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Attractivity properties of oscillator equations with superlinear damping

Pages: 497 - 504, Issue Special, August 2005

 Abstract        Full Text (361.0K)              

János Karsai - Department of Medical Informatics, University of Szeged, Szeged, Korányi fasor 9, 6720, Hungary (email)
John R. Graef - Department of Mathematics, University of Tennessee at Chattanooga, Chattanooga, TN 37403, United States (email)

Abstract: 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:  Second order equations, nonlinear damping, attractivity, monotonicity.
Mathematics Subject Classification:  34C15, 34D05.

Received: September 2004;      Revised: February 2005;      Published: September 2005.