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  • Research Article
  • Open Access

Existence of Solutions for Second-Order Nonlinear Impulsive Differential Equations with Periodic Boundary Value Conditions

Boundary Value Problems20072007:041589

  • Received: 12 February 2007
  • Accepted: 13 April 2007
  • Published:


We are concerned with the nonlinear second-order impulsive periodic boundary value problem , , , , , , new criteria are established based on Schaefer's fixed-point theorem.


  • Differential Equation
  • Partial Differential Equation
  • Ordinary Differential Equation
  • Functional Equation
  • Periodic Boundary


Authors’ Affiliations

Department of Mathematics, Huaiyin Teachers College, Huaian, Jiangsu, 223300, China
Department of Mathematics, Yangzhou University, Yangzhou, 225002, China


  1. Benchohra M, Henderson J, Ntouyas S: Impulsive Differential Equations and Inclusions, Contemporary Mathematics and Its Applications. Volume 2. Hindawi, New York, NY, USA; 2006.View ArticleGoogle Scholar
  2. Liu X (Ed): Advances in Impulsive Differential Equations In Dynamics of Continuous, Discrete & Impulsive Systems. Series A. Mathematical Analysis 2002,9(3):313–462.Google Scholar
  3. Rogovchenko YV: Impulsive evolution systems: main results and new trends. Dynamics of Continuous, Discrete & Impulsive Systems. Series A. Mathematical Analysis 1997,3(1):57–88.MATHMathSciNetGoogle Scholar
  4. Samoĭlenko AM, Perestyuk NA: Impulsive Differential Equations, World Scientific Series on Nonlinear Science. Series A: Monographs and Treatises. Volume 14. World Scientific, River Edge, NJ, USA; 1995:x+462.Google Scholar
  5. Zavalishchin ST, Sesekin AN: Dynamic Impulse Systems. Theory and Applications, Mathematics and Its Applications. Volume 394. Kluwer Academic Publishers, Dordrecht, The Netherlands; 1997:xii+256.Google Scholar
  6. Choisy M, Guégan JF, Rohani P: Dynamics of infectious diseases and pulse vaccination: teasing apart the embedded resonance effects. Physica D: Nonlinear Phenomena 2006,22(1):26–35.View ArticleGoogle Scholar
  7. d'Onofrio A: On pulse vaccination strategy in the SIR epidemic model with vertical transmission. Applied Mathematics Letters 2005,18(7):729–732. 10.1016/j.aml.2004.05.012MATHMathSciNetView ArticleGoogle Scholar
  8. Gao S, Chen L, Nieto JJ, Torres A: Analysis of a delayed epidemic model with pulse vaccination and saturation incidence. Vaccine 2006,24(35–36):6037–6045. 10.1016/j.vaccine.2006.05.018View ArticleGoogle Scholar
  9. He Z, Zhang X: Monotone iterative technique for first order impulsive difference equations with periodic boundary conditions. Applied Mathematics and Computation 2004,156(3):605–620. 10.1016/j.amc.2003.08.013MATHMathSciNetView ArticleGoogle Scholar
  10. Li W-T, Huo H-F: Global attractivity of positive periodic solutions for an impulsive delay periodic model of respiratory dynamics. Journal of Computational and Applied Mathematics 2005,174(2):227–238. 10.1016/ ArticleGoogle Scholar
  11. Tang S, Chen L: Density-dependent birth rate, birth pulses and their population dynamic consequences. Journal of Mathematical Biology 2002,44(2):185–199. 10.1007/s002850100121MathSciNetView ArticleGoogle Scholar
  12. Wang W, Wang H, Li Z: The dynamic complexity of a three-species Beddington-type food chain with impulsive control strategy. Chaos, Solitons & Fractals 2007,32(5):1772–1785. 10.1016/j.chaos.2005.12.025MATHMathSciNetView ArticleGoogle Scholar
  13. Yan J, Zhao A, Nieto JJ: Existence and global attractivity of positive periodic solution of periodic single-species impulsive Lotka-Volterra systems. Mathematical and Computer Modelling 2004,40(5–6):509–518. 10.1016/j.mcm.2003.12.011MATHMathSciNetView ArticleGoogle Scholar
  14. Zhang W, Fan M: Periodicity in a generalized ecological competition system governed by impulsive differential equations with delays. Mathematical and Computer Modelling 2004,39(4–5):479–493. 10.1016/S0895-7177(04)90519-5MATHMathSciNetView ArticleGoogle Scholar
