Electron. J. Diff. Eqns., Vol. 2007(2007), No. 58, pp. 1-9.

Interval oscillation of second-order Emden-Fowler neutral delay differential equations

Mu Chen, Zhiting Xu

Abstract:
Employing Riccati techniques and the integral averaging method, we establish interval oscillation criteria for the second-order Emden-Fowler neutral delay differential equation
$$ [|x'(t)|^{\gamma-1}x'(t)]'+q_1(t)|y(t-\sigma)|^{\alpha-1}y(t-\sigma) +q_2(t)|y(t-\sigma)|^{\beta-1} y(t-\sigma)=0, $$
where $t\geq t_0$ and $x(t)=y(t)+p(t)y(t-\tau)$. The criteria obtained here are different from most known criteria in the sense that they are based on information only on a sequence of subintervals of $[t_0,\infty)$, rather than on the whole half-line. In particular, two interesting examples that illustrate the importance of our results are included.

Submitted January 5, 2007. Published April 22, 2007.
Math Subject Classifications: 34K40, 34K11, 34C10.
Key Words: Interval oscillation; neutral delay differential equation; Emden-Fowler; Riccati technique; integral averaging method.

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Mu Chen
School of Mathematical Sciences
South China Normal University
Guangzhou, 510631, China.
Department of Mathematics, Zhanjiang Education College
Zhanjiang, Guangdong 524037, China
email: zjchenmu03@163.com
Zhiting Xu
School of Mathematical Sciences
South China Normal University
Guangzhou, 510631, China
email: xztxhyyj@pub.guangzhou.gd.cn

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