Suppose that the function $q\left(t\right)$ in the differential equation (1) $y^{\text{'}\text{'}}+q\left(t\right)y=0$ is decreasing on $(b,\infty )$ where $b\ge 0$. We give conditions on $q$ which ensure that (1) has a pair of solutions $y_1\left(t\right),y_2\left(t\right)$ such that the $n$-th derivative ($n\ge 1$) of the function $p\left(t\right)=y_1^2\left(t\right)+y_2^2\left(t\right)$ has the sign ${(-1)}^{n+1}$ for sufficiently large $t$ and that the higher differences of a sequence related to the zeros of solutions of (1) are ultimately regular in sign.

In the paper a comparison theory of Sturm-Picone type is developed for the pair of nonlinear second-order ordinary differential equations first of which is the quasilinear differential equation with an oscillatory forcing term and the second is the so-called half-linear differential equation. Use is made of a new nonlinear version of the Picone’s formula.

Let $L\left(y\right)=y^{\left(n\right)}+q_{n-1}\left(t\right)y^{(n-1)}+\cdots +q_0\left(t\right)y,t\in [a,b)$, be an $n$-th order differential operator, $L^{*}$ be its adjoint and $p,w$ be positive functions. It is proved that the self-adjoint equation $L^{*}\left(p\left(t\right)L\left(y\right)\right)=w\left(t\right)y$ is nonoscillatory at $b$ if and only if the equation $L\left(w^{-1}\left(t\right)L^{*}\left(y\right)\right)=p^{-1}\left(t\right)y$ is nonoscillatory at $b$. Using this result a new necessary condition for property BD of the self-adjoint differential operators with middle terms is obtained.

Oscillatory properties of the second order nonlinear equation $${\left(r\left(t\right)x^{\text{'}}\right)}^{\text{'}}+q\left(t\right)f\left(x\right)=0$$ are investigated. In particular, criteria for the existence of at least one oscillatory solution and for the global monotonicity properties of nonoscillatory solutions are established. The possible coexistence of oscillatory and nonoscillatory solutions is studied too.

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.

We establish Hille-Wintner type comparison 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.

New oscillation criteria are given for the second order sublinear differential equation $${\left[a\left(t\right)\psi \left(x\left(t\right)\right)x^{\text{'}}\left(t\right)\right]}^{\text{'}}+q\left(t\right)f\left(x\left(t\right)\right)=0,t\ge t_0>0,$$ where $a\in C^1\left([t_0,\infty )\right)$ is a nonnegative function, $\psi ,f\in C\left(ℝ\right)$ with $\psi \left(x\right)\ne 0$, $xf\left(x\right)/\psi \left(x\right)>0$ for $x\ne 0$, $\psi$, $f$ have continuous derivative on $ℝ\setminus \left\{0\right\}$ with ${[f\left(x\right)/\psi \left(x\right)]}^{\text{'}}\ge 0$ for $x\ne 0$ and $q\in C\left([t_0,\infty )\right)$ has no restriction on its sign. This oscillation criteria involve integral averages of the coefficients $q$ and $a$ and extend known oscillation criteria for the equation $x^{\text{'}\text{'}}\left(t\right)+q\left(t\right)x\left(t\right)=0$.

This paper is concerned with the oscillatory behavior of the damped half-linear oscillator ${\left(a\left(t\right){\phi }_p\left(x^{\text{'}}\right)\right)}^{\text{'}}+b\left(t\right){\phi }_p\left(x^{\text{'}}\right)+c\left(t\right){\phi }_p\left(x\right)=0$, where ${\phi }_p\left(x\right)={\left|x\right|}^{p-1}\mathrm{sgn}x$ for $x\in ℝ$ and $p>1$. A sufficient condition is established for oscillation of all nontrivial solutions of the damped half-linear oscillator under the integral averaging conditions. The main result can be given by using a generalized Young’s inequality and the Riccati type technique. Some examples are included to illustrate the result. Especially, an example which asserts that all nontrivial solutions are...