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Cím:Integral conditions on the asymptotic stability for the damped linear oscillator with small damping.
Szerző:Hatvani, L.
Forrás:Proc. Am. Math. Soc. 124, No.2, 415-422 (1996). [ISSN 0002-9939; ISSN 1088-6826
Nyelv:English
Absztrakt:The author considers the equation (1) $x'' + h(t)x' + k^2x = 0$ under the assumption (2) $0 \le h(t) \le \overline h < \infty$. He proves that the condition $\limsup_{t \to \infty} (t^{- 2/3} \int^t_0 h(s) ds) > 0$ is sufficient for the asymptotic stability of $x = x' = 0$, and the exponent $2/3$ is best possible. He obtains this as a corollary of a general result on intermittent damping which reads as follows: Suppose that (2) is satisfied. If there exists a sequence $\{I_n\}$ of non-overlapping intervals such that $$\sum^\infty_{n = 1} {1 \over 1 + |I_n |^2} \left( \int^t_0 h(s)ds \right)^3 = \infty,$$ then the zero solution of (1) is asymptotically stable. Moreover, the exponent 3 in the statement is best possible.} \RV{W.M\"uller (Berlin) 
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