PL EN


Preferencje help
Widoczny [Schowaj] Abstrakt
Liczba wyników
2010 | 30 | 1 | 141-161
Tytuł artykułu

The method of upper and lower solutions for partial hyperbolic fractional order differential inclusions with impulses

Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
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.
Twórcy
autor
  • Laboratoire de Mathématiques, Université de Saïda, B.P. 138, 20000, Saïda, Algérie
  • 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
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.bwnjournal-article-doi-10_7151_dmdico_1116
JavaScript jest wyłączony w Twojej przeglądarce internetowej. Włącz go, a następnie odśwież stronę, aby móc w pełni z niej korzystać.