Non-Single-Valley Solutions for p-Order Feigenbaum’s Type Functional Equation f(φ(x))=φp(f(x))

Abstract

This work deals with Feigenbaum’s functional equation $f(\phi (x))={\phi }^p(f(x))$, $\phi (0)=1$, $0\le \phi (x)\le 1$, $x\in [0, 1]$, where $p\ge 2$ is an integer, ${\phi }^p$ is the $p$-fold iteration of $\phi$, and $f(x)$ is a strictly increasing continuous function on $[0, 1]$ that satisfies $f(0)=0$, , $(x\in (0, 1])$. Using a constructive method, we discuss the existence of non-single-valley continuous solutions of the above equation.