On methods for solving composite functional equations

Abstract

In this paper we present some methods for solving a large class of composite functional equations. These methods are then applied to the functional equations

$$\begin{aligned} f(a f(x) f(y) + b(f(x)y+f(y)x) +cxy)= & {} f(x) f(y) \end{aligned}$$

(1)

$$\begin{aligned} f(a f(x)^k y + b f(y)^\ell x + cxy)= & {} f(x) f(y) \end{aligned}$$

(2)

with $a, b, c \in {\mathbb {R}}$ and $k, \ell \in {\mathbb {N}} \cup \{0\}$, for which we obtain all continuous solutions $f: {\mathbb {R}} \rightarrow {\mathbb {R}}$. These equations generalize some well-known composite functional equations.