Oscillation Results for Third Order Nonlinear Delay Dynamic Equations on Time Scales

Tongxing Li ; Zhenlai Han ; Shurong Sun ; Yige Zhao

Bulletin of the Malaysian Mathematical Society, Tome 34 (2011) no. 3

Résumé

In this paper, we consider the third order nonlinear delay dynamic equations

$ (a(t)\{[r(t)x^\Delta(t)]^\Delta\}^\gamma)^\Delta+f(t, x(\tau(t)))=0, $

on a time scale $\mathbb{T}$, where $\gamma>0$ is a quotient of odd positive integers, $a$ and $r$ are positive $rd$-continuous functions on $\mathbb{T},$ and the so-called delay function $\tau:\mathbb{T}\rightarrow\mathbb{T}$ satisfies $\tau(t)\leq t,$ and $\tau(t)\to\infty$ as $t\to\infty$, $f\in C(\mathbb{T}\times\mathbb{R},\mathbb{R})$ is assumed to satisfy $uf(t,u)>0,$ for $u\neq0$ and there exists a positive $rd$-continuous function $p$ on $\mathbb{T}$ such that $f(t,u)/u^\gamma\geq p(t),$ for $u\neq 0$. We establish some new results. Some examples are considered to illustrate the main results.

Détail

Classification : 39A21, 34C10, 34K11, 34N05.