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Resonance and rotation numbers for planar Hamiltonian systems: Multiplicity results via the Poincaré–Birkhoff theorem

TitleResonance and rotation numbers for planar Hamiltonian systems: Multiplicity results via the Poincaré–Birkhoff theorem
Publication TypeJournal Article
Year of Publication2011
AuthorsBoscaggin, A, Garrione, M
JournalNonlinear Analysis: Theory, Methods & Applications
Volume74
Pagination4166 - 4185
ISSN0362-546X
KeywordsMultiple periodic solutions; Poincaré–Birkhoff theorem; Resonance; Rotation number
Abstract

In the general setting of a planar first order system (0.1)u′=G(t,u),u∈R2, with G:[0,T]×R2→R2, we study the relationships between some classical nonresonance conditions (including the Landesman–Lazer one) — at infinity and, in the unforced case, i.e. G(t,0)≡0, at zero — and the rotation numbers of “large” and “small” solutions of (0.1), respectively. Such estimates are then used to establish, via the Poincaré–Birkhoff fixed point theorem, new multiplicity results for T-periodic solutions of unforced planar Hamiltonian systems Ju′=∇uH(t,u) and unforced undamped scalar second order equations x″+g(t,x)=0. In particular, by means of the Landesman–Lazer condition, we obtain sharp conclusions when the system is resonant at infinity.

URLhttp://www.sciencedirect.com/science/article/pii/S0362546X11001817
DOI10.1016/j.na.2011.03.051

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