| Title:
|
Upper and Lower Solutions Method for Darboux Problem for Fractional Order Implicit Impulsive Partial Hyperbolic Differential Equations (English) |
| Author:
|
Abbas, Saïd |
| Author:
|
Benchohra, Mouffak |
| Language:
|
English |
| Journal:
|
Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica |
| ISSN:
|
0231-9721 |
| Volume:
|
51 |
| Issue:
|
2 |
| Year:
|
2012 |
| Pages:
|
5-18 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
In this paper we investigate the existence of solutions for the initial value problems (IVP for short), for a class of implicit impulsive hyperbolic differential equations by using the lower and upper solutions method combined with Schauder’s fixed point theorem. (English) |
| Keyword:
|
partial hyperbolic differential equation |
| Keyword:
|
fractional order |
| Keyword:
|
left-sided mixed |
| Keyword:
|
Riemann–Liouville integral |
| Keyword:
|
mixed regularized derivative |
| Keyword:
|
impulse |
| Keyword:
|
upper solution |
| Keyword:
|
lower solution |
| Keyword:
|
fixed point |
| MSC:
|
26A33 |
| MSC:
|
35A01 |
| MSC:
|
35R11 |
| MSC:
|
35R12 |
| idZBL:
|
Zbl 06204926 |
| idMR:
|
MR3058869 |
| . |
| Date available:
|
2012-11-26T10:12:45Z |
| Last updated:
|
2014-03-12 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/143062 |
| . |
| Reference:
|
[1] Abbas, S., Agarwal, R. P., Benchohra, M.: Darboux problem for impulsive partial hyperbolic differential equations of fractional order with variable times and infinite delay. Nonlinear Anal. Hybrid Syst. 4 (2010), 818–829. Zbl 1204.35171, MR 2680249 |
| Reference:
|
[2] Abbas, S., Benchohra, M.: Partial hyperbolic differential equations with finite delay involving the Caputo fractional derivative. Commun. Math. Anal. 7 (2009), 62–72. Zbl 1178.35371, MR 2535015 |
| Reference:
|
[3] Abbas, S., Benchohra, M.: Upper and lower solutions method for the Darboux problem for fractional order partial differential inclusions. Int. J. Modern Math. 5, 3 (2010), 327–338. Zbl 1244.35157, MR 2779058 |
| Reference:
|
[4] Abbas, S., Benchohra, M.: Upper and lower solutions method for impulsive partial hyperbolic differential equations with fractional order. Nonlinear Anal. Hybrid Syst. 4 (2010), 406–413. Zbl 1202.35340, MR 2645856 |
| Reference:
|
[5] Abbas, S., Benchohra, M.: 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. Zbl 1203.26005, MR 2682404, 10.7151/dmdico.1116 |
| Reference:
|
[6] Abbas, S., Benchohra, M., Górniewicz, L.: Existence theory for impulsive partial hyperbolic functional differential equations involving the Caputo fractional derivative. Sci. Math. Jpn. e-2010 (2010), 271–282, online. Zbl 1200.26004, MR 2666846 |
| Reference:
|
[7] Abbas, S., Benchohra, M., N’Guérékata, G. M.: Topics in Fractional Differential Equations. Developments in Mathematics 27, Springer, New York, 2012. Zbl 1273.35001, MR 2962045 |
| Reference:
|
[8] Abbas, S., Benchohra, M., Nieto, J. J.: Global uniqueness results for fractional order partial hyperbolic functional differential equations. Adv. Differ. Equations ID 379876, doi:10.1155/2011/379876 (2011), 1–25, online. Zbl 1217.35005, MR 2774251 |
| Reference:
|
[9] Abbas, S., Benchohra, M., Vityuk, A. N.: On fractional order derivatives and Darboux problem for implicit differential equations. Fract. Calc. Appl. Anal. 15, 2 (2012), 168–182. MR 2897771 |
| Reference:
|
[10] Agarwal, R. P., O’Regan, D., Staněk, S.: Positive solutions for Dirichlet problems of singular nonlinear fractional differential equations. J. Math. Anal. Appl. 371 (2010), 57–68. Zbl 1206.34009, MR 2660986, 10.1016/j.jmaa.2010.04.034 |
| Reference:
|
[11] Benchohra, M., Ouahab, A.: Uniqueness results for fractional functional differential equations with infinite delay in Fréchet spaces. Appl. Anal. 85 (2006), 1459–1470. Zbl 1175.34080, MR 2282996, 10.1080/00036810601066350 |
| Reference:
|
[12] Benchohra, M., Graef, J. R., Hamani, S.: Existence results for boundary value problems of nonlinear fractional differential equations with integral conditions. Appl. Anal. 87, 7 (2008), 851–863. MR 2458962, 10.1080/00036810802307579 |
| Reference:
|
[13] Benchohra, M., Hamani, S., Ntouyas, S. K.: Boundary value problems for differential equations with fractional order. Surv. Math. Appl. 3 (2008), 1–12. Zbl 1157.26301, MR 2390179 |
| Reference:
|
