Single-valley-extended continuous solutions for the Feigenbaum’s functional equation \(f(\varphi(x))=\varphi^2(f(x))\). (English) Zbl 1298.39021
Summary: This work deals with the Feigenbaum’s functional equation in the broad sense
\[
\begin{cases} f(\varphi(x))=\varphi^2(f(x)),\\ \varphi(0)=1,\;0\leq\varphi(x)\leq1,\;x\in[0,1],\end{cases}
\]
where \(\varphi^2\) is the 2-fold iteration of \(\varphi\), \(f(x)\) is a strictly increasing continuous function on \([0,1]\) and satisfies \(f(0)=0\), \(f(x) <x\) \((x\in(0,1])\). Using a constructive method, we discuss the existence of single-valley-extended continuous solutions of the above equation.
MSC:
| 39B12 | Iteration theory, iterative and composite equations |
| 39B22 | Functional equations for real functions |
Keywords:
Feigenbaum’s functional equation; constructive method; initial function; single-valley-extended continuous solutionsReferences:
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