Sufficient oscillation conditions involving $lim\; sup$ and $lim\; inf$ for first-order differential equations with non-monotone deviating arguments and nonnegative coefficients are obtained. The results are based on the iterative application of the Grönwall inequality. Examples, numerically solved in MATLAB, are also given to illustrate the applicability and strength of the obtained conditions over known ones.

In this paper, we investigate a class of higher order neutral functional differential equations, and obtain some new oscillatory criteria of solutions.

We obtain sufficient conditions for every solution of the differential equation $${[y\left(t\right)-p\left(t\right)y\left(r\left(t\right)\right)]}^{\left(n\right)}+v\left(t\right)G\left(y\left(g\left(t\right)\right)\right)-u\left(t\right)H\left(y\left(h\left(t\right)\right)\right)=f\left(t\right)$$ to oscillate or to tend to zero as $t$ approaches infinity. In particular, we extend the results of Karpuz, Rath and Padhy (2008) to the case when $G$ has sub-linear growth at infinity. Our results also apply to the neutral equation $${[y\left(t\right)-p\left(t\right)y\left(r\left(t\right)\right)]}^{\left(n\right)}+q\left(t\right)G\left(y\left(g\left(t\right)\right)\right)=f\left(t\right)$$ when $q\left(t\right)$ has sign changes. Both bounded and unbounded solutions are consideted here; thus some known results are expanded.

In this paper, we study the oscillatory behavior of the solutions of the delay differential equation of the form $${\left(\frac{1}{r\left(t\right)}{y}^{\text{'}}\left(t\right)\right)}^{\text{'}}+p\left(t\right)y\left(\tau \left(t\right)\right)=0.$$ The obtained results are applied to n-th order delay differential equation with quasi-derivatives of the form $$L_{n}u\left(t\right)+p\left(t\right)u\left(\tau \left(t\right)\right)=0.$$

Some sufficient conditions for oscillation of a first order nonautonomuous delay differential equation with several positive and negative coefficients are obtained.

Our aim in this paper is to present the relationship between property (B) of the third order equation with delay argument y'''(t) - q(t)y(τ(t)) = 0 and the oscillation of the second order delay equation of the form y''(t) + p(t)y(τ(t)) = 0.

Consider the first-order linear delay (advanced) differential equation$$x^{\text{'}}\left(t\right)+p\left(t\right)x\left(\tau \left(t\right)\right)=0(x^{\text{'}}\left(t\right)-q\left(t\right)x\left(\sigma \left(t\right)\right)=0),t\ge t_0,$$ where $p$$\left(q\right)$ is a continuous function of nonnegative real numbers and the argument $\tau \left(t\right)$$...$

We study oscillatory properties of solutions of systems $$\begin{array}{ccc}\hfill {[y_1\left(t\right)-a\left(t\right)y_1\left(g\left(t\right)\right)]}^{\text{'}}=& p_1\left(t\right)y_2\left(t\right),y_2^{\text{'}}\left(t\right)=\hfill & \hfill -p_2\left(t\right)f\left(y_1\left(h\left(t\right)\right)\right),t\ge t_0.\end{array}$$

One of the important methods for studying the oscillation of higher order differential equations is to make a comparison with second order differential equations. The method involves using Taylor's Formula. In this paper we show how such a method can be used for a class of even order delay dynamic equations on time scales via comparison with second order dynamic inequalities. In particular, it is shown that nonexistence of an eventually positive solution of a certain second order delay dynamic inequality...