Oscillation of trinomial differential equations with positive and negative term

In the paper, we offer a new technique for investigation of properties of trinomial differential equations with positive and negative terms \begin{equation*} \left(b(t)\left(a(t)x'(t)\right)'\right)'+p(t)f(x(\tau(t)))-q(t)h(x(\sigma(t)))=0. \end{equation*} We offer criteria for every solution to be oscillatory. We support our results with illustrative examples

Asymptotic and oscillatory behavior of higher order quasilinear delay differential equations

In the paper, we offer such generalization of a lemma due to Philos (and partially Staikos), that yields many applications in the oscillation theory. We present its disposal in the comparison theory and we establish new oscillation criteria for $n-$th order delay differential equation \begin{equation*} \left(r(t)\left[x'(t)\right]^{\gamma}\right)^{(n-1)}+q(t)x^{\gamma}(\tau(t))=0.\tag{$E$} \end{equation*} The presented technique essentially simplifies the examination of the higher order differential equations

Oscillation theorems for second order neutral differential equations

In this paper new oscillation criteria for the second order neutral differential equations of the form \begin{equation*} \left(r(t)\left[x(t)+p(t)x(\tau(t))\right]'\right)'+q(t)x(\sigma(t))+v(t)x(\eta(t))=0 \tag{$E$}\end{equation*} are presented. Gained results are based on the new comparison theorems, that enable us to reduce the problem of the oscillation of the second order equation to the oscillation of the first order equation. Obtained comparison principles essentially simplify the examination of the studied equations. We cover all possible cases when arguments are delayed, advanced or mixed

On the oscillation of higher order delay differential equations

The aim of this paper is to study the asymptotic properties and oscillation of the n-th order delay differential equation r(t)[x⁽ⁿ⁻¹⁾(t)]γ)+ q(t)f(x(τ (t))) = 0. The results obtained are based on some new comparison theorems that reduce the problem of oscillation of an n-th order equation to that of the oscillation of one or more first order equations.Вивчено асимптотичнi властивостi та осциляцiю диференцiального рiвняння n-го порядку з запiзненням r(t)[x⁽ⁿ⁻¹⁾(t)]γ)+ q(t)f(x(τ (t))) = 0. Отриманi результати базуються на деяких нових теоремах порiвняння, якi зводять задачу про осциляцiю рiвняння n-го порядку до такої ж задачi для одного або кiлькох рiвнянь першого порядку