Global structure of solutions to boundary-value problems of impulsive differential equations

Abstract

In this article, we study the structure of global solutions to the boundary-value problem \displaylines{ -x''(t)+f(t,x)=\lambda ax(t),\quad t\in(0,1),\; t\neq\frac{1}{2},\cr \Delta x|_{t=1/2}=\beta_1 x(\frac{1}{2}),\quad \Delta x'|_{t=1/2}=-\beta_{2} x(\frac{1}{2}),\cr x(0)=x(1)=0, } where $\lambda\neq0$, $\beta_1\geq\beta_{2}\geq0$, $\Delta x|_{t=1/2}=x(\frac{1}{2}+0)-x(\frac{1}{2})$, $\Delta x'|_{t=1/2}=x'(\frac{1}{2}+0)-x'(\frac{1}{2}-0)$, and $f:[0,1]\times\mathbb{R}\to\mathbb{R}$, $a:[0,1]\to(0,+\infty)$ are continuous. By a comparison principle and spectral properties of the corresponding linear equations, we prove the existence of solutions by using Rabinowitz-type global bifurcation theorems, and obtain results on the behavior of positive solutions for large $\lambda$ when $f(x)=x^{p+1}$