华东师范大学学报(自然科学版) ›› 2007, Vol. 2007 ›› Issue (5): 34-38.

• 数学 统计学 • 上一篇    下一篇

一类二阶非线性矩阵微分方程的振动性定理 (英)

徐衍聪1, 孟凡伟2   

  1. 1. 华东师范大学 数学系, 上海 200062; 2. 曲阜师范大学 数学系, 山东 曲阜 273165
  • 收稿日期:2006-05-24 修回日期:2006-10-12 出版日期:2007-09-25 发布日期:2007-09-25
  • 通讯作者: 徐衍聪

Oscillation Theorems for Certain Second-Order Nonlinear Matrix Differential Equations(English)

XU Yan-cong1, MENG Fan-wei2   

  1. 1. Department of Mathematics, East China Normal University, Shanghai 200062, China ; 2. Department of Mathematics , Qufu Normal University, Qufu Shandong 273165, China
  • Received:2006-05-24 Revised:2006-10-12 Online:2007-09-25 Published:2007-09-25
  • Contact: XU Yan-cong

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摘要： 建立了二阶非线性矩阵微分系统 $ (a(t)\X’(t))’+b(t)\X’(t)+\Q(t)f(\X(t))= 0,t\geqslant t_ 0 >0$ 的振动性标准, 这里 $\Q(t),$ $f’(\X(t))$ 是 $n \times n$ 矩阵， $f’(\X(t))$ 正定, $a(t)$ 和 $b(t)$ 实值函数.
引进了一个特殊函数 $\phi(t,s,r)=(t-s)^ \alpha (s-r)^ \beta , \alpha,\ \beta > \frac 1 2 $\ 是常数，$ \ r \geqslant t_0,$
得到了形式为 $\lim \sup\lambda_ 1 [.] > $ const 的振动性标准,
改进了一些已知的结果.

关键词: 振动性, 二阶, 矩阵微分方程, 振动性, 二阶, 矩阵微分方程

Abstract: Some new oscillation criteria were established for the
second order nonlinear matrix differential system $
(a(t)\X’(t))’+b(t)\X’(t)+\Q(t)f(\X(t))= 0,t\geqslant t_ 0 >0,$
where $\Q(t),$ $f’(\X(t))$ are $n \times n$ matrices with
$f’(\X(t))$ positive definite, and $a(t),$ $b(t)$ are real-valued
functions. The criteria were presented in the form of $\lim
\sup\lambda_ 1 > const $ by using a particular function
$\phi(t,s,r)$ defined as $\phi(t,s,r)=(t-s)^ \alpha (s-r)^ \beta $,
where $\alpha,\ \beta > \frac 1 2 $ are constants and $r
\geqslant t_0.$ Our results improve many known oscillation
results.

Key words: second order, matrix differential equation, oscillation, second order, matrix differential equation

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