Oscillation of nonlinear second order neutral delay dynamic equations on time scales

Bull. Korean Math. Soc. 2008 Vol. 45, No. 2, 299-312 Printed June 1, 2008

Hassan A. Agwo Ain Shams University

Abstract : In this paper, we establish some oscillation criteria for nonautonomous second order neutral delay dynamic equations \begin{equation*} (x(t)\pm r(t)x(\tau (t)))^{\triangle \triangle }+H(t,x(h_{1}(t)),x^{\triangle }(h_{2}(t)))=0 \end{equation*} on a time scale $\mathbb{T}$. Oscillatory behavior of such equations is not studied before. This is a first paper concerning these equations. The results are not only can be applied on neutral differential equations when $\mathbb{T}=\mathbb{R}$, neutral delay difference equations when $\mathbb{T=N} $ and for neutral delay $q-$difference equations when $\mathbb{T=}q^{\mathbb{N}}$ for $q>1$, but also improved most previous results. Finally, we give some examples to illustrate our main results. These examples are not discussed before and there is no previous theorems determine the oscillatory behavior of such equations.

Keywords : oscillation, time scales, neutral delay, dynamic equation