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

TitleDouble resonance with Landesman–Lazer conditions for planar systems of ordinary differential equations
Publication TypeJournal Article
Year of Publication2011
AuthorsFonda, A, Garrione, M
JournalJournal of Differential Equations
Volume250
Pagination1052 - 1082
ISSN0022-0396
KeywordsDouble resonance; Landesman–Lazer conditions; Nonlinear planar systems
Abstract

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.

URLhttp://www.sciencedirect.com/science/article/pii/S0022039610002901
DOI10.1016/j.jde.2010.08.006

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