Electron. J. Diff. Equ., Vol. 2010(2010), No. 131, pp. 1-11.

Oscillation of solutions for third order functional dynamic equations

Elmetwally M. Elabbasy, Taher S. Hassan

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.

Show me the PDF file (271 KB), TEX file, and other files for this article.

Elmetwally M. Elabbasy
Department of Mathematics, Faculty of Science
Mansoura University, Mansoura, 35516, Egypt
email: emelabbasy@mans.edu.eg
http://www.mans.edu.eg/pcvs/10805/
Taher S. Hassan
Department of Mathematics, Faculty of Science
Mansoura University, Mansoura, 35516, Egypt
email: tshassan@mans.edu.eg
http://www.mans.edu.eg/pcvs/10805/

Return to the EJDE web page