二阶三点边值问题的正解

该文讨论了二阶三点边值问题$-u'(t)=b(t)f(u(t))$满足$u'(0)=0$, $u(1)={\alpha}u({\eta})$ 正解的存在性与多重性, 其中常数$\alpha, \eta\in(0,1)$, $f\in C (0,\infty),0,\infty) )$, $b\in C (0,1],0,\infty) )$且存在$t_0\in0,1]$使$b(t_0)>0$. 利用该问题相应的Green函数, 将其转化为Hammerstein型积分方程, 借助于锥上的不动点指数理论,给出了该问题单个正解和多个正解存在的与其相应线性问题的第一特征值有关的最佳充分性条件.

On Positive Solutions of Second-order Three-point Boundary Value Problem

In this paper, the existence of positive solutions of the second-order three-point boundary value problem $-u'(t)=b(t)f(u(t))$ for all $t\in0,1]$ subject to $u'(0)=0$, $u(1)={\alpha}u({\eta})$ is studied, where $\alpha, \eta\in(0,1)$ are given, $f\in C\big(0,\infty),0,\infty)\big)$, $b\in C\big(0,1],0,\infty)\big)$ and there exists $t_0\in0,1]$ such that $b(t_0)>0$. The problem is transformed into the Hammerstein's integral equation with its corresponding Green's funtion. By applying the fixed point index theory, authors obtain the optimal sufficient conditions for the existence of single and multiple positive solutions of the above mentioned problem concerning the first eigenvalue of the relevant linear problem.