### Comparison theorems for second-order neutral differential equations of mixed type.

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The aim of this paper is to study asymptotic properties of the third-order quasi-linear neutral functional differential equation $${\left[a\left(t\right){\left({[x\left(t\right)+p\left(t\right)x\left(\delta \left(t\right)\right)]}^{\text{'}\text{'}}\right)}^{\alpha}\right]}^{\text{'}}+q\left(t\right){x}^{\alpha}\left(\tau \left(t\right)\right)=0\phantom{\rule{0.166667em}{0ex}},E$$ where $\alpha >0$, $0\le p\left(t\right)\le {p}_{0}<\infty $ and $\delta \left(t\right)\le t$. By using Riccati transformation, we establish some sufficient conditions which ensure that every solution of () is either oscillatory or converges to zero. These results improve some known results in the literature. Two examples are given to illustrate the main results.

By means of Riccati transformation technique, we establish some new oscillation criteria for third-order nonlinear delay dynamic equations ${\left({\left({x}^{\Delta \Delta}\left(t\right)\right)}^{\gamma}\right)}^{\Delta}+p\left(t\right){x}^{\gamma}\left(\tau \left(t\right)\right)=0$ on a time scale ; here γ > 0 is a quotient of odd positive integers and p a real-valued positive rd-continuous function defined on . Our results not only extend and improve the results of T. S. Hassan [Math. Comput. Modelling 49 (2009)] but also unify the results on oscillation of third-order delay differential equations and third-order delay difference...

We study oscillatory behavior of a class of fourth-order quasilinear differential equations without imposing restrictive conditions on the deviated argument. This allows applications to functional differential equations with delayed and advanced arguments, and not only these. New theorems are based on a thorough analysis of possible behavior of nonoscillatory solutions; they complement and improve a number of results reported in the literature. Three illustrative examples are presented.

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