Periodic solutions of forced isochronous oscillators at resonance

Abstract

We study the existence of $2\pi$-periodic solutions for forcednonlinear oscillators at resonance, the nonlinearity being abounded perturbation of a function deriving from an isochronouspotential, i.e. a potential leading to free oscillations that allhave the same period. The family of isochronous oscillatorsconsidered here includes oscillators with jumping nonlinearities,as well as oscillators with a repulsive singularity, to which aparticular attention is paid. The existence results contain, asparticular cases, conditions of Landesman-Lazer type. Even in thecase of perturbed linear oscillators, they improve earlierresults. Multiplicity and non-existence results are also given.