Skip to main content


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

Article metrics

  • 1117 Accesses

  • 3 Citations


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



  1. 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.

  2. 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.

  3. 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.

  4. 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.

  5. 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.

  6. 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.

  7. 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.2004.05.012

  8. 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.2006.05.018

  9. 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.2003.08.013

  10. 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/

  11. 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/s002850100121

  12. 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.2005.12.025

  13. 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.2003.12.011

  14. 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)90519-5

  15. 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)00084-6

  16. 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)00074-0

  17. 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.2005.08.012

  18. 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.2005.03.010

  19. 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.2005.06.014

  20. 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.

  21. 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.2006.09.021

  22. 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.2005.04.005

  23. 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)00889-6

  24. 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.

  25. 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.2006.08.028

  26. 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.2001.7548

  27. 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.2006.01.027

  28. 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.2004.08.034

  29. 29.

    Lloyd NG: Degree Theory, Cambridge Tracts in Mathematics, no. 73. Cambridge University Press, Cambridge, UK; 1978:vi+172.

  30. 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.

  31. 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.1997.5207

Download references

Author information

Correspondence to Chuanzhi Bai.

Rights and permissions

Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Reprints and Permissions

About this article


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