Solutions for the $p$-order Feigenbaum's functional equation $h(g(x))=g^{p}(h(x))$
Volume 111 / 2014
Annales Polonici Mathematici 111 (2014), 183-195
MSC: 39B12.
DOI: 10.4064/ap111-2-6
Abstract
This work deals with Feigenbaum's functional equation $$\left\{ \begin{array}{l} h(g(x))=g^p(h(x)),\\ g(0)=1, \quad -1\leq g(x)\leq1 ,\quad x\in[-1,1], \end{array} \right. $$ where $p\geq 2$ is an integer, $g^p$ is the $p$-fold iteration of $g$, and $h$ is a strictly monotone odd continuous function on $[-1,1]$ with $h(0)=0$ and $|h(x)|<|x|$ ($x\in[-1,1]$, $x\neq 0$). Using a constructive method, we discuss the existence of continuous unimodal even solutions of the above equation.
