Electronic Journal of Differential Equations,
Vol. 2010(2010), No. 131, pp. 1-14.

Title: Oscillation of solutions for third order
       functional dynamic equations

Authors: Elmetwally M. Elabbasy (Mansoura Univ., Egypt)
         Taher S. Hassan (Mansoura Univ., Egypt)

Abstract:
 In this article we study the oscillation of solutions to
 the third order nonlinear functional dynamic equation
 $$
 L_{3}(x(t))+\sum_{i=0}^{n}p_i(t)\Psi_k{\alpha_ki}(x(h_i(t)))=0,
 $$
 on an arbitrary time scale $\mathbb{T}$. Here
 $$
 L_0(x(t))=x(t),\quad L_k(x(t))=\Big(\frac{[
 L_{k-1}x(t)]^{\Delta }}{a_k(t)}\Big)^{\gamma_kk}, \quad k=1,2,3
 $$
 with $a_1, a_2$  positive rd-continuous functions on
 $\mathbb{T}$ and $a_{3}\equiv 1$;
 the functions $p_i$ are nonnegative rd-continuous
 on $\mathbb{T}$ and not all $p_i(t)$ vanish in a neighborhood
 of infinity; $\Psi_k{c}(u)=|u|^{c-1}u$, $c>0$.
 Our main results extend known results and are illustrated by examples.

	 
Submitted April 13, 2010. Published September 14, 2010.
Math Subject Classifications: 34K11, 39A10, 39A99.
Key Words: Oscillation; third order; functional dynamic equations;
           time scales.