Approximately generalized additive functions in several variables via fixed point method

In this paper, we obtain the general solution and the generalized   Hyers-Ulam-Rassias stability in random normed spaces, in non-Archimedean spaces and also in p-Banach spaces and finally the stability via fixed point method for a functional equation\begin{align*}&D_f(x_{۱},.., x_{m}):= \sum^{m}_{k=۲}(\sum^{k}_{i_{۱}=۲}\sum^{k+۱}_{i_{۲}=i_{۱}+۱}... \sum^{m}_{i_{m-k+۱}=i_{m-k}+۱}) f(\sum^{m}_{i=۱, i\neq i_{۱},...,i_{m-k+۱} } x_{i}-\sum^{m-k+۱}_{ r=۱} x_{i_{r}})\\& \hspace {۲.۸cm}+f(\sum^{m}_{ i=۱} x_{i})-۲^{m-۱} f(x_{۱})=۰\end{align*}where m \geq ۲ is an integer number.

Keywords:

Additive function , p-Banach spaces , Random normed spaces , Non-Archimedean spaces , Fixed point method , generalized Hyers-Ulam stability