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,...

The paper investigates the singular initial problem[4pt] ${\left(p\left(t\right)u^{\text{'}}\left(t\right)\right)}^{\text{'}}+q\left(t\right)f\left(u\left(t\right)\right)=0,u\left(0\right)=u_0,u^{\text{'}}\left(0\right)=0$[4pt] on the half-line $[0,\infty )$. Here $u_0\in [L_0,L]$, where $L_0$, $0$ and $L$ are zeros of $f$, which is locally Lipschitz continuous on $ℝ$. Function $p$ is continuous on $[0,\infty )$, has a positive continuous derivative on $(0,\infty )$ and $p\left(0\right)=0$. Function $q$ is continuous on $[0,\infty )$ and positive on $(0,\infty )$. For specific values $u_0$ we prove the existence and uniqueness of damped solutions of this problem. With additional conditions for $f$, $p$ and $q$ it is shown that the problem has for each specified...

We give a sufficient condition for the oscillation of linear homogeneous second order differential equation $y^{\text{'}\text{'}}+p\left(x\right)y^{\text{'}}+q\left(x\right)y=0$, where $p\left(x\right),q\left(x\right)\in C[\alpha ,\infty )$ and $\alpha$ is positive real number.

In this paper we establish some oscillation or nonoscillation criteria for the second order half-linear differential equation $${\left(r\left(t\right)\Phi \left(u^{\text{'}}\left(t\right)\right)\right)}^{\text{'}}+c\left(t\right)\Phi \left(u\left(t\right)\right)=0,$$ where (i) $r,c\in C([t_0,\infty )$, $ℝ:=(-\infty ,\infty ))$ and $r\left(t\right)>0$ on $[t_0,\infty )$ for some $t_0\ge 0$; (ii) $\Phi \left(u\right)={\left|u\right|}^{p-2}u$ for some fixed number $p>1$. We also generalize some results of Hille-Wintner, Leighton and Willet.