Approximately generalized additive functions in several variables via fixed point method

Abstract

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<br />begin{align*}<br />&D_f(x_{1},.., x_{m}):= sum^{m}_{k=2}(sum^{k}_{i_{1}=2}<br />sum^{k+1}_{i_{2}=i_{1}+1}... sum^{m}_{i_{m-k+1}=i_{m-k}+1})<br /> f(sum^{m}_{i=1, ineq i_{1},...,i_{m-k+1} }<br /> x_{i}-sum^{m-k+1}_{ r=1} x_{i_{r}})& hspace {2.8cm}+f(sum^{m}_{ i=1} x_{i})<br />-2^{m-1} f(x_{1})=0<br />end{align*}<br />where $m geq 2$ is an integer number.