A Perron-type theorem for nonautonomous differential equations

We show that the asymptotic exponential behavior of the solutions of a linear equation x′=A(t)xx′=A(t)x is reproduced by the solutions of the nonlinear equation x′=A(t)x+f(t,x)x′=A(t)x+f(t,x) for any sufficiently small perturbation f. More precisely, we show that if the Lyapunov exponents of the linear equation are limits, then the same happens with the Lyapunov exponents of the solutions of the nonlinear equations, in fact without introducing new values. Our approach is based on Lyapunov's theory of regularity, including its generalization to infinite-dimensional spaces.