Oscillation of second-order forced nonlinear dynamic equations on time scales

S. H. Saker, Mansoura University, Mansoura, Egypt

E. J. Qualitative Theory of Diff. Equ., No. 23. (2005), pp. 1-17.

Abstract: In this paper, we discuss the oscillatory behavior of the second-order forced nonlinear dynamic equation
\begin{equation*}
\left( a(t)x^{\Delta }(t)\right) ^{\Delta }+p(t)f(x^{\sigma })=r(t),
\end{equation*}%
on a time scale ${\mathbb{T}}$ when $a(t)>0$. We establish some sufficient conditions which ensure that every solution oscillates or satisfies $\lim \inf_{t\rightarrow \infty }\left\vert x(t)\right\vert =0.$ Our oscillation results when $r(t)=0$ improve the oscillation results for dynamic equations on time scales that has been established by Erbe and Peterson [Proc. Amer. Math. Soc \ 132 (2004), 735-744], Bohner, Erbe and Peterson [J. Math. Anal. Appl. 301 (2005), 491--507] since our results do not require $% \int_{t_{0}}^{\infty }q(t)\Delta t>0$ and $\int_{\pm t_{0}}^{\pm \infty }% \frac{du}{f(u)}<\infty .$ Also, as a special case when ${\mathbb{T=R}}$, and $r(t)=0$ our results improve some oscillation results for differential equations. Some examples are given to illustrate the main results.

You can download the full text of this paper in DVI, PostScript or PDF format.