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中国工业与应用数学学会会刊
主管：中华人民共和国教育部
主办：西安交通大学
ISSN 1005-3085  CN 61-1269/O1
工程数学学报

工程数学学报 ›› 2016, Vol. 33 ›› Issue (4): 402-418.doi: 10.3969/j.issn.1005-3085.2016.04.007

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带混合型非线性项的二阶中立型时标动力方程的振动准则(英)

黄先勇1,  杨启贵2,  曹俊飞1   

  1. 1- 广东第二师范学院数学系，广州 510303
    2- 华南理工大学数学学院，广州 510641
  • 收稿日期:2014-06-26 接受日期:2015-11-19 出版日期:2016-08-15 发布日期:2016-10-15
  • 基金资助:
    国家自然科学基金 (11426066; 11426068; 11301090)；广东教育大学自然科学基金 (2014jcjs03;2015 ybzz01).

Oscillation Criteria for Second-order Neutral Dynamic Equations with Mixed Nonlinearities on Time Scales

HUANG Xian-yong1,  YANG Qi-gui2,  CAO Jun-fei1   

  1. 1- Department of Mathematics, Guangdong University of Education, Guangzhou 510303
    2- School of Mathematics, South China University of Technology, Guangzhou 510641
  • Received:2014-06-26 Accepted:2015-11-19 Online:2016-08-15 Published:2016-10-15
  • Supported by:
    The National Natural Science Foundation of China (11426066; 11426068; 11301090); the Natural Science Foundation of Guangdong University of Education (2014jcjs03; 2015ybzz01).

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摘要： 中立型时标动力方程的振动性在理论上和应用中有着重要的意义．本文研究了一类二阶带混合型非线性项的中立型时标动力方程的振动性．首先，我们定义了中立项系数函数$\pi(t)$．当$\pi(t_{0})=\infty$时，利用广义李卡提变换技巧和均值技巧，建立了二阶中立型动力方程振动的一些新的判据．其次，当$\pi(t_{0})<\infty$时，通过加强假设条件及应用某些不等式和一些分析技巧，我们也得到了该方程振动的几个判据．我们的工作推广并改进了相关文献关于二阶中立型动力方程振动的结果．最后，作为应用给出两个实例说明所获定理的重要性．

关键词: 振动, 中立型动力方程, 二阶, 时标

Abstract:

The oscillation of neutral dynamic equations on time scales has important implications in both theory and application. This paper considers the oscillation of a class of second-order neutral dynamic equations with mixed nonlinearities. Firstly, the neutral coefficient function $\pi(t)$ is defined. By means of the generalized Riccati transformation technique and the averaging technique, we establish some new oscillation criteria for the second-order neutral dynamic equations under the case $\pi(t_{0})=\infty$. Then, when $\pi(t_{0})<\infty$, by strengthening the assumptions and using some inequalities and some analytic techniques, we also obtain several oscillation criteria for the equations. Our work generalizes and improves some known results in the literature for oscillation of second-order neutral dynamic equations. Finally, two examples are presented to illustrate the importance of our results.

Key words: oscillation, neutral dynamic systems, second-order, time scales

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