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We derive a few fundamental estimates for solutions u of weakly nonlinear ODE systems of the form ut = Au + B(t)u + "f(t, u) + g(t), t > t0, where A is a constant n x n matrix all of whose eigenvalues have negative real part and "f is suitably small, with B ∈ Lp(t0,∞), g ∈ Lq(t0,∞) for some 1 ≤ p, q ≤ ∞. Our analysis improves and extends some well known results obtained elsewhere for important families of equations within this class.