数学物理学报(英文版) ›› 2010, Vol. 30 ›› Issue (4): 1291-1300.doi: 10.1016/S0252-9602(10)60125-7

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ON A PROBLEM IN COMPLEX OSCILLATION THEORY OF PERIODIC HIGHER ORDER LINEAR DIFFERENTIAL EQUATIONS

肖丽鹏1, 陈宗煊2   

  1. 1. Institute of Mathematics and Informations, Jiangxi Normal |University, Nanchang 330022, China|2. School of |Mathematical |Science, South China Normal University, Guangzhou 510631, China
  • 收稿日期:2007-12-30 修回日期:2008-10-20 出版日期:2010-07-20 发布日期:2010-07-20
  • 基金资助:

    This work is supported by  the National Natural Foundation of China (10871076) and the Startup Foundation for Doctors of Jiangxi Normal University (2614)

ON A PROBLEM IN COMPLEX OSCILLATION THEORY OF PERIODIC HIGHER ORDER LINEAR DIFFERENTIAL EQUATIONS

 XIAO Li-Peng1, CHEN Zong-Xuan2   

  1. 1. Institute of Mathematics and Informations, Jiangxi Normal |University, Nanchang 330022, China|2. School of |Mathematical |Science, South China Normal University, Guangzhou 510631, China
  • Received:2007-12-30 Revised:2008-10-20 Online:2010-07-20 Published:2010-07-20
  • Supported by:

    This work is supported by  the National Natural Foundation of China (10871076) and the Startup Foundation for Doctors of Jiangxi Normal University (2614)

摘要:

In this article, the zeros of solutions of differential equation

f(k)}(z)+A(z)f(z)=0,                                         (*)
 are studied, where k>2, A(z)=B(ez), B(ς)=g1(1/ς)+g2(ς), gand g2 being  entire functions with g2 transcendental and ο(g2) not equal to a positive integer or infinity. It is shown that any linearly independent solutions f1, f2, …, fk of Eq.(*) satisfy λe(f1… fk) ≥ο(g2) under the condition that fj(z) and fj(z+2πi )(j =1, …, k) are linearly dependent.

关键词: Differential equation, periodic, linearly dependent, complex oscillation

Abstract:

In this article, the zeros of solutions of differential equation

f(k)}(z)+A(z)f(z)=0,                                         (*)
 are studied, where k>2, A(z)=B(ez), B(ς)=g1(1/ς)+g2(ς), gand g2 being  entire functions with g2 transcendental and ο(g2) not equal to a positive integer or infinity. It is shown that any linearly independent solutions f1, f2, …, fk of Eq.(*) satisfy λe(f1… fk) ≥ο(g2) under the condition that fj(z) and fj(z+2πi )(j =1, …, k) are linearly dependent.

Key words: Differential equation, periodic, linearly dependent, complex oscillation

中图分类号: 

  • 30D35