Asymptotic behavior of solutions of neutral nonlinear differential equations

summary:In this paper we study asymptotic behavior of solutions of second order neutral functional differential equation of the form $$\Big (x(t)+px(t-\tau )\Big )^{\prime \prime }+f(t,x(t))=0\,.$$ We present conditions under which all nonoscillatory solutions are asymptotic to $at+b$ as $t\rightarrow \infty$, with $a,b\in R$. The obtained results extend those that are known for equation \[ u^{\prime \prime }+f(t,u)=0\,. \

Property A of differential equations with positive and negative term

In the paper, we elaborate new technique for the investigation of the asymptotic properties for third order differential equations with positive and negative term \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 new easily verifiable criteria for property A. We support our results with illustrative examples

A sharp oscillation result for second-order half-linear non canonical delay differential equations

In the paper, new single-condition criteria for the oscillation of all solutions to a second-order half-linear delay differential equation in noncanonical form are obtained, relaxing a traditionally posed assumption that the delay function is nondecreasing. The oscillation constant is best possible in the sense that the strict inequality cannot be replaced by the nonstrict one without affecting the validity of the theorem. This sharp result is new even in the linear case and, to the best of our knowledge, improves all the existing results reporting in the literature so far. The advantage of our approach is the simplicity of the proof, only based on sequentially improved monotonicities of a positive solution