In the paper the fourth order nonlinear differential equation $y^{\left(4\right)}+{\left(q\left(t\right)y^{\text{'}}\right)}^{\text{'}}+r\left(t\right)f\left(y\right)=0$, where $q\in C^1\left([0,\infty )\right)$, $r\in C^0\left([0,\infty )\right)$, $f\in C^0\left(R\right)$, $r\ge 0$ and $f\left(x\right)x>0$ for $x\ne 0$ is considered. We investigate the asymptotic behaviour of nonoscillatory solutions and give sufficient conditions under which all nonoscillatory solutions either are unbounded or tend to zero for $t\to \infty$.

The paper deals with the quasi-linear ordinary differential equation ${\left(r\left(t\right)\varphi \left(u^{\text{'}}\right)\right)}^{\text{'}}+g(t,u)=0$ with $t\in [0,\infty )$. We treat the case when $g$ is not necessarily monotone in its second argument and assume usual conditions on $r\left(t\right)$ and $\varphi \left(u\right)$. We find necessary and sufficient conditions for the existence of unbounded non-oscillatory solutions. By means of a fixed point technique we investigate their growth, proving the coexistence of solutions with different asymptotic behaviors. The results generalize previous ones due to Elbert–Kusano,...