# American Institute of Mathematical Sciences

November  2002, 8(4): 907-930. doi: 10.3934/dcds.2002.8.907

## Periodic solutions of forced isochronous oscillators at resonance

 1 Université Catholique de Louvain, Institut de Mathématique Pure et Appliquée, Chemin du Cyclotron, 2 , B-1348 Louvain-la-Neuve, Belgium, Belgium 2 Laboratoire Jacques-Louis Lions, Université de Paris 6, 4 place Jussieu BC 187, 75252, Paris, France

Received  March 2001 Revised  April 2002 Published  July 2002

We study the existence of $2\pi$-periodic solutions for forced nonlinear oscillators at resonance, the nonlinearity being a bounded perturbation of a function deriving from an isochronous potential, i.e. a potential leading to free oscillations that all have the same period. The family of isochronous oscillators considered here includes oscillators with jumping nonlinearities, as well as oscillators with a repulsive singularity, to which a particular attention is paid. The existence results contain, as particular cases, conditions of Landesman-Lazer type. Even in the case of perturbed linear oscillators, they improve earlier results. Multiplicity and non-existence results are also given.
Citation: D. Bonheure, C. Fabry, D. Smets. Periodic solutions of forced isochronous oscillators at resonance. Discrete & Continuous Dynamical Systems - A, 2002, 8 (4) : 907-930. doi: 10.3934/dcds.2002.8.907
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