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Zbl 1131.34029
Hatvani, László
The growth condition guaranteeing small solutions for a linear oscillator with an increasing elasticity coefficient. (English)
[J] Georgian Math. J. 14, No. 2, 269-278 (2007). ISSN 1072-947X; ISSN 1572-9176/e

The author investigates a linear second order differential equation. The equation describes the oscillation generated by a restoring force with elasticity coefficient tending to infinity. Using the Armellini-Tonelli-Sansone theorem, the author obtains a growth condition for the function guaranteeing that all solutions of the equation are small. The proof of the main result is based on the asymptotic stability of the zero solution of the damped oscillator. For convenience, a corollary of the main result is formulated, and it is shown that the Armellini-Tonelli-Sansone theorem is a consequence of the corollary. The author shows by an example that the corollary is a real generalization both of the Armellini-Tonelli-Sansone theorem and of the cited result of Pucci and Serrin.
[Iryna Grytsay (Kyiv)]

MSC 2000:: *34C11 Qualitative theory of solutions of ODE: Growth, etc.
34D05 Asymptotic stability of ODE
70J25 Stability of linear oscillatory motions
34A30 Linear ODE and systems
34C15 Nonlinear oscillations of solutions of ODE

Keywords: differential equations; damped oscillator; Armellini-Tonelli-Sansone Theorem; solution tending to zero

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