On The Bounded Oscillation of Certain Second-order Nonlinear Neutral Delay Dynamic Equations with Oscillating Coefficients
- Da-Xue Chen
- Guang-Hui Liu
Abstract
We investigate the bounded oscillation of the second-order nonlinear neutral delay dynamic equation with oscillating coefficients \[
\Big(r(t)\Big|\big[x(t)+p(t)x(\tau(t))\big]^\Delta\Big|^{\alpha-1}\big[x(t)+p(t)x(\tau(t))\big]^\Delta\Big)^\Delta+q(t)|x(t)|^{\beta-1}x(t)=0
\]
on an arbitrary time scale $\mathbb{T}$, where $p$ is an oscillating function defined on $\mathbb{T}$ and $\alpha,\beta>0$ are constants, and obtain several sufficient conditions for the oscillation of all bounded solutions of the equation when $\beta>\alpha, \beta=\alpha$ and $\beta<\alpha$, respectively. Our results extend and complement some known results where $p(t)\equiv 0$ and $\alpha,\beta$ are quotients of odd positive integers.
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- DOI:10.5539/jmr.v3n2p193
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