We consider the equation $$-{\left(r\left(x\right)y^{\text{'}}\left(x\right)\right)}^{\text{'}}+q\left(x\right)y\left(x\right)=f\left(x\right),x\in ℝ(*)$$ where $f\in L_p\left(ℝ\right)$, $p\in (1,\infty )$ and $$\begin{array}{c}r>0,q\ge 0,\frac{1}{r}\in L_1^{\mathrm{loc}}\left(ℝ\right),q\in L_1^{\mathrm{loc}}\left(ℝ\right),\\ \underset{\left|d\right|\to \infty }{lim}{\int }_{x-d}^x\frac{\mathrm{d}t}{r\left(t\right)}·{\int }_{x-d}^{x}q\left(t\right)\mathrm{d}t=\infty .\end{array}$$ In an earlier paper, we obtained a criterion for correct solvability of ($*$) in $L_p\left(ℝ\right),$$p\in (1,\infty )...$

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.