On methods for solving composite functional equations (Q785641)

The author provides some methods for solving a large class of composite functional equations. As an application, these methods are used for the functional equations \[f(af(x)f(y)+b(f(x)y+f(y)x)+cxy)=f(x)f(y),\] \[f(af(x)^{k}y+bf(y)^{l}x+cxy)=f(x)f(y),\] with \(a, b, c \in \mathbb{R}\) and \(k, l \in \mathbb{N}\cup \{0\}\). The first functional equation is a generalization of the Baxter functional equation, while the second one is a generalization of the Golab-Schinzel functional equation. All continuous solutions of both equations \(f:\mathbb{R}\rightarrow \mathbb{R}\) are determined. The author uses various methods to provide rigorous proofs. In particular, she takes some special values of the parameters \(a, b, c \in \mathbb{R}\) and \(k, l \in \mathbb{N}\cup \{0\}\). For instance, assuming \(b^{2}=ac+b\), the first equation admits one of the following nonconstant continuous functions as solutions: \(f(x)=-\frac{c}{b}x\) and \(f(x)=-\frac{c}{b}x\), \(x \in \mathbb{R}\). Many other cases are considered as well.