We study the generalized half-linear second order differential equation via the associated Riccati type differential equation and Prüfer transformation. We establish a de la Vallée Poussin type inequality for the distance of consecutive zeros of a nontrivial solution and this result we apply to the “classical” half-linear differential equation regarded as a perturbation of the half-linear Euler differential equation with the so-called critical oscillation constant. In the second part of the paper...

The existence of decaying positive solutions in ${ℝ}_{+}$ of the equations $\left(E_{\lambda }\right)$ and $\left(E_{\lambda }^1\right)$ displayed below is considered. From the existence of such solutions for the subhomogeneous cases (i.e. $t^{1-p}F(r,tU,t|U^{\text{'}}\left|\right)\searrow 0$ as $t\nearrow \infty$), a super-sub-solutions method (see § 2.2) enables us to obtain existence theorems for more general cases.

We establish Vallée Poussin type disconjugacy and disfocality criteria for the half-linear second order differential equation $u^{\text{'}\text{'}}={p\left(t\right)\left|u\right|}^{\alpha }{\left|u^{\text{'}}\right|}^{1-\alpha }sgnu+g\left(t\right)u^{\text{'}}$, where α ∈ (0,1] and the functions $p,g\in L_{loc}(a,b)$ are allowed to have singularities at the end points t = a, t = b of the interval under consideration.

For linear differential equations of the second order in the Jacobi form $$y^{\text{'}\text{'}}+p\left(x\right)y=0$$ O. Borvka introduced a notion of dispersion. Here we generalize this notion to certain classes of linear differential equations of arbitrary order. Connection with Abel’s functional equation is derived. Relations between asymptotic behaviour of solutions of these equations and distribution of zeros of their solutions are also investigated.