Existence of positive solutions for a second-order periodic boundary value problem with impulsive effects. (English) Zbl 1360.34062

Xu, Jiafa; Wei, Zhongli; Ding, Youzheng

Topol. Methods Nonlinear Anal. 43, No. 1, 11-21 (2014).

Summary: 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 \[ \begin{aligned} -u''+ \rho^2u= f(t,u),\quad & t\in J',\\ -\Delta u'|_{t=t_k}= I_k(u(t_k)),\quad & k=1,\dots,m\end{aligned} \]
\[ u(0)= u(2\pi)=0,\quad u'(0)- u'(2\pi)= 0. \] Here \(J'=J\setminus\{t_1,\dots,t_m\}\), \(f\in C(J\times\mathbb{R}^+,\mathbb{R}^+)\), \(I_k\in C(\mathbb{R}^+, \mathbb{R}^+)\), 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.

MSC:

34B37	Boundary value problems with impulses for ordinary differential equations
34B15	Nonlinear boundary value problems for ordinary differential equations
34B18	Positive solutions to nonlinear boundary value problems for ordinary differential equations
47H10	Fixed-point theorems

Keywords:

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