一类非线性四阶常微分方程边值问题解的存在唯一性  

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

作　　者：杨丽娟 YANG Li-juan(College of Mathematics and Statistics,Northwest Normal University,Lanzhou 730070,Gansu,China)

机构地区：[1]西北师范大学数学与统计学院,甘肃兰州730070

出　　处：《山东大学学报（理学版）》2020年第6期101-108,共8页Journal of Shandong University(Natural Science)

基　　金：国家自然科学基金资助项目(11671322)。

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]×R 3→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。This article considers the existence and uniqueness of solutions of boundary value problems of nonlinear fourth-order ordi-nary 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]×R 3→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 sat-isfies the Lipschitz condition.

关 键 词：LERAY-SCHAUDER原理 WIRTINGER不等式 解 存在性 唯一性 

分 类 号：O175.8[理学—数学] O175.14[理学—基础数学]