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Journal of Dynamical and Control Systems

, Volume 20, Issue 4, pp 559–574 | Cite as

Sign-Changing and Multiple Solutions of Impulsive Boundary Value Problems Via Critical Point Methods

  • Yu Tian
  • Weigao Ge
  • Donal O’Regan
Article

Abstract

In this paper, we study the second-order impulsive boundary value problem
$$\left\{\begin{array}{ll} -Lu=f(x, u), \, \, x\in [0, 1]\backslash\{x_{1}, x_{2}, \cdots, x_{l}\}, \\ -{\Delta} (p(x_{i}) u'(x_{i}))=I_{i}(u(x_{i})), \quad i=1, 2, \cdots, l, \\ R_{1}(u)=0, \, \, \, R_{2}(u)=0, \end{array}\right.$$
where Lu = (p(x)u′)′ − q(x)u is a Sturm-Liouville operator, R 1(u) = αu′(0) − βu(0) and R 2(u) = γu′(1) + σu(1). The existence of sign-changing and multiple solutions is obtained. The technical approach is based on minimax methods and invariant sets of descending flow.

Keywords

Sign-changing solution Impulsive boundary value problem Invariant set of descending flow Critical point 

Mathematics Subject Classifications (2010)

34B15 35A15 

Notes

Acknowledgments

Project 11001028 is supported by the National Science Foundation for Young Scholars, Project 11071014 is supported by the National Science Foundation of P.R. China, and Project YETP0458 is supported by the Beijing Higher Education Young Elite Teacher.

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Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.School of ScienceBeijing University of Posts and TelecommunicationsBeijingPeople’s Republic of China
  2. 2.Department of Applied MathematicsBeijing Institute of TechnologyBeijingPeople’s Republic of China
  3. 3.School of Mathematics, Statistics and Applied MathematicsNational University of IrelandGalwayIreland

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