Spectral representation theory and stability of the multiplicative Dhombres functional equation in \(f\)-algebras (Q2350257)

The paper presents a non-standard method for stability of functional equations theory. The main task is to transfer already known stability results for real-valued functions into the case of mappings taking values in some Riesz spaces (\(f\)-algebras). To reach this task, the author applies the Ogasawara-Maeda spectral representation theorem. Then, this method is particularly used for proving stability of the Dhombres' equation \[ (F(x)+F(y))(F(x+y)-F(x)-F(y))=0 \] in the realm of mappings \(F: G\to L\), where \(G\) is an abelian group and \(L\) stands for an Archimedean, uniformly complete \(f\)-algebra with a multiplicative identity. Roughly speaking, it is proved that any approximate solution of the above equation can be approximated by a unique additive mapping \(A: G\to L\).