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三阶半线性时滞微分方程的振动性和渐近性

林全文1,俞元洪2   

  1. 1.广东石油化工学院数学系, 茂名 525000; 2.中国科学院数学与系统科学研究院, 北京 100190
  • 出版日期:2015-02-25 发布日期:2015-05-19

林全文,俞元洪. 三阶半线性时滞微分方程的振动性和渐近性[J]. 系统科学与数学, 2015, 35(2): 233-244.

LIN Quanwen,YU Yuanhong. OSCILLATORY AND ASYMPTOTIC PROPERTIES FOR THIRD ORDER HALF-LINEAR DELAY DIFFERENTIAL EQUATIONS[J]. Journal of Systems Science and Mathematical Sciences, 2015, 35(2): 233-244.

OSCILLATORY AND ASYMPTOTIC PROPERTIES FOR THIRD ORDER HALF-LINEAR DELAY DIFFERENTIAL EQUATIONS

LIN Quanwen1 , YU Yuanhong2   

  1. 1.Department of  Mathematics, Guangdong University of Petrochemical Technology, Maoming  525000; 2.Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190
  • Online:2015-02-25 Published:2015-05-19

虑三阶半线性时滞微分方程 $$([x''(t)]^{\alpha})'+q(t)x^{\alpha} (\sigma(t))=0,\quad t\geq t_{0},\eqno{(*)}$$ 其中$q(t)$是正函数, $\alpha>0$是奇正整数之商,时滞函 数$0<\sigma(t)\leq t$, $\sigma'(t)>0$满足$\lim_{t\rightarrow\infty}\sigma(t)=\infty$. 文章建立了保证方程$(\ast)$振动或者解收敛到零的Hille型和 Nehari型充分条件.文章的结果即使在时滞不存在的情况也是新的.为说明主要结果给出了例子.

The objective of this paper is to study the oscillation and asymptotic behavior of the third order half-linear delay differential equation $$([x''(t)]^{\alpha})'+q(t)x^{\alpha}(\sigma(t))=0, \quad t\geq t_{0}, \eqno{(*)}$$ where $q(t)$ is a positive function, $\alpha>0$ is a quotient of odd positive integers and the delay function $0<\sigma(t)\leq t$, $\sigma'(t)>0~$ satisfies $\lim_{t\rightarrow\infty}\sigma(t)=\infty$. We establish some sufficient conditions of Hille and Nehari type, which ensure that $(\ast)$ is oscillatory or the solutions converge to zero. Our results are new even in the nondelay case. Some examples are considered to illustrate the main results.

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