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

Slyusarchuk, V. E.

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

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