Title: | Asymptotic behaviour of solutions of real two-dimensional differential system with nonconstant delay (English) |

Author: | Rebenda, Josef |

Language: | English |

Journal: | Archivum Mathematicum |

ISSN: | 0044-8753 (print) |

ISSN: | 1212-5059 (online) |

Volume: | 45 |

Issue: | 3 |

Year: | 2009 |

Pages: | 223-236 |

Summary lang: | English |

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Category: | math |

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Summary: | In this article, stability and asymptotic properties of solutions of a real two-dimensional system $x^{\prime }(t) = \mathbf{A} (t) x(t) + \mathbf{B} (t) x (\tau (t)) + \mathbf{h} (t, x(t), x(\tau (t)))$ are studied, where $\mathbf{A}$, $\mathbf{B}$ are matrix functions, $\mathbf{h}$ is a vector function and $\tau (t) \le t$ is a nonconstant delay which is absolutely continuous and satisfies $\lim \limits _{t \rightarrow \infty } \tau (t) = \infty $. Generalization of results on stability of a two-dimensional differential system with one constant delay is obtained using the methods of complexification and Lyapunov-Krasovskii functional and some new corollaries and examples are presented. (English) |

Keyword: | stability |

Keyword: | asymptotic behaviour |

Keyword: | differential system |

Keyword: | nonconstant delay |

Keyword: | Lyapunov method |

MSC: | 34K12 |

MSC: | 34K20 |

MSC: | 34K25 |

idZBL: | Zbl 1212.34235 |

idMR: | MR2591678 |

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Date available: | 2009-09-18T11:22:43Z |

Last updated: | 2013-09-19 |

Stable URL: | http://hdl.handle.net/10338.dmlcz/134228 |

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Reference: | [1] Kalas, J.: Asymptotic behaviour of a two-dimensional differential systems with nonconstant delay.accepted in Math. Nachr. |

Reference: | [2] Kalas, J., Baráková, L.: Stability and asymptotic behaviour of a two-dimensional differential system with delay.J. Math. Anal. Appl. 269 (2002), 278–300. Zbl 1008.34064, MR 1907886, 10.1016/S0022-247X(02)00023-9 |

Reference: | [3] Ráb, M., Kalas, J.: Stability of dynamical systems in the plane.Differential Integral Equations 3 (1990), 124–144. MR 1014730 |

Reference: | [4] Rebenda, J.: Asymptotic properties of solutions of real two-dimensional differential systems with a finite number of constant delays.Mem. Differential Equations Math. Phys. 41 (2007), 97–114. Zbl 1157.34356, MR 2391945 |

Reference: | [5] Rebenda, J.: Stability of the trivial solution of real two-dimensional differential system with nonconstant delay.In 6. matematický workshop - sborník, FAST VUT Brno 2007, 2007, 49–50 (abstract). Fulltext available at http://math.fce.vutbr.cz/~pribyl/workshop_2007/prispevky/Rebenda.pdf. |

Reference: | [6] Rebenda, J.: Asymptotic behaviour of real two-dimensional differential system with a finite number of constant delays.Demonstratio Math. 41 (4) (2008), 845–857. Zbl 1169.34051, MR 2484509 |

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