We investigate functional equations $f\left(p\right(x\left)\right)=q\left(f\right(x\left)\right)$ where $p$ and $q$ are given real functions defined on the set $ℝ$ of all real numbers. For these investigations, we can use methods for constructions of homomorphisms of mono-unary algebras. Our considerations will be confined to functions $p,q$ which are strictly increasing and continuous on $ℝ$. In this case, there is a simple characterization for the existence of a solution of the above equation. First, we give such a characterization. Further, we present a construction...