We investigate an asymptotic behaviour of damped non-oscillatory solutions of the initial value problem with a time singularity ${\left(p\left(t\right)u^{\text{'}}\left(t\right)\right)}^{\text{'}}+p\left(t\right)f\left(u\left(t\right)\right)=0$, $u\left(0\right)=u_0$, $u^{\text{'}}\left(0\right)=0$ on the unbounded domain $[0,\infty )$. Function $f$ is locally Lipschitz continuous on $ℝ$ and has at least three zeros $L_0<0$, $0$ and $L>0$. The initial value $u_0\in (L_0,L)\setminus \left\{0\right\}$. Function $p$ is continuous on $[0,\infty ),$ has a positive continuous derivative on $(0,\infty )$ and $p\left(0\right)=0$. Asymptotic formulas for damped non-oscillatory solutions and their first derivatives are derived under some additional assumptions. Further,...

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 $t\to \infty$.