Differential Equations and Dynamical Systems

, Volume 22, Issue 4, pp 427–439 | Cite as

Positive Solutions for a \(n\)th-Order Impulsive Differential Equation with Integral Boundary Conditions

Original Research
First Online:

Abstract

In this paper we study the existence of positive solutions for the following \(n\)th-order impulsive boundary value problem
$$\begin{aligned} \left\{ \begin{array}{l} u^{(n)}(t)+f(t,u(t))=0,\quad t\in [0,1],t\not =t_k,\\ -\Delta u^{(n-1)}|_{t=t_k}=I_k(u(t_k)),\quad k=1,2,\ldots ,m,\\ u(0)=\int _0^1 u(t)\mathrm d \alpha (t),\qquad u(1)=\int _0^1 u(t)\mathrm d \beta (t),\\ u^{\prime }(0)=\cdots =u^{(n-3)}(0)=u^{(n-2)}(0)=0. \end{array}\right. \end{aligned}$$
Here \(f\in C([0,1]\times \mathbb{R }^+,\mathbb{R }^+), I_k\in C(\mathbb{R }^+,\mathbb{R }^+) ( {\mathbb{R }^+:=[0,\infty )})\) and \(\int _0^1 u(t)\mathrm d \alpha (t), \int _0^1 u(t)\mathrm d \beta (t)\) are Riemann–Stieltjes integrals (i.e., \(\alpha (t)\) and \(\beta (t)\) have bounded variation). We use the Krasnoselskii–Zabreiko fixed point theorem to establish our main results. Furthermore, our nonlinear term \(f\) is allowed to grow superlinearly and sublinearly.

Keywords

Boundary value problem Impulsive effect Positive solution  Riemann–Stieltjes integral Krasnoselskii–Zabreiko fixed point theorem 
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Notes

Acknowledgments

Research was supported by the NNSF-China (10971046), the NSF of Shandong Province (ZR2012AQ007) and Graduate Independent Innovation Foundation of Shandong University (yzc12063).

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Copyright information

© Foundation for Scientific Research and Technological Innovation 2013

Authors and Affiliations

  1. 1.School of Mathematics, Shandong UniversityJinanChina
  2. 2.School of Mathematics, Statistics and Applied MathematicsNational University of IrelandGalwayIreland
  3. 3.Department of MathematicsQingdao Technological UniversityQingdaoChina

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