Hartman’s linearization on nonautonomous unbounded system

We extend Hartman’s linearization theorem to the uniformly asymptotic stable and unbounded case. We get the following conclusion: there is a constant δ>0δ>0 such that the nonlinear system dxdt=A(t)x+f(x,t) and its linear part dxdt=A(t)x are topologically equivalent if the linear system is uniformly asymptotically stable and f(x,t)f(x,t) satisfies Lipschitz’ condition with constant δδ.