Conjugacy and disconjugacy criteria are established for the equation $$u^{\text{'}\text{'}}+p\left(t\right)u=0,$$ where $p:]-\infty ,+\infty [\to ]-\infty ,+\infty [$ is a locally summable function.

Oscillation properties of the self-adjoint, two term, differential equation $${(-1)}^n{\left(p\left(x\right)y^{\left(n\right)}\right)}^{\left(n\right)}+q\left(x\right)y=0(*)$$ are investigated. Using the variational method and the concept of the principal system of solutions it is proved that (*) is conjugate on $R=(-\infty ,\infty )$ if there exist an integer $m\in \{0,1,\cdots ,n-1\}$ and $c_0,\cdots ,c_m\in R$ such that $${\int }_{\infty }^0{x}^{2(n-m-1)}{p}^{-1}\left(x\right)dx=\infty ={\int }_0^{\infty }{x}^{2(n-m-1)}{p}^{-1}\left(x\right)dx$$ and $$\underset{x_1\downarrow -\infty ,x_2\uparrow \infty }{lim\; sup}{\int }_{x_1}^{x_2}q\left(x\right){(c_0+c_{1}x+\cdots +c_m{x}^m)}^{2}dx\le 0,q\left(x\right)\neg \equiv 0.$$ Some extensions of this criterion are suggested.

Sufficient conditions on the function $c\left(t\right)$ ensuring that the half-linear second order differential equation $$\left(\right|u^{\text{'}}{|}^{p-2}{u}^{\text{'}}{)}^{\text{'}}+c\left(t\right){\left|u\left(t\right)\right|}^{p-2}u\left(t\right)=0,p>1$$ possesses a nontrivial solution having at least two zeros in a given interval are obtained. These conditions extend some recently proved conjugacy criteria for linear equations which correspond to the case $p=2$.

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.