带参数的一阶周期边值问题正解的全局结构

Global structure of positive solutions for first-order periodic boundary value problem with parameter

摘要:

本文运用Dancer全局分歧定理研究了带参数的一阶周期边值问题\$$ \left\{\begin{array}{ll} u'(t)+a(t)u(t)=r f(u),~~\ \ \ t\in (0,1),\\[2ex] u(0)=u(1), \end{array} \right. $$ 正解的全局结构,获得了正解存在的最优区间，其中$r$为正参数,$f:\mathbb{R}\rightarrow \mathbb{R}$且$sf(s)>0,s\neq0,a:[0,1]\rightarrow[0,\infty)$连续,且在[0,1]的任意子区间内不恒为0.

Abstract:

In this paper, we use the Dancer's global bifurcation theorem to study the global structure of positive solutions for the following first-order periodic boundary value problem with parameter $$ \left\{\begin{array}{ll} u'(t)+a(t)u(t)=r f(u),~~\ \ \ t\in (0,1),\\[2ex] u(0)=u(1). \end{array} \right. $$ where $r$ is a posotive parameter, $f:\mathbb{R}\rightarrow \mathbb{R}$ and $ sf(s)>0,~s\neq0,~a:[0,1]\rightarrow[0,\infty)$~and ~$a(t)\not\equiv0$~on any subinterval of [0,1].