On asymptotics of oscillatory solutions to nth-order delay differential equations

We study bounded and decaying to zero solutions of the delay differential equation x⁽ⁿ⁾(t)+∑i=1mp_i(t)x(t−τ_i(t))=0fort∈[0,∞),t≥0,x(ξ)=φ(ξ)for ξ<0. Kondrat'ev and Kiguradze introduced and defined principles of asymptotic behavior for its solution in the sense of the trichotomy: oscillatory, non-oscillatory with absolute values monotonically decaying to zero or monotonically increasing to ∞. Expanding upon such studies, we estimate the oscillation amplitudes of solutions. Decay to zero is established through fast oscillation: once distances between zeros are small enough, the Grönwall inequality growth estimate implies the amplitudes decrease to zero as t→∞. Exact growth estimates and calculation of these distances between zeros are proposed through evaluation for the spectral radii of some compact operators associated with the Green's function for an n-point problem.