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Periodic boundary value problems for fractional differential equations involving a Riemann-Liouville fractional derivative. (English) Zbl 1202.26017
Summary: We shall discuss the properties of the well-known Mittag-Leffler function, and consider the existence and uniqueness of the solution of the periodic boundary value problem for a fractional differential equation involving a Riemann-Liouville fractional derivative by using the monotone iterative method.

MSC:
26A33Fractional derivatives and integrals (real functions)
34A08Fractional differential equations
34B15Nonlinear boundary value problems for ODE
34B99Boundary value problems for ODE
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References:
[1] Kilbas, A. A.; Srivastava, H. M.; Trujillo, J. J.: Theory and applications of fractional differential equations, (2006) · Zbl 1092.45003
[2] Pitcher, E.; Sewell, W. E.: Existence theorems for solutions of differential equations of non-integral order, Bull. amer. Math. soc. 44, No. 2, 100-107 (1938) · Zbl 0018.30701 · doi:10.1090/S0002-9904-1938-06695-5
[3] Al-Bassam, M. A.: Some existence theorems on differential equations of generalized order, J. reine angew. Math. 218, No. 1, 70-78 (1965) · Zbl 0156.30804 · doi:10.1515/crll.1965.218.70 · crelle:GDZPPN002181150
[4] Delbosco, D.; Rodino, L.: Existence and uniqueness for a nonlinear fractional differential equation, J. math. Anal. appl. 204, No. 2, 609-625 (1996) · Zbl 0881.34005 · doi:10.1006/jmaa.1996.0456
[5] Kilbas, A. A.; Marzan, S. A.: Nonlinear differential equations in weighted spaces of continuous functions, Dokl. nats. Akad. nauk belarusi 47, No. 1, 29-35 (2003) · Zbl 1204.26009 · http://www.ac.by/publications/dan/dan47_1.html
[6] Rivero, M.; Rodríguez-Germá, L.; Trujillo, J. J.: Linear fractional differential equations with variable coefficients, Appl. math. Lett. 21, 892-897 (2008) · Zbl 1152.34305 · doi:10.1016/j.aml.2007.09.010
[7] Ibrahim, Rabha W.; Darus, Maslina: Subordination and superordination for univalent solutions for fractional differential equations, J. math. Anal. appl. 345, 871-879 (2008) · Zbl 1147.30009 · doi:10.1016/j.jmaa.2008.05.017
[8] Zhang, Shuqin: The existence of a positive solution for a nonlinear fractional differential equation, J. math. Anal. appl. 252, 804-812 (2000) · Zbl 0972.34004 · doi:10.1006/jmaa.2000.7123
[9] Zhang, Shuqin: Positive solution for some class of nonlinear fractional differential equation, J. math. Anal. appl. 278, No. 1, 136-148 (2003) · Zbl 1026.34008 · doi:10.1016/S0022-247X(02)00583-8
[10] Devi, J. V.; Lakshmikantham, V.: Nonsmooth analysis and fractional differential equations, Nonlinear anal. IMA 70, No. 12, 4151-4157 (2009) · Zbl 1237.49022
[11] Kosmatov, N.: Integral equations and initial value problems for nonlinear differential equations of fractional order, Nonlinear anal. TMA 70, No. 7, 2521-2529 (2009) · Zbl 1169.34302 · doi:10.1016/j.na.2008.03.037
[12] Shuqin, Zhang: 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
[13] Belmekki, Mohammed; Nieto, Juan J.; Rodríguez-López, Rosana: Existence of periodic solutions for a nonlinear fractional differential equation, Boundary value problems 2009 (2009) · Zbl 1181.34006 · doi:10.1155/2009/324561
[14] Nieto, J. J.: Maximum principles for fractional differential equations derived from Mittag--Leffler functions, Appl. math. Lett. 23, No. 10, 1248-1251 (2010) · Zbl 1202.34019 · doi:10.1016/j.aml.2010.06.007
[15] Chang, Y. -K.; Nieto, J. J.: Some new existence results for fractional differential inclusions with boundary conditions, Math. comput. Modelling 49, 605-609 (2009) · Zbl 1165.34313 · doi:10.1016/j.mcm.2008.03.014
[16] Bonilla, B.; Rivero, M.; Rodríguez-Germá, L.; Trujillo, J. J.: Fractional differential equations as alternative models to nonlinear differential equations, Appl. math. Comput. 187, 79-88 (2007) · Zbl 1120.34323 · doi:10.1016/j.amc.2006.08.105
[17] Jumarie, G.: An approach via fractional analysis to non-linearity induced by coarse-graining in space, Nonlinear anal. RWA 11, 535-546 (2010) · Zbl 1195.37054 · doi:10.1016/j.nonrwa.2009.01.003
[18] 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, 1838-1843 (2009) · Zbl 1205.34003 · doi:10.1016/j.camwa.2009.07.091
[19] Agarwal, Ravi P.; Lakshmikantham, V.; Nieto, Juan 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
[20] Luchko, Y. F.; Rivero, M.; Trujillo, J. J.; Velasco, M. P.: Fractional models, non-locality, and complex systems, Comput. math. Appl. 59, 1048-1056 (2010) · Zbl 1189.37095 · doi:10.1016/j.camwa.2009.05.018
[21] Wei, Zhongli; Li, Qingdong; Che, Junling: Initial value problems for fractional differential equations involving Riemann--Liouville sequential fractional derivative, J. math. Anal. appl. 367, No. 1, 260-272 (2010) · Zbl 1191.34008 · doi:10.1016/j.jmaa.2010.01.023
[22] Ladde, G. S.; Lakshmikantham, V.; Vatsala, A. S.: Monotone iterative techniques for nonlinear differential equations, (1985) · Zbl 0658.35003
[23] Ahmad, B.; Sivasundaram, S.: Existence results and monotone iterative technique for impulsive hybrid functional differential systems with anticipation and retardation, Appl. math. Comput. 197, 515-524 (2008) · Zbl 1142.34049 · doi:10.1016/j.amc.2007.07.065
[24] Nieto, J. J.; Rodríguez-López, R.: Monotone method for first-order functional differential equations, Comput. math. Appl. 52, 471-484 (2006) · Zbl 1140.34406 · doi:10.1016/j.camwa.2006.01.012
[25] Jiang, D.; Nieto, J. J.; Zuo, W.: On monotone method for first and second order periodic boundary value problems and periodic solutions of functional differential equations, J. math. Anal. appl. 289, 691-699 (2004) · Zbl 1134.34322 · doi:10.1016/j.jmaa.2003.09.020
[26] Nieto, J. J.; Rodríguez-López, R.: Boundary value problems for a class of impulsive functional equations, Comput. math. Appl. 55, 2715-2731 (2008) · Zbl 1142.34362 · doi:10.1016/j.camwa.2007.10.019
[27] Lakshmikanthan, V.; Vatsala, A. S.: General uniqueness and monotone iterative technique for fractional differential equations, Appl. math. Lett. 21, 828-834 (2008) · Zbl 1161.34031 · doi:10.1016/j.aml.2007.09.006
[28] Lakshmikanthan, V.; Vatsala, A. S.: Basic theory of fractional differential equations, Nonlinear anal. TMA 69, No. 8, 2677-2682 (2008) · Zbl 1161.34001 · doi:10.1016/j.na.2007.08.042
[29] Podlubny, I.: Fractional differential equations, (1999) · Zbl 0924.34008