Authors

• R. Farokhzad Rostami ¹
• S.A.R. Hoseinioun ²

¹ Department of Mathematics, Gonbad Kavous University, Gonbad Kavous, Golestan, Iran

² Department of Mathematical Sciences, University of Arkansas, Fayetteville, Arkansas 72701, USA

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

Keywords

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