In this paper, we shall establish sufficient conditions for the existence of solutions for a boundary value problem for fractional differential inclusions. Both cases of convex valued and nonconvex valued right hand sides are considered.
@article{bwmeta1.element.bwnjournal-article-doi-10_7151_dmdico_1099,
author = {Mouffak Benchohra and Samira Hamani},
title = {Boundary value problems for differential inclusions with fractional order},
journal = {Discussiones Mathematicae, Differential Inclusions, Control and Optimization},
volume = {28},
year = {2008},
pages = {147-164},
zbl = {1181.26012},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmdico_1099}
}
Mouffak Benchohra; Samira Hamani. Boundary value problems for differential inclusions with fractional order. Discussiones Mathematicae, Differential Inclusions, Control and Optimization, Tome 28 (2008) pp. 147-164. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmdico_1099/
[000] [1] J.P. Aubin and A. Cellina, Differential Inclusions, Springer-Verlag, Berlin-Heidelberg, New York, 1984. | Zbl 0538.34007
[001] [2] J.P. Aubin and H. Frankowska, Set-Valued Analysis, Birkhauser, Boston, 1990.
[002] [3] 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. | Zbl 1148.34042
[003] [4] 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. | Zbl 1175.34080
[004] [5] M. Benchohra, S. Hamani and S.K. Ntouyas, Boundary value problems for differential equations with fractional order, Surv. Math. Appl. 3 (2008), 1-12. | Zbl 1157.26301
[005] [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. | Zbl 1209.34096
[006] [7] C. Castaing and M. Valadier, Convex Analysis and Measurable Multifunctions, Lecture Notes in Mathematics 580, Springer-Verlag, Berlin-Heidelberg-New York, 1977.
[007] [8] H. Covitz and S. B. Nadler Jr., Multivalued contraction mappings in generalized metric spaces, Israel J. Math. 8 (1970), 5-11. | Zbl 0192.59802
[008] [9] K. Deimling, Multivalued Differential Equations, Walter De Gruyter, Berlin-New York, 1992.
[009] [10] D. Delbosco and L. Rodino, Existence and uniqueness for a nonlinear fractional differential equation, J. Math. Anal. Appl. 204 (1996), 609-625. | Zbl 0881.34005
[010] [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.
[011] [12] K. Diethelm and N.J. Ford, Analysis of fractional differential equations, J. Math. Anal. Appl. 265 (2002), 229-248. | Zbl 1014.34003
[012] [13] K. Diethelm and G. Walz, Numerical solution of fractional order differential equations by extrapolation, Numer. Algorithms 16 (1997), 231-253. | Zbl 0926.65070
[013] [14] A. Granas and J. Dugundji, Fixed Point Theory, Springer-Verlag, New York, 2003. | Zbl 1025.47002
[014] [15] A.M.A. El-Sayed, Fractional order evolution equations, J. Fract. Calc. 7 (1995), 89-100.
[015] [16] A.M.A. El-Sayed, Fractional order diffusion-wave equations, Intern. J. Theoretical Physics 35 (1996), 311-322.
[016] [17] A.M.A. El-Sayed, Nonlinear functional differential equations of arbitrary orders, Nonlinear Anal. 33 (1998), 181-186. | Zbl 0934.34055
[017] [18] A.M.A. El-Sayed and A.G. Ibrahim, Multivalued fractional differential equations, Appl. Math. Comput. 68 (1995), 15-25. | Zbl 0830.34012
[018] [19] L. Gaul, P. Klein and S. Kempfle, Damping description involving fractional operators, Mech. Systems Signal Processing 5 (1991), 81-88.
[019] [20] W.G. Glockle and T.F. Nonnenmacher, A fractional calculus approach of self-similar protein dynamics, Biophys. J. 68 (1995), 46-53.
[020] [21] Sh. Hu and N. Papageorgiou, Handbook of Multivalued Analysis, Theory I, Kluwer, Dordrecht, 1997.
[021] [22] N. Heymans and I. Podlubny, Physical interpretation of initial conditions for fractional differential equations with Riemann-Liouville fractional derivatives, Rheologica Acta 45 (5) (2006), 765-772.
[022] [23] R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific, Singapore, 2000. | Zbl 0998.26002
[023] [24] M. Kisielewicz, Differential Inclusions and Optimal Control, Kluwer, Dordrecht, The Netherlands, 1991.
[024] [25] 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. | Zbl 1160.34301
[025] [26] 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. | Zbl 1092.45003
[026] [27] V. Lakshmikantham and J.V. Devi, Theory of fractional differential equations in a Banach space, Eur. J. Pure Appl. Math. 1 (1) (2008), 38-45. | Zbl 1146.34042
[027] [28] F. Mainardi, Fractional calculus: Some basic problems in continuum and statistical mechanics, in: 'Fractals and Fractional Calculus in Continuum Mechanics' (A. Carpinteri and F. Mainardi, Eds), pp. 291-348, Springer-Verlag, Wien, 1997. | Zbl 0917.73004
[028] [29] F. Metzler, W. Schick, H.G. Kilian and T.F. Nonnenmacher, Relaxation in filled polymers: A fractional calculus approach, J. Chem. Phys. 103 (1995), 7180-7186.
[029] [30] K.S. Miller and B. Ross, An Introduction to the Fractional Calculus and Differential Equations, John Wiley, New York, 1993. | Zbl 0789.26002
[030] [31] K.B. Oldham and J. Spanier, The Fractional Calculus, Academic Press, New York, London, 1974. | Zbl 0292.26011
[031] [32] A. Ouahab, Some results for fractional boundary value problem of differential inclusions, Nonlinear Anal. (2007), doi:10.1016/j.na.2007.10.021. | Zbl 1169.34006
[032] [33] I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999.
[033] [34] I. Podlubny, Geometric and physical interpretation of fractional integration and fractional differentiation, Fract. Calculus Appl. Anal. 5 (2002), 367-386. | Zbl 1042.26003
[034] [35] I. Podlubny, I. Petraš, B.M. Vinagre, P. O'Leary and L. Dorčak, Analogue realizations of fractional-order controllers. Fractional order calculus and its applications, Nonlinear Dynam. 29 (2002), 281-296. | Zbl 1041.93022
[035] [36] S.G. Samko, A.A. Kilbas and O.I. Marichev, Fractional Integrals and Derivatives. Theory and Applications, Gordon and Breach, Yverdon, 1993.
[036] [37] K. Yosida, Functional Analysis, 6th edn. Springer-Verlag, Berlin, 1980.
[037] [38] C. Yu and G. Gao, Existence of fractional differential equations, J. Math. Anal. Appl. 310 (2005), 26-29. | Zbl 1088.34501