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
$$\begin{array}{rl}& D_f(x_1,..,x_m):=\sum _{k=2}^m(\sum _{i_1=2}^k\sum _{i_2=i_1+1}^{k+1}...\sum _{i_{m-k+1}=i_{m-k}+1}^m)f(\sum _{i=1,i\ne i_1,...,i_{m-k+1}}^m{x}_i-\sum _{r=1}^{m-k+1}{x}_{i_r})\\ & +f(\sum _{i=1}^m{x}_i)-2^{m-1}f(x_1)=0\end{array}$$
where $m\ge 2$ is an integer number.

Keywords

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