Hartman's linearization on nonautonomous unbounded system (Q860717)

Assume that the linear system \[ x'=A(t)x\tag{1} \] is uniformly asymptotically stable. The author shows that if a nonlinearity \(f(t,x)\) has a small Lipschitz constant, then there exists a function \(h(t,x)\) such that \(h(t,\cdot)\) is a homeomorphism for any fixed \(t\), and \(h(x(t),t)\) is a solution of (1) if and only if \(x(t)\) is a solution of the perturbed system \[ x'=A(t)x+f(t,x). \]