SpringerOpen Newsletter

Receive periodic news and updates relating to SpringerOpen.

Open Access Research Article

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

Chuanzhi Bai1* and Dandan Yang2

Author Affiliations

1 Department of Mathematics, Huaiyin Teachers College, Huaian, Jiangsu 223300, China

2 Department of Mathematics, Yangzhou University, Yangzhou 225002, China

For all author emails, please log on.

Boundary Value Problems 2007, 2007:041589  doi:10.1155/2007/41589

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2007/1/041589


Received:12 February 2007
Revisions received:19 March 2007
Accepted:13 April 2007
Published:21 May 2007

© 2007 Bai and Yang

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.

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

References

  1. Benchohra, M, Henderson, J, Ntouyas, S: Impulsive Differential Equations and Inclusions, Contemporary Mathematics and Its Applications, Hindawi, New York, NY, USA (2006)

  2. Liu X (ed.): Advances in Impulsive Differential Equations. Dynamics of Continuous, Discrete & Impulsive Systems. Series A. Mathematical Analysis. 9(3), 313–462 (2002). PubMed Abstract | Publisher Full Text OpenURL

  3. Rogovchenko, YV: Impulsive evolution systems: main results and new trends. Dynamics of Continuous, Discrete & Impulsive Systems. Series A. Mathematical Analysis. 3(1), 57–88 (1997). PubMed Abstract | Publisher Full Text OpenURL

  4. Samoĭlenko, AM, Perestyuk, NA: Impulsive Differential Equations, World Scientific Series on Nonlinear Science. Series A: Monographs and Treatises,p. x+462. World Scientific, River Edge, NJ, USA (1995)

  5. Zavalishchin, ST, Sesekin, AN: Dynamic Impulse Systems. Theory and Applications, Mathematics and Its Applications,p. xii+256. Kluwer Academic Publishers, Dordrecht, The Netherlands (1997)

  6. Choisy, M, Guégan, JF, Rohani, P: Dynamics of infectious diseases and pulse vaccination: teasing apart the embedded resonance effects. Physica D: Nonlinear Phenomena. 22(1), 26–35 (2006)

  7. d'Onofrio, A: On pulse vaccination strategy in the SIR epidemic model with vertical transmission. Applied Mathematics Letters. 18(7), 729–732 (2005). Publisher Full Text OpenURL

  8. Gao, S, Chen, L, Nieto, JJ, Torres, A: Analysis of a delayed epidemic model with pulse vaccination and saturation incidence. Vaccine. 24(35-36), 6037–6045 (2006). PubMed Abstract | Publisher Full Text OpenURL

  9. He, Z, Zhang, X: Monotone iterative technique for first order impulsive difference equations with periodic boundary conditions. Applied Mathematics and Computation. 156(3), 605–620 (2004). Publisher Full Text OpenURL

  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. 174(2), 227–238 (2005). Publisher Full Text OpenURL

  11. Tang, S, Chen, L: Density-dependent birth rate, birth pulses and their population dynamic consequences. Journal of Mathematical Biology. 44(2), 185–199 (2002). PubMed Abstract | Publisher Full Text OpenURL

  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. 32(5), 1772–1785 (2007). PubMed Abstract | Publisher Full Text OpenURL

  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. 40(5-6), 509–518 (2004). Publisher Full Text OpenURL

  14. Zhang, W, Fan, M: Periodicity in a generalized ecological competition system governed by impulsive differential equations with delays. Mathematical and Computer Modelling. 39(4-5), 479–493 (2004). Publisher Full Text OpenURL

  15. Zhang, X, Shuai, Z, Wang, K: Optimal impulsive harvesting policy for single population. Nonlinear Analysis: Real World Applications. 4(4), 639–651 (2003). Publisher Full Text OpenURL

  16. Agarwal, RP, O'Regan, D: Multiple nonnegative solutions for second order impulsive differential equations. Applied Mathematics and Computation. 114(1), 51–59 (2000). Publisher Full Text OpenURL

  17. Chen, L, Sun, J: Nonlinear boundary value problem of first order impulsive functional differential equations. Journal of Mathematical Analysis and Applications. 318(2), 726–741 (2006). Publisher Full Text OpenURL

  18. Ding, W, Han, M, Mi, J: Periodic boundary value problem for the second-order impulsive functional differential equations. Computers & Mathematics with Applications. 50(3-4), 491–507 (2005). PubMed Abstract | Publisher Full Text OpenURL

  19. Nieto, JJ, Rodríguez-López, R: Periodic boundary value problem for non-Lipschitzian impulsive functional differential equations. Journal of Mathematical Analysis and Applications. 318(2), 593–610 (2006). Publisher Full Text OpenURL

  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. 12(3-4), 397–415 (2005). PubMed Abstract | Publisher Full Text OpenURL

  21. Chen, J, Tisdell, CC, Yuan, R: On the solvability of periodic boundary value problems with impulse. Journal of Mathematical Analysis and Applications. 331(2), 902–912 (2007). Publisher Full Text OpenURL

  22. Li, J, Nieto, JJ, Shen, J: Impulsive periodic boundary value problems of first-order differential equations. Journal of Mathematical Analysis and Applications. 325(1), 226–236 (2007). Publisher Full Text OpenURL

  23. Nieto, JJ: Periodic boundary value problems for first-order impulsive ordinary differential equations. Nonlinear Analysis. 51(7), 1223–1232 (2002). Publisher Full Text OpenURL

  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. 16(4), 451–462 (2004)

  25. Rudd, M, Tisdell, CC: On the solvability of two-point, second-order boundary value problems. Applied Mathematics Letters. 20(7), 824–828 (2007). Publisher Full Text OpenURL

  26. Dong, Y: Sublinear impulse effects and solvability of boundary value problems for differential equations with impulses. Journal of Mathematical Analysis and Applications. 264(1), 32–48 (2001). Publisher Full Text OpenURL

  27. Liu, Y: Further results on periodic boundary value problems for nonlinear first order impulsive functional differential equations. Journal of Mathematical Analysis and Applications. 327(1), 435–452 (2007). Publisher Full Text OpenURL

  28. Qian, D, Li, X: Periodic solutions for ordinary differential equations with sublinear impulsive effects. Journal of Mathematical Analysis and Applications. 303(1), 288–303 (2005). Publisher Full Text OpenURL

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

  30. Lakshmikantham, V, Baĭnov, DD, Simeonov, PS: Theory of Impulsive Differential Equations, Series in Modern Applied Mathematics,p. xii+273. World Scientific, Teaneck, NJ, USA (1989)

  31. Nieto, JJ: Basic theory for nonresonance impulsive periodic problems of first order. Journal of Mathematical Analysis and Applications. 205(2), 423–433 (1997). Publisher Full Text OpenURL