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Maximum principles for fractional differential equations derived from Mittag-Leffler functions. (English) Zbl 1202.34019
Summary: We present two new maximum principles for a linear fractional differential equation with initial or periodic boundary conditions. Some properties of the classical Mittag-Leffler functions are crucial in our arguments. These comparison results allow us to study the corresponding nonlinear fractional differential equations and to obtain approximate solutions.
34A08Fractional differential equations
34A30Linear ODE and systems, general
34B15Nonlinear boundary value problems for ODE
34C10Qualitative theory of oscillations of ODE: zeros, disconjugacy and comparison theory
[1]Kilbas, A. A.; Srivastava, H. M.; Trujillo, J. J.: Theory and applications of fractional differential equations, (2006)
[2]Kiryakova, V.: Generalized fractional calculus and applications, (1994)
[3]Miller, K. S.; Ross, B.: An introduction to the fractional calculus and differential equations, (1993)
[4]Oldham, K. B.; Spanier, J.: The fractional calculus, (1974)
[5]Podlubny, I.: Fractional differential equation, (1999)
[6]Samko, S. G.; Kilbas, A. A.; Marichev, O. I.: Fractional integrals and derivatives. Theory and applications, (1993) · Zbl 0818.26003
[7]Agarwal, R. P.; Lakshmikantham, V.; Nieto, J. J.: On the concept of solution for fractional differential equations with uncertainty, Nonlinear anal. 72, 2859-2862 (2010) · Zbl 1188.34005 · doi:10.1016/j.na.2009.11.029
[8]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
[9]Araya, D.; Lizama, C.: Almost automorphic mild solutions to fractional differential equations, Nonlinear anal. 69, 3692-3705 (2008) · Zbl 1166.34033 · doi:10.1016/j.na.2007.10.004
[10]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
[11]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
[12]Diethelm, K.; Ford, N. J.: Analysis of fractional differential equations, J. math. Anal. appl. 265, 229-248 (2002) · Zbl 1014.34003 · doi:10.1006/jmaa.2001.7194
[13]J.J. Nieto, Comparison results for periodic boundary value problem of fractional differential equations, Fract. Dyn. Syst. (in press).
[14]Shuqin, Z.: 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
[15]Jumarie, G.: Laplace’s transform of fractional order via the Mittag–Leffler function and modified Riemann–Liouville derivative, Appl. math. Lett. 22, 1659-1664 (2009) · Zbl 1181.44001 · doi:10.1016/j.aml.2009.05.011
[16]Kiryakova, V.: The multi-index Mittag–Leffler functions as an important class of special functions of fractional calculus, Comput. math. Appl. 59, 1885-1895 (2010) · Zbl 1189.33034 · doi:10.1016/j.camwa.2009.08.025
[17]Libertiaux, V.; Pascon, F.: Differential versus integral formulation of fractional hyperviscoelastic constitutive laws for brain tissue modelling, J. comput. Appl. math. 234, 2029-2035 (2010) · Zbl 1191.92009 · doi:10.1016/j.cam.2009.08.060
[18]Miller, K. S.; Samko, S. G.: Completely monotonic functions, Integral transforms spec. Funct. 12, 389-402 (2001) · Zbl 1035.26012 · doi:10.1080/10652460108819360
[19]Pollard, H.: The complete monotonic character of the Mittag–Leffler function Eα(-x), Bull. amer. Math. soc. 54, 1115-1116 (1948) · Zbl 0033.35902 · doi:10.1090/S0002-9904-1948-09132-7
[20]Nieto, J. J.: Differential inequalities for functional perturbations first-order ordinary differential equations, Appl. math. Lett. 15, 173-179 (2002) · Zbl 1014.34060 · doi:10.1016/S0893-9659(01)00114-8
[21]Belmekki, M.; Nieto, J. J.; Rodríguez-López, R.: Existence of periodic solutions for a nonlinear fractional differential equation, Bound. value probl. 2009 (2009) · Zbl 1181.34006 · doi:10.1155/2009/324561