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

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
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