非线性三点共轭边值问题正解存在性的特征值准则

Eigenvalue Criteria of Existence of Positive Solutions for the Nonlinear Three-Point Conjugate Boundary Value Problem

本文提出了三点边值问题-v″(t)=b(t)f(v(t)),满足v′(0)=0及v(1)=αv(η)的共轭问题-u″(t)=b(t)f(u(t)),u′(0)=u(1)=0及u′_+(η)-u′_-(η)=αu′(1),得到了相应的Green函数．将其转化为Hammertein型积分方程,借助于其相应线性问题的第一特征值,利用锥上的不动点指数理论,给出了共轭问题单个正解及多个正解存在的特征值准则．

In this paper,the conjugate problem-u″（t）=b（t）f（u（t）） subject to u′（0）= u（1）=0 and u′_＋（η）-u′_（η）=αu′（1） of the three-point boundary value problem -v″（t）=b（t）f（v（t）） subject to v′（0）=0,v（1）=αv（η） is put forward and investigated. The problem is translated to Hammertein＇s integral equation with the use of Green＇s function.Then,by using the first eigenvalue of the relevant linear problem and applying the fixed point index theory,the eigenvalue criteria of the existence for single and multiple positive solutions of the conjugate problem are given under some conditions concerning the first eigenvalue of the linear problem corresponding to it.