A condition for the existence of a unique Green-Samoĭlenko function for the problem of invariant torus (Q2754814)

The author deals with the system of equations NEWLINE\[NEWLINE\dot \varphi =a(\varphi),\quad \dot x=P(\varphi)x,\tag{1}NEWLINE\]NEWLINE with \(a(\varphi)\in C^1(T^m \mapsto \mathbb R^m),\;P(\varphi)\in C(T^m \mapsto \mathbb R^{n\times n}),\;T^m:=\mathbb R^m/2\pi \mathbb Z^m\). 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). As it was shown by \textit{A. M. Samoilenko} [Elements of the mathematical theory of multi-frequency oscillations. Dordrecht etc.: Kluwer Academic Publishers (1991; Zbl 0732.34043)], system (1) has a unique Green-Samoilenko function of the invariant torus problem provided that system (2) is exponentially dichotomic on the whole real axis. NEWLINENEWLINENEWLINEThe author considers the case where system (2) is exponentially dichotomic on each semi-axis \(\mathbb R_+\) and \(\mathbb R_-\) with projectors \(C_{\pm }(\varphi)\) satisfying for some positive \(K,\;\alpha \) the following conditions NEWLINE\[NEWLINE\begin{aligned} \|\Omega_0^t(\varphi)C_{\pm }(\varphi)\Omega_\tau ^0(\varphi)\|&\leq Ke^{-\alpha (t-\tau)},\;\text{for}\quad \mathbb R_{\pm }\ni t\geq\tau ,\\ \|\Omega_0^t(\varphi)(\text{Id}-C_{\pm }(\varphi))\Omega_\tau ^0(\varphi)\|&\leq Ke^{-\alpha(\tau - t)},\;\text{for} \mathbb R_{\pm }\ni t<\tau.\end{aligned}NEWLINE\]NEWLINE It is shown that if \(\det(C_{+}(\varphi)+C_-(\varphi)-\text{Id})\neq 0\), then system (2) is exponentially dichotomic on the whole real axis and thus, system (1) has a unique Green-Samoilenko function of the invariant torus problem.