New proofs of the existence of the Feigenbaum functions (Q1086385)

The following problem is considered: find real constants \(\lambda\in (0,1)\) and \(r>1\), and an even solution g of the Cvitanović-Feigenbaum functional equation \[ \lambda g(x)=-g(g(-\lambda x)),\quad x\in (-1,1), \] with the following properties: (i) there exists a function f, analytic with \(f'<0\) on [0,1] such that \(g(x)=f(x^ r)\) for all \(x\in (0,1)\), \(f(0)=1\). (ii) The inverse function U of f extends to an anti-Herglotz function, i.e. -U is holomorphic in the upper (lower) half-plane and maps it into itself. Solutions are sought as fixed points of suitable maps and the Schauder-Tikhonov theorem is applied. The existence of solutions is obtained for all \(\lambda\in (0,1)\), and the existence of the Eckmann- Wittwer functions is recovered. The method also provides the existence of solutions for certain given values of r, and in particular, for \(r=2\), a proof requiring no computer.