Table of Contents
ISRN Applied Mathematics
Volume 2014 (2014), Article ID 657629, 10 pages
http://dx.doi.org/10.1155/2014/657629
Research Article

Periodic Boundary Value Problems for a Class of Impulsive Functional Differential Equations of Hybrid Type

Department of Mathematics, Jishou University, Jishou, Hunan 416000, China

Received 16 October 2013; Accepted 13 November 2013; Published 4 February 2014

Academic Editors: T. Y. Kam, M. Mei, and X.-S. Yang

Copyright © 2014 Guoping Chen. 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.

Linked References

  1. D. D. Baĭnov and P. S. Simeonov, Syetems with impulse Effect: Stability Theory and Applications, Chichester, UK, 1989. View at MathSciNet
  2. V. D. Mill'man and A. D. Myshkis, “On the stability of motion in the presence of impulsive,” Siberian Mathematical Journal, vol. 1, pp. 233–237, 1960. View at Google Scholar
  3. A. M. Samoilenko and N. A. Perestyuk, Differential Equations with Impulsive Effect, Kiev State University, 1980, (Russian).
  4. V. Lakshmikantham, D. D. Baĭnov, and P. S. Simeonov, Theory of Impulsive Differential Equations, vol. 6, World Scientific, Singapore, 1989. View at MathSciNet
  5. D. D. Bainov and P. S. Simeonov, Impulsive Differential Equations: Periodic Solutions and Applications, Longman Scientific and Technical, Harlow, UK, 1993.
  6. J. Shen and J. Li, “Impulsive control for stability of Volterra functional differential equations,” Journal for Analysis and its Applications, vol. 24, no. 4, pp. 721–734, 2005. View at Publisher · View at Google Scholar · View at MathSciNet
  7. I. Rachunkova and M. Tvrdy, “Existence results for impulsive second-order periodic problems,” Nonlinear Analysis. Theory, Methods & Applications A, vol. 59, no. 1-2, pp. 133–146, 2004. View at Publisher · View at Google Scholar · View at MathSciNet
  8. J. Shen, J. Li, and Q. Wang, “Boundedness and periodicity in impulsive ordinary and functional differential equations,” Nonlinear Analysis. Theory, Methods & Applications A, vol. 65, no. 10, pp. 1986–2002, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  9. J. L. Li, Boundary value problems and periodic solutions of impulsive differential equations [Ph.D. thesis], Hunan Normal University, 2006, Chinese.
  10. R. Liang and J. Shen, “Periodic boundary value problem for the first order impulsive functional differential equations,” Journal of Computational and Applied Mathematics, vol. 202, no. 2, pp. 498–510, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. Y. Tian and W. Ge, “Applications of variational methods to boundary-value problem for impulsive differential equations,” Proceedings of the Edinburgh Mathematical Society II, vol. 51, no. 2, pp. 509–527, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  12. J. Shen and J. Li, “Existence and global attractivity of positive periodic solutions for impulsive predator-prey model with dispersion and time delays,” Nonlinear Analysis. Real World Applications, vol. 10, no. 1, pp. 227–243, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  13. J. J. Nieto and D. O'Regan, “Variational approach to impulsive differential equations,” Nonlinear Analysis. Real World Applications, vol. 10, no. 2, pp. 680–690, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  14. Y. Shao and B. Dai, “The existence of exponential periodic attractor of impulsive BAM neural network with periodic coefficients and distributed delays,” Neurocomputing, vol. 73, pp. 3123–3131, 2010. View at Google Scholar
  15. Z. Zhang and R. Yuan, “An application of variational methods to Dirichlet boundary value problem with impulses,” Nonlinear Analysis. Real World Applications, vol. 11, no. 1, pp. 155–162, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  16. J. Zhou and Y. Li, “Existence of solutions for a class of second-order Hamiltonian systems with impulsive effects,” Nonlinear Analysis. Theory, Methods & Applications A, vol. 72, no. 3-4, pp. 1594–1603, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  17. A. V. Arutyunov, D. Yu. Karamzin, and F. Pereira, “Pontryagin's maximum principle for constrained impulsive control problems,” Nonlinear Analysis. Theory, Methods & Applications A, vol. 75, no. 3, pp. 1045–1057, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  18. L. Nie, Z. Teng, and A. Torres, “Dynamic analysis of an SIR epidemic model with state dependent pulse vaccination,” Nonlinear Analysis. Real World Applications, vol. 13, no. 4, pp. 1621–1629, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  19. G. S. Ladde, V. Lakshmikantham, and A. S. Vatsala, Monotone Iterative Techniques for Nonlinear Differential Equations, Pitman, London, UK, 1985. View at MathSciNet
  20. J. Li and J. Shen, “Periodic boundary value problems for impulsive differential-difference equations,” Indian Journal of Pure and Applied Mathematics, vol. 35, no. 11, pp. 1265–1277, 2004. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  21. I. Rachunkova and M. Tvrdy, “Non-ordered lower and upper functions in second order impulsive periodic problems,” Dynamics of Continuous, Discrete & Impulsive Systems A, vol. 12, no. 3-4, pp. 397–415, 2005. View at Google Scholar · View at MathSciNet
  22. Z. He and J. Yu, “Periodic boundary value problem for first-order impulsive functional differential equations,” Journal of Computational and Applied Mathematics, vol. 138, no. 2, pp. 205–217, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  23. W. Ding, J. Mi, and M. Han, “Periodic boundary value problems for the first order impulsive functional differential equations,” Applied Mathematics and Computation, vol. 165, no. 2, pp. 433–446, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  24. X. Z. Liu and D. J. Guo, “Periodic boundary value problems for class of impulsive integrodifferential equations in Banach spaces,” Journal of Mathematical Analysis and Applications, vol. 216, pp. 284–302, 1995. View at Google Scholar
  25. Z. He and X. He, “Monotone iterative technique for impulsive integro-differential equations with periodic boundary conditions,” Computers & Mathematics with Applications, vol. 48, no. 1-2, pp. 73–84, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  26. Z. He and X. He, “Periodic boundary value problems for first order impulsive integro-differential equations of mixed type,” Journal of Mathematical Analysis and Applications, vol. 296, no. 1, pp. 8–20, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  27. J. J. Nieto and R. Rodríguez-López, “Existence and approximation of solutions for nonlinear functional differential equations with periodic boundary value conditions,” Computers & Mathematics with Applications, vol. 40, no. 4-5, pp. 433–442, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  28. J. J. Nieto and R. Rodríguez-López, “Remarks on periodic boundary value problems for functional differential equations,” Journal of Computational and Applied Mathematics, vol. 158, no. 2, pp. 339–353, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet