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

Resonant Sturm–Liouville Boundary Value Problems for Differential Systems in the Plane

“…The central point of our proof is given by Lemma 2.4, which provides a sharp interpretation of the Landesman-Lazer condition in terms of the (noninteger) number of windings of large solutions to (2.17) around the origin. This property was already highlighted (for complete windings) in [3,Section 4], and then applied, together with the Poincaré-Birkhoff fixed point theorem, to obtain multiplicity of T -periodic solutions for planar Hamiltonian systems like Jz = ∇ z H(t, z), with ∇ z H(t, 0) ≡ 0 and exhibiting a "gap" between zero and infinity. In the same spirit, we can use our Landesman-Lazer conditions to obtain multiple solutions for resonant Sturm-Liouville problems, improving the results in [11,Section 5].…”

“…J − ) to make sense, a suitable L 1 -control from above (resp. below) on R(t, ·) is in this case needed (see, for instance, [3]). For the sake of simplicity, we have preferred to assume the two-sided boundedness condition (2.12).…”

“…The corresponding statement in the periodic framework was obtained in [10], using a degree theoretical approach. Here, instead, we take advantage of a sharp dynamical interpretation of the Landesman-Lazer condition first developed in [3] (see Remark 2.7), and we tackle our problem by means of an elementary planar shooting technique. This is done in detail in Section 2, while in Section 3 we give some corollaries dealing with Sturm-Liouville boundary value problems associated with (1.3), to be compared with the results in [1,5,6].…”

Lower bound on the number of periodic solutions for asymptotically linear planar Hamiltonian systems

“…We remark that similar conditions at infinity have been proposed for higher dimensional Hamiltonian systems in [10,11,16], and are used to estimate rotations in second order ODEs, e.g. in [3,8].…”

A systematic approach to nonresonance conditions for periodically forced planar Hamiltonian systems

“…As a first example of application we can show that, when H satisfies (2.9), the system (2.1) is isochronous. Indeed, the homogeneity property implies that Ω(E) = √ E Ω(1) , for any E ≥ 0, so that a(E) = E a (1). Consequently, the period is given by T(E) = a � (E) = a(1).…”