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Impulsive problems for fractional differential equations with boundary value conditions. (English) Zbl 1268.34014
Summary: We consider impulsive problems for a fractional differential equation with boundary value conditions. Natural formulae of a solution for our problem are introduced, which can be regarded as a novelty item. Some related new existence results are established by applying fixed point methods via a generalized singular Gronwall inequality. Examples are given to illustrate the results.
MSC:
34A08Fractional differential equations
34B10Nonlocal and multipoint boundary value problems for ODE
34B37Boundary value problems for ODE with impulses
References:
[1]Bainov, D. D.; Simeonov, P. S.: Impulsive differential equations: periodic solutions and applications, (1993)
[2]Lakshmikantham, V.; Bainov, D. D.; Simeonov, P. S.: Theory of impulsive differential equations, (1989) · Zbl 0718.34011
[3]Benchohra, M.; Henderson, J.; Ntouyas, S. K.: Impulsive differential equations and inclusions, vol. 2, (2006) · Zbl 1130.34003
[4]Diethelm, K.: The analysis of fractional differential equations, Lecture notes in math. (2010)
[5]Kilbas, A. A.; Srivastava, H. M.; Trujillo, J. J.: Theory and applications of fractional differential equations, North-holland mathematics studies 204 (2006)
[6]Lakshmikantham, V.; Leela, S.; Devi, J. Vasundhara: Theory of fractional dynamic systems, (2009)
[7]Miller, K. S.; Ross, B.: An introduction to the fractional calculus and differential equations, (1993)
[8]M.W. Michalski, Derivatives of noninteger order and their applications, Dissertationes Mathematicae, Polska Akademia Nauk., Instytut Matematyczny, Warszawa, 1993. · Zbl 0880.26007
[9]Podlubny, I.: Fractional differential equations, (1999)
[10]Tarasov, V. E.: Fractional dynamics: application of fractional calculus to dynamics of particles, fields and media, (2010)
[11]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
[12]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)
[13]Bai, Z. B.: On positive solutions of a nonlocal fractional boundary value problem, Nonlinear anal. 72, 916-924 (2010) · Zbl 1187.34026 · doi:10.1016/j.na.2009.07.033
[14]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
[15]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
[16]Wang, J.; Zhou, Y.: A class of fractional evolution equations and optimal controls, Nonlinear anal. Real world appl. 12, 262-272 (2011) · Zbl 1214.34010 · doi:10.1016/j.nonrwa.2010.06.013
[17]Wang, J.; Zhou, Y.; Wei, W.: A class of fractional delay nonlinear integrodifferential controlled systems in Banach spaces, Commun. nonlinear sci. Numer. simul. 16, 4049-4059 (2011) · Zbl 1223.45007 · doi:10.1016/j.cnsns.2011.02.003
[18]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
[19]Zhou, Y.; Jiao, F.: Nonlocal Cauchy problem for fractional evolution equations, Nonlinear anal. Real world appl. 11, 4465-4475 (2010)
[20]Zhou, Y.; Jiao, F.; Li, J.: Existence and uniqueness for p-type fractional neutral differential equations, Nonlinear anal. 71, 2724-2733 (2009) · Zbl 1175.34082 · doi:10.1016/j.na.2009.01.105
[21]Zhou, Y.; Jiao, F.; Li, J.: Existence and uniqueness for fractional neutral differential equations with infinite delay, Nonlinear anal. 71, 3249-3256 (2009) · Zbl 1177.34084 · doi:10.1016/j.na.2009.01.202
[22]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
[23]Ahmad, B.; Sivasundaram, S.: Existence results for nonlinear impulsive hybrid boundary value problems involving fractional differential equations, Nonlinear anal. Hybrid syst. 3, 251-258 (2009) · Zbl 1193.34056 · doi:10.1016/j.nahs.2009.01.008
[24]Benchohra, M.; Seba, D.: Impulsive fractional differential equations in Banach spaces, Electron. J. Qual. theory differ. Equ., No. 8, 1-14 (2009) · Zbl 1189.26005 · doi:emis:journals/EJQTDE/sped1/108.pdf
[25]Balachandran, K.; Kiruthika, S.: Existence of solutions of abstract fractional impulsive semilinear evolution equations, Electron. J. Qual. theory differ. Equ., No. 4, 1-12 (2010) · Zbl 1201.34091 · doi:emis:journals/EJQTDE/2010/201004.pdf
[26]Wang, G.; Zhang, L.; Song, G.: Systems of first order impulsive functional differential equations with deviating arguments and nonlinear boundary conditions, Nonlinear anal. 74, 974-982 (2011) · Zbl 1223.34091 · doi:10.1016/j.na.2010.09.054
[27]Lv, L.; Wang, J.; Wei, W.: Existence and uniqueness results for fractional differential equations with boundary value conditions, Opuscula math. 31, 629-643 (2011) · Zbl 1225.26010 · doi:http://www.opuscula.agh.edu.pl/vol31/4/art/opuscula_math_3142.pdf