In the general setting of a planar first order system u′= G(t,u),u∈R2, with G:[0,T] ×R2→R2, we study the relationships between some classical nonresonance conditions (including the LandesmanLazer 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 PoincarBirkhoff 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 LandesmanLazer condition, we obtain sharp conclusions when the system is resonant at infinity.
Boscaggin, A., Garrione, M. (2011). Resonance and rotation numbers for planar Hamiltonian systems: Multiplicity results via the PoincarBirkhoff theorem. NONLINEAR ANALYSIS, 74(12), 4166-4185 [10.1016/j.na.2011.03.051].
Resonance and rotation numbers for planar Hamiltonian systems: Multiplicity results via the PoincarBirkhoff theorem
BOSCAGGIN, ALBERTO;GARRIONE, MAURIZIO
2011
Abstract
In the general setting of a planar first order system u′= G(t,u),u∈R2, with G:[0,T] ×R2→R2, we study the relationships between some classical nonresonance conditions (including the LandesmanLazer 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 PoincarBirkhoff 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 LandesmanLazer condition, we obtain sharp conclusions when the system is resonant at infinity.
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