We obtain conditions for existence and (almost) non-oscillation of solutions of a second order linear homogeneous functional differential equations $$u^{\text{'}\text{'}}\left(x\right)+\sum _i{p}_i\left(x\right)u^{\text{'}}\left(h_i\left(x\right)\right)+\sum _i{q}_i\left(x\right)u\left(g_i\left(x\right)\right)=0$$ without the delay conditions $h_i\left(x\right),g_i\left(x\right)\le x$, $i=1,2,...$, and $$u^{\text{'}\text{'}}\left(x\right)+{\int }_0^{\infty }{u}^{\text{'}}\left(s\right){\mathrm{d}}_s{r}_1(x,s)+{\int }_0^{\infty }u\left(s\right){\mathrm{d}}_s{r}_0(x,s)=0.$$

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 $c\left(t\right)<0$.

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

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.