一类二阶常微分方程组边值问题解的存在性与唯一性  

Existence and Uniqueness of Solutions for Boundary Value Problems of a Class of Second-Order Ordinary Differential Equation System

作　　者：王丹 李永祥 WANG Dan;LI Yongxiang(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：《武汉大学学报（理学版）》2021年第5期433-440,共8页Journal of Wuhan University:Natural Science Edition

基　　金：国家自然科学基金(11661071,12061062)。

摘　　要：本文讨论二阶常微分方程组边值问题{-u″(t)=f(t,u(t),v(t),u′(t)),t∈[0,1]-v″(t)=g(t,u(t),v(t),v′(t)),t∈[0,1]u(0)=u(1)=0,v(0)=v(1)=0解的存在性与唯一性,其中f,g:[0,1]×ℝ×ℝ×ℝ→ℝ连续。在非线性项f(t,x,y,p)与g(t,x,y,p)关联的不等式条件,以及f(t,x,y,p)与g(t,x,y,p)关于p满足Nagumo型增长条件下,运用Leray-Schauder不动点定理,获得了该问题解的存在性及唯一性。This paper discusses the existence and uniqueness of solutions to the system of second-order ordinary differential equations boundary value problems{-u″(t)=f(t,u(t),v(t),u′(t)),t∈[0,1]-v″(t)=g(t,u(t),v(t),v′(t)),t∈[0,1]u(0)=u(1)=0,v(0)=v(1)=0 where f,g:[0,1]×ℝ×ℝ×ℝ→ℝcontinuous.Applying the Leray-Schauder fixed point theorem,the existence and uniqueness of solutions is obtained under the condition that the nonlinear term f(t,x,y,p)and g(t,x,y,p)satisfy the associated inequality condition,and f(t,x,y,p)and g(t,x,y,p)satisfy the Nagumo-type growth condition respectively on p.

关 键 词：二阶常微分方程组 边值问题 存在性与唯一性 Nagumo型增长条件 LERAY-SCHAUDER不动点定理 

分 类 号：O175.15[理学—数学]