Our aim in this paper is to present sufficient conditions for the oscillation of the second order neutral differential equation (x(t)-px(t-))"+q(t)x((t))=0.

Some results concerning oscillation of second order self-adjoint matrix differential equations are obtained. These may be regarded as a generalization of results for the corresponding scalar equations.

In this work, necessary and sufficient conditions for the oscillation of solutions of 2-dimensional linear neutral delay difference systems of the form $$\Delta \left[\begin{array}{c}x\left(n\right)+p\left(n\right)x(n-m)\\ y\left(n\right)+p\left(n\right)y(n-m)\end{array}\right]=\left[\begin{array}{cc}a\left(n\right)& b\left(n\right)\\ c\left(n\right)& d\left(n\right)\end{array}\right]\left[\begin{array}{c}x(n-\alpha )\\ y(n-\beta )\end{array}\right]$$ are established, where $m>0$, $\alpha \ge 0$, $\beta \ge 0$ are integers and $a\left(n\right)$, $b\left(n\right)$, $c\left(n\right)$, $d\left(n\right)$, $p\left(n\right)$ are sequences of real numbers.

We state and prove two oscillation results which deal with bounded solutions of a forced higher order differential equation. One proof involves the use of a nonlinear functional.

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 $$L_{n}u\left(t\right)+p\left(t\right)u\left(\tau \left(t\right)\right)=0.$$

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.