Two-Parametric Conditionally Oscillatory Half-Linear Differential Equations

Abstract

We study perturbations of the nonoscillatory half-linear differential equation $\left(r\right(t\left)\Phi \right(x\text{'}\left)\right)\text{'}+c\left(t\right)\Phi \left(x\right)=0$, $\Phi \left(x\right):=|x{|}^{p-2}x$, . We find explicit formulas for the functions $\dot{r}$, $\dot{c}$ such that the equation $\left[\left(r\right(t)+\lambda \dot{r}(t\left)\right)\Phi (x\text{'})\right]\text{'}+[c\left(t\right)+\mu \dot{c}\left(t\right)]\Phi \left(x\right)=0$ is conditionally oscillatory, that is, there exists a constant $\gamma$such that the previous equation is oscillatory if and nonoscillatory if $\mu -\lambda <\gamma$. The obtained results extend the previous results concerning two-parametric perturbations of the half-linear Euler differential equation.