Some results for fractional boundary value problem of differential inclusions

Nonlinear Analysis, Theory, Methods and Applications

Volumn 69, Issue 11, 2008, Pages 3877-3896

  Ouahab, Abdelghani ^a  

^a UNIVERSITY OF SIDI BEL ABBES   (Algeria)

Author keywords

Boundary; Compactness; Continuous selection; Decomposable; Fixed point; Fractional derivative; Fractional differential inclusions; Fractional integral; Functional differential inclusions; Infinite delay; Relaxation theorem

Indexed keywords

BOUNDARY VALUE PROBLEMS; DIFFERENTIAL EQUATIONS; INITIAL VALUE PROBLEMS; OPTIMAL CONTROL SYSTEMS; SOLUTIONS; THEOREM PROVING;

BOUNDARY; COMPACTNESS; CONTINUOUS SELECTION; DECOMPOSABLE; FIXED POINT; FRACTIONAL DERIVATIVE; FRACTIONAL DIFFERENTIAL INCLUSIONS; FRACTIONAL INTEGRAL; FUNCTIONAL DIFFERENTIAL INCLUSIONS; INFINITE DELAY; RELAXATION THEOREM;

BOUNDARY CONDITIONS;

EID: 53949105887     PISSN: 0362546X     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.na.2007.10.021     Document Type: Article

References (51)

