The basic idea of this paper is to give the existence theorem and the method of averaging for the system of functional-differential inclusions of the form ⎧$ẋ\left(t\right)\in F(t,x_t,y_t)$ (0) ⎨ ⎩$ẋ\left(t\right)\in G(t,x_t,y_t)$ (1)

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.