Asymptotic positivity of solutions of second order differential equations
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A body in a damped oscillator eventually stops at the origin. Can we drag the body to the positive side by giving a positive driving force? Unfortunately, due to the oscillatory motion of the body, it is not true in general. In this paper, we give a sufficient condition on the driving force guaranteeing the asymptotic positivity of the position of the body, which means the negative part of the position vanishes in time. Also the result will be extended to a wider class of differential equations including the damped oscillator.
KeywordsSecond order linear differential equation Bounds Oscillating solutions Asymptotic positivity Gronwall’s inequality
Mathematics Subject Classification34C11 34E99
This work was partially done when the author was in Laboratoire J.-L. Lions at Université Pierre-et-Marie Curie; the author would like to thank the Fondation Sciences Mathématiques de Paris for financial support.
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