Mild solution of fractional order differential equations with not instantaneous impulses
Pei-Luan Li ; Chang-Jin Xu
Open Mathematics, Tome 13 (2015), / Harvested from The Polish Digital Mathematics Library

In this paper, we investigate the boundary value problems of fractional order differential equations with not instantaneous impulse. By some fixed-point theorems, the existence results of mild solution are established. At last, one example is also given to illustrate the results.

Publié le : 2015-01-01
EUDML-ID : urn:eudml:doc:271049
@article{bwmeta1.element.doi-10_1515_math-2015-0042,
     author = {Pei-Luan Li and Chang-Jin Xu},
     title = {Mild solution of fractional order differential equations with not instantaneous impulses},
     journal = {Open Mathematics},
     volume = {13},
     year = {2015},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_1515_math-2015-0042}
}
Pei-Luan Li; Chang-Jin Xu. Mild solution of fractional order differential equations with not instantaneous impulses. Open Mathematics, Tome 13 (2015) . http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_1515_math-2015-0042/

[1] Mophou G. M., Existence and uniqueness of mild solutions to impulsive fractional differential equations, Nonlinear Anal., 2010,72 (3-4), 1604–1615

[2] Tai Z., Wang X., Controllability of fractional-order impulsive neutral functional infinite delay integrodifferential systems in Banach spaces, Appl. Math. Lett., 2009, 22 (11), 1760–1765 [Crossref] | Zbl 1181.34078

[3] Shu X., Lai Y., Chen Y., The existence of mild solutions for impulsive fractional partial differential equations, Nonlinear Anal., 2011, 74, 2003–2011 | Zbl 1227.34009

[4] Zhang X., Huang X., Liu Z., The existence and uniqueness of mild solutions for impulsive fractional equations with nonlocal conditions and infinite delay, Nonlinear Anal. Hybrid Syst., 2010, 4 , 775–781 [WoS] | Zbl 1207.34101

[5] Hernandez E., O’Regan D., On a new class of abstract impulsive differential equations, Proc. Amer. Math. Soc., 2013, 141, 1641– 1649 | Zbl 1266.34101

[6] Pierri M., O’Regan D., Rolnik V., Existence of solutions for semi-linear abstract differential equations with not instantaneous impulses, Appl. Math. Comput., 2013, 219, 6743–6749 | Zbl 1293.34019

[7] Zhou Y., Jiao F.,Nonlocal Cauchy problem for fractional evolution equations, Nonlinear Anal: RWA, 2010, 11, 4465–4475 | Zbl 1260.34017

[8] Zhou Y., Jiao F., Li J., Existence and uniqueness for fractional neutral differential equations with infinite delay, Nonlinear Anal: TMA., 2009,7, 3249–3256 | Zbl 1177.34084

[9] Diethelm K., The analysis of fractional differential equations, Lect. Notes Math., 2010 [Crossref] | Zbl 1215.34001

[10] Kilbas A.A., Srivastava M.H., Trujillo J.J., Theory and Applications of Fractional Differential Equations, in: North-Holland Mathematics studies, vol.204, Elsevier Science B.V., Amsterdam, 2006

[11] Lakshmikantham V., Leela S., Vasundhara Devi J., Theory of fractional dynamic systems, Cambridge Scientific Publishers, Cambridge, 2009 | Zbl 1188.37002

[12] Miller K.S., Ross B., An introduction to the fractional calculus and differential equations, John Wiley, New York, 1993 | Zbl 0789.26002

[13] Podlubny I., Fractional differential equations,. Academic Press, New York, 1999

[14] Tarasov VE., Fractional dynamics: application of fractional calculus to dynamics of particles, fields and media, Springer, HEP, 2011

[15] Agarwal R. P., Benchohra M., Hamani S.,A survey on existence results for boundary valueproblems of nonlinear fractional differential equations and inclusions, Acta. Appl. Math., 2010, 109, 973–1033 [WoS] | Zbl 1198.26004

[16] Benchohra M., Henderson J., Ntouyas S.K., Ouahab A., Existence results for fractional order functional differential equations with infinite delay, J. Math. Anal. Appl., 2008,338,1340–1350 | Zbl 1209.34096

[17] Wang J., Zhou Y.,A class of fractional evolution equations and optimal controls, Nonlinear Anal: RWA., 2011,12, 262–272 | Zbl 1214.34010

[18] Wang J., Zhou Y., Wei W.,A class of fractional delay nonlinear integrodifferential controlled systems in Banach spaces, Commun Nonlinear Sci Numer Simulat, 2011, 16 , 4049–4059 [WoS][Crossref] | Zbl 1223.45007

[19] Zhang S., Existence of positive solution for some class of nonlinear fractional differential equations, J. Math. Anal. Appl., 2003, 278, 136–148 | Zbl 1026.34008

[20] Guo T., Jiang W., Impulsive problems for fractional differential equations with boundary valueconditions, Comput. Math. Appl., 2012, 64 , 3281–3291 [Crossref] | Zbl 1268.34014

[21] Krasnoselskii Ma., Topological methods in the theory of nonlinear integral equation., Pergamon Press, New York, 1964