##
**Fractional differential equations in terms of comparison results and Lyapunov stability with initial time difference.**
*(English)*
Zbl 1196.34010

The paper considers how Lyapunov stability concepts can be applied to fractional differential equations of Riemann-Liouville or Caputo type. Following a review which explains why this idea is not straightforward, the author discusses basic theory on existence and uniqueness. The concepts of ITD-stability and Lyapunov-like functions are introduced and shown to provide a useful way of understanding classical fractional stability. The paper concludes by describing some comparison results and showing how the various definitions of stability are inter-related.

Reviewer: Neville Ford (Chester)

### MSC:

34A08 | Fractional ordinary differential equations |

34D20 | Stability of solutions to ordinary differential equations |

### Keywords:

Lyapunov stability; fractional differential equations; Riemann-Liouville derivatives; Caputo derivatives; initial time difference
PDF
BibTeX
XML
Cite

\textit{C. Yakar}, Abstr. Appl. Anal. 2010, Article ID 762857, 16 p. (2010; Zbl 1196.34010)

### References:

[1] | M. Caputo, “Linear models of dissipation whose Q is almost independent, II,” Geophysical Journal of the Royal Astronomical Society, vol. 13, pp. 529-539, 1967. |

[2] | A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland, Amsterdam, The Netherlands, 2006. · Zbl 1092.45003 |

[3] | F. Brauer and J. Nohel, The Qualitative Theory of Ordinary Differential Equations, W.A. Benjamin, New York, NY, USA, 1969. · Zbl 0179.13202 |

[4] | V. Lakshmikantham and S. Leela, Differential and Integral Inequalities, vol. 1, Academic Press, New York, NY, USA, 1969. · Zbl 0177.12403 |

[5] | V. Lakshmikantham, S. Leela, and A. A. Martynyuk, Stability Analysis of Nonlinear Systems, vol. 125 of Monographs and Textbooks in Pure and Applied Mathematics, Marcel Dekker, New York, NY, USA, 1989. · Zbl 0676.34003 |

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

[7] | M. D. Shaw and C. Yakar, “Stability criteria and slowly growing motions with initial time difference,” Problems of Nonlinear Analysis in Engineering Systems, vol. 1, pp. 50-66, 2000. |

[8] | I. Podlubny, Fractional Differential Equations, vol. 198 of Mathematics in Science and Engineering, Academic Press, San Diego, Calif, USA, 1999. · Zbl 0924.34008 |

[9] | V. Lakshmikantham and A. S. Vatsala, “Differential inequalities with initial time difference and applications,” Journal of Inequalities and Applications, vol. 3, no. 3, pp. 233-244, 1999. · Zbl 0988.34008 |

[10] | S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach, Yverdon, Switzerland, 1993. · Zbl 0818.26003 |

[11] | M. D. Shaw and C. Yakar, “Generalized variation of parameters with initial time difference and a comparison result in term Lyapunov-like functions,” International Journal of Non-Linear Differential Equations-Theory Methods and Applications, vol. 5, pp. 86-108, 1999. |

[12] | C. Yakar, “Boundedness criteria with initial time difference in terms of two measures,” Dynamics of Continuous, Discrete & Impulsive Systems. Series A, vol. 14, supplement 2, pp. 270-274, 2007. |

[13] | C. Yakar, “Strict stability criteria of perturbed systems with respect to unperturbed systems in terms of initial time difference,” in Complex Analysis and Potential Theory, pp. 239-248, World Scientific, Hackensack, NJ, USA, 2007. · Zbl 1159.34040 |

[14] | C. Yakar and M. D. Shaw, “A comparison result and Lyapunov stability criteria with initial time difference,” Dynamics of Continuous, Discrete & Impulsive Systems. Series A, vol. 12, no. 6, pp. 731-737, 2005. · Zbl 1172.34323 |

[15] | C. Yakar and M. D. Shaw, “Initial time difference stability in terms of two measures and a variational comparison result,” Dynamics of Continuous, Discrete & Impulsive Systems. Series A, vol. 15, no. 3, pp. 417-425, 2008. · Zbl 1149.34034 |

[16] | C. Yakar and M. D. Shaw, “Practical stability in terms of two measures with initial time difference,” Nonlinear Analysis: Theory, Methods & Applications, vol. 71, no. 12, pp. e781-e785, 2009. · Zbl 1238.93073 |

[17] | V. Lakshmikantham and A. S. Vatsala, “Basic theory of fractional differential equations,” Nonlinear Analysis: Theory, Methods & Applications, vol. 69, no. 8, pp. 2677-2682, 2008. · Zbl 1161.34001 |

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.