Strongly topological linearization with generalized exponential dichotomy

It has been proved that if x′=A(t)xx′=A(t)x has a generalized exponential dichotomy and f(t,x)f(t,x) satisfies certain conditions, then the nonlinear system x′=A(t)x+f(t,x)x′=A(t)x+f(t,x) is topologically equivalent to its linear system x′=A(t)xx′=A(t)x. In this paper, we prove that if the condition |A(t)|≤M⋅a(t)|A(t)|≤M⋅a(t) is added, then x′=A(t)x+f(t,x)x′=A(t)x+f(t,x) is strongly topologically equivalent to x′=A(t)xx′=A(t)x, where MM is some positive number and a(t)a(t) is the eigenfunction of the generalized exponential dichotomy, and therefore the corresponding solutions of x′=A(t)x+f(t,x)x′=A(t)x+f(t,x) and x′=A(t)xx′=A(t)x have the same stability.