On the linearization theorem for nonautonomous differential equations (Q888463)

In [J. Math. Anal. Appl. 41, 753--758 (1973; Zbl 0272.34056)], \textit{K. J. Palmer} proposed some generalization of the Hartman and Grobman's linearization theorem [\textit{P. Hartman}, Ordinary differential equations. York-London-Sydney: John Wiley and Sons, Inc (1964; Zbl 0125.32102)] to the nonautonomous case. Here, the authors consider the system \[ x_1'=A_1(t)x_1+f(t,x),\;\; x_2'=A_2(t)x_2,\;\;x=(x_1,x_2)\in \mathbb R^{n+m} \] and extend Palmer's result in two directions: (1) the nonlinear term might be unbounded or not Lipschitz continuous, and (2) only \(x_1'=A_1(t)x_1\) possesses an exponential dichotomy.