| Title:
|
A global uniqueness result for fractional order implicit differential equations (English) |
| Author:
|
Abbas, Said |
| Author:
|
Benchohra, Mouffak |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
53 |
| Issue:
|
4 |
| Year:
|
2012 |
| Pages:
|
605-614 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
In this paper we investigate the global existence and uniqueness of solutions for the initial value problems (IVP for short), for a class of implicit hyperbolic fractional order differential equations by using a nonlinear alternative of Leray-Schauder type for contraction maps on Fréchet spaces. (English) |
| Keyword:
|
partial hyperbolic differential equation |
| Keyword:
|
fractional order |
| Keyword:
|
left-sided mixed Riemann-Liouville integral |
| Keyword:
|
mixed regularized derivative |
| Keyword:
|
solution |
| Keyword:
|
Fréchet space |
| Keyword:
|
fixed point |
| MSC:
|
26A33 |
| idMR:
|
MR3016429 |
| . |
| Date available:
|
2013-03-02T13:47:18Z |
| Last updated:
|
2015-02-11 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/143193 |
| . |
| 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.: The method of upper and lower solutions for partial hyperbolic fractional order differential inclusions with impulses.Discuss. Math. Differ. Incl. Control Optim. 30 (2010), no. 1, 141–161. Zbl 1203.26005, MR 2682404, 10.7151/dmdico.1116 |
| Reference:
|
[4] Abbas S., Benchohra M., Gorniewicz L.: Existence theory for impulsive partial hyperbolic functional differential equations involving the Caputo fractional derivative.Sci. Math. Jpn. 72 (2010), 49–60. Zbl 1200.26004, MR 2666846 |
| Reference:
|
[5] Abbas S., Benchohra M., N'Guérékata G.M.: Topics in Fractional Differential Equations.Developments in Mathematics, 27, Springer, New York, 2012. MR 2962045, 10.1007/978-1-4614-4036-9 |
| Reference:
|
[6] 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:
|
[7] Belarbi A., 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:
|
[8] Benchohra M., Graef J.R., Hamani S.: Existence results for boundary value problems of nonlinear fractional differential equations with integral conditions.Appl. Anal. 87 (2008), no. 7, 851–863. MR 2458962, 10.1080/00036810802307579 |
| Reference:
|
[9] Benchohra M., Hamani S., Ntouyas S.K.: Boundary value problems for differential equations with fractional order.Surv. Math. Appl. 3 (2008), 1–12. Zbl 1198.26007, MR 2390179 |
| Reference:
|
[10] 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:
|
[11] Frigon M., Granas A.: Résultats de type Leray-Schauder pour des contractions sur des espaces de Fréchet.Ann. Sci. Math. Québec 22 (1998), no. 2, 161–168. Zbl 1100.47514, MR 1677243 |
| Reference:
|
[12] Henry D.: Geometric Theory of Semilinear Parabolic Partial Differential Equations.Springer, Berlin-New York, 1989. |
| Reference:
|
[13] Hilfer R.: Applications of Fractional Calculus in Physics.World Scientific, Singapore, 2000. Zbl 0998.26002, MR 1890104 |
| Reference:
|
[14] 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:
|
[15] Kilbas A.A., Marzan S.A.: Nonlinear differential equations with the Caputo fractional derivative in the space of continuously differentiable functions.Differ. Equ. 41 (2005), 84–89. Zbl 1160.34301, MR 2213269, 10.1007/s10625-005-0137-y |
| Reference:
|
[16] Miller K.S., Ross B.: An Introduction to the Fractional Calculus and Differential Equations.John Wiley, New York, 1993. MR 1219954 |
| Reference:
|
[17] Oldham K.B., Spanier J.: The Fractional Calculus.Academic Press, New York, London, 1974. Zbl 0292.26011, MR 0361633 |
| Reference:
|
[18] Podlubny I.: Fractional Differential Equation.Academic Press, San Diego, 1999. |
| Reference:
|
[19] 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:
|
[20] Vityuk A.N.: Existence of solutions of partial differential inclusions of fractional order.Izv. Vyssh. Uchebn. Zaved. Mat. 8 (1997), 13–19. MR 1485123 |
| Reference:
|
[21] Vityuk A.N., Golushkov A.V.: Existence of solutions of systems of partial differential equations of fractional order.Nonlinear Oscil. 7 (2004), no. 3, 318–325. MR 2151816, 10.1007/s11072-005-0015-9 |
| Reference:
|
[22] Vityuk A.N., Mykhailenko A.V.: On a class of fractional-order differential equation.Nonlinear Oscil. 11 (2008), no. 3, 307–319. MR 2512745, 10.1007/s11072-009-0032-1 |
| Reference:
|
[23] Vityuk A.N., Mykhailenko A.V.: The Darboux problem for an implicit fractional-order differential equation.J. Math. Sci. 175 (4) (2011), 391–401. MR 2977138, 10.1007/s10958-011-0353-3 |
| . |
