A dynamical systems result on asymptotic integration of linear differential systems

We are interested in the asymptotic integration of linear differential systems of the form x′=[Λ(t)+R(t)]x, where Λ is diagonal and R Lp[t0,∞) for p [1,2]. Our dichotomy condition is in terms of the spectrum of the omega-limit set ωΛ. Our results include examples that are not covered by the Hartman–Wintner theorem.