Asymptotic properties of noncanonical third order differential equations

Abstract

Abstract The purpose of the paper is to show that noncanonical operator $$\begin{array}{} \displaystyle \mathcal {L}\,y=\left(r_2(t)\left(r_1(t)y'(t)\right)'\right)' \end{array}$$ can be easily written in essentially unique canonical form $$\begin{array}{} \displaystyle \mathcal {L}\,y = q_3(t)\left(q_2(t)\left(q_1(t)\left(q_0(t)y(t)\right)'\right)'\right)' \end{array}$$ such that $$\begin{array}{} \displaystyle \int\limits^\infty \frac{1}{q_i(s)}\,\text{d}{s}=\infty, \quad i=1,2. \end{array}$$ The canonical representation is applied for examination of the third order noncanonical equations $$\begin{array}{} \displaystyle \left(r_2(t)\left(r_1(t)y'(t)\right)'\right)'+p(t)y(\tau(t))=0. \end{array}$$