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On the concept and existence of solution for impulsive fractional differential equations. (English) Zbl 1252.35277
Summary: This paper is motivated from some recent papers treating the problem of the existence of a solution for impulsive differential equations with fractional derivative. We firstly show that the formula of solutions in cited papers are incorrect. Secondly, we reconsider a class of impulsive fractional differential equations and introduce a correct formula of solutions for a impulsive Cauchy problem with Caputo fractional derivative. Further, some sufficient conditions for existence of the solutions are established by applying fixed point methods. Some examples are given to illustrate the results.
35R12Impulsive partial differential equations
35R11Fractional partial differential equations
35A01Existence problems for PDE: global existence, local existence, non-existence
[1]Diethelm, K.: The analysis of fractional differential equations, Lect notes math (2010)
[2]Kilbas, A. A.; Srivastava, H. M.; Trujillo, J. J.: Theory and applications of fractional differential equations, (2006)
[3]Lakshmikantham, V.; Leela, S.; Devi, J. Vasundhara: Theory of fractional dynamic systems, (2009)
[4]Miller, K. S.; Ross, B.: An introduction to the fractional calculus and differential equations, (1993)
[5]Michalski, M. W.: Derivatives of noninteger order and their applications, dissertationes mathematicae, CCCXXVIII, Inst math Polish acad sci (1993) · Zbl 0880.26007
[6]Podlubny, I.: Fractional differential equations, (1999)
[7]Tarasov, V. E.: Fractional dynamics: application of fractional calculus to dynamics of particles, fields and media, (2011)
[8]Agarwal, R. P.; Benchohra, M.; Hamani, S.: A survey on existence results for boundary value problems of nonlinear fractional differential equations and inclusions, Acta appl math 109, 973-1033 (2010) · Zbl 1198.26004 · doi:10.1007/s10440-008-9356-6
[9]Ahmad, B.; Nieto, J. J.: Existence of solutions for anti-periodic boundary value problems involving fractional differential equations via Leray – Schauder degree theory, Topol methods nonlinear anal 35, 295-304 (2010)
[10]Bai, Z.: On positive solutions of a nonlocal fractional boundary value problem, Nonlinear anal: TMA 72, 916-924 (2010) · Zbl 1187.34026 · doi:10.1016/j.na.2009.07.033
[11]Benchohra, M.; Henderson, J.; Ntouyas, S. K.; Ouahab, A.: Existence results for fractional order functional differential equations with infinite delay, J math anal appl 338, 1340-1350 (2008) · Zbl 1209.34096 · doi:10.1016/j.jmaa.2007.06.021
[12]Mophou, G. M.; N’guérékata, G. M.: Existence of mild solutions of some semilinear neutral fractional functional evolution equations with infinite delay, Appl math comput 216, 61-69 (2010) · Zbl 1191.34098 · doi:10.1016/j.amc.2009.12.062
[13]Wang, J.; Zhou, Y.: A class of fractional evolution equations and optimal controls, Nonlinear anal: RWA 12, 262-272 (2011) · Zbl 1214.34010 · doi:10.1016/j.nonrwa.2010.06.013
[14]Wang, J.; Zhou, Y.; Wei, W.: A class of fractional delay nonlinear integrodifferential controlled systems in Banach spaces, Commun nonlinear sci numer simulat 16, 4049-4059 (2011) · Zbl 1223.45007 · doi:10.1016/j.cnsns.2011.02.003
[15]Zhang, S.: Existence of positive solution for some class of nonlinear fractional differential equations, J math anal appl 278, 136-148 (2003) · Zbl 1026.34008 · doi:10.1016/S0022-247X(02)00583-8
[16]Zhou, Y.; Jiao, F.: Nonlocal Cauchy problem for fractional evolution equations, Nonlinear anal: RWA 11, 4465-4475 (2010)
[17]Zhou, Y.; Jiao, F.; Li, J.: Existence and uniqueness for p-type fractional neutral differential equations, Nonlinear anal: TMA 71, 2724-2733 (2009) · Zbl 1175.34082 · doi:10.1016/j.na.2009.01.105
[18]Zhou, Y.; Jiao, F.; Li, J.: Existence and uniqueness for fractional neutral differential equations with infinite delay, Nonlinear anal: TMA 71, 3249-3256 (2009) · Zbl 1177.34084 · doi:10.1016/j.na.2009.01.202
[19]Agarwal, R. P.; Benchohra, M.; Hamani, S.: A survey on existence results for boundary value problems of nonlinear fractional differential equations and inclusions, Acta appl math 109, 973-1033 (2010) · Zbl 1198.26004 · doi:10.1007/s10440-008-9356-6
[20]Ahmad, B.; Sivasundaram, S.: Existence results for nonlinear impulsive hybrid boundary value problems involving fractional differential equations, Nonlinear anal: HS 3, 251-258 (2009) · Zbl 1193.34056 · doi:10.1016/j.nahs.2009.01.008
[21]Benchohra, M.; Seba, D.: Impulsive fractional differential equations in Banach spaces, EJ qualitative theor differ equat spec ed I 2009, No. 8, 1-14 (2009) · Zbl 1189.26005 · doi:emis:journals/EJQTDE/sped1/108.pdf
[22]Balachandran, K.; Kiruthika, S.: Existence of solutions of abstract fractional impulsive semilinear evolution equations, EJ qualitative theor differ equat 2010, No. 4, 1-12 (2010) · Zbl 1201.34091 · doi:emis:journals/EJQTDE/2010/201004.pdf
[23]Wang, G.; Zhang, L.; Song, G.: Systems of first order impulsive functional differential equations with deviating arguments and nonlinear boundary conditions, Nonlinear anal: TMA 74, 974-982 (2011) · Zbl 1223.34091 · doi:10.1016/j.na.2010.09.054
[24]Ye, H.; Gao, J.; Ding, Y.: A generalized Gronwall inequality and its application to a fractional differential equation, J math anal appl 328, 1075-1081 (2007) · Zbl 1120.26003 · doi:10.1016/j.jmaa.2006.05.061
[25]Wei, W.; Xiang, X.; Peng, Y.: Nonlinear impulsive integro-differential equation of mixed type and optimal controls, Optimization 55, 141-156 (2006) · Zbl 1101.45002 · doi:10.1080/02331930500530401
[26]Tarasov VE. Theoretical physics models with integro-differentiation of fractional order. IKI, RCD; 2011 (in Russian).
[27]Westerlund, S.: Dead matter has memory!, Causal consult. (2002)