Table of Contents Author Guidelines Submit a Manuscript
Abstract and Applied Analysis
Volume 2013, Article ID 869837, 17 pages
http://dx.doi.org/10.1155/2013/869837
Research Article

Existence Results for Langevin Fractional Differential Inclusions Involving Two Fractional Orders with Four-Point Multiterm Fractional Integral Boundary Conditions

1Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia
2Department of Mathematics, University of Ioannina, 451 10 Ioannina, Greece

Received 2 January 2013; Accepted 22 March 2013

Academic Editor: Juan J. Nieto

Copyright © 2013 Ahmed Alsaedi 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. S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives, Gordon and Breach Science Publishers, Yverdon, Switzerland, 1993. View at Zentralblatt MATH · View at MathSciNet
  2. I. Podlubny, Fractional Differential Equations, vol. 198 of Mathematics in Science and Engineering, Academic Press, San Diego, CA, USA, 1999. View at Zentralblatt MATH · View at MathSciNet
  3. A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, vol. 204 of North-Holland Mathematics Studies, Elsevier Science, Amsterdam, The Netherlands, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  4. J. Sabatier, O. P. Agrawal, and J. A. T. Machado, Eds., Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering, Springer, Dordrecht, The Netherlands, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  5. V. Lakshmikantham, S. Leela, and J. Vasundhara Devi, Theory of Fractional Dynamic Systems, Cambridge Academic Publishers, Cambridge, UK, 2009.
  6. D. Baleanu, A. K. Golmankhaneh, and A. K. Golmankhaneh, “Fractional Nambu mechanics,” International Journal of Theoretical Physics, vol. 48, no. 4, pp. 1044–1052, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  7. B. Ahmad and S. K. Ntouyas, “Nonlinear fractional differential equations and inclusions of arbitrary order and multi-strip boundary conditions,” Electronic Journal of Differential Equations, vol. 2012, no. 98, pp. 1–22, 2012. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  8. Nyamoradi N. and M. Javidi, “Existence of multiple positive solutions for fractional differential inclusion with m-point boundary conditions and two fractional orders,” Electronic Journal of Differential Equations, vol. 2012, no. 187, pp. 1–26, 2012. View at Google Scholar
  9. B. Ahmad and J. J. Nieto, “Sequential fractional differential equations with three-point boundary conditions,” Computers & Mathematics with Applications, vol. 64, no. 10, pp. 3046–3052, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  10. M. R. Ubriaco, “Entropies based on fractional calculus,” Physics Letters A, vol. 373, no. 30, pp. 2516–2519, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. 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 · View at MathSciNet
  12. 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, Article ID 981728, p. 47, 2009. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  13. 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 · View at MathSciNet
  14. 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, vol. 2010, no. 20, pp. 1–16, 2010. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  15. A. Cernea, “On the existence of solutions for nonconvex fractional hyperbolic differential inclusions,” Communications in Mathematical Analysis, vol. 9, no. 1, pp. 109–120, 2010. View at Google Scholar · View at MathSciNet
  16. B. Ahmad and S. K. Ntouyas, “Some existence results for boundary value problems of fractional differential inclusions with non-separated boundary conditions,” Electronic Journal of Qualitative Theory of Differential Equations, no. 71, pp. 1–17, 2010. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  17. W. Sudsutad and J. Tariboon, “Existence results of fractional integro-differential equations with m-point multi-term fractional order integral boundary conditions,” Boundary Value Problems, vol. 2012, article 94, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  18. W. T. Coffey, Y. P. Kalmykov, and J. T. Waldron, The Langevin Equation, vol. 14 of World Scientific Series in Contemporary Chemical Physics, World Scientific Publishing, River Edge, NJ, USA, 2nd edition, 2004. View at MathSciNet
  19. S. C. Lim, M. Li, and L. P. Teo, “Langevin equation with two fractional orders,” Physics Letters A, vol. 372, no. 42, pp. 6309–6320, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  20. S. C. Lim and L. P. Teo, “The fractional oscillator process with two indices,” Journal of Physics A, vol. 42, no. 6, Article ID 065208, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  21. M. Uranagase and T. Munakata, “Generalized Langevin equation revisited: mechanical random force and self-consistent structure,” Journal of Physics A, vol. 43, no. 45, Article ID 455003, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  22. S. I. Denisov, H. Kantz, and P. Hänggi, “Langevin equation with super-heavy-tailed noise,” Journal of Physics A, vol. 43, no. 28, Article ID 285004, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  23. B. Ahmad and J. J. Nieto, “Solvability of nonlinear Langevin equation involving two fractional orders with Dirichlet boundary conditions,” International Journal of Differential Equations, Article ID 649486, 10 pages, 2010. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  24. A. Lozinski, R. G. Owens, and T. N. Phillips, “The Langevin and Fokker-Planck equations in polymer rheology,” Handbook of Numerical Analysis, vol. 16, pp. 211–303, 2011. View at Publisher · View at Google Scholar · View at Scopus
  25. L. Lizana, T. Ambjörnsson, A. Taloni, E. Barkai, and M. A. Lomholt, “Foundation of fractional Langevin equation: harmonization of a many-body problem,” Physical Review E, vol. 81, no. 5, 8 pages, 2010. View at Google Scholar
  26. B. Ahmad, J. J. Nieto, A. Alsaedi, and M. El-Shahed, “A study of nonlinear Langevin equation involving two fractional orders in different intervals,” Nonlinear Analysis. Real World Applications, vol. 13, no. 2, pp. 599–606, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  27. K. Deimling, Multivalued Differential Equations, vol. 1 of de Gruyter Series in Nonlinear Analysis and Applications, Walter de Gruyter, Berlin, Germany, 1992. View at Publisher · View at Google Scholar · View at MathSciNet
  28. S. Hu and N. S. Papageorgiou, Handbook of Multivalued Analysis, vol. I of Mathematics and its Applications, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1997. View at MathSciNet
  29. G. V. Smirnov, Introduction to the Theory of Differential Inclusions, vol. 41 of Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, USA, 2002. View at MathSciNet
  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 Sciences. Série des Sciences Mathématiques, vol. 13, pp. 781–786, 1965. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  31. A. Granas and J. Dugundji, Fixed Point Theory, Springer Monographs in Mathematics, Springer, New York, NY, USA, 2005. View at MathSciNet
  32. A. Bressan and G. Colombo, “Extensions and selections of maps with decomposable values,” Studia Mathematica, vol. 90, no. 1, pp. 69–86, 1988. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  33. M. Kisielewicz, Differential Inclusions and Optimal Control, vol. 44 of Mathematics and its Applications (East European Series), Kluwer Academic Publishers Group, Dordrecht, The Netherlands, 1991. View at MathSciNet
  34. H. Covitz and S. B. Nadler Jr., “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 · View at MathSciNet
  35. M. Frigon, “Théorèmes d'existence de solutions d'inclusions différentielles,” in Topological Methods in Differential Equations and Inclusions, A. Granas and M. Frigon, Eds., vol. 472 of NATO ASI Series C, pp. 51–87, Kluwer Academic, Dordrecht, The Netherlands, 1995. View at Google Scholar · View at MathSciNet
  36. C. Castaing and M. Valadier, Convex Analysis and Measurable Multifunctions, vol. 580 of Lecture Notes in Mathematics, Springer, Berlin, Germany, 1977. View at MathSciNet