Basic theory of fractional differential equations. (English) Zbl 1161.34001

This short paper addresses two interesting problems: establishing the comparison principle for fractional differential equations (FDEs), establishing the global existence theorem of FDEs. Here, the fractional derivative is in the sense of Riemann-Louville. Local existence results are also included but they are not new.


34A12 Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations
34A99 General theory for ordinary differential equations
26A33 Fractional derivatives and integrals
34C11 Growth and boundedness of solutions to ordinary differential equations
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[1] Caputo, M., Linear models of dissipation whose Q is almost independent, II, Geophy. J. Roy. Astronom., 13, 529-539 (1967)
[2] Glöckle, W. G.; Nonnenmacher, T. F., A fractional calculus approach to self similar protein dynamics, Biophys. J., 68, 46-53 (1995)
[3] Diethelm, K.; Ford, N. J., Analysis of fractional differential equations, J. Math. Anal. Appl., 265, 229-248 (2002) · Zbl 1014.34003
[4] Diethelm, K.; Ford, N. J., Multi-order fractional differential equations and their numerical solution, Appl. Math. Comput., 154, 621-640 (2004) · Zbl 1060.65070
[5] Diethelm, K.; Freed, A. D., On the solution of nonlinear fractional differential equations used in the modeling of viscoplasticity, (Keil, F.; Mackens, W.; Vob, H.; Werther, J., Scientific Computing in Chemical Engineering II: Computational Fluid Dynamics, Reaction Engineering, and Molecular Properties (1999), Springer: Springer Heidelberg), 217-224
[6] Kiryakova, V., Generalized fractional calculus and applications, (Pitman Res. Notes Math. Ser., vol. 301 (1994), Longman-Wiley: Longman-Wiley New York) · Zbl 1189.33034
[7] Lakshmikantham, V.; Leela, S., Differential and Integral Inequalities, vol. I (1969), Academic Press: Academic Press New York · Zbl 0177.12403
[8] Metzler, R.; Schick, W.; Kilian, H. G.; Nonnenmacher, T. F., Relaxation in filled polymers: A fractional calculus approach, J. Chem. Phys., 103, 7180-7186 (1995)
[9] Podlubny, I., Fractional Differential Equations (1999), Academic Press: Academic Press San Diego · Zbl 0918.34010
[10] Samko, S. G.; Kilbas, A. A.; Marichev, O. I., Fractional Integrals and Derivatives, Theory and Applications (1993), Gordon and Breach: Gordon and Breach Yverdon · Zbl 0818.26003
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