Solutions for the $p$-order Feigenbaum's functional equation $h(g(x))=g^{p}(h(x))$

Min Zhang¹ ; Jianguo Si²

¹School of Science China University of Petroleum 266555 Qingdao, Shandong People's Republic of China
²School of Mathematics Shandong University 250100 Jinan, Shandong People's Republic of China

Annales Polonici Mathematici, Tome 111 (2014) no. 2, pp. 183-195

Résumé

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.

Détail

DOI : 10.4064/ap111-2-6

Keywords: work deals feigenbaums functional equation begin array x quad leq leq quad end array right where geq integer p fold iteration strictly monotone odd continuous function neq using constructive method discuss existence continuous unimodal even solutions above equation