Properties of the third order trinomial functional differential equations

Abstract

The purpose of the paper is to study asymptotic properties of the third-order delay differential equation \begin{equation*} {\left(r_2(t)\left(r_1(t)\left(y'(t)\right)^\gamma\right)'\right)'}+p(t){\left(y'(t)\right)^\gamma}+q(t)f\left(y\left(\tau(t)\right)\right)=0. \tag{$E$} \end{equation*} Employing comparison principles with a suitable first order delay differential equation we shall establish criteria for all nonoscillatory solutions of ($E$) to converge to zero, while oscillation of a couple of first order delay differential equations yields oscillation of ($E$). An example is provided to illustrate the main results.

Keywords

• third-order
• functional
• trinomial
• differential equations
• transformation
• oscillation
• canonical form