In this paper, oscillation and asymptotic behaviour of solutions of $$y^{\text{'}\text{'}\text{'}}+a\left(t\right)y^{\text{'}\text{'}}+b\left(t\right)y^{\text{'}}+c\left(t\right)y=0$$ have been studied under suitable assumptions on the coefficient functions $a,b,c\in C\left(\right[\sigma ,\infty ),R)$, $\sigma \in R$, such that $a\left(t\right)\ge 0$, $b\left(t\right)\le 0$ and $c\left(t\right)<0$.

In this paper we are concerned with the oscillation of third order nonlinear delay differential equations of the form $${\left(r_2\left(t\right){\left(r_1\left(t\right)x^{\text{'}}\right)}^{\text{'}}\right)}^{\text{'}}+p\left(t\right)x^{\text{'}}+q\left(t\right)f\left(x\left(g\left(t\right)\right)\right)=0.$$ We establish some new sufficient conditions which insure that every solution of this equation either oscillates or converges to zero.

The paper deals with oscillation criteria of fourth order linear differential equations with quasi-derivatives.

We study oscillatory properties of solutions of the systems of differential equations of neutral type.

Sufficient conditions are obtained so that every solution of ${[y\left(t\right)-p\left(t\right)y(t-\tau )]}^{\left(n\right)}+Q\left(t\right)G\left(y(t-\sigma )\right)=f\left(t\right)$ where n ≥ 2, p,f ∈ C([0,∞),ℝ), Q ∈ C([0,∞),[0,∞)), G ∈ C(ℝ,ℝ), τ > 0 and σ ≥ 0, oscillates or tends to zero as $t\to \infty$. Various ranges of p(t) are considered. In order to accommodate sublinear cases, it is assumed that ${\int }_0^{\infty }Q\left(t\right)dt=\infty$. Through examples it is shown that if the condition on Q is weakened, then there are sublinear equations whose solutions tend to ±∞ as t → ∞.

In this paper, necessary and sufficient conditions are obtained for every bounded solution of $${[y\left(t\right)-p\left(t\right)y(t-\tau )]}^{\left(n\right)}+Q\left(t\right)G\left(y(t-\sigma )\right)=f\left(t\right),t\ge 0,(*)$$ to oscillate or tend to zero as $t\to \infty$ for different ranges of $p\left(t\right)$. It is shown, under some stronger conditions, that every solution of $(*)$ oscillates or tends to zero as $t\to \infty$. Our results hold for linear, a class of superlinear and other nonlinear equations and answer a conjecture by Ladas and Sficas, Austral. Math. Soc. Ser. B 27 (1986), 502–511, and generalize some known results.