Table of Contents Author Guidelines Submit a Manuscript
Discrete Dynamics in Nature and Society
Volume 2012 (2012), Article ID 363562, 14 pages
http://dx.doi.org/10.1155/2012/363562
Research Article

Existence of Solutions for Fractional-Order Neutral Differential Inclusions with Impulsive and Nonlocal Conditions

1Department of Basic Courses, Nanjing Institute of Technology, Nanjing 211167, China
2Research Center for Complex Systems and Network Sciences and Department of Mathematics, Southeast University, Nanjing 210096, China

Received 21 March 2012; Accepted 19 May 2012

Academic Editor: Zengji Du

Copyright © 2012 Chao Song et al. 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. L. Byszewski, “Theorems about the existence and uniqueness of solutions of a semilinear evolution nonlocal Cauchy problem,” Journal of Mathematical Analysis and Applications, vol. 162, no. 2, pp. 494–505, 1991. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  2. L. Byszewski and V. Lakshmikantham, “Theorem about the existence and uniqueness of a solution of a nonlocal abstract Cauchy problem in a Banach space,” Applicable Analysis. An International Journal, vol. 40, no. 1, pp. 11–19, 1991. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  3. S. K. Ntouyas and P. Ch. Tsamatos, “Global existence for semilinear evolution equations with nonlocal conditions,” Journal of Mathematical Analysis and Applications, vol. 210, no. 2, pp. 679–687, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  4. M. Benchohra and S. K. Ntouyas, “Nonlocal Cauchy problems for neutral functional differential and integrodifferential inclusions in Banach spaces,” Journal of Mathematical Analysis and Applications, vol. 258, no. 2, pp. 573–590, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  5. J. Liang, J. Liu, and T.-J. Xiao, “Nonlocal Cauchy problems governed by compact operator families,” Nonlinear Analysis. Theory, Methods & Applications, vol. 57, no. 2, pp. 183–189, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  6. X. Xue, “Existence of solutions for semilinear nonlocal Cauchy problems in Banach spaces,” Electronic Journal of Differential Equations, vol. 64, pp. 1–7, 2005. View at Google Scholar · View at Zentralblatt MATH
  7. X. Xue, “Existence of semilinear differential equations with nonlocal initial conditions,” Acta Mathematica Sinica, vol. 23, no. 6, pp. 983–988, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  8. V. Lakshmikantham, D. D. Bainov, and P. S. Simeonov, Theory of Impulsive Differential Equations, World Scientific Publishing, Teaneck, NJ, USA, 1989.
  9. A. M. Samoilenko and N. A. Perestyuk, Impulsive Differential Equations, World Scientific Publishing, River Edge, NJ, USA, 1995. View at Publisher · View at Google Scholar
  10. N. Abada, M. Benchohra, and H. Hammouche, “Existence and controllability results for impulsive partial functional differential inclusions,” Nonlinear Analysis. Theory, Methods & Applications, vol. 69, no. 9, pp. 2892–2909, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  11. 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
  12. G. M. N'Guérékata, “A Cauchy problem for some fractional abstract differential equation with non local conditions,” Nonlinear Analysis. Theory, Methods & Applications, vol. 70, no. 5, pp. 1873–1876, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  13. G. M. Mophou, “Existence and uniqueness of mild solutions to impulsive fractional differential equations,” Nonlinear Analysis. Theory, Methods & Applications, vol. 72, no. 3-4, pp. 1604–1615, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  14. F. L. Chen, A. Chen, and X. Wang, “On the solutions for impulsive fractional functional differential equations,” Differential Equations and Dynamical Systems, vol. 17, no. 4, pp. 379–391, 2009. View at Publisher · View at Google Scholar
  15. Y. Zhou and F. Jiao, “Existence of mild solutions for fractional neutral evolution equations,” Computers & Mathematics with Applications. An International Journal, vol. 59, no. 3, pp. 1063–1077, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  16. Y. Zhou and F. Jiao, “Nonlocal Cauchy problem for fractional evolution equations,” Nonlinear Analysis. Real World Applications. An International Multidisciplinary Journal, vol. 11, no. 5, pp. 4465–4475, 2010. View at Publisher · View at Google Scholar
