### 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.

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

#### Full Text:

FULL TEXT### Refbacks

- There are currently no refbacks.