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《山东大学学报(理学版)》 ›› 2020, Vol. 55 ›› Issue (6): 101-108.doi: 10.6040/j.issn.1671-9352.0.2020.062

• • 上一篇    

一类非线性四阶常微分方程边值问题解的存在唯一性

杨丽娟   

  1. 西北师范大学数学与统计学院, 甘肃 兰州 730070
  • 发布日期:2020-06-01
  • 作者简介:杨丽娟(1997— ), 女, 硕士研究生, 研究方向为常微分方程边值问题. E-mail:18419068954@163.com
  • 基金资助:
    国家自然科学基金资助项目(11671322)

Existence and uniqueness of solutions for a class of boundary value problems of nonlinear fourth-order ordinary differential equations

YANG Li-juan   

  1. College of Mathematics and Statistics, Northwest Normal University, Lanzhou 730070, Gansu, China
  • Published:2020-06-01

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摘要： 研究了非线性四阶常微分方程边值问题{u(4)(t)=f(t,u(t),u'(t),u″(t)), a.e. t∈(0,1),u(0)=u″(0)=u(1)=u″(1)=0,其中非线性项f:［0,1］×R3→R为Carathéodory函数。运用Leray-Schauder原理,在f满足适当的至多线性增长性条件时,获得了该问题解的存在性。进一步,在f满足Lipschitz条件时, 得到了该问题解的存在唯一性。

关键词: Leray-Schauder原理, Wirtinger不等式, 解, 存在性, 唯一性

Abstract: This article considers the existence and uniqueness of solutions of boundary value problems of nonlinear fourth-order ordinary differential equations{u(4)(t)=f(t,u(t),u'(t),u″(t)), a.e. t∈(0,1),u(0)=u″(0)=u(1)=u″(1)=0,where nonlinearity f:［0,1］×R3→R is a Carathéodory function. The existence of solutions is obtained when f satisfies the condition of proper utmost linear growth by using the Leray-Schauder principle. Furthermore, the uniqueness of solutions is proved when f satisfies the Lipschitz condition.

Key words: Leray-Schauder principle, Wirtingers inequality, solution, existence, uniqueness

中图分类号: 

  • O175.8
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