| Title:
|
Initial value problems for fractional functional differential inclusions with Hadamard type derivative (English) |
| Author:
|
Guerraiche, Nassim |
| Author:
|
Hamani, Samira |
| Author:
|
Henderson, Johnny |
| Language:
|
English |
| Journal:
|
Archivum Mathematicum |
| ISSN:
|
0044-8753 (print) |
| ISSN:
|
1212-5059 (online) |
| Volume:
|
52 |
| Issue:
|
4 |
| Year:
|
2016 |
| Pages:
|
263-273 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
We establish sufficient conditions for the existence of solutions of a class of fractional functional differential inclusions involving the Hadamard fractional derivative with order $\alpha \in (0,1]$. Both cases of convex and nonconvex valued right hand side are considered. (English) |
| Keyword:
|
fractional differential inclusion |
| Keyword:
|
Hadamard-type fractional derivative |
| Keyword:
|
fractional integral |
| Keyword:
|
fixed point |
| Keyword:
|
convex |
| MSC:
|
26A33 |
| MSC:
|
34A60 |
| idZBL:
|
Zbl 06674903 |
| idMR:
|
MR3610653 |
| DOI:
|
10.5817/AM2016-4-263 |
| . |
| Date available:
|
2016-12-20T21:53:06Z |
| Last updated:
|
2018-01-10 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/145932 |
| . |
| Reference:
|
[1] Agarwal, R.P, Benchohra, M., Hamani, S.: A survey on existence results for boundary value problems for nonlinear fractional differential equations and inclusions.Acta Appl. Math. 109 (3) (2010), 973–1033. MR 2596185, 10.1007/s10440-008-9356-6 |
| Reference:
|
[2] Ahmed, B., Ntouyas, S.K.: Initial value problems for hybrid Hadamard fractional equations.EJDE 2014 (161) (2014), 1–8. MR 3239404 |
| Reference:
|
[3] Aubin, J.P., Cellina, A.: Differential Inclusions.Springer-Verlag, Berlin-Heidelberg, New York, 1984. Zbl 0538.34007, MR 0755330 |
| Reference:
|
[4] Aubin, J.P., Frankowska, H.: Set-Valued Analysis.Birkhäuser, Boston, 1990. Zbl 0713.49021, MR 1048347 |
| Reference:
|
[5] Benchohra, M., Djebali, S., Hamani, S.: Boundary value problems of differential inclusions with Riemann-Liouville fractional derivative.Nonlinear Oscillation 14 (1) (2011), 7–20. MR 2976264, 10.1007/s11072-011-0137-1 |
| Reference:
|
[6] Benchohra, M., Hamani, S.: Boundary value problems for differential inclusions with fractional order.Diss. Math. Diff. Inclusions. Control and Optimization 28 (2008), 147–164. Zbl 1181.26012, MR 2548782, 10.7151/dmdico.1099 |
| Reference:
|
[7] Benchohra, M., Hamani, S.: Nonlinear boundary value problems for differential inclusions with Caputo fractional derivative.Topol. Meth. Nonlinear Anal. 32 (1) (2008), 115–130. Zbl 1180.26002, MR 2466806 |
| Reference:
|
[8] Bressan, A., Colombo, G.: Extensions and selections of maps with decomposable values.Studia Math. 90 (1988), 69–86. Zbl 0677.54013, MR 0947921, 10.4064/sm-90-1-69-86 |
| Reference:
|
[9] Burton, T.A., Kirk, C.: A fixed point theorem of Krasnoselskii-Schaefer type.Math. Nachr. 189 (1998), 23–31. Zbl 0896.47042, MR 1492921, 10.1002/mana.19981890103 |
| Reference:
|
[10] Butzer, P.L., Kilbas, A.A., Trujillo, J.J.: Composition of Hadamard-type fractional integration operators and the semigroup property.J. Math. Anal. Appl. 269 (2002), 387–400. MR 1907120, 10.1016/S0022-247X(02)00049-5 |
| Reference:
|
[11] Butzer, P.L., Kilbas, A.A., Trujillo, J.J.: Fractional calculus in the Mellin setting and Hadamard-type fractional integrals.J. Math. Anal. Appl. 269 (2002), 1–27. Zbl 0995.26007, MR 1907871, 10.1016/S0022-247X(02)00001-X |
| Reference:
|
[12] Butzer, P.L., Kilbas, A.A., Trujillo, J.J.: Mellin transform analysis and integration by parts for Hadamard-type fractional integrals.J. Math. Anal. Appl. 270 (2002), 1–15. Zbl 1022.26011, MR 1911748, 10.1016/S0022-247X(02)00066-5 |
| Reference:
|