1. Aubin J.P., and Cellina A. Differential Inclusions (1984), Springer-Verlag, Berlin, Heidelberg, New York
2. Aubin J.P., and Frankowska H. Set-Valued Analysis (1990), Birkhauser, Boston
3. Bai Z., and Lu H. Positive solutions for boundary value problem of nonlinear fractional differential equations. J. Math. Anal. Appl. 311 (2005) 495-505
4. Benchohra M., Henderson J., Ntouyas S.K., and Ouahab A. Existence results for fractional order functional differential equations with infinite delay. J. Math. Anal. Appl. 338 (2008) 1340-1350
5. Bressan A., and Colombo G. Extensions and selections of maps with decomposable values. Studia Math. 90 (1988) 70-85
6. Castaing C., and Valadier M. Convex Analysis and Measurable Multifunctions. Lecture Notes in Mathematics vol. 580 (1977), Springer-Verlag, Berlin, Heidelberg, New York
7. Covitz H., and Nadler Jr. S.B. Multivalued contraction mappings in generalized metric spaces. Israel J. Math. 8 (1970) 5-11
8. El-Sayed A.M.A. Fractional order evolution equations. J. Fract. Calc. 7 (1995) 89-100
9. El-Sayed A.M.A. Fractional order diffusion-wave equations. Internat. J. Theoret. Phys. 35 (1996) 311-322
10. El-Sayed A.M.A. Nonlinear functional differential equations of arbitrary orders. Nonlinear Anal. 33 (1998) 181-186
11. El-Sayed A.M.A., and Ibrahim A.G. Multivalued fractional differential equations. Appl. Math. Comput. 68 (1995) 15-25
12. Frigon M., and Granas A. Théorèmes d'existence pour des inclusions différentielles sans convexité. C. R. Acad. Sci. Paris, Ser. I 310 (1990) 819-822
13. Delbosco D., and Rodino L. Existence and uniqueness for a nonlinear fractional differential equation. J. Math. Anal. Appl. 204 (1996) 609-625
14. Diethelm K., and Freed A.D. On the solution of nonlinear fractional order differential equations used in the modeling of viscoplasticity. In: Keil F., Mackens W., Voss H., and Werther J. (Eds). Scientific Computing in Chemical Engineering II-Computational Fluid Dynamics, Reaction Engineering and Molecular Properties (1999), Springer-Verlag, Heidelberg 217-224
15. Deimling K. Multivalued Differential Equations (1992), Walter De Gruyter, Berlin, New York
16. Diethelm K., and Ford N.J. Analysis of fractional differential equations. J. Math. Anal. Appl. 265 (2002) 229-248
17. Diethelm K., and Walz G. Numerical solution of fractional order differential equations by extrapolation. Numer. Algorithms 16 (1997) 231-253
18. Granas A., and Dugundji J. Fixed Point Theory (2003), Springer-Verlag, New York
19. Gaul L., Klein P., and Kempfle S. Damping description involving fractional operators. Mech. Syst. Signal Process. 5 (1991) 81-88
20. Glockle W.G., and Nonnenmacher T.F. A fractional calculus approach of self-similar protein dynamics. Biophys. J. 68 (1995) 46-53
21. Górniewicz L. Topological Fixed Point Theory of Multivalued Mappings. Mathematics and its Applications vol. 495 (1999), Kluwer Academic Publishers, Dordrecht
22. Hale J.K. Retarded equations with infinite delay. Functional and Approximation of Fixed Points, Proceedings, Bonn. Lecture Notes in Mathematics vol. 1223 (1987), Springer, Berlin 61-63
23. Hale J.K., and Kato J. Phase space for retarded equations with infinite delay. Funkcial. Ekvac. 21 (1978) 11-41
24. Hale J.K., and Lunel S.M.V. Introduction to Functional Differential Equations. Applied Mathematical Sciences. vol. 99 (1993), Springer-Verlag, New York
25. Hu Sh., and Papageorgiou N. Handbook of Multivalued Analysis, Volume I: Theory (1997), Kluwer Academic Publishers, Dordrecht, Boston, London
26. Hiai F., and Umegaki H. Integrals conditional expectations, and martingales of multivalued functions. J. Multivariate Anal. 7 (1977) 149-182
27. Hino Y., Murakani S., and Naito T. Functional Differential Equations with Infinite Delay. Lecture Notes in Mathematics vol. 1473 (1991), Springer-Verlag, Berlin
28. Kappel F., and Schappacher W. Some considerations to the fundamental theory of infinite delay equations. J. Differential Equations 37 (1980) 141-183
29. Kilbas A.A., and Trujillo J.J. Differential equations of fractional order: Methods, results and problems II. Appl. Anal. 81 (2002) 435-493
30. Kisielewicz M. Differential Inclusions and Optimal Control (1991), Kluwer, Dordrecht, The Netherlands
31. Lakshmikantham V. Theory of fractional functional differential equations. Nonlinear Anal. TMA 69 10 (2008) 3337-3343
32. Lakshmikantham V., and Vatsala A.S. Basic theory of fractional differential equations. Nonlinear Anal. TMA 69 8 (2008) 2677-2682
33. V. Lakshmikantham, A.S. Vatsala, Theory of Fractional Differential Inequalities and Applications, Commun. Appl. Anal. (in press)
34. V. Lakshmikantham, A.S. Vatsala, General uniqueness and monotone iterative technique for fractional differential equations, Math. Lett. (in press)
35. Lasota A., and Opial Z. An application of the Kakutani-Ky Fan theorem in the theory of ordinary differential equations. Bull. Acad. Pol. Sci. Ser. Sci. Math. Astronom. Phys. 13 (1965) 781-786
36. Lakshmikantham V., Wen L., and Zhang B. Theory of Differential Equations with Unbounded Delay. Mathematics and its Applications (1994), Kluwer Academic Publishers, Dordrecht
37. Mainardi F. Fractional calculus: Some basic problems in continuum and statistical mechanics. In: Carpinteri A., and Mainardi F. (Eds). Fractals and Fractional Calculus in Continuum Mechanics (1997), Springer-Verlag, Wien 291-348
38. Metzler F., Schick W., Kilian H.G., and Nonnenmacher T.F. Relaxation in filled polymers: A fractional calculus approach. J. Chem. Phys. 103 (1995) 7180-7186
39. Momani S.M., and Hadid S.B. Some comparison results for integro-fractional differential inequalities. J. Fract. Calc. 24 (2003) 37-44
40. Momani S.M., Hadid S.B., and Alawenh Z.M. Some analytical properties of solutions of differential equations of noninteger order. Int. J. Math. Math. Sci. (2004) 697-701
41. Miller K.S., and Ross B. An Introduction to the Fractional Calculus and Differential Equations (1993), John Wiley, New York
42. Nakhushev A.M. The stum-Liouville problems for a second order ordinary equations with fractional derivatives in the lower. Dokl. Akad. Nauk SSSR 234 (1977) 308-311
43. Podlubny I. Fractional Differential Equation (1999), Academic Press, San Diego
44. Podlubny I., Petraš I., Vinagre B.M., O'Leary P., and Dorčak L. Analogue realizations of fractional-order controllers. Fractional order calculus and its applications. Nonlinear Dynam. 29 (2002) 281-296
45. Kamenskii M., Obukhovskii V., and Zecca P. Condensing multi-valued Maps and Semilinear Differential Inclusions in Banach Spaces (2001), Walter de Gruyter & Co., Berlin
46. Samko S.G., Kilbas A.A., and Marichev O.I. Fractional Integrals and Derivatives Theory and Applications (1993), Gordon and Breach, Yverdon
47. Sawano K. Some consideration on the fundamental theorems for functional differential equations with infinite delay. Funkcial. Ekvac. 25 (1982) 97-104
48. Schumacher K. Existence and continuous dependence for differential equations with unbounded delay. Arch. Ration. Mech. Anal. 64 (1978) 315-335
49. Shin J.S. An existence of functional differential equations, with infinity delay in a Banach space. Funkcial. Ekvac. 30 (1987) 19-29
50. Tolstonogov A.A. Differential Inclusions in a Banach Space (2000), Kluwer, Dordrecht, The Netherlands
51. Yu C., and Gao G. Existence of fractional differential equations. J. Math. Anal. Appl. 310 (2005) 26-29