Abstract
There are many studies on the asymptotic behavior of solutions of differential equations. In the present paper, we consider another aspect of this problem, namely, the rate of the asymptotic convergence of solutions. Let ϕ (t) be a scalar continuous monotonically increasing positive function tending to ∞ as t → ∞. It is established that if all solutions of a differential system satisfy the inequality
then the solution x(t; t 0, x 0) of this differential system tends to 0 faster than \(M\frac{{\varphi (t_0 )}}{{\varphi (t)}}\).
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REFERENCES
J. Kurzweil and I. Vrkoc, “The converse theorems of Lyapunov and Persidskij concerning the stability of motion,” Czech. Math. J., 7(82), 254–274 (1957).
N. Levinson, “The asymptotic behavior of a system of linear differential equations,” Amer. J. Math., 68, 1–6 (1946).
I. G. Malkin, Theory of Stability of Motion [in Russian], Gostekhizdat, Moscow (1952).
A. A. Martynyuk, Stability Analysis: Nonlinear Mechanic Equations (Stability and Control: Theory, Methods and Applications), Vol. 2, New York-London (1995).
A. A. Martynyuk, V. Lakshmikantham, and S. Leela, Stability of Motion: Method of Integral Inequalities [in Russian], Naukova Dumka, Kiev (1989).
A. A. Martynyuk, “A survey of some classical and modern developments of stability theory,” Nonlin. Analysis, 40, 483–496 (2000).
N. Rouche, P. Habets, and M. Laloy, Stability Theory by Liapunov’s Direct Method, Springer, New York (1977).
V. V. Rumyantsev and A. S. Oziraner, Partial Stability and Stabilization of Motion [in Russian], Nauka, Moscow (1987).
T. Yoshizawa, Stability Theory by Liapunov’s Second Method, Math. Soc. Jpn., Tokyo (1966).
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Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 57, No. 1, pp. 137–142, January, 2005.
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Vu, T., Pham, V. On the Asymptotic Behavior of Solutions of Differential Systems. Ukr Math J 57, 165–172 (2005). https://doi.org/10.1007/s11253-005-0179-6
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DOI: https://doi.org/10.1007/s11253-005-0179-6