Abstract
There has been much interest during recent years in boundedness questions linked to forced oscillator equations of the type Here the case we are interested in is a bounded nonlinearityf:R - R for which the limits f (∞) =(x)and f (—∞) =1(x)both exist. Moreover, p is a 27r-periodic forcing, and we ask for sharp conditions which allow us to decide whether all solutions of (1.1) are bounded in the(x¡À)-phase plane.
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References
V.I. Arnold, Mathematical Methods of Classical Mechanics, 2nd edition, Springer-Verlag, Berlin, Heidelberg, New York, Tokyo, 1989.
A.C. Lazer and D.E. Leach, Bounded perturbations of forced harmonic oscillators at resonance, Ann. Math. Pura Appl. 82 (1969), 49–68.
M. Levi, Quasiperiodic motions in superquadratic time-periodic potentials, Comm. Math. Phys. 143 (1991), 43–83.
B. Liu, Boundedness in nonlinear oscillations at resonance, J. Diff. Eq. 153 (1999), 142–174.
J.L. Massera, The existence of periodic solutions of systems of differential equations, Duke Math J. 17 (1950), 457–475.
R. Ortega, Boundedness in a piecewise linear oscillator and a variant of the small twist theorem, Proc. London Math. Soc. 79 (1999), 381–413.
[7] R. Ortega, Twist mappings, invariant curves and periodic differential equations, pp. 79–112.
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Kunze, M. (2001). Remarks on Boundedness of Semilinear Oscillators. In: Grossinho, M.R., Ramos, M., Rebelo, C., Sanchez, L. (eds) Nonlinear Analysis and its Applications to Differential Equations. Progress in Nonlinear Differential Equations and Their Applications, vol 43. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4612-0191-5_22
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DOI: https://doi.org/10.1007/978-1-4612-0191-5_22
Publisher Name: Birkhäuser, Boston, MA
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