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具有分布时滞的二阶非线性中立型时标动力方程的振动定理

陈大学(1), 刘洁纯(2)   

  1. (1)湖南工程学院理学院, 湖南 411104;(2)湖南工程学院成人教育学院, 湖南 411104
  • 收稿日期:2008-07-03 修回日期:2009-02-04 出版日期:2010-09-25 发布日期:2010-09-25

陈大学;刘洁纯. 具有分布时滞的二阶非线性中立型时标动力方程的振动定理[J]. 系统科学与数学, 2010, 30(9): 1191-1205.

CHEN Daxue;LIU Jiechun. Oscillation Theorems for Second-Order Nonlinear Neutral Dynamic Equations on Time Scales with Distributed Delay[J]. Journal of Systems Science and Mathematical Sciences, 2010, 30(9): 1191-1205.

Oscillation Theorems for Second-Order Nonlinear Neutral Dynamic Equations on Time Scales with Distributed Delay

CHEN Daxue(1), LIU Jiechun(2)   

  1. (1)College of Science, Hunan Institute of Engineering, Hunan 411104;(2)Adult Education College, Hunan Institute of Engineering, Hunan 411104
  • Received:2008-07-03 Revised:2009-02-04 Online:2010-09-25 Published:2010-09-25
利用广义Riccati变换技术, 研究时标上具有分布时滞的二阶非线性中立型动力方程
\[
\big(r(t)\big(\big(y(t)+p(t)y(\tau(t))\big)^{\it \Delta}\big)^\beta\big)^{\it \Delta}+\int_c^dF(t,\xi,y(\delta(t,\xi))){\it \Delta}\xi=0
\]
的振动性, 其中$\beta>0$是两个正奇数之比, 获得了方程所有解振动的几个充分条件,
推广和改进了一些已知的结果, 并给出了几个应用实例.
This paper is concerned with the oscillation of the second-order nonlinear neutral dynamic equation with distributed delay
\[
\big(r(t)\big(\big(y(t)+p(t)y(\tau(t))\big)^{\it \Delta}\big)^\beta\big)^{\it \Delta}+\int_c^dF(t,\xi,y(\delta(t,\xi))){\it \Delta}\xi=0,
\]
on an arbitrary time scale $\mathbb{T}$, where $\beta>0$ is a quotient of odd positive integers. By employing the generalized Riccati transformation technique,
several sufficient conditions for all solutions of the equation be oscillatory. The results extend and improve some known results. Some examples to illustrate the main results are given.

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