Table of Contents Author Guidelines Submit a Manuscript
The Scientific World Journal
Volume 2014, Article ID 978519, 13 pages
http://dx.doi.org/10.1155/2014/978519
Research Article

About Positivity of Green's Functions for Nonlocal Boundary Value Problems with Impulsive Delay Equations

Department of Mathematics and Computer Science, Ariel University, Ariel, Israel

Received 18 August 2013; Accepted 22 October 2013; Published 13 February 2014

Academic Editors: K. Ammari, F. J. Garcia-Pacheco, and K. Zhu

Copyright © 2014 Alexander Domoshnitsky and Irina Volinsky. 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. N. V. Azbelev, V. P. Maksimov, and L. F. Rakhmatullina, Introduction to the Theory of Functional Differential Equations, Advanced Series in Mathematical Science and Engineering, World Federation, Atlanta, Ga, USA, 1995.
  2. D. Baĭnov and P. Simeonov, Impulsive Differential Equations, vol. 66 of Pitman Monographs and Surveys in Pure and Applied Mathematics, Longman Scientific & Technical, Harlow, UK, 1993. View at MathSciNet
  3. V. Lakshmikantham, D. D. Baĭnov, and P. S. Simeonov, Theory of Impulsive Differential Equations, vol. 6 of Series in Modern Applied Mathematics, World Scientific, Singapore, 1989. View at MathSciNet
  4. S. G. Pandit and S. G. Deo, Differential Systems Involving Impulses, vol. 954 of Lecture Notes in Mathematics, Springer, Berlin, Germany, 1982. View at MathSciNet
  5. A. M. Samoilenko and N. A. Perestyuk, Differential Equations with Impulse Effect, Villa Skola, Kiev, Russia, 1987, (Russian).
  6. A. M. Samoilenko and A. N. Perstyuk, Impulsive Differential Equations, World Scientific, Singapore, 1992.
  7. S. T. Zavalishchin and A. N. Sesekin, Dynamic Impulse Systems: Theory and Applications, vol. 394 of Mathematics and Its Applications, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1997. View at MathSciNet
  8. M. Federson and S. Schwabik, “Generalized ODE approach to impulsive retarded functional differential equations,” Differential and Integral Equations, vol. 19, no. 11, pp. 1201–1234, 2006. View at Google Scholar · View at MathSciNet
  9. M. Federson and S. Schwabik, “A new approach to impulsive retarded differential equations: stability results,” Functional Differential Equations, vol. 16, no. 4, pp. 583–607, 2009. View at Google Scholar · View at MathSciNet
  10. Z. Halas and M. Tvrdý, “Continuous dependence of solutions of generalized linear differential equations on a parameter,” Functional Differential Equations, vol. 16, no. 2, pp. 299–313, 2009. View at Google Scholar · View at MathSciNet
  11. J. Kurzweil, “Generalized ordinary differential equations and continuous dependence on a parameter,” Czechoslovak Mathematical Journal, vol. 7, no. 82, pp. 418–449, 1957. View at Google Scholar · View at MathSciNet
  12. S. Schwabik, Generalized Ordinary Differential Equations, Eorkd Scientific, Singapore, 1992.
  13. M. Tvrdý, “Generalized differential equations in the space of regulated functions (boundary value problems and controllability),” Mathematica Bohemica, vol. 116, no. 3, pp. 225–244, 1991. View at Google Scholar · View at MathSciNet
  14. M. Tvrdý, “Differential and integral equations in the space of regulated functions,” Memoirs on Differential Equations and Mathematical Physics, vol. 25, pp. 1–104, 2002. View at Google Scholar · View at MathSciNet
  15. M. Ashordia, “Criteria of correctness of linear boundary value problems for systems of generalized ordinary differential equations,” Czechoslovak Mathematical Journal, vol. 46, no. 121, pp. 385–404, 1996. View at Google Scholar · View at MathSciNet
  16. A. Domoshnitsky and M. Drakhlin, “On boundary value problems for first order impulse functional-differential equations,” in Boundary Value Problems for Functional Differential Equations, J. Henderson, Ed., pp. 107–117, World Scientific, Singapore, 1995. View at Publisher · View at Google Scholar · View at MathSciNet
  17. A. Domoshnitsky and M. Drakhlin, “Nonoscillation of first order impulse differential equations with delay,” Journal of Mathematical Analysis and Applications, vol. 206, no. 1, pp. 254–269, 1997. View at Publisher · View at Google Scholar · View at MathSciNet
