About this Journal Submit a Manuscript Table of Contents
Abstract and Applied Analysis
Volume 2012 (2012), Article ID 359452, 13 pages
http://dx.doi.org/10.1155/2012/359452
Research Article

A Note on Impulsive Fractional Evolution Equations with Nondense Domain

1School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan 430074, China
2School of Mathematics and Statistics, Suzhou University, Suzhou 234000, China

Received 22 March 2012; Revised 28 July 2012; Accepted 29 July 2012

Academic Editor: Juan J. Trujillo

Copyright © 2012 Zufeng Zhang and Bin Liu. 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. K. B. Oldham and J. Spanier, The Fractional Calculus, Academic Press, London, UK, 1974. View at Zentralblatt MATH
  2. K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley & Sons, New York, NY, USA, 1993. View at Zentralblatt MATH
  3. I. Podlubny, Fractional Differential Equations, vol. 198, Academic Press, San Diego, Calif, USA, 1999. View at Zentralblatt MATH
  4. D. Delbosco and L. Rodino, “Existence and uniqueness for a nonlinear fractional differential equation,” Journal of Mathematical Analysis and Applications, vol. 204, no. 2, pp. 609–625, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  5. S. Aizicovici and M. McKibben, “Existence results for a class of abstract nonlocal Cauchy problems,” Nonlinear Analysis A, vol. 39, no. 5, pp. 649–668, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  6. K. Diethelm and N. J. Ford, “Analysis of fractional differential equations,” Journal of Mathematical Analysis and Applications, vol. 265, no. 2, pp. 229–248, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  7. S. D. Eidelman and A. N. Kochubei, “Cauchy problem for fractional diffusion equations,” Journal of Differential Equations, vol. 199, no. 2, pp. 211–255, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  8. M. Benchohra and B. A. Slimani, “Existence and uniqueness of solutions to impulsive fractional differential equations,” Electronic Journal of Differential Equations, vol. 10, pp. 1–11, 2009. View at Zentralblatt MATH
  9. G. M. Mophou and G. M. N'Guérékata, “On integral solutions of some nonlocal fractional differential equations with nondense domain,” Nonlinear Analysis A, vol. 71, no. 10, pp. 4668–4675, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  10. V. Daftardar-Gejji and H. Jafari, “Analysis of a system of nonautonomous fractional differential equations involving Caputo derivatives,” Journal of Mathematical Analysis and Applications, vol. 328, no. 2, pp. 1026–1033, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  11. M. M. El-Borai, “Some probability densities and fundamental solutions of fractional evolution equations,” Chaos, Solitons and Fractals, vol. 14, no. 3, pp. 433–440, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  12. Y. Zhou and F. Jiao, “Nonlocal Cauchy problem for fractional evolution equations,” Nonlinear Analysis, vol. 11, no. 5, pp. 4465–4475, 2010. View at Publisher · View at Google Scholar
  13. J. Cao, Q. Yang, and Z. Huang, “Optimal mild solutions and weighted pseudo-almost periodic classical solutions of fractional integro-differential equations,” Nonlinear Analysis A, vol. 74, no. 1, pp. 224–234, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  14. X.-B. Shu, Y. Lai, and Y. Chen, “The existence of mild solutions for impulsive fractional partial differential equations,” Nonlinear Analysis A, vol. 74, no. 5, pp. 2003–2011, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  15. Z. Tai and S. Lun, “On controllability of fractional impulsive neutral infinite delay evolution integrodifferential systems in Banach spaces,” Applied Mathematics Letters, vol. 25, no. 2, pp. 104–110, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  16. J. Wang and Y. Zhou, “A class of fractional evolution equations and optimal controls,” Nonlinear Analysis, vol. 12, no. 1, pp. 262–272, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  17. G. Da Prato and E. Sinestrari, “Differential operators with nondense domain,” Annali della Scuola Normale Superiore di Pisa, vol. 14, no. 2, pp. 285–344, 1987.
  18. H. R. Thieme, ““Integrated semigroups” and integrated solutions to abstract Cauchy problems,” Journal of Mathematical Analysis and Applications, vol. 152, no. 2, pp. 416–447, 1990. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  19. H. R. Thieme, “Semiflows generated by Lipschitz perturbations of non-densely defined operators,” Differential and Integral Equations, vol. 3, no. 6, pp. 1035–1066, 1990. View at Zentralblatt MATH
  20. M. Adimy, H. Bouzahir, and K. Ezzinbi, “Existence for a class of partial functional differential equations with infinite delay,” Nonlinear Analysis A, vol. 46, no. 1, pp. 91–112, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  21. K. Ezzinbi and J. H. Liu, “Nondensely defined evolution equations with nonlocal conditions,” Mathematical and Computer Modelling, vol. 36, no. 9-10, pp. 1027–1038, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  22. M. Benchohra, E. P. Gatsori, J. Henderson, and S. K. Ntouyas, “Nondensely defined evolution impulsive differential inclusions with nonlocal conditions,” Journal of Mathematical Analysis and Applications, vol. 286, no. 1, pp. 307–325, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  23. E. P. Gatsori, “Controllability results for nondensely defined evolution differential inclusions with nonlocal conditions,” Journal of Mathematical Analysis and Applications, vol. 297, no. 1, pp. 194–211, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  24. N. Abada, M. Benchohra, and H. Hammouche, “Existence and controllability results for nondensely defined impulsive semilinear functional differential inclusions,” Journal of Differential Equations, vol. 246, no. 10, pp. 3834–3863, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  25. V. Kavitha and M. Mallika Arjunan, “Controllability of non-densely defined impulsive neutral functional differential systems with infinite delay in Banach spaces,” Nonlinear Analysis, vol. 4, no. 3, pp. 441–450, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  26. M. Belmekki and M. Benchohra, “Existence results for fractional order semilinear functional differential equations with nondense domain,” Nonlinear Analysis A, vol. 72, no. 2, pp. 925–932, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  27. W. Arendt, “Vector-valued Laplace transforms and Cauchy problems,” Israel Journal of Mathematics, vol. 59, no. 3, pp. 327–352, 1987. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  28. H. Kellerman and M. Hieber, “Integrated semigroups,” Journal of Functional Analysis, vol. 84, no. 1, pp. 160–180, 1989. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  29. K. Yosida, Functional Analysis, vol. 123, Springer, Berlin, Germany, 6th edition, 1980.
  30. W. Arendt, “Resolvent positive operators,” Proceedings of the London Mathematical Society, vol. 54, no. 2, pp. 321–349, 1987. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  31. F. Mainardi, P. Paradisi, and R. Gorenflo, “Probability distributions generated by fractional diffusion equations,” in Econophysics: An Emerging Science, J. Kertesz and I. Kondor, Eds., Kluwer Academic Publishers, Dordrecht, The Netherlands, 2000.