##
**Existence and uniqueness of solutions for the Cauchy-type problems of fractional differential equations.**
*(English)*
Zbl 1197.34004

The authors are concerned with presenting existence and uniqueness results for Cauchy-type problems for fractional differential equations based on the Riemann-Liouville or Hadamard fractional derivative. The authors present a sequence of lemmas providing basic theory and draw attention to new features in their work that highlight differences from the published results in the monograph by A. A. Kilbas, H. M. Srivastava and J. J. Trujillo [Theory and applications of fractional differential equations. Amsterdam: Elsevier (2006; Zbl 1092.45003)]. Examples are presented that highlight these differences in the theoretical results.

Reviewer: Neville Ford (Chester)

### MSC:

34A08 | Fractional ordinary differential equations |

34A12 | Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations |

### Keywords:

existence; uniqueness; Cauchy problem; fractional differential equations; Riemann-Liouville derivative### Citations:

Zbl 1092.45003
PDF
BibTeX
XML
Cite

\textit{C. Kou} et al., Discrete Dyn. Nat. Soc. 2010, Article ID 142175, 15 p. (2010; Zbl 1197.34004)

### References:

[1] | B. Bonilla, M. Rivero, L. Rodríguez-Germá, and J. J. Trujillo, “Fractional differential equations as alternative models to nonlinear differential equations,” Applied Mathematics and Computation, vol. 187, no. 1, pp. 79-88, 2007. · Zbl 1120.34323 |

[2] | A. D. Fitt, A. R. H. Goodwin, K. A. Ronaldson, and W. A. Wakeham, “A fractional differential equation for a MEMS viscometer used in the oil industry,” Journal of Computational and Applied Mathematics, vol. 229, no. 2, pp. 373-381, 2009. · Zbl 1235.34201 |

[3] | E. Ahmed and A. S. Elgazzar, “On fractional order differential equations model for nonlocal epidemics,” Physica A: Statistical Mechanics and its Applications, vol. 379, no. 2, pp. 607-614, 2007. |

[4] | Y. Ding and H. Ye, “A fractional-order differential equation model of HIV infection of CD4+ T-cells,” Mathematical and Computer Modelling, vol. 50, no. 3-4, pp. 386-392, 2009. · Zbl 1185.34005 |

[5] | H. Xu, “Analytical approximations for a population growth model with fractional order,” Communications in Nonlinear Science and Numerical Simulation, vol. 14, no. 5, pp. 1978-1983, 2009. · Zbl 1221.65210 |

[6] | K. M. Furati and N.-E. Tatar, “On chaos synchronization of fractional differential equations,” Journal of Mathematical Analysis and Applications, vol. 332, no. 1, pp. 441-454, 2007. · Zbl 1121.34055 |

[7] | J. Yan and C. Li, “Long time behavior for a nonlinear fractional model,” Chaos, Solitons & Fractals, vol. 32, no. 2, pp. 725-735, 2007. · Zbl 1132.37308 |

[8] | C. Kou, Y. Yan, and J. Liu, “Stability analysis for fractional differential equations and their applications in the models of HIV-1 infection,” Computer Modeling in Engineering & Sciences, vol. 39, no. 3, pp. 301-317, 2009. · Zbl 1257.92033 |

[9] | F. Mainardi, “Fractional calculus: some basic problems in continuum and statistical mechanics,” in Fractals and Fractional Calculus in Continuum Mechanics (Udine, 1996), vol. 378 of CISM Courses and Lectures, pp. 291-348, Springer, Vienna, Austria, 1997. · Zbl 0917.73004 |

[10] | R. Metzler and T. F. Nonnenmacher, “Fractional diffusion: exact representations of spectral functions,” Journal of Physics A, vol. 30, no. 4, pp. 1089-1093, 1997. · Zbl 1001.82538 |

[11] | R. Metzler and T. F. Nonnenmacher, “Fractional diffusion, waiting-time distributions, and Cattaneo-type equations,” Physical Review E, vol. 57, no. 6, pp. 6409-6414, 1998. |