  17. M. Benchohra, J. Henderson, S. K. Ntouyas, and A. Ouahab, “Existence results for fractional functional differential inclusions with infinite delay and applications to control theory,” Fractional Calculus & Applied Analysis. An International Journal for Theory and Applications, vol. 11, no. 1, pp. 35–56, 2008. View at Google Scholar · View at Zentralblatt MATH
  18. M. Benchohra and B. A. Slimani, “Existence and uniqueness of solutions to impulsive fractional differential equations,” Electronic Journal of Differential Equations, vol. 120, pp. 1–11, 2009. View at Google Scholar
  19. A. P. Chen and Y. Chen, “Existence of solutions to anti-periodic boundary value problem for nonlinear fractional differential equations with impulses,” Advances in Difference Equations, Article ID 915689, 17 pages, 2011. View at Publisher · View at Google Scholar
  20. S. Hamani, M. Benchohra, and J. R. Graef, “Existence results for boundary-value problems with nonlinear fractional differential inclusions and integral conditions,” Electronic Journal of Differential Equations, p. No. 20, 16, 2010. View at Google Scholar · View at Zentralblatt MATH
  21. Y.-K. Chang and J. J. Nieto, “Some new existence results for fractional differential inclusions with boundary conditions,” Mathematical and Computer Modelling, vol. 49, no. 3-4, pp. 605–609, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  22. B. Ahmad, J. J. Nieto, and J. Pimentel, “Some boundary value problems of fractional differential equations and inclusions,” Computers & Mathematics with Applications. An International Journal, vol. 62, no. 3, pp. 1238–1250, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  23. B. Ahmad and S. Sivasundaram, “Existence results for nonlinear impulsive hybrid boundary value problems involving fractional differential equations,” Nonlinear Analysis. Hybrid Systems. An International Multidisciplinary Journal, vol. 3, no. 3, pp. 251–258, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  24. K. Balachandran and S. Kiruthika, “Existence of solutions of abstract fractional impulsive semilinear evolution equations,” Electronic Journal of Qualitative Theory of Differential Equations, p. No. 4, 12, 2010. View at Google Scholar · View at Zentralblatt MATH
  25. K. Balachandran, S. Kiruthika, and J. J. Trujillo, “Existence results for fractional impulsive integrodifferential equations in Banach spaces,” Communications in Nonlinear Science and Numerical Simulation, vol. 16, no. 4, pp. 1970–1977, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  26. R. P. Agarwal, M. Benchohra, and S. Hamani, “A survey on existence results for boundary value problems of nonlinear fractional differential equations and inclusions,” Acta Applicandae Mathematicae. An International Survey Journal on Applying Mathematics and Mathematical Applications, vol. 109, no. 3, pp. 973–1033, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  27. R. P. Agarwal and B. Ahmad, “Existence theory for anti-periodic boundary value problems of fractional differential equations and inclusions,” Computers & Mathematics with Applications. An International Journal, vol. 62, no. 3, pp. 1200–1214, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  28. A. P. Chen and Y. Chen, “Existence of solutions to anti-periodic boundary value problem for nonlinear fractional differential equations,” Differential Equations and Dynamical Systems, vol. 19, no. 3, pp. 237–252, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  29. B. C. Dhage, A. Boucherif, and S. K. Ntouyas, “On periodic boundary value problems of first-order perturbed impulsive differential inclusions,” Electronic Journal of Differential Equations, vol. 84, pp. 1–9, 2004. View at Google Scholar · View at Zentralblatt MATH
  30. A. Lasota and Z. Opial, “An application of the Kakutani-Ky-Fan theorem in the theory of ordinary differential equations,” Bulletin de l'Académie Polonaise des Sciencess, vol. 13, pp. 781–786, 1965. View at Google Scholar · View at Zentralblatt MATH
  31. J.-P. Aubin and A. Cellina, Differential Inclusions, Springer-Verlag, Berlin, Germany, 1984.
  32. K. Deimling, Multivalued Differential Equations, Walter de Gruyter, Berlin, Germany, 1992. View at Publisher · View at Google Scholar
  33. I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, Calif, USA, 1999.
  34. S. Zhang, “Positive solutions for boundary-value problems of nonlinear fractional differential equations,” Electronic Journal of Differential Equations, vol. 36, pp. 1–12, 2006. View at Google Scholar · View at Zentralblatt MATH