Nonautonomous difference equations and a Perron-type theorem

We show that if the Lyapunov exponents of a linear difference equation x(m+1)=Lmxm are limits, then the same happens with the Lyapunov exponents of the solutions of the nonlinear equation x(m+1)=Lmxm+fm(xm) for any sufficiently small sequence fm. We consider the general case of infinite delay.