Functional equations and group substitutions (Q627955)

Let \(G=\{g_1,g_2,\dots, g_n\}\) be a group of diffeomorphisms of the open interval \(I\subseteq\mathbb{R}\) such that \(\xi\in I\) is a common fixed point of all \(g_j\in G\). Let \(\eta\) be real and let \(F\colon\mathbb{R}^n\times\mathbb{R}\to\mathbb{R}\) be a \(C^1\)-function such that \(F(\eta,\dots,\eta,\xi)=0\). Furthermore it is assumed that certain matrix functions derived from the partial derivatives of \(F\) satisfy certain conditions. Then it is proved that there is some open subinterval \(J\) of \(I\) containing \(\xi\) such that the functional equation \(F(f(g_1(x)),\dots,f(g_n(x)),x)=0\) has a unique differentiable solution \(f\colon J\to\mathbb{R}\) satisfying \(f(\xi)=\eta\). There are also some applications of the main result.