[12] | W. M. Glöckle, R. Metzler, and T. F. Nonnenmacher, “Fractional model equation for anomalous diffusion,” Physica A, vol. 211, pp. 13-24, 1994. |

[13] | B. J. West, P. Grigolini, R. Metzler, and T. F. Nonnenmacher, “Fractional diffusion and Levy stable processes,” Physical Review E, vol. 55, no. 1, part A, pp. 99-106, 1997. |

[14] | H. E. Roman and M. Giona, “Fractional diffusion equation on fractals: three-dimensional case and scattering function,” Journal of Physics A, vol. 25, no. 8, pp. 2107-2117, 1992. · Zbl 0755.60068 |

[15] | T. F. Nonnenmacher and D. J. F. Nonnenmacher, “Towards the formulation of a nonlinear fractional extended irreversible thermodynamics,” Acta Physica Hungarica, vol. 66, no. 1-4, pp. 145-154, 1989. |

[16] | E. Pitcher and W. E. Sewell, “Existence theorems for solutions of differential equations of non-integral order,” Bulletin of the American Mathematical Society, vol. 44, no. 2, pp. 100-107, 1938. · Zbl 0019.40801 |

[17] | J. H. Barrett, “Differential equations of non-integer order,” Canadian Journal of Mathematics, vol. 6, no. 4, pp. 529-541, 1954. · Zbl 0058.10702 |

[18] | P. L. Butzer and A. A. Kilbas, “Mellin transform analysis and integration by parts for Hadamard-type fractional integrals,” Journal of Mathematical Analysis and Applications, vol. 270, no. 1, pp. 1-15, 2002. · Zbl 1022.26011 |

[19] | P. L. Butzer, A. A. Kilbas, and J. J. Trujillo, “Fractional calculus in the Mellin setting and Hadamard-type fractional integrals,” Journal of Mathematical Analysis and Applications, vol. 269, no. 1, pp. 1-27, 2002. · Zbl 0995.26007 |

[20] | P. L. Butzer, A. A. Kilbas, and J. J. Trujillo, “Compositions of Hadamard-type fractional integration operators and the semigroup property,” Journal of Mathematical Analysis and Applications, vol. 269, no. 2, pp. 387-400, 2002. · Zbl 1027.26004 |

[21] | A. A. Kilbas, “Hadamard-type fractional calculus,” Journal of the Korean Mathematical Society, vol. 38, no. 6, pp. 1191-1204, 2001. · Zbl 1018.26003 |

[22] | A. A. Kilbas, O. I. Marichev, and S. G. Samko, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach Science Publishers, Amsterdam, The Netherlands, 1993. · Zbl 0818.26003 |

[23] | A. A. Kilbas and J. J. Trujillo, Hadamard-Type Fractional Integrals and Derivatives, vol. 11, Trudy Instituta Matematiki, Minsk, Russia, 2002. · Zbl 0995.26007 |

[24] | H. A. H. Salem, “On the existence of continuous solutions for a singular system of non-linear fractional differential equations,” Applied Mathematics and Computation, vol. 198, no. 1, pp. 445-452, 2008. · Zbl 1153.26004 |

[25] | A. Bashir and J. N. Juan, “Existence results for a coupled system of nonlinear fractional differential equations with three-point boundary conditions,” Computers & Mathematics with Applications, vol. 58, no. 9, pp. 1838-1843, 2009. · Zbl 1205.34003 |

[26] | G. Mehdi, “Solution of nonlinear fractional differential equations using homotopy analysis method,” Applied Mathematical Modelling, vol. 58, no. 9, pp. 1838-1843, 2009. · Zbl 1205.34003 |

[27] | O. Abdulaziz, I. Hashim, and S. Momani, “Solving systems of fractional differential equations by homotopy-perturbation method,” Physics Letters A, vol. 372, no. 4, pp. 451-459, 2008. · Zbl 1217.81080 |

[28] | A. A. Kilbas, H. M. Sprivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, San Diego, Calif, USA, 2006. · Zbl 1092.45003 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.