一类非线性二阶三点边值问题正解的全局结构

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

摘要:

本文考虑二阶常微分方程三点边值问题 \[\begin{cases} u''(t)+h(t)f(u)=0,~~\ \ \ t\in (0,1),\\[2ex] u'(0)=0, ~u(1)=\lambda u(\eta), \end{cases} \] 其中~$\eta\in[0,1)$,~参数$\lambda\in[0,1)$,~函数~$f\in C( [0,\infty),[0,\infty))$~满足~$f(s)>0,~s>0$,~$h\in C( [0,1],[0,\infty))$~在~$[0,1]$~的任意子区间内不恒为零.~在满足条件 ~$f_{0}=0,~f_{\infty}=\infty$~时,~讨论了该边值问题解所构成的连通分支随着参数~$\lambda$~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.

Abstract:

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~$\eta\in[0,1)$,~$\lambda\in[0,1)$~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.