On the hyperstability of the generalized class of Drygas functional equations on semigroups. (English) Zbl 1470.39061
Summary: The aim of this paper is to offer some hyperstability results for the following functional equation
\[ \sum_{\lambda \in \Lambda }f(x\lambda .y)=Lf(x)+\sum_{\lambda \in \Lambda }f(\lambda .y)\quad (x,y\in S), \]
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
\[ \sum_{\lambda \in \Lambda }f(x\lambda .y)=Lf(x)+\sum_{\lambda \in \Lambda }f(\lambda .y)+F(x,y)\quad (x,y \in S), \]
where \(F:S\times S \rightarrow G\).
MSC:
| 39B82 | Stability, separation, extension, and related topics for functional equations |
| 39B52 | Functional equations for functions with more general domains and/or ranges |
| 47H14 | Perturbations of nonlinear operators |
| 47J20 | Variational and other types of inequalities involving nonlinear operators (general) |
| 47H10 | Fixed-point theorems |
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