  15. Zhang X, Shuai Z, Wang K: Optimal impulsive harvesting policy for single population. Nonlinear Analysis: Real World Applications 2003,4(4):639–651. 10.1016/S1468-1218(02)00084-6MATHMathSciNetView ArticleGoogle Scholar
  16. Agarwal RP, O'Regan D: Multiple nonnegative solutions for second order impulsive differential equations. Applied Mathematics and Computation 2000,114(1):51–59. 10.1016/S0096-3003(99)00074-0MATHMathSciNetView ArticleGoogle Scholar
  17. Chen L, Sun J: Nonlinear boundary value problem of first order impulsive functional differential equations. Journal of Mathematical Analysis and Applications 2006,318(2):726–741. 10.1016/j.jmaa.2005.08.012MATHMathSciNetView ArticleGoogle Scholar
  18. Ding W, Han M, Mi J: Periodic boundary value problem for the second-order impulsive functional differential equations. Computers & Mathematics with Applications 2005,50(3–4):491–507. 10.1016/j.camwa.2005.03.010MATHMathSciNetView ArticleGoogle Scholar
  19. Nieto JJ, Rodríguez-López R: Periodic boundary value problem for non-Lipschitzian impulsive functional differential equations. Journal of Mathematical Analysis and Applications 2006,318(2):593–610. 10.1016/j.jmaa.2005.06.014MATHMathSciNetView ArticleGoogle Scholar
  20. Rachůnková I, Tvrdý M: Non-ordered lower and upper functions in second order impulsive periodic problems. Dynamics of Continuous, Discrete & Impulsive Systems. Series A. Mathematical Analysis 2005,12(3–4):397–415.MATHMathSciNetGoogle Scholar
  21. Chen J, Tisdell CC, Yuan R: On the solvability of periodic boundary value problems with impulse. Journal of Mathematical Analysis and Applications 2007,331(2):902–912. 10.1016/j.jmaa.2006.09.021MATHMathSciNetView ArticleGoogle Scholar
  22. Li J, Nieto JJ, Shen J: Impulsive periodic boundary value problems of first-order differential equations. Journal of Mathematical Analysis and Applications 2007,325(1):226–236. 10.1016/j.jmaa.2005.04.005MATHMathSciNetView ArticleGoogle Scholar
  23. Nieto JJ: Periodic boundary value problems for first-order impulsive ordinary differential equations. Nonlinear Analysis 2002,51(7):1223–1232. 10.1016/S0362-546X(01)00889-6MATHMathSciNetView ArticleGoogle Scholar
  24. Bai C: Existence of solutions for second order nonlinear functional differential equations with periodic boundary value conditions. International Journal of Pure and Applied Mathematics 2004,16(4):451–462.MATHMathSciNetGoogle Scholar
  25. Rudd M, Tisdell CC: On the solvability of two-point, second-order boundary value problems. Applied Mathematics Letters 2007,20(7):824–828. 10.1016/j.aml.2006.08.028MATHMathSciNetView ArticleGoogle Scholar
  26. Dong Y: Sublinear impulse effects and solvability of boundary value problems for differential equations with impulses. Journal of Mathematical Analysis and Applications 2001,264(1):32–48. 10.1006/jmaa.2001.7548MATHMathSciNetView ArticleGoogle Scholar
  27. Liu Y: Further results on periodic boundary value problems for nonlinear first order impulsive functional differential equations. Journal of Mathematical Analysis and Applications 2007,327(1):435–452. 10.1016/j.jmaa.2006.01.027MATHMathSciNetView ArticleGoogle Scholar
  28. Qian D, Li X: Periodic solutions for ordinary differential equations with sublinear impulsive effects. Journal of Mathematical Analysis and Applications 2005,303(1):288–303. 10.1016/j.jmaa.2004.08.034MATHMathSciNetView ArticleGoogle Scholar
  29. Lloyd NG: Degree Theory, Cambridge Tracts in Mathematics, no. 73. Cambridge University Press, Cambridge, UK; 1978:vi+172.Google Scholar
  30. Lakshmikantham V, Baĭnov DD, Simeonov PS: Theory of Impulsive Differential Equations, Series in Modern Applied Mathematics. Volume 6. World Scientific, Teaneck, NJ, USA; 1989:xii+273.View ArticleGoogle Scholar
  31. Nieto JJ: Basic theory for nonresonance impulsive periodic problems of first order. Journal of Mathematical Analysis and Applications 1997,205(2):423–433. 10.1006/jmaa.1997.5207MATHMathSciNetView ArticleGoogle Scholar


© C. Bai and D. Yang 2007

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