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.