The continual differentiable peak-unimodal solutions of Feigenbaum's functional equations (Q1179308)

The author studies the two functional equations of Feigenbaum: \(g(x)=- 1/\lambda g(g(-\lambda x))\) and \(g(x)=1/\lambda g(g(\lambda x))\). He gives iterative techniques for constructing solutions for the additional equations \(f(\lambda x)=\lambda f^{-1}(x)\) and \(\phi(x)=- 1/\lambda\phi(\phi(f(\lambda x)))\). He then establishes a constructive existence theorem for \(C^ 1\) unimodal solutions of the Feigenbaum equations under suitable initial conditions.