A sufficient integral condition for the absence of eventually positive solutions of a first order stable type differential inequality with one nondecreasing retarded argument is given. In the special case of equality the result becomes an oscillation criterion.

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 $p=2$.