Fractional order impulsive partial hyperbolic differential inclusions with variable times
Saïd Abbas; Mouffak Benchohra; Lech Górniewicz
Discussiones Mathematicae, Differential Inclusions, Control and Optimization (2011)
- Volume: 31, Issue: 1, page 91-114
- ISSN: 1509-9407
Access Full Article
topAbstract
topHow to cite
topSaïd Abbas, Mouffak Benchohra, and Lech Górniewicz. "Fractional order impulsive partial hyperbolic differential inclusions with variable times." Discussiones Mathematicae, Differential Inclusions, Control and Optimization 31.1 (2011): 91-114. <http://eudml.org/doc/271200>.
@article{SaïdAbbas2011,
abstract = {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.},
author = {Saïd Abbas, Mouffak Benchohra, Lech Górniewicz},
journal = {Discussiones Mathematicae, Differential Inclusions, Control and Optimization},
keywords = {impulsive functional differential inclusions; fractional order; solution; left-sided mixed Riemann-Liouville integral; Caputo fractional-order derivative; variable times; fixed point},
language = {eng},
number = {1},
pages = {91-114},
title = {Fractional order impulsive partial hyperbolic differential inclusions with variable times},
url = {http://eudml.org/doc/271200},
volume = {31},
year = {2011},
}
TY - JOUR
AU - Saïd Abbas
AU - Mouffak Benchohra
AU - Lech Górniewicz
TI - Fractional order impulsive partial hyperbolic differential inclusions with variable times
JO - Discussiones Mathematicae, Differential Inclusions, Control and Optimization
PY - 2011
VL - 31
IS - 1
SP - 91
EP - 114
AB - 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.
LA - eng
KW - impulsive functional differential inclusions; fractional order; solution; left-sided mixed Riemann-Liouville integral; Caputo fractional-order derivative; variable times; fixed point
UR - http://eudml.org/doc/271200
ER -
References
top- [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. Zbl1178.35371
- [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 Zbl1219.35345
- [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 Zbl1202.35340
- [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 Zbl1203.26005
- [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). Zbl1211.35269
- [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. Zbl1200.26004
- [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 Zbl1198.26004
- [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 Zbl1175.34080
- [9] M. Belmekki, M. Benchohra and L. Górniewicz, Functional differential equations with fractional order and infinite delay, Fixed Point Theory 9 (2008), 423-439.
- [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 Zbl1198.26008
- [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. Zbl1157.26301
- [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 Zbl1130.34003
- [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 Zbl1209.34096
- [14] K. Deimling, Multivalued Differential Equations, Walter De Gruyter, Berlin-New York, 1992. doi: 10.1515/9783110874228 Zbl0760.34002
- [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.
- [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 Zbl1014.34003
- [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
- [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
- [19] A. Granas and J. Dugundji, Fixed Point Theory, Springer-Verlag, New York, 2003. Zbl1025.47002
- [20] L. Górniewicz, Topological Fixed Point Theory of Multivalued Mappings, Mathematics and its Applications, 495, Kluwer Academic Publishers, Dordrecht, 1999. Zbl0937.55001
- [21] R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific, Singapore, 2000. empty Zbl0998.26002
- [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. Zbl1137.34308
- [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 Zbl1160.34301
- [24] M. Kisielewicz, Differential Inclusions and Optimal Control, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1991. Zbl0731.49001
- [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.
- [26] V. Lakshmikantham, D.D. Bainov and P.S. Simeonov, Theory of Impulsive Differential Equations, World Scientific, Singapore, 1989. Zbl0719.34002
- [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. Zbl0917.73004
- [28] K.S. Miller and B. Ross, An Introduction to the Fractional Calculus and Differential Equations, John Wiley, New York, 1993. Zbl0789.26002
- [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 Zbl1041.93022
- [30] S.G. Samko, A.A. Kilbas and O.I. Marichev, Fractional Integrals and Derivatives. Theory and Applications, Gordon and Breach, Yverdon, 1993.
- [31] A.M. Samoilenko and N.A. Perestyuk, Impulsive Differential Equations, World Scientific, Singapore, 1995. doi: 10.1142/9789812798664 Zbl0837.34003
- [32] N.P. Semenchuk, On one class of differential equations of noninteger order, Differents. Uravn. 10 (1982), 1831-1833. Zbl0522.34012
- [33] A.N. Vityuk, Existence of solutions of partial differential inclusions of fractional order, Izv. Vyssh. Uchebn., Ser. Mat. 8 (1997), 13-19. Zbl0905.35102
- [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 Zbl1092.35500
NotesEmbed ?
topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.