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 $u_0$ a unique...

In this paper we shall study some oscillatory and nonoscillatory properties of solutions of a nonlinear third order differential equation, using the results and methods of the linear differential equation of the third order.

The aim of the paper is to study the structure of oscillatory solutions of a nonlinear third order differential equation $y^{\text{'}\text{'}\text{'}}+p{y}^{\text{'}\text{'}}+q{y}^{\text{'}}+rf(y,y^{\text{'}},y^{\text{'}\text{'}})=0$.

The aim of this paper is to study the global structure of solutions of three differential inequalities with respect to their zeros. New information for the differential equation of the third order with quasiderivatives is obtained, too.

Oscillation and nonoscillation criteria are established for the equation $$u^{\text{'}\text{'}}+{p\left(t\right)\left|u\right|}^{\alpha }{\left|u^{\text{'}}\right|}^{1-\alpha }sgnu=0,$$ where $\alpha \in ]0,1]$, and $p:[0,+\infty [\to [0,+\infty [$ is a locally summable function.

We obtain sufficient conditions for every solution of the differential equation $${[y\left(t\right)-p\left(t\right)y\left(r\left(t\right)\right)]}^{\left(n\right)}+v\left(t\right)G\left(y\left(g\left(t\right)\right)\right)-u\left(t\right)H\left(y\left(h\left(t\right)\right)\right)=f\left(t\right)$$ to oscillate or to tend to zero as $t$ approaches infinity. In particular, we extend the results of Karpuz, Rath and Padhy (2008) to the case when $G$ has sub-linear growth at infinity. Our results also apply to the neutral equation $${[y\left(t\right)-p\left(t\right)y\left(r\left(t\right)\right)]}^{\left(n\right)}+q\left(t\right)G\left(y\left(g\left(t\right)\right)\right)=f\left(t\right)$$ when $q\left(t\right)$ has sign changes. Both bounded and unbounded solutions are consideted here; thus some known results are expanded.