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Impulsive periodic boundary value problems for fractional differential equation involving Riemann-Liouville sequential fractional derivative. (English) Zbl 1234.34005

The author considers nonlinear impulsive fractional differential equations involving Riemann-Liouville fractional derivatives with periodic boundary conditions:

𝒟 2α u(t)=f(t,u,𝒟 α u),t(0,1]{t 1 ,,t m },0<α1,
lim t0 + t 1-α u(t)=u(1),lim t0 + t 1-α 𝒟 α u(t)=𝒟 α u(1),
lim tt j + (t-t j ) 1-α (u(t)-u(t j ))=I j (u(t j )),
lim tt j + (t-t j ) 1-α (𝒟 α u(t)-𝒟 α u(t j ))=I ¯ j (u(t j ))·

The author establishes the existence of solutions to the periodic boundary value problem using upper and lower solutions and the monotone iterative method.

MSC:
34A08Fractional differential equations
34B37Boundary value problems for ODE with impulses
34A45Theoretical approximation of solutions of ODE
References:
[1]Miller, K. S.; Ross, B.: An introduction to the fractional calculus and fractional differential equations, (1993)
[2]Samko, S. G.; Kilbas, A. A.; Marichev, O. I.: Fractional integrals and derivatives: theory and applications, (1993) · Zbl 0818.26003
[3]Podlubny, I.: Fractional differential equations, (1999)
[4]Kilbas, A. A.; Srivastava, H. M.; Trujillo, J. J.: Theory and applications of fractional differential equations, (2006)
[5]Lakshmikantham, V.; Leela, S.; Devi, J. Vasundhara: Theory of fractional dynamic systems, (2009)
[6]Delbosco, D.; Rodino, L.: Existence and uniqueness for a nonlinear fractional differential equation, J. math. Anal. appl. 204, 609-625 (1996) · Zbl 0881.34005 · doi:10.1006/jmaa.1996.0456
[7]El-Sayed, A. M. A.: Nonlinear functional differential equations of arbitrary orders, Nonlinear anal. 33, 181-186 (1998) · Zbl 0934.34055 · doi:10.1016/S0362-546X(97)00525-7
[8]Kilbas, A. A.; Marzan, S. A.: Nonlinear differential equations with the Caputo fractional derivative in the space of continuously differentiable functions, Differ. equ. 41, 84-89 (2005) · Zbl 1160.34301 · doi:10.1007/s10625-005-0137-y
[9]Furati, K. M.; Tatar, N-E.: Behavior of solutions for a weighted Cauchy-type fractional differential problem, J. fract. Calc. 28, 23-42 (2005) · Zbl 1131.26304
[10]Kilbas, A. A.; Trujillo, J. J.: Differential equations of fractional order: methods, results, and problems. I, Appl. anal. 78, 153-192 (2001) · Zbl 1031.34002 · doi:10.1080/00036810108840931
[11]Bai, C.; Fang, J.: The existence of a positive solution for a singular coupled system of nonlinear fractional differential equations, Appl. math. Comput. 150, 611-621 (2004) · Zbl 1061.34001 · doi:10.1016/S0096-3003(03)00294-7
[12]Bai, C.: Positive solutions for nonlinear fractional differential equations with coefficient that changes sign, Nonlinear anal. 64, 677-685 (2006) · Zbl 1152.34304 · doi:10.1016/j.na.2005.04.047
[13]Lakshmikantham, V.; Leela, S.: A Krasnoselskii-Krein-type uniqueness result for fractional differential equations, Nonlinear anal. 71, 3421-3424 (2009) · Zbl 1177.34004 · doi:10.1016/j.na.2009.02.008
[14]Arara, A.; Benchohra, M.; Hamidi, N.; Nieto, J. J.: Fractional order differential equations on an unbounded domain, Nonlinear anal. 72, 580-586 (2010) · Zbl 1179.26015 · doi:10.1016/j.na.2009.06.106
[15]Agarwal, R. P.; Lakshmikantham, V.; Nieto, J. J.: On the concept of solution for fractional differential equations with uncertainty, Nonlinear anal. 72, No. 6, 2859-2862 (2010) · Zbl 1188.34005 · doi:10.1016/j.na.2009.11.029
[16]Shi, K.; Wang, Y.: On a stochastic fractional partial differential equation driven by a Lévy space-time white noise, J. math. Anal. appl. 364, No. 1, 119-129 (2010) · Zbl 1185.60071 · doi:10.1016/j.jmaa.2009.11.010
[17]Devi, J. V.; Lakshmikantham, V.: Nonsmooth analysis and fractional differential equations, Nonlinear anal. 70, No. 12, 4151-4157 (2009)
[18]Zhang, S.: Monotone iterative method for initial value problem involving Riemann-Liouville fractional derivatives, Nonlinear anal. 71, 2087-2093 (2009) · Zbl 1172.26307 · doi:10.1016/j.na.2009.01.043
