Oscillation of Second-Order Nonlinear Delay Dynamic Equations with Damping on Time Scales

Abstract

We use the generalized Riccati transformation and the inequality technique to establish some new oscillation criteria for the second-order nonlinear delay dynamic equation with damping on a time scale $\mathbb{T}$$(r(t)\mathrm{g}(x(t)$, $x^{\mathrm{\Delta }}(t)){)}^{\mathrm{\Delta }}+p(t)g(x(t)$, $x^{\mathrm{\Delta }}(t))\hspace{0.17em}+\hspace{0.17em}q(t)f(x(\tau (t)))=0,$ where $r(t)$, $p(t)$, and $q(t)$ are positive right dense continuous (rd-continuous) functions on $\mathbb{T}$. Our results improve and extend some results established by Zhang et al., 2011. Also, our results unify the oscillation of the second-order nonlinear delay differential equation with damping and the second-order nonlinear delay difference equation with damping. Finally, we give some examples to illustrate our main results.