HYERS-ULAM STABILITY FOR A SPECIAL CLASS OF FUNCTIONAL EQUATIONS

Abstract

In this paper, we investigate the stability in the sense of Hyers-Ulam for a
class of the following type functional equations:

$$\sum_{\lambda \in \Phi}{f(x+\lambda y+a_{\lambda})}=Nf(x)+h(y),\ x,y\in S$$

where $\mathbb{K}$ is a complete valued field of characteristic zero, $F$ is
a complete normed space (Archimedean or ultrametric) over
$\mathbb{K}$, $(S,+)$ is an abelian monoid, $f,h\colon S\to F$,
$\Phi$ is a finite automorphism group of $S$, $N$ is the
cardinality of $\Phi$ and $a_{\lambda}\in S$, $\lambda \in \Phi$.