二阶微分方程三点边值问题定号解的存在性  

Existence of one-signed solutions for three-point boundary value problems of second-order differential equations

作　　者：刘慧娟 LIU Huijuan(School of Mathematics and Statistics,Xidian University,Xi'an 710126,Shaanxi,China)

机构地区：[1]西安电子科技大学数学与统计学院,陕西西安710126

出　　处：《山东大学学报(理学版)》2024年第12期79-86,共8页Journal of Shandong University(Natural Science)

基　　金：国家自然科学基金资助项目(12061064)。

摘　　要：研究二阶微分方程三点边值问题{u″+a(t)f(u)=0,t∈[0,1],u(0)=0,u(1)=u(ε)的定号解的存在性,其中ε∈(0,1),a∈C([0,1],(0,∞)),f∈C(R,R)且当s≠0时,sf(s)>0,λ_(1)为线性特征问题u″+λ_(a)(t)u=0,u(0)=0,u(1)=u(ε),t∈[0,1]的主特征值。当λ_(1)f_(∞)<1<λ_(1)f_(0)或λ_(1)f_(0)<1<λ_(1)f_(∞)时,问题至少存在一个正解u(t)和一个负解v(t)。主要结果的证明基于分歧理论。In this paper,we study the existence of one-signed solutions for three-point boundary value problems of nonlinear second-order differential equations {u″+a(t)f(u)=0,t∈[0,1],u(0)=0,u(1)=u(ε)whereε∈(0,1),a∈C([0,1],(0,∞)),f∈C(R,R)with sf(s)>0 for s≠0,λ_(1)is the principal eigenvalue of the linear eigen-value problem:u″+λ_(a)(t)u=0,u(0)=0,u(1)=u(ε),t∈[0,1].Assume that eitherλ_(1)f_(∞)<1<λ_(1)f_(0)orλ_(1)f_(0)<1<λ_(1)f_(∞),the problem has at least one positive solution u(t)and one negative solution v(t).The proof of main results is based on bifurcation techniques.

关 键 词：二阶微分方程 三点边值问题 格林函数 分歧理论 定号解 

分 类 号：O175.8[理学—数学]