[14] Benchohra, M., Henderson, J., Ntouyas, S. K.: Impulsive Differential Equations and Inclusions. Hindawi Publishing Corporation, Vol 2, New York, 2006. Zbl 1130.34003, MR 2322133 |
| Reference:
|
[15] Benchohra, M., Henderson, J., Ntouyas, S. K., Ouahab, A.: Existence results for functional differential equations of fractional order. J. Math. Anal. Appl. 338 (2008), 1340–1350. MR 2386501, 10.1016/j.jmaa.2007.06.021 |
| Reference:
|
[16] Diethelm, K., Ford, N. J.: Analysis of fractional differential equations. J. Math. Anal. Appl. 265 (2002), 229–248. Zbl 1014.34003, MR 1876137, 10.1006/jmaa.2000.7194 |
| Reference:
|
[17] Glockle, W. G., Nonnenmacher, T. F.: A fractional calculus approach of selfsimilar protein dynamics. Biophys. J. 68 (1995), 46–53. 10.1016/S0006-3495(95)80157-8 |
| Reference:
|
[18] Granas, A., Dugundji, J.: Fixed Point Theory. Springer-Verlag, New York, 2003. Zbl 1025.47002, MR 1987179 |
| Reference:
|
[19] Hilfer, R.: Applications of Fractional Calculus in Physics. World Scientific, Singapore, 2000. Zbl 0998.26002, MR 1890104 |
| Reference:
|
[20] Kilbas, A. A., Srivastava, H. M., Trujillo, J. J.: Theory and Applications of Fractional Differential Equations. North-Holland Mathematics Studies 204, Elsevier Science B.V., Amsterdam, 2006. Zbl 1092.45003, MR 2218073 |
| Reference:
|
[21] Kilbas, A. A., Marzan, S. A.: Nonlinear differential equations with the Caputo fractional derivative in the space of continuously differentiable functions. Differential Equations 41 (2005), 84–89. Zbl 1160.34301, MR 2213269, 10.1007/s10625-005-0137-y |
| Reference:
|
[22] Lakshmikantham, V., Bainov D. D., Simeonov, P. S.: Theory of Impulsive Differntial Equations. World Scientific, Singapore, 1989. MR 1082551 |
| Reference:
|
[23] Lakshmikantham, V., Pandit, S. G.: The Method of upper, lower solutions and hyperbolic partial differential equations. J. Math. Anal. Appl. 105 (1985), 466–477. Zbl 0569.35056, MR 0778480, 10.1016/0022-247X(85)90062-9 |
| Reference:
|
[24] Mainardi, F.: Fractional calculus: Some basic problems in continuum and statistical mechanics. In: Carpinteri, A., Mainardi, F. (eds) Fractional Calculus in Continuum Mechanics Springer-Verlag, Wien, 1997, 291–348. MR 1611587 |
| Reference:
|
[25] Metzler, F., Schick, W., Kilian, H. G., Nonnenmacher, T. F.: Relaxation in filled polymers: A fractional calculus approach. J. Chem. Phys. 103 (1995), 7180–7186. 10.1063/1.470346 |
| Reference:
|
[26] Miller, K. S., Ross, B.: An Introduction to the Fractional Calculus and Differential Equations. John Wiley, New York, 1993. MR 1219954 |
| Reference:
|
[27] Oldham, K. B., Spanier, J.: The Fractional Calculus. Academic Press, New York, London, 1974. Zbl 0292.26011, MR 0361633 |
| Reference:
|
[28] Pandit, S. G.: Monotone methods for systems of nonlinear hyperbolic problems in two independent variables. Nonlinear Anal. 30 (1997), 235–272. Zbl 0892.35095, MR 1602940, 10.1016/S0362-546X(96)00265-9 |
| Reference:
|
[29] Podlubny, I., Petraš, I., Vinagre, B. M., O’Leary, P., Dorčak, L.: Analogue realizations of fractional-order controllers. fractional order calculus and its applications. Nonlinear Dynam. 29 (2002), 281–296. MR 1926477 |
| Reference:
|
[30] Samko, S. G., Kilbas, A. A., Marichev, O. I.: Fractional Integrals and Derivatives. Theory and Applications. Gordon and Breach, Yverdon, 1993. Zbl 0818.26003, MR 1347689 |
| Reference:
|
[31] Staněk, S.: The existence of positive solutions of singular fractional boundary value problems. Comput. Math. Appl. 62 (2011), 1379–1388. Zbl 1228.34020, MR 2824725, 10.1016/j.camwa.2011.04.048 |
| Reference:
|
[32] Vityuk, A. N.: Existence of solutions of partial differential inclusions of fractional order. Izv. Vyssh. Uchebn., Ser. Mat. 8 (1997), 13–19. MR 1485123 |
| Reference:
|
[33] Vityuk, A. N., Golushkov, A. V.: Existence of solutions of systems of partial differential equations of fractional order. Nonlinear Oscil. 7, 3 (2004), 318–325. MR 2151816, 10.1007/s11072-005-0015-9 |
| Reference:
|
[34] Vityuk, A. N., Mykhailenko, A. V.: On a class of fractional-order differential equation. Nonlinear Oscil. 11, 3 (2008), 307–319. MR 2512745, 10.1007/s11072-009-0032-1 |
| Reference:
|
[35] Vityuk, A. N., Mykhailenko, A. V.: The Darboux problem for an implicit fractional-order differential equation. J. Math. Sci. 175, 4 (2011), 391–401. Zbl 1279.35092, MR 2977138, 10.1007/s10958-011-0353-3 |
| . |
