On the hyperstability of the generalized class of Drygas functional equations on semigroups

Abstract

The aim of this paper is to offer some hyperstability results for the following functional equation

$$\begin{aligned} \sum _{\lambda \in \Lambda }f(x\lambda .y)=Lf(x)+\sum _{\lambda \in \Lambda }f(\lambda .y)\;\;\;\; (x,y\in S), \end{aligned}$$

where S is a semigroup, $\Lambda$ is a finite subgroup of the group of endomorphisms of S, L is the cardinality of $\Lambda$ (i.e. $L=card(\Lambda )$) and $f:S\rightarrow G$ such that $(G,+)$ is a L-cancellative abelian group with a metric d. Moreover, we discuss some remarks concerning particular cases of the considered equation and the inhomogeneous equation

$$\begin{aligned} \sum _{\lambda \in \Lambda }f(x\lambda .y)=Lf(x)+\sum _{\lambda \in \Lambda }f(\lambda .y)+F(x,y)\;\;\; (x,y \in S), \end{aligned}$$

where $F:S\times S \rightarrow G$.