Table of Contents Author Guidelines Submit a Manuscript
Discrete Dynamics in Nature and Society
Volume 2013, Article ID 259730, 10 pages
http://dx.doi.org/10.1155/2013/259730
Research Article

Existence and Uniqueness of Positive Solutions to Nonlinear Second Order Impulsive Differential Equations with Concave or Convex Nonlinearities

1Department of Mathematics, Taiyuan University of Technology, Taiyuan, Shanxi 030024, China
2School of Mathematical Sciences, Shanxi University, Taiyuan, Shanxi 030006, China

Received 29 January 2013; Accepted 15 May 2013

Academic Editor: Yanbin Sang

Copyright © 2013 Lingling Zhang and Chengbo Zhai. 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. A. Cabada and E. Liz, “Boundary value problems for higher order ordinary differential equations with impulses,” Nonlinear Analysis: Theory, Methods and Applications, vol. 32, no. 6, pp. 775–786, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  2. A. Cabada, J. J. Nieto, D. Franco, and S. I. Trofimchuk, “A generalization of the monotone method for second order periodic boundary value problem with impulses at fixed points,” Dynamics of Continuous, Discrete and Impulsive Systems, vol. 7, no. 1, pp. 145–158, 2000. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  3. L. Chen and J. Sun, “Boundary value problem of second order impulsive functional differential equations,” Journal of Mathematical Analysis and Applications, vol. 323, no. 1, pp. 708–720, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  4. W. Ding, M. Han, and J. Mi, “Periodic boundary value problem for the second-order impulsive functional differential equations,” Computers & Mathematics with Applications, vol. 50, no. 3-4, pp. 491–507, 2005. View at Publisher · View at Google Scholar · View at MathSciNet
  5. L. H. Erbe and X. Liu, “Existence results for boundary value problems of second order impulsive differential equations,” Journal of Mathematical Analysis and Applications, vol. 149, no. 1, pp. 56–69, 1990. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  6. M. Feng and D. Xie, “Multiple positive solutions of multi-point boundary value problem for second-order impulsive differential equations,” Journal of Computational and Applied Mathematics, vol. 223, no. 1, pp. 438–448, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  7. D. J. Guo, “Existence of solutions of boundary value problems for nonlinear second order impulsive differential equations in Banach spaces,” Journal of Mathematical Analysis and Applications, vol. 181, no. 2, pp. 407–421, 1994. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  8. L. Hu, L. Liu, and Y. Wu, “Positive solutions of nonlinear singular two-point boundary value problems for second-order impulsive differential equations,” Applied Mathematics and Computation, vol. 196, no. 2, pp. 550–562, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  9. T. Jankowski, “Existence of solutions for second order impulsive differential equations with deviating arguments,” Nonlinear Analysis: Theory, Methods and Applications, vol. 67, no. 6, pp. 1764–1774, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  10. T. Jankowski, “Positive solutions to second order four-point boundary value problems for impulsive differential equations,” Applied Mathematics and Computation, vol. 202, no. 2, pp. 550–561, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. T. Jankowski, “Existence of positive solutions to second order four-point impulsive differential problems with deviating arguments,” Computers & Mathematics with Applications, vol. 58, no. 4, pp. 805–817, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  12. E. K. Lee and Y.-H. Lee, “Multiple positive solutions of singular two point boundary value problems for second order impulsive differential equations,” Applied Mathematics and Computation, vol. 158, no. 3, pp. 745–759, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  13. X. Lin and D. Jiang, “Multiple positive solutions of Dirichlet boundary value problems for second order impulsive differential equations,” Journal of Mathematical Analysis and Applications, vol. 321, no. 2, pp. 501–514, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  14. L. Liu, L. Hu, and Y. Wu, “Positive solutions of two-point boundary value problems for systems of nonlinear second-order singular and impulsive differential equations,” Nonlinear Analysis: Theory, Methods and Applications, vol. 69, no. 11, pp. 3774–3789, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  15. E. Liz and J. J. Nieto, “The monotone iterative technique for periodic boundary value problems of second order impulsive differential equations,” Commentationes Mathematicae Universitatis Carolinae, vol. 34, no. 3, pp. 405–411, 1993. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  16. J. J. Nieto and R. Rodríguez-López, “Periodic boundary value problem for non-Lipschitzian impulsive functional differential equations,” Journal of Mathematical Analysis and Applications, vol. 318, no. 2, pp. 593–610, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  17. I. Rachůnková and M. Tvrdý, “Existence results for impulsive second-order periodic problems,” Nonlinear Analysis: Theory, Methods and Applications, vol. 59, no. 1-2, pp. 133–146, 2004. View at Publisher · View at Google Scholar · View at MathSciNet
  18. I. Rachůnková and M. Tvrdý, “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
  19. M. U. Akhmet, G. A. Bekmukhambetova, and Y. Serinagaoglu, The Dynamics of the Systemic Arterial Pressure Through Impulsive Differential Equations, Institute for Mathematics and its Applications, Middle East Technical University, Ankara, Turkey, 2005.
  20. A. d'Onofrio, “On pulse vaccination strategy in the SIR epidemic model with vertical transmission,” Applied Mathematics Letters, vol. 18, no. 7, pp. 729–732, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  21. S. Gao, L. Chen, J. J. Nieto, and A. Torres, “Analysis of a delayed epidemic model with pulse vaccination and saturation incidence,” Vaccine, vol. 24, no. 35-36, pp. 6037–6045, 2006. View at Google Scholar
  22. W.-T. Li and H.-F. Huo, “Global attractivity of positive periodic solutions for an impulsive delay periodic model of respiratory dynamics,” Journal of Computational and Applied Mathematics, vol. 174, no. 2, pp. 227–238, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  23. S. Tang and L. Chen, “Density-dependent birth rate, birth pulses and their population dynamic consequences,” Journal of Mathematical Biology, vol. 44, no. 2, pp. 185–199, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  24. W. Zhang and M. Fan, “Periodicity in a generalized ecological competition system governed by impulsive differential equations with delays,” Mathematical and Computer Modelling, vol. 39, no. 4-5, pp. 479–493, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  25. X. Zhang, Z. Shuai, and K. Wang, “Optimal impulsive harvesting policy for single population,” Nonlinear Analysis: Theory, Methods and Applications, vol. 4, no. 4, pp. 639–651, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  26. R. P. Agarwal and D. O'Regan, “A multiplicity result for second order impulsive differential equations via the Leggett Williams fixed point theorem,” Applied Mathematics and Computation, vol. 161, no. 2, pp. 433–439, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  27. W. Wang, X. Fu, and X. Yang, “Positive solutions of periodic boundary value problems for impulsive differential equations,” Computers & Mathematics with Applications, vol. 58, no. 8, pp. 1623–1630, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  28. X. Zhang, “Existence of positive solution for second-order nonlinear impulsive singular differential equations of mixed type in Banach spaces,” Nonlinear Analysis: Theory, Methods and Applications, vol. 70, no. 4, pp. 1620–1628, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  29. J. Chu, X. Lin, D. Jiang, D. O'Regan, and R. P. Agarwal, “Positive solutions for second-order superlinear repulsive singular Neumann boundary value problems,” Positivity, vol. 12, no. 3, pp. 555–569, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  30. C.-B. Zhai, C. Yang, and X.-Q. Zhang, “Positive solutions for nonlinear operator equations and several classes of applications,” Mathematische Zeitschrift, vol. 266, no. 1, pp. 43–63, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet