Table of Contents
ISRN Mathematical Analysis
Volume 2011, Article ID 436045, 20 pages
http://dx.doi.org/10.5402/2011/436045
Research Article

Some Results for Nonlinear (𝑛+1)-Term Fractional Integrodifferential Inclusions with Multipoint Boundary Conditions

Department of Applied Mathematics, Donghua University, Shanghai 201620, China

Received 18 April 2011; Accepted 24 May 2011

Academic Editors: B. Djafari-Rouhani, G. L. Karakostas, and W. Kryszewski

Copyright © 2011 Huacheng Zhou 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. R. W. Ibrahim and S. Momani, “On the existence and uniqueness of solutions of a class of fractional differential equations,” Journal of Mathematical Analysis and Applications, vol. 334, no. 1, pp. 1–10, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  2. A. Babakhani and V. Daftardar-Gejji, “Existence of positive solutions of nonlinear fractional differential equations,” Journal of Mathematical Analysis and Applications, vol. 278, no. 2, pp. 434–442, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  3. K. Deimling, Multivalued Differential Equations, Walter de Gruyter, Berlin, Germany, 1992.
  4. V. Lakshmikantham, S. Leela, and J. Vasundhara, Theory of Fractional Dynamic Systems, Cambridge Academic, Cambridge, UK, 2009.
  5. V. Lakshmikantham and A. S. Vatsala, “Basic theory of fractional differential equations,” Nonlinear Analysis. Theory, Methods & Applications, vol. 69, no. 8, pp. 2677–2682, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  6. C. Yu and G. Gao, “Existence of fractional differential equations,” Journal of Mathematical Analysis and Applications, vol. 310, no. 1, pp. 26–29, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  7. A. Babakhani and V. D. Gejji, “Existence of positive solutions of nonlinear fractional differential equations,” Journal of Mathematical Analysis and Applications, vol. 278, no. 2, pp. 434–442, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  8. K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley & Sons, New York, NY, USA, 1993.
  9. I. Podlubny, Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Application, vol. 198, Academic Press, San Diego, Calif, USA, 1999.
  10. 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
  11. M. Stojanović, “Existence-uniqueness result for a nonlinear -term fractional equation,” Journal of Mathematical Analysis and Applications, vol. 353, no. 1, pp. 244–255, 2009. View at Publisher · View at Google Scholar
  12. A. Arara, M. Benchohra, N. Hamidi, and J. J. Nieto, “Fractional order differential equations on an unbounded domain,” Nonlinear Analysis. Theory, Methods & Applications, vol. 72, no. 2, pp. 580–586, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  13. A. M. A. El-Sayed and A. G. Ibrahim, “Multivalued fractional differential equations,” Applied Mathematics and Computation, vol. 68, no. 1, pp. 15–25, 1995. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  14. R. P. Agarwal, M. Belmekki, and M. Benchohra, “A survey on semilinear differential equations and inclusions involving Riemann-Liouville fractional derivative,” Advances in Difference Equations, vol. 2009, Article ID 981728, 47 pages, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  15. R. P. Agarwal, M. Belmekki, and M. Benchohra, “Existence results for semilinear functional differential inclusions involving Riemann-Liouville fractional derivative,” Dynamics of Continuous, Discrete & Impulsive Systems. Series A, vol. 17, no. 3, pp. 347–361, 2010. View at Google Scholar · View at Zentralblatt MATH
  16. 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, vol. 109, no. 3, pp. 973–1033, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  17. M. Benchohra, J. R. Graef, and S. Hamani, “Existence results for boundary value problems with non-linear fractional differential equations,” Applicable Analysis, vol. 87, no. 7, pp. 851–863, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  18. 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, vol. 11, no. 1, pp. 35–56, 2008. View at Google Scholar · View at Zentralblatt MATH
  19. M. Benchohra, J. R. Graef, and S. Hamani, “Existence results for boundary value problems with non-linear fractional differential equations,” Applicable Analysis, vol. 87, no. 7, pp. 851–863, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  20. R. P. Agarwal, M. Benchohra, and S. Hamani, “Boundary value problems for differential inclusions with fractional order,” Advanced Studies in Contemporary Mathematics, vol. 16, no. 2, pp. 181–196, 2008. View at Google Scholar · View at Zentralblatt MATH
  21. B. Ahmad, “Existence of solutions for fractional differential equations of order q(2,3] with anti-periodic boundary conditions,” Journal of Applied Mathematics and Computing, vol. 34, no. 1-2, pp. 385–391, 2010. View at Publisher · View at Google Scholar
  22. J. Henderson and A. Ouahab, “Fractional functional differential inclusions with finite delay,” Nonlinear Analysis. Theory, Methods & Applications, vol. 70, no. 5, pp. 2091–2105, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  23. A. Lin and L. Hu, “Existence results for impulsive neutral stochastic functional integro-differential inclusions with nonlocal initial conditions,” Computers & Mathematics with Applications, vol. 59, no. 1, pp. 64–73, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  24. J. Henderson and A. Ouahab, “Impulsive differential inclusions with fractional order,” Computers & Mathematics with Applications, vol. 59, no. 3, pp. 1191–1226, 2010. View at Google Scholar · View at Zentralblatt MATH
  25. 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
  26. A. Ouahab, “Some results for fractional boundary value problem of differential inclusions,” Nonlinear Analysis. Theory, Methods & Applications., vol. 69, no. 11, pp. 3877–3896, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  27. 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
  28. G. Li and X. Xue, “On the existence of periodic solutions for differential inclusions,” Journal of Mathematical Analysis and Applications, vol. 276, no. 1, pp. 168–183, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  29. R. M. Colombo, A. Fryszkowski, T. Rzeżuchowski, and V. Staicu, “Continuous selections of solution sets of Lipschitzean differential inclusions,” Funkcialaj Ekvacioj. Serio Internacia, vol. 34, no. 2, pp. 321–330, 1991. View at Google Scholar · View at Zentralblatt MATH
  30. H. F. Bohnenblust and S. Karlin, “On a theorem of Ville,” in Contributions to the Theory of Games, vol. 1, pp. 155–160, Princeton University, Princeton, NJ, USA, 1950. View at Google Scholar · View at Zentralblatt MATH
  31. M. Kisielewicz, Differential Inclusions and Optimal Control, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1991.
  32. L. Górniewicz, “Topological fixed point theory of multivalued mappings,” in Mathematics and Its Applications, vol. 495, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1999. View at Google Scholar · View at Zentralblatt MATH
  33. S. Hu and N. S. Papageorgiou, Handbook of Multivalued Analysis. Vol. I, vol. 419 of Mathematics and Its Applications, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1997.
  34. A. A. Tolstonogov, Differential Inclusions in a Banach Space, Kluwer Academic Publishers, Dordrecht, The Netherlands, 2000.
  35. 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 Sciences. Série des Sciences Mathématiques, Astronomiques et Physiques, vol. 13, pp. 781–786, 1965. View at Google Scholar · View at Zentralblatt MATH
  36. H. Covitz and S. B. Nadler,, “Multi-valued contraction mappings in generalized metric spaces,” Israel Journal of Mathematics, vol. 8, pp. 5–11, 1970. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  37. P. D. Lax, Functional Analysis, John Wiley & Sons, New York, NY, USA, 2002.
  38. C. Castaing and M. Valadier, “Convex analysis and measurable multifunctions,” in Lecture Notes in Mathematics, vol. 580, Springer, Berlin, Germany, 1977. View at Google Scholar