Table of Contents Author Guidelines Submit a Manuscript
Journal of Function Spaces and Applications
Volume 2013, Article ID 479049, 9 pages
http://dx.doi.org/10.1155/2013/479049
Research Article

Second-Order Impulsive Differential Equations with Functional Initial Conditions on Unbounded Intervals

Dipartimento di Matematica e Informatica, Università della Calabria, 87036 Arcavacata di Rende, Cosenza, Italy

Received 29 December 2012; Accepted 27 February 2013

Academic Editor: Feliz Minhós

Copyright © 2013 Luigi Muglia and Paolamaria Pietramala. 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. Baĭnov and P. Simeonov, Impulsive Differential Equations: Periodic Solutions and Applications, vol. 66 of Pitman Monographs and Surveys in Pure and Applied Mathematics, Longman Scientific & Technical, New York, NY, USA, 1993. View at MathSciNet
  2. 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
  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, Teaneck, NJ, USA, 1989. View at MathSciNet
  4. A. M. Samoĭlenko and N. A. Perestyuk, Impulsive Differential Equations, vol. 14, World Scientific, River Edge, NJ, USA, 1995. View at Publisher · View at Google Scholar · View at MathSciNet
  5. X. Liu and A. R. Willms, “Impulsive controllability of linear dynamical systems with applications to maneuvers of spacecraft,” Mathematical Problems in Engineering, vol. 2, pp. 277–299, 1996. View at Google Scholar
  6. R. Sakthivel, E. R. Anandhi, and N. I. Mahmudov, “Approximate controllability of second-order systems with state-dependent delay,” Numerical Functional Analysis and Optimization, vol. 29, no. 11-12, pp. 1347–1362, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  7. G. Arthi and K. Balachandran, “Controllability of damped second-order impulsive neutral functional differential systems with infinite delay,” Journal of Optimization Theory and Applications, vol. 152, no. 3, pp. 799–813, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  8. D. N. Chalishajar, “Controllability of second order impulsive neutral functional differential inclusions with infinite delay,” Journal of Optimization Theory and Applications, vol. 154, no. 2, pp. 672–684, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  9. E. Hernández, M. Rabello, and H. R. Henríquez, “Existence of solutions for impulsive partial neutral functional differential equations,” Journal of Mathematical Analysis and Applications, vol. 331, no. 2, pp. 1135–1158, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  10. G. Infante and P. Pietramala, “Nonlocal impulsive boundary value problems with solutions that change sign, CP1124, Mathematical Models in Engineering, Biology, and Medicine,” in Proceedings of the International Conference on Boundary Value Problems, A. Cabada, E. Liz, and J. J. Nieto, Eds., pp. 205–213, 2009.
  11. G. Infante, P. Pietramala, and M. Zima, “Positive solutions for a class of nonlocal impulsive BVPs via fixed point index,” Topological Methods in Nonlinear Analysis, vol. 36, no. 2, pp. 263–284, 2010. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  12. T. Jankowski, “Positive solutions of three-point boundary value problems for second order impulsive differential equations with advanced arguments,” Applied Mathematics and Computation, vol. 197, no. 1, pp. 179–189, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  13. R. Sakthivel, N. I. Mahmudov, and J. H. Kim, “On controllability of second order nonlinear impulsive differential systems,” Nonlinear Analysis. Theory, Methods & Applications, vol. 71, no. 1-2, pp. 45–52, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  14. S. Sivasankaran, M. M. Arjunan, and V. Vijayakumar, “Existence of global solutions for second order impulsive abstract partial differential equations,” Nonlinear Analysis. Theory, Methods & Applications, vol. 74, no. 17, pp. 6747–6757, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  15. D. N. Chalishajar and F. S. Acharya, “Controllability of second order semi-linear neutral impulsive differential inclusions on unbounded domain with infinite delay in Banach spaces,” Bulletin of the Korean Mathematical Society, vol. 48, no. 4, pp. 813–838, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  16. G. Marino, P. Pietramala, and L. Muglia, “Impulsive neutral semilinear equations on unbounded intervals,” Nonlinear Functional Analysis and Applications, vol. 9, no. 4, pp. 527–543, 2004. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  17. G. Marino, P. Pietramala, and L. Muglia, “Impulsive neutral integrodifferential equations on unbounded intervals,” Mediterranean Journal of Mathematics, vol. 1, no. 1, pp. 93–108, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  18. S. K. Ntouyas and D. O'Regan, “Existence results on semi-infinite intervals for nonlocal evolutions equations and inclusions via semigroup theory,” Numerical Functional Analysis and Optimization, vol. 29, no. 3-4, pp. 419–444, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  19. J. P. Dauer and K. Balachandran, “Existence of solutions of nonlinear neutral integrodifferential equations in Banach spaces,” Journal of Mathematical Analysis and Applications, vol. 251, no. 1, pp. 93–105, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  20. G. Marino, P. Pietramala, and H.-K. Xu, “Nonlinear neutral integrodifferential equations on unbounded intervals,” International Mathematical Forum, vol. 1, no. 17–20, pp. 933–946, 2006. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  21. H. O. Fattorini, Second Order Linear Differential Equations in Banach Spaces, vol. 108 of North-Holland Mathematics Studies, North-Holland, Amsterdam, The Netherlands, 1985. View at MathSciNet
  22. J. A. Goldstein, Semigroups of Linear Operators and Applications, The Clarendon Press Oxford University Press, New York, NY, USA, 1985. View at MathSciNet
  23. C. C. Travis and G. F. Webb, “Cosine families and abstract nonlinear second order differential equations,” Acta Mathematica Academiae Scientiarum Hungaricae, vol. 32, no. 1-2, pp. 75–96, 1978. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  24. H. Schaefer, “Über die Methode der a priori-Schranken,” Mathematische Annalen, vol. 129, pp. 415–416, 1955. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  25. E. De Pascale, G. Lewicki, and G. Marino, “Some conditions for compactness in BC(Q) and their application to boundary value problems,” Analysis, vol. 22, no. 1, pp. 21–32, 2002. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet