一类非线性四阶边值问题解的存在唯一性

doi:10.6040/j.issn.1671-9352.0.2020.003

《山东大学学报(理学版)》 ›› 2020, Vol. 55 ›› Issue (6): 93-100.doi: 10.6040/j.issn.1671-9352.0.2020.003

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一类非线性四阶边值问题解的存在唯一性

李朝倩

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:李朝倩(1995— ), 女, 硕士研究生, 研究方向为常微分方程边值问题. E-mail:2386616217@qq.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 nonlinear fourth-order boundary value problem

LI Zhao-qian

College of Mathematics and Statistics, Northwest Normal University, Lanzhou 730070, Gansu, China

Published:2020-06-01

摘要/Abstract

摘要： 考察了一类非线性四阶边值问题{u⁽⁴⁾(t)=f(t,u(t),u'(t),u″(t),u(t)), t∈(0,1),u(0)=u'(0)=u'(1)=u″(1)=0解的存在唯一性,其中f:［0,1］×R⁴→R为Carathéodory函数。当非线性项f满足至多线性增长条件时,获得了该问题解的存在性。而当f满足Lipschitz型条件时, 进一步得到了该问题解的存在唯一性。主要结果的证明基于Leray-Schauder不动点定理。

关键词: Carathéodory函数, Wirtinger不等式, Leray-Schauder不动点定理, 存在唯一性

Abstract: This paper is devoted to investigate the existence and uniqueness of solutions for a class of nonlinear fourth-order boundary value problem{u⁽⁴⁾(t)=f(t,u(t),u'(t),u″(t),u(t)), t∈(0,1),u(0)=u'(0)=u'(1)=u″(1)=0,where f:［0,1］×R⁴→R is a Carathéodory function. The existence of the solution is proved in the case f is utmost linear growth. Furthermore the existence and uniqueness of the solution is also obtained when f is Lipschitz type. The proof of the main results is based on Leray-Schauder fixed point theorem.

Key words: Carathéodory function, Wirtinger inequality, Leray-Schauder fixed point theorem, existence and uniqueness

中图分类号:

O175.8

李朝倩. 一类非线性四阶边值问题解的存在唯一性[J]. 《山东大学学报(理学版)》, 2020, 55(6): 93-100.

LI Zhao-qian. Existence and uniqueness of solutions for a class of nonlinear fourth-order boundary value problem[J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2020, 55(6): 93-100.

参考文献

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一类非线性四阶边值问题解的存在唯一性

Existence and uniqueness of solutions for a class of nonlinear fourth-order boundary value problem

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