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2005, 2005(Special): 497-504. doi: 10.3934/proc.2005.2005.497

Attractivity properties of oscillator equations with superlinear damping

János Karsai 1, and  John R. Graef 2,

1. 

Department of Medical Informatics, University of Szeged, Szeged, Korányi fasor 9, 6720, Hungary
2. 

Department of Mathematics, University of Tennessee at Chattanooga, Chattanooga, TN 37403, United States

Received  September 2004 Revised  February 2005 Published  September 2005

The authors investigate the asymptotic behavior of solutions of the damped nonlinear oscillator equation $$x''+ a(t)|x'|^\alpha \sgn(x') + f(x)=0, $$ where $uf(u) > 0$ for $u \neq 0$, $a(t)\geq 0$, and $\alpha\geq 1$. The case $\alpha=1$ has been investigated by a number of other authors. There are also some results for the case $\alpha>1$, but they are not really based on the power $\alpha$, although it plays an essential role in the behavior of the solutions. In this paper, we give new attractivity results for the large damping case, $a(t) \geq a_0 > 0$, that improve previously known results. Our conditions involve the power $\alpha$ in such a way that our results reduce directly to known conditions in the case $\alpha=1$. Some open problems for future research are also indicated.
Keywords: attractivity, nonlinear damping, monotonicity., Second order equations.
Mathematics Subject Classification: 34C15, 34D0.
Citation: János Karsai, John R. Graef. Attractivity properties of oscillator equations with superlinear damping. Conference Publications, 2005, 2005 (Special) : 497-504. doi: 10.3934/proc.2005.2005.497
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