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**Stability for impulsive differential systems in terms of two measures.**
*(English)*
Zbl 0669.34056

Using different measures for the “deviations” of the initial value and the value at the moment t of a solution, the authors introduce new concepts of stability for impulsive differential systems. Sufficient conditions for stability, uniform stability, asymptotic stability, and uniform asymptotic stability in this new sense are established by the aid of Lyapunov functions, which are assumed to be piecewise continuous. The theorems have interesting corollaries involving and improving a variety of stability results found in the literature.

Reviewer: L.Hatvani

### MSC:

34D20 | Stability of solutions to ordinary differential equations |

34D05 | Asymptotic properties of solutions to ordinary differential equations |

### Keywords:

impulsive differential systems; uniform stability; asymptotic stability; uniform asymptotic stability; Lyapunov functions
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\textit{V. Lakshmikantham} and \textit{X. Liu}, Appl. Math. Comput. 29, No. 1, 89--98 (1989; Zbl 0669.34056)

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### References:

[1] | Perestyuk, N. A.; Samoilenko, A. M., Differential Equations with Impulsive Effects (1987), Central Publishers: Central Publishers Kiev, U.S.S.R · Zbl 0837.34003 |

[3] | Simeonov, P. S.; Bainov, D. D., Stability with respect to part of the variables in systems with impulse effect, J. Math. Anal. Appl., 117, 1, 247-263 (1986) · Zbl 0588.34044 |

[6] | Salvadori, L., Sul problema della stabilita, Rend. Accad. Naz. Lincei, 53, 35-38 (1972) · Zbl 0257.34057 |

[7] | Marachkov, M., On a theorem on stability, Bull. Soc. Phys.-Math Karan, 12, 171-174 (1940) |

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