[13] Castaing, C., Valadier, M.: Convex Analysis and Measurable Multifunctions.Lecture Notes in Math., vol. 580, Springer-Verlag, Berlin-Heidelberg-New York, 1977. Zbl 0346.46038, MR 0467310, 10.1007/BFb0087688 |
| Reference:
|
[14] Covitz, H., Nadler, Jr., S.B.: Multivalued contraction mappings in generalized metric spaces.Israel J. Math. 8 (1970), 5–11. MR 0263062, 10.1007/BF02771543 |
| Reference:
|
[15] Deimling, K.: Multivalued Differential Equations.Walter De Gruyter, Berlin-New York, 1992. Zbl 0820.34009, MR 1189795 |
| Reference:
|
[16] Delbosco, D., Rodino, L.: Existence and uniqueness for a nonlinear fractional differential equation.J. Math. Anal. Appl. 204 (1996), 609–625. Zbl 0881.34005, MR 1421467, 10.1006/jmaa.1996.0456 |
| Reference:
|
[17] 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:
|
[18] Fryszkowski, A.: Continuous selections for a class of nonconvex multivalued maps.Studia Math. 76 (1983), 163–174. MR 0730018, 10.4064/sm-76-2-163-174 |
| Reference:
|
[19] Fryszkowski, A.: Fixed Point Theory for Decomposable Sets. Topological Fixed Point Theory and Its Applications.vol. 2, Kluwer Academic Publishers, Dordrecht, 2004. MR 2084131 |
| Reference:
|
[20] Hadamard, J.: Essai sur l’etude des fonctions donnees par leur development de Taylor.J. Math. Pure Appl. 8 (1892), 101–186. |
| Reference:
|
[21] Hamani, S., Benchohra, M., Graef, J.R.: Existence results for boundary-value problems with nonlinear fractional differential inclusions and integral conditions.EJDE 2010 No. 20 (2010), 1–16. Zbl 1185.26010, MR 2592005 |
| Reference:
|
[22] Heymans, N., Podlubny, I.: Physical interpretation of initial conditions for fractional differential equations with Riemann-Liouville fractional derivatives.Rheologica Acta 45 (5) (2006), 765–772. MR 1658022, 10.1007/s00397-005-0043-5 |
| Reference:
|
[23] Hilfer, R.: Applications of Fractional Calculus in Physics.World Scientific, Singapore, 2000. Zbl 0998.26002, MR 1890104 |
| Reference:
|
[24] Kaufmann, E.R., Mboumi, E.: Positive solutions of a boundary value problem for a nonlinear fractional differential equation.EJQTDE 2007 (3) (2007), 11pp. MR 2369417 |
| Reference:
|
[25] Kilbas, A.A., Marzan, S.A.: Nonlinear differential equations with the Caputo fractional derivative in the space of continuously differentiable functions.Differential Equ. 41 (2005), 84–89. Zbl 1160.34301, MR 2213269, 10.1007/s10625-005-0137-y |
| Reference:
|
[26] Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations.North-Holland Mathematics Studies, Elsevier Science B.V., Amsterdam, 2006. Zbl 1092.45003, MR 2218073 |
| Reference:
|
[27] Miller, K.S., Ross, B.: An Introduction to the Fractional Calculus and Differential Equations.John Wiley, New York, 1993. MR 1219954 |
| Reference:
|
[28] Momani, S.M., Hadid, S.B., Alawenh, Z.M.: Some analytical properties of solutions of differential equations of noninteger order.Int. J. Math. Math. Sci. 2004 (2004), 697–701. Zbl 1069.34002, MR 2054178, 10.1155/S0161171204302231 |
| Reference:
|
[29] Podlubny, I.: Fractional Differential Equations.Academic Press, San Diego, 1999. Zbl 0924.34008, MR 1658022 |
| Reference:
|
[30] Podlubny, I.: Geometric and physical interpretation of fractional integration and fractional differentiation.Fract. Calculus Appl. Anal. 5 (2002), 367–386. Zbl 1042.26003, MR 1967839 |
| Reference:
|
[31] Thiramanus, P., Ntouyas, S.K., Tariboon, J.: Existence and uniqueness results for Hadamard-type fractional differential equations with nonlocal fractional integral boundary conditions.Abstr. Appl. Anal. (2014), Art ID 902054, 9 pp. MR 3228094 |
| Reference:
|
[32] Zhang, S.: Positive solutions for boundary-value problems of nonlinear fractional differential equations.EJDE (2006), no. 36, 1–12. Zbl 1096.34016, MR 2213580 |
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