Double resonance with Landesman–Lazer conditions for planar systems of ordinary differential equations

We prove the existence of periodic solutions for first order planar systems at resonance. The nonlinearity is indeed allowed to interact with two positively homogeneous Hamiltonians, both at resonance, and some kind of Landesman–Lazer conditions are assumed at both sides. We are thus able to obtain,...

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Bibliographic Details
Published in:Journal of Differential Equations 2011-01, Vol.250 (2), p.1052-1082
Main Authors: Fonda, Alessandro, Garrione, Maurizio
Format: Article
Language:English
Subjects:
Double resonance
Landesman–Lazer conditions
Nonlinear planar systems
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Summary:We prove the existence of periodic solutions for first order planar systems at resonance. The nonlinearity is indeed allowed to interact with two positively homogeneous Hamiltonians, both at resonance, and some kind of Landesman–Lazer conditions are assumed at both sides. We are thus able to obtain, as particular cases, the existence results proposed in the pioneering papers by Lazer and Leach (1969) [27], and by Frederickson and Lazer (1969) [18]. Our theorem also applies in the case of asymptotically piecewise linear systems, and in particular generalizes Fabry's results in Fabry (1995) [10], for scalar equations with double resonance with respect to the Dancer–Fučik spectrum.
ISSN:0022-0396
1090-2732
DOI:10.1016/j.jde.2010.08.006