Open Access

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

Boundary Value Problems20072007:041589

DOI: 10.1155/2007/41589

Received: 12 February 2007

Accepted: 13 April 2007

Published: 21 May 2007

Abstract

We are concerned with the nonlinear second-order impulsive periodic boundary value problem https://static-content.springer.com/image/art%3A10.1155%2F2007%2F41589/MediaObjects/13661_2007_Article_645_IEq1_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2007%2F41589/MediaObjects/13661_2007_Article_645_IEq2_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2007%2F41589/MediaObjects/13661_2007_Article_645_IEq3_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2007%2F41589/MediaObjects/13661_2007_Article_645_IEq4_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2007%2F41589/MediaObjects/13661_2007_Article_645_IEq5_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2007%2F41589/MediaObjects/13661_2007_Article_645_IEq6_HTML.gif , new criteria are established based on Schaefer's fixed-point theorem.

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Authors’ Affiliations

(1)
Department of Mathematics, Huaiyin Teachers College
(2)
Department of Mathematics, Yangzhou University

References

  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/j.cam.2004.04.010MATHMathSciNetView 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

Copyright

© C. Bai and D. Yang 2007

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.