一类非线性二阶三点边值问题正解的全局结构

本文考虑二阶常微分方程三点边值问题{u″(t)+h(t)f(u)=0,t∈(0,1),u′(0)=0,u(1)=λu(η),其中η∈0,1),参数λ∈0,1),函数f∈C(0,∞),0,∞))满足f(s)0,s0,h∈C(0,1],0,∞))在0,1]的任意子区间内不恒为零.在满足条件f0=0,f∞=∞时,本文讨论了该边值问题解所构成的连通分支随着参数λ在0,1]内的变化而变化的情形,建立了正解的全局结构.主要结果的证明基于锥上的不动点指数定理以及解集连通性质.

Global structure of positive solutions for a class of nonlinear second order three point boundary value problems

In this paper we consider the second-order three-point boundary value Problem~ \\begin{cases} u''(t)+h(t)f(u)=0,~~\ \ \ t\in (0,1),\\2ex] u''(0)=0, ~u(1)=\lambda u(\eta), \end{cases} \] where~$\eta\in0,1)$,~$\lambda\in0,1)$~is a parameter,~$f\in C( 0,\infty),0,\infty))$~satisfies~$f(s)>0$~for $s>0$, and $h\in C( 0,1],0,\infty))$~is not identically zero on any subinterval of 0,1]. We give information on the interesting problem as to what happens to the norms of positive solutions as $\lambda$ varies in $0,1)$ under the conditions of~$f_{0}=0,~f_{\infty}=\infty$.~The proof of main result is based upon the fixed point index theory on cone and connectivity properties of the solution set.