[19]Belmekki, M.; Nieto, J. J.; Rodriguez-López, R.: Existence of periodic solution for a nonlinear fractional differential equation, Bound. value probl. 2009 (2009) · Zbl 1181.34006 · doi:10.1155/2009/324561
[20]Ahmad, B.; Nieto, J. J.: Existence results for a coupled system of nonlinear fractional differential equations with three-point boundary conditions, Comput. math. Appl. 58, No. 9, 1838-1843 (2009) · Zbl 1205.34003 · doi:10.1016/j.camwa.2009.07.091
[21]Kosmatov, N.: Integral equations and initial value problems for nonlinear differential equations of fractional order, Nonlinear anal. 70, No. 7, 2521-2529 (2009) · Zbl 1169.34302 · doi:10.1016/j.na.2008.03.037
[22]Kaufmann, E.; Mboumi, E.: Positive solutions of a boundary value problem for a nonlinear fractional differential equation, Electron. J. Qual. theory differ. Equ. 2008 (2008) · Zbl 1183.34007 · doi:emis:journals/EJQTDE/2008/200803.html
[23]Bai, Z.; Lü, H.: Positive solutions for boundary value problem of nonlinear fractional differential equation, J. math. Anal. appl. 311, 495-505 (2005) · Zbl 1079.34048 · doi:10.1016/j.jmaa.2005.02.052
[24]Bai, C.: Triple positive solutions for a boundary value problem of nonlinear fractional differential equation, Electron. J. Qual. theory differ. Equ. 2008, No. 24, 1-10 (2008) · Zbl 1183.34005 · doi:emis:journals/EJQTDE/2008/200824.html
[25]Wei, Z.; Li, Q.; Che, J.: Initial value problems for fractional differential equations involving Riemann-Liouville sequential fractional derivative, J. math. Anal. appl. 367, 260-272 (2010) · Zbl 1191.34008 · doi:10.1016/j.jmaa.2010.01.023
[26]Agarwal, R. P.; O’regan, D.; Staněk, S.: Positive solutions for Dirichlet problems of singular nonlinear fractional differential equations, J. math. Anal. appl. 371, 57-68 (2010) · Zbl 1206.34009 · doi:10.1016/j.jmaa.2010.04.034
[27]B. Ahmad, J.J. Nieto, Riemann-Liouville fractional differential equations with fractional boundary conditions, Fixed Point Theory, in press.
[28]Nieto, J. J.: Maximum principles for fractional differential equations derived from Mittag-Leffler functions, Appl. math. Lett. 23, 1248-1251 (2010) · Zbl 1202.34019 · doi:10.1016/j.aml.2010.06.007
[29]Girejko, E.; Mozyrska, D.; Wyrwas, M.: A sufficient condition of viability for fractional differential equations with the Caputo derivative, J. math. Anal. appl. 381, No. 1, 146-154 (2011) · Zbl 1222.34007 · doi:10.1016/j.jmaa.2011.04.004
[30]Wei, Z.; Dong, W.; Che, J.: Periodic boundary value problems for fractional differential equations involving a Riemann-Liouville fractional derivative, Nonlinear anal. 73, 3232-3238 (2010) · Zbl 1202.26017 · doi:10.1016/j.na.2010.07.003
[31]Lakshmikantham, V.; Bainov, D. D.; Simeonov, P. S.: Theory of impulsive differential equations, (1989) · Zbl 0718.34011
[32]Benchohra, M.; Henderson, J.; Ntouyas, S. K.: Impulsive differential equations and inclusions, (2007)
[33]Nieto, J. J.: Impulsive resonance periodic problems of first order, Appl. math. Lett. 15, 489-493 (2002) · Zbl 1022.34025 · doi:10.1016/S0893-9659(01)00163-X
[34]Tisdell, C. C.: Existence of solutions to first-order periodic boundary value problems, J. math. Anal. appl. 323, 1325-1332 (2006) · Zbl 1109.34016 · doi:10.1016/j.jmaa.2005.11.047
[35]Luo, Z.; Nieto, J. J.: New results for the periodic boundary value problem for impulsive integro-differential equations, Nonlinear anal. 70, 2248-2260 (2009) · Zbl 1166.45002 · doi:10.1016/j.na.2008.03.004
[36]Agarwal, R. P.; Benchohra, M.; Slimani, B. A.: Existence results for differential equations with fractional order and impulses, Mem. differential equations math. Phys. 44, 1-21 (2008) · Zbl 1178.26006
[37]Benchohra, M.; Slimani, B. A.: Impulsive fractional differential equations, Electron. J. Differential equations 2009, No. 10, 1-11 (2009)
[38]Nieto, J. J.: Basic theory for nonresonance impulsive periodic problems of first order, J. math. Anal. appl. 205, 423-433 (1997) · Zbl 0870.34009 · doi:10.1006/jmaa.1997.5207
[39]Guo, D.: Existence of solutions of boundary value problems for second order impulsive differential equations in Banach spaces, J. math. Anal. appl. 181, 407-421 (1994) · Zbl 0807.34076 · doi:10.1006/jmaa.1994.1031