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

Jiafa Xu, Zhongli Wei, Youzheng Ding

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.

Keywords


Periodic boundary value problem; fixed point theorem; positive solution; cone

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