On asymptotic properties of solutions to third-order delay differential equations

The purpose of the paper is to show that the canonical operator $L_3$ given by $$L_3(\cdot) = \left(r_2\left(r_1(\cdot)'\right)'\right)'$$ where the functions $r_i(t)\in \mathcal{C}([t_0,\infty), [0,\infty))$ satisfy \[ \int_{t_0}^{\infty}\frac{\mathrm{d} s}{r_i(s)} = \infty, \quad (i = 1,2), \] can be written in a certain \textit{strongly noncanonical} form \begin{equation*} L_3(\cdot) \equiv b_3\left(b_2\left(b_1\left(b_0(\cdot)\right)'\right)'\right)', \end{equation*} such that the functions $b_i(t)\in \mathcal{C}([t_0,\infty), [0,\infty))$ satisfy \[ \int_{t_0}^{\infty}\frac{\mathrm{d} s}{b_i(s)} < \infty, \quad (i = 1,2). \] We study some relations between canonical and strongly noncanonical operators, showing the advantage of this reverse approach based on the use of a noncanonical representation of $L_3$ in the study of oscillatory and asymptotic properties of third-order delay differential equations