  18. A. Domoshnitsky, M. Drakhlin, and E. Litsyn, “On N-th order functional-differential equations with impulses,” in Memoirs on Differential Equations and Mathematical Physics, vol. 12, pp. 50–56, 1997. View at Google Scholar
  19. A. Domoshnitsky, “Boundary value problem for first order impulsive functional-di¤erential nonlinear equations,” Functional Differential Equations, vol. 4, pp. 37–45, 1997. View at Google Scholar
  20. A. Domoshnitsky, M. Drakhlin, and E. Litsyn, “On boundary value problems for N-th order functional differential equations with impulses,” Advances in Mathematical Sciences and Applications, vol. 8, no. 2, pp. 987–996, 1998. View at Google Scholar
  21. D. D. Baĭnov, S. G. Hristova, S. C. Hu, and V. Lakshmikantham, “Periodic boundary value problems for systems of first order impulsive differential equations,” Differential and Integral Equations, vol. 2, no. 1, pp. 37–43, 1989. View at Google Scholar · View at MathSciNet
  22. J. Li, J. J. Nieto, and J. Shen, “Impulsive periodic boundary value problems of first-order differential equations,” Journal of Mathematical Analysis and Applications, vol. 325, no. 1, pp. 226–236, 2007. View at Publisher · View at Google Scholar · View at MathSciNet
  23. 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 MathSciNet
  24. S. C. Hu and V. Lakshmikantham, “Periodic boundary value problems for second order impulsive differential systems,” Nonlinear Analysis. Theory, Methods & Applications, vol. 13, no. 1, pp. 75–85, 1989. View at Publisher · View at Google Scholar · View at MathSciNet
  25. R. P. Agarwal, L. Berezansky, E. Braverman, and A. Domoshnitsky, Nonoscillation Theory of Functional Differential Equations with Applications, Springer, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  26. V. Lakshmikantham and X. Liu, “On quasistability for impulsive differential systems,” Nonlinear Analysis. Theory, Methods & Applications, vol. 13, no. 7, pp. 819–828, 1989. View at Publisher · View at Google Scholar · View at MathSciNet
  27. M. Benchohra, J. Henderson, and S. K. Ntouyas, Impulsive Differential Equations and Inclusions, vol. 2 of Contemporary Mathematics and Its Applications, Hindawi Publishing Corporation, New York, NY, USA, 2006. View at Publisher · View at Google Scholar · View at MathSciNet
  28. A. L. Skubachevskii, “Nonclassical boundary value problems,” Journal of Mathematical Sciences, vol. 155, no. 2, pp. 199–330, 2008. View at Google Scholar
  29. M. Benchohra, J. Henderson, and S. K. Ntouyas, “An existence result for first-order impulsive functional differential equations in Banach spaces,” Computers & Mathematics with Applications, vol. 42, no. 10-11, pp. 1303–1310, 2001. View at Publisher · View at Google Scholar · View at MathSciNet
  30. Z. Fan, “Impulsive problems for semilinear differential equations with nonlocal conditions,” Nonlinear Analysis. Theory, Methods & Applications, vol. 72, no. 2, pp. 1104–1109, 2010. View at Publisher · View at Google Scholar · View at MathSciNet
  31. Z. Fan and G. Li, “Existence results for semilinear differential equations with nonlocal and impulsive conditions,” Journal of Functional Analysis, vol. 258, no. 5, pp. 1709–1727, 2010. View at Publisher · View at Google Scholar · View at MathSciNet
  32. S. Ji and G. Li, “Existence results for impulsive differential inclusions with nonlocal conditions,” Computers & Mathematics with Applications, vol. 62, no. 4, pp. 1908–1915, 2011. View at Publisher · View at Google Scholar · View at MathSciNet
  33. S. Ji and S. Wen, “Nonlocal Cauchy problem for impulsive differential equations in Banach spaces,” International Journal of Nonlinear Science, vol. 10, no. 1, pp. 88–95, 2010. View at Google Scholar · View at MathSciNet
  34. S. Ji and G. Li, “A unified approach to nonlocal impulsive differential equations with the measure of noncompactness,” Advances in Difference Equations, vol. 2012, article 182, 2012. View at Publisher · View at Google Scholar
  35. L. Zhu, Q. Dong, and G. Li, “Impulsive differential equations with nonlocal conditions in general Banach spaces,” Advances in Difference Equations, vol. 2012, article 10, 2012. View at Google Scholar