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

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].