Iterative functional equation

Summary

An iterative functional equation is a functional equation in one variable in which the unknown function is composed with a given map or with itself. A typical form is $\varphi(f(x)) = g(x, \varphi(x))$, where $f$ and $g$ are given and $\varphi$ is unknown. Such equations describe how a quantity changes under one step of an iterated map. The Abel equation, the Schröder equation and the equation $\varphi(\varphi(x)) = f(x)$ for a functional square root are the main examples.