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山东大学学报(理学版) ›› 2016, Vol. 51 ›› Issue (12): 47-53.doi: 10.6040/j.issn.1671-9352.0.2016.243

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非线性微分方程三阶三点边值问题正解的存在性

郭丽君   

  1. 兰州交通大学博文学院电信工程系, 甘肃 兰州 730101
  • 收稿日期:2016-05-31 出版日期:2016-12-20 发布日期:2016-12-20
  • 作者简介:郭丽君(1980— ),女,硕士,讲师,研究方向为微分方程边值问题. E-mail:5148806@qq.com

Existence of positive solutions for a third-order three-point boundary value problem of nonlinear differential equations

GUO Li-jun   

  1. Department of Electro and Information Engineering, Lanzhou Jiaotong University Bowen College, Lanzhou 730101, Gansu, China
  • Received:2016-05-31 Online:2016-12-20 Published:2016-12-20

摘要: 三阶微分方程有着广泛的应用背景和重要的理论价值,格林函数在三阶三点边值问题的正解存在性理论中有着重要作用,考虑三阶三点边值问题{u(t)+a(t)f(u(t))=0, t∈(0,1),u(0)=u″(0)=0, u'(1)=αu(η),其中0<η<1, 0<α<1/η。 通过建立相关线性边值问题的格林函数得到解的形式,运用不动点指数理论建立上述边值问题至少两个正解的若干存在性准则。

关键词: 正解, 三阶三点边值问题, 存在性, 格林函数, 不动点指数理论,

Abstract: The third order differential equation has a wide application background and an important theoretical value, Green function plays an important role in the existence of positive solutions for the third order three point boundary value problems. This paper is concerned with the following boundary value problem{u(t)+a(t)f(u(t))=0, t∈(0,1),u(0)=u″(0)=0, u'(1)=αu(η),where 0<η<1 and 0<α<1. By establishing Green function for the associated linear boundary value problem, the solution for the above boundary value problem is obtained. Then some existence criteria of at least two positive solutions are obtained by using the fixed point index theorem.

Key words: fixed point index theorem, existence, Green function, third-order three-point boundary value problem, cone, positive solution

中图分类号: 

  • O175.8
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