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Abstrakty
In this paper we use the upper and lower solutions method to investigate the existence of solutions of a class of impulsive partial hyperbolic differential inclusions at fixed moments of impulse involving the Caputo fractional derivative. These results are obtained upon suitable fixed point theorems.
Kategorie tematyczne
Rocznik
Tom
Numer
Strony
141-161
Opis fizyczny
Daty
wydano
2010
otrzymano
2009-09-14
Twórcy
autor
- Laboratoire de Mathématiques, Université de Saïda, B.P. 138, 20000, Saïda, Algérie
autor
- Laboratoire de Mathématiques, Université de Sidi Bel-Abbès, B.P. 89, 22000, Sidi Bel-Abbès, Algérie
Bibliografia
- [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.
- [2] S. Abbas and M. Benchohra, Darboux problem for perturbed partial differential equations of fractional order with finite delay, Nonlinear Anal.: Hybrid Systems 3 (2009), 597-604.
- [3] 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. doi:10.1007/s10440-008-9356-6
- [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.
- [5] M. Benchohra, J.R. Graef and S. Hamani, Existence results for boundary value problems of nonlinear fractional differential equations with integral conditions, Appl. Anal. 87 (7) (2008), 851-863.
- [6] M. Benchohra, S. Hamani and S.K. Ntouyas, Boundary value problems for differential equations with fractional order, Surv. Math. Appl. 3 (2008), 1-12.
- [7] M. Benchohra, J. Henderson and S.K. Ntouyas, Impulsive Differential Equations and Inclusions, Hindawi Publishing Corporation, Vol. 2, New York, 2006. doi:10.1155/9789775945501
- [8] 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.
- [9] H.F. Bohnenblust and S. Karlin, On a theorem of ville. Contribution to the theory of games, Annals of Mathematics Studies, no. 24, Priceton University Press, Princeton N.G. (1950), 155-160.
- [10] M. Dawidowski and I. Kubiaczyk, An existence theorem for the generalized hyperbolic equation $z''_{xy} ∈ F(x,y,z)$ in Banach space, Ann. Soc. Math. Pol. Ser. I, Comment. Math. 30 (1) (1990), 41-49.
- [11] K. Deimling, Multivalued Differential Equations, Walter De Gruyter, Berlin-New York, 1992. doi:10.1515/9783110874228
- [12] K. Diethelm and A.D. Freed, On the solution of nonlinear fractional order differential equations used in the modelling 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.
- [13] K. Diethelm and N.J. Ford, Analysis of fractional differential equations, J. Math. Anal. Appl. 265 (2002), 229-248.
- [14] L. Gaul, P. Klein and S. Kempfle, Damping description involving fractional operators, Mech. Systems Signal Processing 5 (1991), 81-88.
- [15] W.G. Glockle and T.F. Nonnenmacher, A fractional calculus approach of selfsimilar protein dynamics, Biophys. J. 68 (1995), 46-53.
- [16] L. Górniewicz, Topological Fixed Point Theory of Multivalued Mappings, Mathematics and its Applications, 495, Kluwer Academic Publishers, Dordrecht, 1999.
- [17] S. Heikkila and V. Lakshmikantham, Monotone Iterative Technique for Nonlinear Discontinuous Differential Equations, Marcel Dekker Inc., New York, 1994.
- [18] R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific, Singapore, 2000.
- [19] Sh. Hu and N. Papageorgiou, Handbook of Multivalued Analysis, Volume I: Theory, Kluwer, Dordrecht, Boston, London, 1997.
- [20] Z. Kamont, Hyperbolic Functional Differential Inequalities and Applications, Kluwer Academic Publishers, Dordrecht, 1999. doi:10.1007/978-94-011-4635-7
- [21] Z. Kamont and K. Kropielnicka, Differential difference inequalities related to hyperbolic functional differential systems and applications, Math. Inequal. Appl. 8 (4) (2005), 655-674.
- [22] A.A. Kilbas, B. Bonilla and J. Trujillo, Nonlinear differential equations of fractional order in a space of integrable functions, Dokl. Russ. Akad. Nauk 374 (4) (2000), 445-449.
- [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.
- [24] 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.
- [25] M. Kisielewicz, Differential Inclusions and Optimal Control, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1991.
- [26] V. Lakshmikantham, S. Leela and J. Vasundhara, Theory of Fractional Dynamic Systems, Cambridge Academic Publishers, Cambridge, 2009.
- [27] V. Lakshmikantham and S.G. Pandit, The method of upper, lower solutions and hyperbolic partial differential equations, J. Math. Anal. Appl. 105 (1985), 466-477.
- [28] G.S. Ladde, V. Lakshmikanthan and A.S. Vatsala, Monotone Iterative Techniques for Nonliner Differential Equations, Pitman Advanced Publishing Program, London, 1985.
- [29] 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.
- [30] 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. doi:10.1063/1.470346
- [31] K.S. Miller and B. Ross, An Introduction to the Fractional Calculus and Differential Equations, John Wiley, New York, 1993.
- [32] S.G. Pandit, Monotone methods for systems of nonlinear hyperbolic problems in two independent variables, Nonlinear Anal. 30 (1997), 235-272. doi:10.1016/S0362-546X(96)00265-9
- [33] I. Podlubny, Fractional Differential Equation, Academic Press, San Diego, 1999.
- [34] 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
- [35] S.G. Samko, A.A. Kilbas and O.I. Marichev, Fractional Integrals and Derivatives. Theory and Applications, Gordon and Breach, Yverdon, 1993.
- [36] N.P. Semenchuk, On one class of differential equations of noninteger order, Differents. Uravn. 10 (1982), 1831-1833.
- [37] A.A. Tolstonogov, Differential Inclusions in a Banach Space, Kluwer Academic Publishers, Dordrecht, 2000. doi:10.1007/978-94-015-9490-5
- [38] A.N. Vityuk, Existence of solutions of partial differential inclusions of fractional order, Izv. Vyssh. Uchebn., Ser. Mat. 8 (1997), 13-19.
- [39] 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
- [40] C. Yu and G. Gao, Existence of fractional differential equations, J. Math. Anal. Appl. 310 (2005), 26-29.
- [41] S. Zhang, Positive solutions for boundary-value problems of nonlinear fractional diffrential equations, Electron. J. Differential Equations 36 (2006), 1-12. doi:10.1155/ADE/2006/90479
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Bibliografia
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bwmeta1.element.bwnjournal-article-doi-10_7151_dmdico_1116