On the asymptotics of solutions of a class of linear functional-differential equations

A sharp estimate of the growth of solutions of the initial value problem for systems of the form u̇(t) = ∑^l_j=1 C_j(t) u(λ_j t), t ≥ t₀ > 0, 0 < λ_j < 1, u(t) = φ(t) for λt₀ ≤ t ≤ t₀, λ = min λ_j, where C_j(t) are matrices with elements of power growth, is found. As a corollary of this result, it follows, for instance, that each solution of the initial value problem satisfies the estimate ∥u(t)∥ ≤ C exp{γln²(1 + |t|)} for some C > 0 and γ > 0.