On the oscillation of higher-order delay differential equations

Abstract

The aim of this paper is to study the asymptotic properties and oscillation of the nth-order delay differential equation E $$ {{\left( {r(t){{{\left[ {{z^{{\left( {n-1} \right)}}}(t)} \right]}}^{\gamma }}} \right)}^{\prime }}+q(t)f\left( {x\left( {\tau (t)} \right)} \right)=0. $$ The results obtained are based on some new comparison theorems that reduce the problem of oscillation of an nth-order equation to the problem of oscillation of one or more first-order equations. We handle both cases $$ \begin{array}{*{20}{c}} {\int\limits^{\infty } {{r^{{-1/\gamma }}}(t)dt=\infty } } & {\mathrm{and}} & {\int\limits^{\infty } {{r^{{-1/\gamma }}}(t)dt<\infty .} } \end{array} $$ The comparison principles simplify the analysis of Eq. (E).