This paper deals with the existence of solutions to some classes of partial impulsive hyperbolic differential inclusions with variable times involving the Caputo fractional derivative. Our works will be considered by using the nonlinear alternative of Leray-Schauder type.
@article{bwmeta1.element.bwnjournal-article-doi-10_7151_dmdico_1129, author = {Sa\"\i d Abbas and Mouffak Benchohra and Lech G\'orniewicz}, title = {Fractional order impulsive partial hyperbolic differential inclusions with variable times}, journal = {Discussiones Mathematicae, Differential Inclusions, Control and Optimization}, volume = {31}, year = {2011}, pages = {91-114}, zbl = {1269.26002}, language = {en}, url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmdico_1129} }
Saïd Abbas; Mouffak Benchohra; Lech Górniewicz. Fractional order impulsive partial hyperbolic differential inclusions with variable times. Discussiones Mathematicae, Differential Inclusions, Control and Optimization, Tome 31 (2011) pp. 91-114. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmdico_1129/
[000] [1] S. Abbas and M. Benchohra, Partial hyperbolic differential equations with finite delay involving the Caputo fractional derivative, Commun. Math. Anal. 7 (2009), 62-72. | Zbl 1178.35371
[001] [2] S. Abbas and M. Benchohra, Darboux problem for perturbed partial differential equations of fractional order with finite delay, Nonlinear Anal. Hybrid Syst. 3 (2009), 597-604. doi: 10.1016/j.nahs9.05.001.200 | Zbl 1219.35345
[002] [3] S. Abbas and M. Benchohra, Upper and lower solutions method for impulsive partial hyperbolic differential equations with fractional order, Nonlinear Anal. Hybrid Syst. 4 (2010), 406-413. doi: 10.1016/j.nahs.2009.10.004 | Zbl 1202.35340
[003] [4] S. Abbas and M. Benchohra, The method of upper and lower solutions for partial hyperbolic fractional order differential inclusions with impulses, Discuss. Math. Differ. Incl. Control Optim. 30 (1) (2010), 141-161. empty | Zbl 1203.26005
[004] [5] S. Abbas and M. Benchohra, Existence theory for impulsive partial hyperbolic differential equations of fractional order at variable times, Fixed Point Theory, (to appear). | Zbl 1211.35269
[005] [6] S. Abbas, M. Benchohra and L. Górniewicz, Existence theory for impulsive partial hyperbolic functional differential equations involving the Caputo fractional derivative, Sci. Math. Jpn. online e- 2010, 271-282. | Zbl 1200.26004
[006] [7] R.P Agarwal, M. Benchohra and S. Hamani, A survey on existence result for boundary value problems of nonlinear fractional differential equations and inclusions, Acta. Appl. Math. 109 (3) (2010), 973-1033. doi: 10.1007/s10440-008-9356-6 | Zbl 1198.26004
[007] [8] 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. doi: 10.1080/00036810601066350 | Zbl 1175.34080
[008] [9] M. Belmekki, M. Benchohra and L. Górniewicz, Functional differential equations with fractional order and infinite delay, Fixed Point Theory 9 (2008), 423-439.
[009] [10] M. Benchohra, J.R. Graef and S. Hamani, Existence results for boundary value problems with non-linear fractional differential equations, Appl. Anal. 87 (7) (2008), 851-863. doi: 10.1080/00036810802307579 | Zbl 1198.26008
[010] [11] 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
[011] [12] M. Benchohra, J. Henderson and S. Ntouyas, Impulsive Differential Equations and Inclusions, Hindawi Publishing Corporation, New York, NY, USA, 2006. doi: 10.1155/9789775945501 | Zbl 1130.34003
[012] [13] M. Benchohra, J. Henderson, S.K. Ntouyas and A. Ouahab, Existence results for functional differential equations of fractional order, J. Math. Anal. Appl. 338 (2008), 1340-1350. doi: 10.1016/j.jmaa.2007.06.021 | Zbl 1209.34096
[013] [14] K. Deimling, Multivalued Differential Equations, Walter De Gruyter, Berlin-New York, 1992. doi: 10.1515/9783110874228 | Zbl 0760.34002
[014] [15] 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.
[015] [16] K. Diethelm and N.J. Ford, Analysis of fractional differential equations, J. Math. Anal. Appl. 265 (2002), 229-248. doi: 10.1006/jmaa.2000.7194 | Zbl 1014.34003
[016] [17] L. Gaul, P. Klein and S. Kempfle, Damping description involving fractional operators, Mech. Systems Signal Processing 5 (1991), 81-88. doi: 10.1016/0888-3270(91)90016-X
[017] [18] W.G. Glockle and T.F. Nonnenmacher, A fractional calculus approach of selfsimilar protein dynamics, Biophys. J. 68 (1995), 46-53. doi: 10.1016/S0006-3495(95)80157-8
[018] [19] A. Granas and J. Dugundji, Fixed Point Theory, Springer-Verlag, New York, 2003. | Zbl 1025.47002
[019] [20] L. Górniewicz, Topological Fixed Point Theory of Multivalued Mappings, Mathematics and its Applications, 495, Kluwer Academic Publishers, Dordrecht, 1999. | Zbl 0937.55001
[020] [21] R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific, Singapore, 2000. empty | Zbl 0998.26002
[021] [22] A.A. Kilbas, B. Bonilla and J. Trujillo, Nonlinear differential equations of fractional order in a space of integrable functions, Dokl. Ross. Akad. Nauk 374 (4) (2000), 445-449. | Zbl 1137.34308
[022] [23] 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. doi: 10.1007/s10625-005-0137-y | Zbl 1160.34301
[023] [24] M. Kisielewicz, Differential Inclusions and Optimal Control, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1991. | Zbl 0731.49001
[024] [25] A.A. Kilbas, H.M. Srivastava and J.J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, 204. Elsevier Science B.V., Amsterdam, 2006.
[025] [26] V. Lakshmikantham, D.D. Bainov and P.S. Simeonov, Theory of Impulsive Differential Equations, World Scientific, Singapore, 1989. | Zbl 0719.34002
[026] [27] 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
[027] [28] K.S. Miller and B. Ross, An Introduction to the Fractional Calculus and Differential Equations, John Wiley, New York, 1993. | Zbl 0789.26002
[028] [29] 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. doi: 10.1023/A:1016556604320 | Zbl 1041.93022
[029] [30] S.G. Samko, A.A. Kilbas and O.I. Marichev, Fractional Integrals and Derivatives. Theory and Applications, Gordon and Breach, Yverdon, 1993.
[030] [31] A.M. Samoilenko and N.A. Perestyuk, Impulsive Differential Equations, World Scientific, Singapore, 1995. doi: 10.1142/9789812798664 | Zbl 0837.34003
[031] [32] N.P. Semenchuk, On one class of differential equations of noninteger order, Differents. Uravn. 10 (1982), 1831-1833. | Zbl 0522.34012
[032] [33] A.N. Vityuk, Existence of solutions of partial differential inclusions of fractional order, Izv. Vyssh. Uchebn., Ser. Mat. 8 (1997), 13-19. | Zbl 0905.35102
[033] [34] A.N. Vityuk and A.V. Golushkov, Existence of solutions of systems of partial differential equations of fractional order, Nonlinear Oscil. 7 (3) (2004), 318-325. doi: 10.1007/s11072-005-0015-9 | Zbl 1092.35500