In this paper we investigate oscillatory properties of the second order half-linear equation $${\left(r\left(t\right)\Phi \left(y^{\text{'}}\right)\right)}^{\text{'}}+c\left(t\right)\Phi \left(y\right)=0,\Phi \left(s\right):={\left|s\right|}^{p-2}s.(*)$$ Using the Riccati technique, the variational method and the reciprocity principle we establish new oscillation and nonoscillation criteria for (*). We also offer alternative methods of proofs of some recent oscillation results.

In this paper we are concerned with sufficient conditions for the existence of minimal and maximal solutions of differential equations of the form $$L_{4}y+f(t,y)=0,$$ where $L_{4}y$ is the iterated linear differential operator of order $4$ and $f:[a,\infty )\times (0,\infty )\to (0,\infty )$ is a continuous function.

We obtain monotonicity results concerning the oscillatory solutions of the differential equation $\left(a\left(t\right)\right|y^{\text{'}}{{|}^{p-1}{y}^{\text{'}})}^{\text{'}}+f(t,y,y^{\text{'}})=0$. The obtained results generalize the results given by the first author in [1] (1976). We also give some results concerning a special case of the above differential equation.