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

作　　者：李朝倩 LI Zhao-qian(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93-100,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：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]×R 4→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。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]×R 4→R is a Carathéodory function.The existence of the solution is proved in the case f is utmost linear growth.Fur-thermore 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.

关 键 词：Carathéodory函数 WIRTINGER不等式 LERAY-SCHAUDER不动点定理 存在唯一性 

分 类 号：O175.8[理学—数学] O175.14[理学—基础数学]