Abstract
In the first part of the paper we consider periodic perturbations of some planar Hamiltonian systems. In a general setting, we detect conditions ensuring the existence and multiplicity of periodic solutions. In the second part, the same ideas are used to deal with some more general planar differential systems.
Highlights
The meaning of the word resonance is well understood for a linear equation of the type x + x = q(t) where λ is a positive constant and q(t) is a 2π-periodic forcing: resonance occurs when all the solutions are unbounded, both in the past and in the future
It seems to be commonly accepted to consider as nonresonance conditions on the function g(x) those ensuring that
Even for the particular case of the scalar equation (1.1), it has been shown in [7] that a nonresonance condition of type (1.5) is sufficient for the existence of a 2 -periodic solution provided that g is differentiable, with a globally bounded derivative
Summary
[13] and the references therein) In this case, “nonresonance conditions” necessarily involve both the functions g(x) and q(t, x), and they are supposed to guarantee the existence of at least one 2 -periodic solution of the differential equation. Even for the particular case of the scalar equation (1.1), it has been shown in [7] that a nonresonance condition of type (1.5) is sufficient for the existence of a 2 -periodic solution provided that g is differentiable, with a globally bounded derivative. We always denote by ⟨⋅ , ⋅⟩ the Euclidean scalar product in R2 , with associated norm| ⋅ |
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