Multidimensional functional equations and normalization of local matrix- functions
This is a continuation of interesting and important investigations of the existence of \(C^{\infty}\)-solutions of functional equations with a given Taylor series. In the present paper the functional equation \((1)\quad \phi [F(x)]=Q(x)\phi (x)+\gamma (x)\) is considered under the assumption that F, Q and \(\gamma\) are defined and of the class \(C^{\infty}\) on a neighbourhood of the origin in \({\mathbb{R}}^ n\) with values in \({\mathbb{R}}^ n\), \({\mathbb{C}}^{m^ 2}\) and \({\mathbb{C}}^ m\), respectively, and F is a diffeomorphism. The author begins with a more general equation \((2)\quad \phi [F(x)]=H(x,\phi (x))\) assuming that the given function H is defined and of the class \(C^{\infty}\) on a neighbourhood of the origin in \({\mathbb{R}}^ n\times {\mathbb{R}}^ m\) with values in \({\mathbb{R}}^ m\), and has the form \(Q(x)y+\gamma (x)+h(x,y)\) with \(h(x,0)=\partial h/\partial y(x,0)=0;\) if \({\mathbb{R}}^ n=L_ 1\oplus L_ 2\) with locally invariant under F subspaces \(L_ 1\) and \(L_ 2\) and \(dist(F(x),L_ 2)<dist(x,L_ 2)\) for \(x\not\in L_ 2\), \(dist(F^{- 1}(x),L_ 1)<dist(x,L_ 1)\) for \(x\not\in L_ 1\), \(\gamma\) is flat at the origin and the matrix Q(0) is nilpotent, then there exists a solution \(\phi\) of (2) defined and of the class \(C^{\infty}\) on a neighbourhood of the origin in \({\mathbb{R}}^ n\) with values in \({\mathbb{R}}^ m\) and flat at the origin. In the next step the existence of a locally invertible \(C^{\infty}\)-map T: \({\mathbb{R}}^ n\to {\mathbb{C}}^{m^ 2}\) which reduces the matrix Q(x) to a diagonal hypermatrix \(diag[Q_ 1(x),Q_ 2(x)]\) with non-singular matrix \(Q_ 1(0)\) and a nilpotent \(Q_ 2(0)\) is proved. These main steps (and another reduction theorem proved in the paper) lead to a theorem which guarantees that to each formal series which fulfils (1) formally there exists a solution \(\phi\) of (1) defined and of the class \(C^{\infty}\) on a neighbourhood of the origin in \({\mathbb{R}}^ n\) with values in \({\mathbb{C}}^ m\). Three further theorems deal with the homogeneous case and bring necessary and sufficient conditions for the existence of non-zero and flat at the origin local \(C^{\infty}\)-solutions.
scientific article; zbMATH DE number 4088191