New proofs of the existence of the Feigenbaum functions

A new proof of the existence of analytic, unimodal solutions of the Cvitanovi-Feigenbaum functional equation g(x) = –g(g(–x)),g(x) 1 - const|x|^r at 0, valid for all in (0, 1), is given, and the existence of the Eckmann-Wittwer functions 8] is recovered. The method also provides the existence of solutions for certain given values ofr, and in particular, forr=2, a proof requiring no computer.