Oscillation of Second-Order Sublinear Impulsive Differential Equations

Abstract

Oscillation criteria obtained by Kusano and Onose (1973) and by Belohorec (1969) are extended to second-order sublinear impulsive differential equations of Emden-Fowler type: $x\left(t\right)+p\left(t\right)|x\left(\tau \right(t\left)\right){|}^{\alpha -1}x\left(\tau \right(t\left)\right)=0$ , $t\ne {\theta }_k$; $\Delta x\text{'}\left(t\right){|}_{t={\theta }_k}+q_k|x\left(\tau \right({\theta }_k\left)\right){|}^{\alpha -1}x\left(\tau \right({\theta }_k\left)\right)=0$; $\Delta x\left(t\right){|}_{t={\theta }_k}=0$, $(0<\alpha <1)$ by considering the cases $\tau \left(t\right)\le t$ and $\tau \left(t\right)=t$, respectively. Examples are inserted to show how impulsive perturbations greatly affect the oscillation behavior of the solutions.