Smoothness of quasiperiodic solutions to linear systems of ordinary differential equations with degenerating symmetric matrix at derivatives (Q2768859)

The authors study the following system of ODEs NEWLINE\[NEWLINE\dot \varphi =\omega,\quad A(\varphi)\dot x+B(\varphi)=f(\varphi).NEWLINE\]NEWLINE Here, \(\omega \in \mathbb{R}^m\) is a constant vector, \(x\in \mathbb{R}^n,\) \(A(\varphi),B(\varphi)\in C(T^m \mapsto \mathbb{R}^{n\times n}), f(\varphi)\in C(T^m \mapsto \mathbb{R}^n), T^m:=\mathbb{R}^m/2\pi \mathbb{Z}^m\). The symmetric matrix \(A(\varphi)\) becomes degenerate on an arbitrary subset of the torus \(T^m\). The existence problem for a quasiperiodic solution \(x=u(\omega t)\), with \(u:T^m \mapsto \mathbb{R}^n\), is reduced to the system of PDEs NEWLINE\[NEWLINEA(\varphi)\sum_{i=1}^{m}\omega_i\frac{\partial u}{\partial \varphi_i}+B(\varphi)u=f(\varphi).NEWLINE\]NEWLINE The authors generalize sufficient conditions for the existence of a solution \(u(\varphi)\in C^k(T^m \mapsto \mathbb{R}^n)\), which were obtained in the book of \textit{A. M. Samojlenko} [Elements of the mathematical theory of multi-frequency oscillations. Dordrecht etc.: Kluwer Academic Publishers (1991; Zbl 0732.34043)].