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.