i=1}^n a_i x_{1 i}\right.  \left., \sum_{i=1}^n a_i x_{2 i}, \ldots \ldots, \sum_{i=1}^n a_i x_{n i}\right) \\
= & \left(\sum_{i=1}^n a_i\right) h\left(\sum_{i=1}^n x_{1 i}, \sum_{i=1}^n x_{2 i}, \ldots \ldots, \sum_{i=1}^n x_{n i}\right) \\
& +\left(a_1-\sum_{i=2}^n a_i\right) h\left(x_{11}-\sum_{i=2}^n x_{1 i}, x_{21}-\sum_{i=2}^n x_{2 i}, \ldots \ldots, x_{n 1}-\sum_{i=2}^n x_{n i}\right)
\end{aligned}
$$
where \(a_i(i=1,2, \ldots n)\) are different integers greater than 1, using two different technique.