Stability Behavior of the Zero Solution for Nonlinear Damped Vectorial Second Order Differential Equation

DOI: 10.4236/ijmnta.2012.14019   PDF   HTML     3,287 Downloads   5,721 Views  

Abstract

In this paper, a theoretical treatment of the stability behavior of the zero solution of nonlinear damped oscillator in the vectorial case is investigated. We study the sufficient conditions for the boundedness of solution of the nonlinear damped vectorial oscillator and the conditions for the stability of the zero solution to be uniformly stable as well as asymptotically stable.

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Ramadan, M. and El-Kholy, S. (2012) Stability Behavior of the Zero Solution for Nonlinear Damped Vectorial Second Order Differential Equation. International Journal of Modern Nonlinear Theory and Application, 1, 125-129. doi: 10.4236/ijmnta.2012.14019.

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1] T. A. Burton and T. Furumochi, “A Note on Stability by Schauder’s Theorem,” Funkcialaj Ekvacioj, Vol. 44, No. 1, 2001, pp. 73-82.
[2] V. A. Coppel, “Stability and Asymptotic Behavior of Differential Equations,” D. C. Heath and Company, Lexington, 1965.
[3] D. Grossman, “Introduction to Differential Equations with Boundary Value Problems,” Wiley-Interscience, John Wiley & Sons, Inc., Hoboken, 1987.
[4] J. Hale, “Ordinary Differential Equations,” Wiley Interscience, John Wiley & Sons, Inc., Hoboken, 1969.
[5] G. H. Morosanu and C. Vladimirescu, “Stability for a Nonlinear Second Order ODE,” Funkcialaj Ekvacioj, Vol. 48, No. 1, 2005, pp. 49-56. doi:10.1619/fesi.48.49
[6] G. H. Morosanu and C. Vladimirescu, “Stability for a Damped Nonlinear Oscillator,” Nonlinear Analysis, Vol. 60, No. 2, 2005, pp. 303-310.?
[7] J. Awrejcewicz, “Classical Mechanics. Dynamics,” Springer-Verlag, New York, 2012,
[8] R. F. Curtain and A. J. Pritchard, “Functional Analysis in Modern Applied Mathematics,” Academic Press Inc. Ltd., London, 1977.
[9] R. Bellman, “Stability Theory of Differential Equations,” Dover Publications Inc., Mineola, 1953.
[10] V. Lakshmikantham and S. Leela, “Differential and Integral Inequalities. Theory and Applications,” Academic Press, New York and London, 1969.

  
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