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

We study bounded and decaying to zero solutions of the delay differential equation$x^{\left(n\right)}\left(t\right)+\sum _{i=1}^m{p}_i\left(t\right)x(t-{\tau }_i(t\left)\right)=0\text{for}t\in [0,\infty ),t\ge 0,x\left(\xi \right)=\phi \left(\xi \right)\text{for }\xi <0.$ 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 $t\to \infty$. 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.