Approximately generalized additive functions in several variables via fixed point method

Farokhzad Rostami, R.; Hoseinioun, S.A.R.

doi:10.22075/ijnaa.2015.304

Approximately generalized additive functions in several variables via fixed point method

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

10.22075/ijnaa.2015.304

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