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Cím:Asymptotic stability of the equilibrium of the damped oscillator.
Szerző:Hatvani, L\'aszl\'o; Totik, Vilmos
Forrás:Differ. Integral Equ. 6, No.4, 835-848 (1993).
Nyelv:English
Absztrakt:Conditions are given guaranteeing the property $x(t) \to 0$, $\dot x(t) \to 0$ $(t \to \infty)$ for every solution of the equation $\ddot x+h(t) \dot x+k\sp 2x=0$ $(t \ge 0, 0<k=\text{const.})$, where $h$ is a nonnegative function. It is known that this property requires that in the average the damping coefficient $h$ is not ``too small'' or ``too large''. In the first part we give a necessary and sufficient growth condition on $h$, provided that $h$ is not ``too small'' in some integral sense. Then, considering the case of small $h$, we show that not only the size, but the distribution of the damping ``bumps'' is important. The main theorem takes into account both of them. Finally, we formulate theorems for the general case when $h$ can be both small and large. It is pointed out that the conditions restricting $h$ above and below are interdependent. 
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