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**Theorems on instability of systems with respect to linear approximation.**
*(English.
Russian original)*
Zbl 0941.34061

Ukr. Math. J. 48, No. 8, 1251-1262 (1996); translation from Ukr. Mat. Zh. 48, No. 8, 1104-1113 (1996).

Summary: The author studies the instability of solutions to differential equations with a stationary linear part and a nonstationary nonlinear compact part in a Banach space.

### MSC:

34G20 | Nonlinear differential equations in abstract spaces |

34D30 | Structural stability and analogous concepts of solutions to ordinary differential equations |

34D10 | Perturbations of ordinary differential equations |

47N20 | Applications of operator theory to differential and integral equations |

### Keywords:

instability; solutions; differential equations; stationary linear part; nonstationary nonlinear compact part
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\textit{V. E. Slyusarchuk}, Ukr. Math. J. 48, No. 8, 1251--1262 (1996; Zbl 0941.34061); translation from Ukr. Mat. Zh. 48, No. 8, 1104--1113 (1996)

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### References:

[1] | Yu. L. Daletskii and M. G. Krein,Stability of Solutions of Differential Equations in Banach Spaces [in Russian] Nauka, Moscow 1970. |

[2] | D. Henry,Geometric Theory of Semilinear Parabolic Equations, Springer, Berlin 1981. · Zbl 0456.35001 |

[3] | V. E. Slyusarchuk, ”Difference equations in functional spaces,” in: D. I. Martynyuk,Addition II, Lectures on Qualitative Theory of Difference Equations [in Russian], Naukova Dumka, Kiev (1972), pp. 197–222. |

[4] | V. E. Slyusarchuk, ”Problem of instability with respect to the first approximation,”Mat. Zametki,23, No. 5, 721–723 (1978). · Zbl 0393.34029 |

[5] | V. E. Slyusarchuk, ”First approximation instability of systems,”Differents. Uravn.,16, No. 4, 760–761 (1980). · Zbl 0436.34046 |

[6] | V. E. Slyusarchuk, ”On the theory of stability of systems with respect to the first approximation,”Dokl. Akad. Nauk Ukr. SSR, Ser.A, No. 9, 27–30 (1981). · Zbl 0496.34029 |

[7] | V. E. Slyusarchuk, ”The statement on instability with respect to the first approximation,”Ukr. Mat. Zh.,34, No. 2, 241–244 (1982). · Zbl 0508.34045 |

[8] | V. E. Slyusarchuk, ”Several additions to the theory of stability of systems with respect to the first approximation,”Mat Zametki,33, No. 4, 595–603 (1983). · Zbl 0517.34046 |

[9] | V. E. Slyusarchuk, ”New theorems on stability of difference systems with respect to the first approximation,”Differents. Uravn.,19, No. 5, 906–908 (1983). · Zbl 0522.34070 |

[10] | V. E. Slyusarchuk, ”Instability of differential equations with respect to the first approximation,”Mat. Zametki,37, No. 1, 72–77 (1985). · Zbl 0577.34054 |

[11] | V. E. Slyusarchuk, ”Instability of difference equations with respect to the first approximation,”Differents. Uravn.,22, No. 4, 722–723 (1986). · Zbl 0606.39003 |

[12] | V. E. Slyusarchuk,Bounded Solutions of Functional and Functional-Difference Equations [in Russian], Author’s Abstract of the Doctoral Degree Thesis (Physics and Mathematics), Rovno (1983). |

[13] | V. E. Slyusarchuk, ”Instability of autonomous systems with respect to the linear approximation,” in:Asymptotic Methods and Their Application to Problems of Mathematical Physics [in Russian], Institute of Mathematics, Ukrainian Academy of Sciences, Kiev (1990), pp. 112–114. |

[14] | V. D. Mil’man, ”Geometrical theory of Banach spaces,”Usp. Mat. Nauk,25, No. 3, 113–174 (1970). |

[15] | S. G. Krein (editor),Functional Analysis [in Russian] Nauka, Moscow 1972. |

[16] | W. Rudin,Functional Analysis, McGraw-Hill, New York 1973. · Zbl 0253.46001 |

[17] | B. P. Demidovich,Lectures on Mathematical Theory of Stability [in Russian] Nauka, Moscow 1967. · Zbl 0155.41601 |

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