Existence of positive solutions for a second order periodic boundary value problem with impulsive effects

Abstract

In this paper, we are mainly concerned with the existence and multiplicity of positive solutions for the following second order periodic boundary value problem involving impulsive effects $$ \begin{cases} -u''+\rho^2u=f(t,u), & t\in J',\\ -\Delta u'|_{t=t_k}=I_k(u(t_k)), & k=1,\ldots,m,\\ u(0)-u(2\pi)=0,\quad u'(0)-u'(2\pi)=0. \end{cases} $$ Here $J'=J\setminus \{t_1,\ldots, t_m\}$, $f\in C(J\times \mathbb{R}^+, \mathbb{R}^+)$, $I_k\in C( \mathbb{R}^+, \mathbb{R}^+)$, where $ \mathbb{R}^+=[0,\infty)$, $J=[0,2\pi]$. The proof of our main results relies on the fixed point theorem on cones. The paper extends some previous results and reports some new results about impulsive differential equations.