Existence and uniqueness of local solutions of functional equations (Q1091545)

The author considers local \(C^{\infty}\) solutions \(\phi\) of the following iterative functional equation \((1)\;\phi (f(x))=g(x)\phi (x)+h(x),\) where f is a real \(C^{\infty}\) selfmapping of a neighbourhood of the origin and \(g: {\mathbb{R}}\to {\mathcal C}^{m^ 2}\), \(h: {\mathbb{R}}\to {\mathcal C}^ m\) are also \(C^{\infty}\) mappings. One of the results obtained reads: If F is a local diffeomorphism satisfying either \(| f'(0)| \neq 1\) or \(| f'(0)| =1\) and \((f\circ f)(x)=x+\hat f(x),\hat f(x)\neq 0\), then for every formal solution \(\phi\) of (1) there exists a local \(C^{\infty}\) solution of (1) whose Taylor series at the origin coincides with \(\phi\). No proofs are given.