Interval Oscillation Criteria of Second Order Mixed Nonlinear Impulsive Differential Equations with Delay

Abstract

We study the following second order mixed nonlinear impulsive differential equations with delay ${\left(r\left(t\right){\Phi }_{\alpha }\left(x^{\prime }\left(t\right)\right)\right)}^{\prime }+p_0\left(t\right){\Phi }_{\alpha }\left(x\left(t\right)\right)+{\sum }_{i=1}^n{p}_i\left(t\right){\Phi }_{{\beta }_i}\left(x\left(t-\sigma \right)\right)=e\left(t\right),t\ge t_0,t\ne {\tau }_k,x\left({\tau }_k^{+}\right)=a_{k}x\left({\tau }_k\right),x\text{'}\left({\tau }_k^{+}\right)=b_{k}x\text{'}\left({\tau }_k\right),k=\mathrm{1,2},\dots$, where ${\Phi }_{*}\left(u\right)=|u{|}^{*-1}u$, $\sigma$ is a nonnegative constant, $\left\{{\tau }_k\right\}$ denotes the impulsive moments sequence, and ${\tau }_{k+1}-{\tau }_k>\sigma$. Some sufficient conditions for the interval oscillation criteria of the equations are obtained. The results obtained generalize and improve earlier ones. Two examples are considered to illustrate the main results.