Interval Oscillation Criteria for Forced Second-Order Nonlinear Delay Dynamic Equations with Damping and Oscillatory Potential on Time Scales

Abstract

We are concerned with the interval oscillation of general type of forced second-order nonlinear dynamic equation with oscillatory potential of the form ${\left(r\left(t\right)g_{\mathrm{1}}\left(x\left(t\right),x^{\mathrm{\Delta }}\left(t\right)\right)\right)}^{\mathrm{\Delta }}+p(t)g_{\mathrm{2}}(x(t),x^{\mathrm{\Delta }}(t))x^{\mathrm{\Delta }}(t)+q(t)f(x(\tau (t)))=e(t)$, on a time scale $\mathbb{T}$. We will use a unified approach on time scales and employ the Riccati technique to establish some oscillation criteria for this type of equations. Our results are more general and extend the oscillation criteria of Erbe et al. (2010). Also our results unify the oscillation of the forced second-order nonlinear delay differential equation and the forced second-order nonlinear delay difference equation. Finally, we give some examples to illustrate our results.