On the Feigenbaum's functional equation \(f^ p(\lambda x)=\lambda f(x)\)
The author considers the generalization of Feigenbaum's functional equation \(f^ p (\lambda x) = \lambda f(x)\), where \(p \geq 2\) is an integer, \(x \in [-1,1]\), \(f^ p\) denotes the \(p\)-fold iteration of \(f\), \(\lambda \in (0,1)\) is to be determined and \(f(0) = 1\), \(-1 \leq f(x) \leq 1\). For the case \(p=2\) the existence of even unimodal \(C^ 1\) solution is well-known, the case \(p=3\) was recently studied and the author's results give a feasible method to construct even unimodal \(C^ 1\) solutions for any \(p \geq 2\). Other classes of solutions like continuous single-valley solutions or piecewise smooth single-valley solutions are studied.
scientific article; zbMATH DE number 589871