Boundary value problems for differential inclusions with fractional order

Advanced Studies in Contemporary Mathematics (Kyungshang)

Volumn 16, Issue 2, 2008, Pages 181-196

  Agarwal, Ravi P ^a   Benchohra, Mouffak ^b   Hamani, Samira ^b  

^a Florida Institute of Technology ^*  (United States)

^b UNIVERSITY OF SIDI BEL ABBES   (Algeria)

Author keywords

Caputo fractional derivative; Differential inclusion; Existence; Fixed point; Fractional integral

Indexed keywords

EID: 42649131330     PISSN: 12293067     EISSN: None     Source Type: Journal    
DOI: None     Document Type: Article

References (40)

1. R.P Agarwal, M. Benchohra and S. Hamani, Boundary value problems for fractional differential equations (submitted).
2. J. P. Aubin and A. Cellina, Differential Inclusions, Springer-Verlag, Berlin-Heidelberg, New York, 1984.
3. J. P. Aubin and H. Frankowska, Set-Valued Analysis, Birkhauser, Boston, 1990.
4. A. Belarbi, M. Benchohra, S. Hamani and S.K. Ntouyas, Perturbed functional differential equations with fractional order. Commun. Appl. Anal. 11 (3-4) (2007), 429-440.
5. A. Belarbi, M. Benchohra and A. Ouahab, Uniqueness results for fractional functional differential equations with infinite delay in Fréchet spaces, Appl. Anal. 85 (2006), 1459-1470.
6. M. Benchohra, J. Henderson, S.K. Ntouyas and A. Ouahab, Existence results for fractional order functional differential equations with infinite delay, J. Math. Anal. Appl. 338 (2) (2008), 1340-1350.
7. C. Castaing and M. Valadier, Convex Analysis and Measurable Multifunctions, Lecture Notes in Mathematics 580, Springer-Verlag, Berlin-Heidelberg-New York, 1977.
8. H. Covitz and S. B. Nadler Jr., Multivalued contraction mappings in generalized metric spaces, Israel J. Math. 8 (1970), 5-11.
9. K. Deimling, Multivalued Differential Equations, Walter De Gruyter, Berlin-New York, 1992.
10. D. Delbosco and L. Rodino, Existence and uniqueness for a nonlinear fractional differential equation, J. Math. Anal. Appl. 204 (1996), 609-625.
11. K. Diethelm and A.D. Freed, On the solution of nonlinear fractional order differential equations used in the modeling of viscoplasticity, in "Scientifice Computing in Chemical Engineering II-Computational Fluid Dynamics, Reaction Engineering and Molecular Properties" (F. Keil, W. Mackens, H. Voss, and J. Werther, Eds), pp 217-224, Springer-Verlag, Heidelberg, 1999.
12. K. Diethelm and N. J. Ford, Analysis of fractional differential equations, J. Math. Anal. Appl. 265 (2002), 229-248.
13. K. Diethelm and G. Walz, Numerical solution of fractional order differential equations by extrapolation, Numer. Algorithms 16 (1997), 231-253.
14. A. Granas and J. Dugundji, Fixed Point Theory, Springer-Verlag, New York, 2003.
15. A. M. A. El-Sayed, Fractional order evolution equations, J. Fract. Calc. 7 (1995), 89-100.
16. A. M. A. El-Sayed, Fractional order diffusion-wave equations, Intern. J. Theo. Physics 35 (1996), 311-322.
17. A. M. A. El-Sayed, Nonlinear functional differential equations of arbitrary orders, Nonlinear Anal. 33 (1998), 181-186.
18. A. M. A. El-Sayed and A. G. Ibrahim, Multivalued fractional differential equations, Appl. Math. Comput. 68 (1995), 15-25.
19. L. Gaul, P. Klein and S. Kempfle, Damping description involving fractional operators, Mech. Systems Signal Processing 5 (1991), 81-88.
20. W. G. Glockle and T. F. Nonnenmacher, A fractional calculus approach of self-similar protein dynamics, Biophys. J. 68 (1995), 46-53.
21. Sh. Hu and N. Papageorgiou, Handbook of Multivalued Analysis, Theory I, Kluwer, Dordrecht, 1997.
22. R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific, Singapore, 2000.
23. M. Kisielewicz, Differential Inclusions and Optimal Control, Kluwer, Dordrecht, The Netherlands, 1991.
24. A. A. Kilbas and S. A. Marzan, Nonlinear differential equations with the Caputo fractional derivative in the space of continuously differentiable functions, Differential Equations 41 (2005), 84-89.
25. A.A. Kilbas, Hari M. Srivastava, and Juan J. Trujillo, Theory and Applications of Fractional Differential Equations. North-Holland Mathematics Studies, 204. Elsevier Science B.V., Amsterdam, 2006.
26. V. Lakshmikantham, and A.S. Vatsala, Basic theory of fractional differential equations, Nonlinear Anal. Theory, Methods (2007), doi:10.1016/j.na.2007.09.025.
27. V. Lakshmikantham, and A.S. Vatsala, Theory of fractional differential inequalities and applications, Commun. Appl. Anal. 11 (3&4) (2007), 395-402.
28. V. Lakshmikantham and A.S. Vatsala, General uniqueness and monotone iterative technique for fractional differential equations, Appl. Math. Letters (2007), doi:10.1016/j.aml.2007.09.006.
29. F. Mainardi, Fractional calculus: Some basic problems in continuum and statistical mechanics, in "Fractals and Ractional Calculus in Continuum Mechanics" (A. Carpinteri and F. Mainardi, Eds), pp. 291-348, Springer-Verlag, Wien, 1997.
30. F. Metzler, W. Schick, H. G. Kilian and T. F. Nonnenmacher, Relaxation in filled polymers: A fractional calculus approarch, J. Chem. Phys. 103 (1995), 7180-7186.
31. K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Differential Equations, John Wiley, New York, 1993.
32. S. M. Momani and S. B. Hadid, Some comparison results for integro-fractional differential inequalities. J. Fract. Calc. 24 (2003), 37-44.
33. S. M. Momani, S. B. Hadid and Z. M. Alawenh, Some analytical properties of solutions of differential equations of noninteger order, Int. J. Math. Math. Sci. 2004 (2004), 697-701.
34. K.B. Oldham and J. Spanier, The Fractional Calculus, Academic Press, New York, London, 1974.
35. A. Ouahab, Some results for fractional boundary value problem of differential inclusions, Nonlinear Anal. (2007), doi:10.1016/ j.na.2007.10.021
36. I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999.
37. I. Podlubny, I. Petraš, B. M. Vinagre, P. O'Leary and L. Dorčak, Analogue realizations of fractional-order controllers. Factional order calculus and its applications, Nonlinear Dynam. 29 (2002), 281-296.
38. S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives. Theory and Applications, Gordon and Breach, Yverdon, 1993.
39. C. Yu and G. Gao, Existence of fractional differential equations, J. Math. Anal. Appl. 310 (2005), 26-29.
40. S. Zhang, Positive solutions for boundary-value problems of nonlinear fractional diffrential equations, Electron. J. Diff. Equ. 2006, No. 36, pp, 1-12.