Local equivalence of functional equations (Q1077646)

Let \(F: R^ n\to R^ n\) and G, H, \(\phi\) : \(R^ n\times R^ p\to R^ p\) be local \(C^{\infty}\)-mappings such that \(F(t)=\Lambda t+f(t)\), \(G(t,z)=A(t)z+g(t,z)\), \(H(t,z)=A_ 1(t)z+h(t,z)\), \(\phi (t,y)=y+\phi (t,y)\), where \(g(0,z)=O(\| z\|^ 2)\), \(h(0,z)=O(\| z\|^ 2)\), \(\phi (0,z)=O(\| y\|^ 2)\) and satisfy the following relation \[ (1)\quad \Phi (F(t),H(t,y))=G(t,\Phi (t,y)). \] Then the functional equations (2) \(x(F(t))=G(t,x(t))\) and (3) \(y(F(t))=H(t,y(t))\), where \(x(t)=\Phi (t,x(t))\), are locally \(C^{\infty}\) equivalent. If the matrices \(\Lambda\) and A(0) have no points of their spectra in zero and on the unit circle and equation (1) has a solution \(\Phi\) in formal power series, then (1) has a local \(C^{\infty}\) solution and consequently equations (2) and (3) are locally \(C^{\infty}\) equivalent. Let \(\{\lambda_ 1,...,\lambda_ n\}\) be the eigenvalues of \(\Lambda\) and \(\{\alpha_ 1,...,\alpha_ p\}\) the eigenvalues of A(0), \(| \lambda_ i| \neq 0,1\) and \(\alpha_ i\neq \alpha_ 1^{k_ 1}...\alpha_ p^{k_ p} \lambda_ 1^{m_ 1}...\lambda_ n^{m_ n}\) for \(k_ j,m_ j\geq 0\), \(i=1,...,p\) and \(A_ 1(0)=A(0)\), then (2) and (3) are locally \(C^{\infty}\) equivalent.