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

Abstract

In this work, we use the generalized Riccati transformation and the inequality technique to establish some new oscillation criteria for the second-order nonlinear delay dynamic equation ${(p(t){(x^{\mathrm{\Delta }}(t))}^{\gamma })}^{\mathrm{\Delta }}+q(t)f(x(\tau (t)))=0$, on a time scale $,$ where $\gamma$ is the quotient of odd positive integers and p(t) and q(t) are positive right-dense continuous (rd-continuous) functions on $𝕋$. Our results improve and extend some results established by Sun et al. 2009. Also our results unify the oscillation of the second-order nonlinear delay differential equation and the second-order nonlinear delay difference equation. Finally, we give some examples to illustrate our main results.