On the existence of the unique Green function for the linear extension of a dynamical system on a torus (Q2754816)

This paper presents the system NEWLINE\[NEWLINE\dot \varphi =a(\varphi),\quad \dot x=P(\varphi)x,\tag{1}NEWLINE\]NEWLINE with \(a(\varphi)\in C_{Lip}(T^m \mapsto \mathbb{R}^m), P(\varphi)\in C(T^m \mapsto \mathbb{R}^{n\times n}), T^m:=\mathbb{R}^m/2\pi \mathbb{Z}^m\). Let \(\Omega_0^t(\varphi)\) stand for the evolutionary operator of the system NEWLINE\[NEWLINE\dot x=P(\varphi_t(\varphi))x\tag{2}NEWLINE\]NEWLINE where \(\{T^m,\{\varphi_t(\varphi)\}_{t\in \mathbb{R}}\}\) is the flow of the first subsystem in (1). NEWLINENEWLINENEWLINEThe author shows that NEWLINENEWLINENEWLINEif (a) for some \(K>0, \gamma >0\), and \(n\times n\)-matrix \(H(\varphi)\) the following inequalities hold true NEWLINE\[NEWLINE\|\Omega_0^t(\varphi)H(\varphi)\|\leq Ke^{-\gamma t}, \text{for} t\geq 0,\quad \|\Omega_0^t(\varphi)(\text{Id}-H(\varphi)\|\leq Ke^{-\gamma t} \text{for} t<0,NEWLINE\]NEWLINE and (b) the only solution to (2) which satisfies \(\lim_{|t|\to \infty}x(t)=0\) is the trivial one, NEWLINENEWLINENEWLINEthen \(H(\varphi)\in C(T^m \mapsto \mathbb{R}^{n\times n})\) and this matrix function is the unique one for which the matrix function NEWLINE\[NEWLINEG_{0}(\tau,\varphi)=\begin{cases} \Omega_\tau^0(\varphi)H(\varphi_\tau(\varphi)),&\text{for} \tau \leq 0, \\ \Omega_\tau^0(\varphi)(H(\varphi_\tau(\varphi))-\text{Id}),& \text{for} \tau \leq 0, \end{cases} NEWLINE\]NEWLINE satisfies the inequality \(\|G_{0}(\tau,\varphi)\|\leq Ke^{-\gamma|\tau|}\). The matrix function \(G_{0}(\tau,\varphi)\) represents the Green function for (1).NEWLINENEWLINENEWLINEThis result is then used to prove the following assertion from \textit{Yu. A. Mitropol'skij, A. M. Samojlenko} and \textit{V. L. Kulik} [Studies in dichotomy of linear systems of differential equations by means of Lyapunov functions. Kiev: Naukova dumka (1990; Zbl 0776.34041)]: if there exists such a nondegenerate symmetric matrix \(S(\varphi)\in C^1(T^m \mapsto \mathbb{R}^{n\times n})\) that \(S'(\varphi)a(\varphi)+S(\varphi)P(\varphi)+P^*(\varphi)S(\varphi)<0\) for all \(\varphi \in T^m\), then system (1) has a unique Green function.