LAUSR

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