数学物理学报(英文版) ›› 2013, Vol. 33 ›› Issue (5): 1398-1406.doi: 10.1016/S0252-9602(13)60091-0

• 论文 • 上一篇    下一篇

PROPERTY (X+) FOR SECOND-ORDER NONLINEAR DIFFERENTIAL EQUATIONS OF GENERALIZED EULER TYPE

Asadollah AGHAJANI*| Vahid ROOMI   

  1. School of Mathematics, Iran University of Science and Technology, Narmak, Tehran, Iran
  • 收稿日期:2011-06-08 修回日期:2013-01-21 出版日期:2013-09-20 发布日期:2013-09-20
  • 通讯作者: Asadollah AGHAJANI,aghajani@iust.ac.ir E-mail:aghajani@iust.ac.ir; roomi@iust.ac.ir

PROPERTY (X+) FOR SECOND-ORDER NONLINEAR DIFFERENTIAL EQUATIONS OF GENERALIZED EULER TYPE

Asadollah AGHAJANI*| Vahid ROOMI   

  1. School of Mathematics, Iran University of Science and Technology, Narmak, Tehran, Iran
  • Received:2011-06-08 Revised:2013-01-21 Online:2013-09-20 Published:2013-09-20
  • Contact: Asadollah AGHAJANI,aghajani@iust.ac.ir E-mail:aghajani@iust.ac.ir; roomi@iust.ac.ir

摘要:

In this paper the generalized nonlinear Euler differential equation t2k(tu´)u´´+t(f(u) + k(tu´)) + g(u) = 0 is considered. Here the functions f(u), g(u) and k(u) sat-isfy smoothness conditions which guarantee the uniqueness of solutions of initial value problems, however, no conditions of sub(super) linearity are assumed. We present some necessary and sufficient conditions and some tests for the equivalent planar system to have or fail to have property (X+), which is very important for the existence of periodic solutions and oscillation theory.

关键词: property (X+), nonlinear differential equations, Li´enard system

Abstract:

In this paper the generalized nonlinear Euler differential equation t2k(tu´)u´´+t(f(u) + k(tu´)) + g(u) = 0 is considered. Here the functions f(u), g(u) and k(u) sat-isfy smoothness conditions which guarantee the uniqueness of solutions of initial value problems, however, no conditions of sub(super) linearity are assumed. We present some necessary and sufficient conditions and some tests for the equivalent planar system to have or fail to have property (X+), which is very important for the existence of periodic solutions and oscillation theory.

Key words: property (X+), nonlinear differential equations, Li´enard system

中图分类号: 

  • 34C10