We prove existence and multiplicity results for periodic solutions of Hamiltonian systems, by the use of a higher dimensional version of the Poincaré–Birkhoff fixed point theorem. The first part of the paper deals with periodic perturbations of a completely integrable system, while in the second part we focus on some suitable global conditions, so to deal with weakly coupled systems.

1 aFonda, Alessandro1 aGarrione, Maurizio1 aGidoni, Paolo uhttps://www.math.sissa.it/publication/periodic-perturbations-hamiltonian-systems01258nas a2200145 4500008004100000245010200041210006900143260008500212300001400297490000700311520069200318100002201010700002301032856005701055 2013 eng d00aGeneralized Sturm-Liouville boundary conditions for first order differential systems in the plane0 aGeneralized SturmLiouville boundary conditions for first order d bNicolaus Copernicus University, Juliusz P. Schauder Centre for Nonlinear Studies a293–3250 v423 aWe study asymptotically positively homogeneous first order systems in the plane, with boundary conditions which are positively homogeneous, as well. Defining a generalized concept of Fučík spectrum which extends the usual one for the scalar second order equation, we prove existence and multiplicity of solutions. In this way, on one hand we extend to the plane some known results for scalar second order equations (with Dirichlet, Neumann or Sturm-Liouville boundary conditions), while, on the other hand, we investigate some other kinds of boundary value problems, where the boundary points are chosen on a polygonal line, or in a cone. Our proofs rely on the shooting method.

1 aFonda, Alessandro1 aGarrione, Maurizio uhttps://projecteuclid.org:443/euclid.tmna/146124898101133nas a2200157 4500008004100000022001400041245008000055210007300135260000800208300001400216490000700230520064600237100002300883700002300906856004600929 2013 eng d a1420-900400aPlanar Hamiltonian systems at resonance: the Ahmad–Lazer–Paul condition0 aPlanar Hamiltonian systems at resonance the Ahmad–Lazer–Paul con cJun a825–8430 v203 aWe consider the planar Hamiltonian system\$\$Ju^{\backslashprime} = \backslashnabla F(u) + \backslashnabla_u R(t,u), \backslashquad t \backslashin [0,T], \backslash,u \backslashin \backslashmathbb{R}^2,\$\$with F(u) positive and positively 2-homogeneous and \$\${\backslashnabla_{u}R(t, u)}\$\$sublinear in u. By means of an Ahmad-Lazer-Paul type condition, we prove the existence of a T-periodic solution when the system is at resonance. The proof exploits a symplectic change of coordinates which transforms the problem into a perturbation of a linear one. The relationship with the Landesman–Lazer condition is analyzed, as well.

1 aBoscaggin, Alberto1 aGarrione, Maurizio uhttps://doi.org/10.1007/s00030-012-0181-200498nas a2200121 4500008004100000245013300041210006900174260003300243300001400276490000700290100002300297856005600320 2012 eng d00aResonance at the first eigenvalue for first-order systems in the plane: vanishing Hamiltonians and the Landesman-Lazer condition0 aResonance at the first eigenvalue for firstorder systems in the bKhayyam Publishing, Inc.c05 a505–5260 v251 aGarrione, Maurizio uhttps://projecteuclid.org:443/euclid.die/135601267601344nas a2200181 4500008004100000022001400041245010900055210007100164300001600235490000800251520070400259653002100963653003300984653002901017100002201046700002301068856007101091 2011 eng d a0022-039600aDouble resonance with Landesman–Lazer conditions for planar systems of ordinary differential equations0 aDouble resonance with Landesman–Lazer conditions for planar syst a1052 - 10820 v2503 aWe 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.

10aDouble resonance10aLandesman–Lazer conditions10aNonlinear planar systems1 aFonda, Alessandro1 aGarrione, Maurizio uhttp://www.sciencedirect.com/science/article/pii/S002203961000290100992nas a2200145 4500008004100000245009400041210006900135260003700204300001400241490000700255520041000262100002200672700002300694856012900717 2011 eng d00aNonlinear resonance: a comparison between Landesman-Lazer and Ahmad-Lazer-Paul conditions0 aNonlinear resonance a comparison between LandesmanLazer and Ahma bAdvanced Nonlinear Studies, Inc. a391–4040 v113 aWe show that the Ahmad-Lazer-Paul condition for resonant problems is more general than the Landesman-Lazer one, discussing some relations with other existence conditions, as well. As a consequence, such a relation holds, for example, when considering resonant boundary value problems associated with linear elliptic operators, the p-Laplacian and, in the scalar case, with an asymmetric oscillator.

1 aFonda, Alessandro1 aGarrione, Maurizio uhttps://www.math.sissa.it/publication/nonlinear-resonance-comparison-between-landesman-lazer-and-ahmad-lazer-paul-conditions01079nas a2200121 4500008004100000245007600041210006900117300001400186490000700200520062100207100002300828856010600851 2011 eng d00aResonance and Landesman-Lazer conditions for first order systems in R^20 aResonance and LandesmanLazer conditions for first order systems a153–1600 v663 aThe first part of the paper surveys the concept of resonance for $T$-periodic nonlinear problems. In the second part, some new results about existence conditions for nonlinear planar systems are presented. In particular, the Landesman-Lazer conditions are generalized to systems in $\mathbbR^2$ where the nonlinearity interacts with two resonant Hamiltonians. Such results apply to second order equations, generalizing previous theorems by Fabry [4] (for the undamped case), and Frederickson-Lazer [9] (for the case with friction). The results have been obtained with A. Fonda, and have been published in [8].

1 aGarrione, Maurizio uhttps://www.math.sissa.it/publication/resonance-and-landesman-lazer-conditions-first-order-systems-r201464nas a2200193 4500008004100000022001400041245012500055210006900180300001600249490000700265520078200272653003201054653003301086653001401119653002001133100002301153700002301176856007101199 2011 eng d a0362-546X00aResonance and rotation numbers for planar Hamiltonian systems: Multiplicity results via the Poincaré–Birkhoff theorem0 aResonance and rotation numbers for planar Hamiltonian systems Mu a4166 - 41850 v743 aIn 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.

10aMultiple periodic solutions10aPoincaré–Birkhoff theorem10aResonance10aRotation number1 aBoscaggin, Alberto1 aGarrione, Maurizio uhttp://www.sciencedirect.com/science/article/pii/S0362546X11001817