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The growth condition guaranteeing small solutions for a linear oscillator with an increasing elasticity coefficient. (English) Zbl 1131.34029

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.

MSC:
34C11Qualitative theory of solutions of ODE: growth, boundedness
34D05Asymptotic stability of ODE
70J25Stability of linear oscillatory motions
34A30Linear ODE and systems, general
34C15Nonlinear oscillations, coupled oscillators (ODE)