We establish Hartman-Wintner type criteria for the half-linear second order differential equation $${\left(r\left(t\right)\Phi \left(x^{\text{'}}\right)\right)}^{\text{'}}+c\left(t\right)\Phi \left(x\right)=0,\Phi \left(x\right)={\left|x\right|}^{p-2}x,p>1,$$ where this equation is viewed as a perturbation of another equation of the same form.

Sufficient conditions are established for the oscillation of proper solutions of the system $$\begin{array}{cc}\hfill u_1^{\text{'}}\left(t\right)& =p\left(t\right)u_2\left(\sigma \left(t\right)\right),\hfill \\ \hfill u_2^{\text{'}}\left(t\right)& =-q\left(t\right)u_1\left(\tau \left(t\right)\right),\hfill \end{array}$$ where $p,q:R_{+}\to R_{+}$ are locally summable functions, while $\tau$ and $\sigma :R_{+}\to R_{+}$ are continuous and continuously differentiable functions, respectively, and $\underset{t\to +\infty }{lim}\tau \left(t\right)=+\infty$, $\underset{t\to +\infty }{lim}\sigma \left(t\right)=+\infty$.

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