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**A refinement of quasilinearization method for Caputo’s sense fractional-order differential equations.**
*(English)*
Zbl 1204.34011

Summary: The method of quasilinearization for fractional-order differential equationin Caputo’s sense is applied to obtain lower and upper sequences in terms of the solutions of linear fractional-order differential equations in Caputo’s sense. It is also shown that these sequences converge to the unique solution of the nonlinear fractional-order differential equation in Caputo’s sense uniformly and semiquadratically with less restrictive assumptions.

### MSC:

34A08 | Fractional ordinary differential equations |

34A45 | Theoretical approximation of solutions to ordinary differential equations |

### References:

[1] | V. Lakshmikantham and A. S. Vatsala, Generalized Quasilinearization for Nonlinear Problems, vol. 440 of Mathematics and Its Applications, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1998. · Zbl 0997.34501 |

[2] | V. Lakshmikantham, S. Leela, and J. Vasundhara Devi, Theory of Fractional Dynamic Systems, Cambridge Academic, Cambridge, UK, 2009. · Zbl 1188.37002 |

[3] | S. Köksal and C. Yakar, “Generalized quasilinearization method with initial time difference,” Simulation, vol. 24, no. 5, 2002. |

[4] | J. Vasundhara Devi and C. Suseela, “Quasilinearization for fractional differential equations,” Communications in Applied Analysis, vol. 12, no. 4, pp. 407-418, 2008. · Zbl 1184.34015 |

[5] | J. Vasundhara Devi, F. A. McRae, and Z. Drici, “Generalized quasilinearization for fractional differential equations,” Computers & Mathematics with Applications, vol. 59, no. 3, pp. 1057-1062, 2010. · Zbl 1189.34010 · doi:10.1016/j.camwa.2009.05.017 |

[6] | C. Yakar and A. Yakar, “An extension of the quasilinearization method with initial time difference,” Dynamics of Continuous, Discrete & Impulsive Systems. Series A, vol. 14, supplement 2, pp. 1-305, 2007. · Zbl 1200.34011 |

[7] | C. Yakar and A. Yakar, “Further generalization of quasilinearization method with initial time difference,” Journal of Applied Functional Analysis, vol. 4, no. 4, pp. 714-727, 2009. · Zbl 1200.34011 |

[8] | T. C. Hu, D. L. Qian, and C. P. Li, “Comparison theorems for fractional differential equations,” Communication on Applied Mathematics and Computation, vol. 23, no. 1, pp. 97-103, 2009. |

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