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.

%B Advances in Nonlinear Analysis %I De Gruyter %V 5 %P 367–382 %G eng %R 10.1515/anona-2015-0122 %0 Journal Article %J Topol. Methods Nonlinear Anal. %D 2013 %T Generalized Sturm-Liouville boundary conditions for first order differential systems in the plane %A Alessandro Fonda %A Maurizio Garrione %XWe 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.

%B Topol. Methods Nonlinear Anal. %I Nicolaus Copernicus University, Juliusz P. Schauder Centre for Nonlinear Studies %V 42 %P 293–325 %G eng %U https://projecteuclid.org:443/euclid.tmna/1461248981 %0 Journal Article %J Nonlinear Differential Equations and Applications NoDEA %D 2013 %T Planar Hamiltonian systems at resonance: the Ahmad–Lazer–Paul condition %A Alberto Boscaggin %A Maurizio Garrione %XWe 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.

%B Nonlinear Differential Equations and Applications NoDEA %V 20 %P 825–843 %8 Jun %G eng %U https://doi.org/10.1007/s00030-012-0181-2 %R 10.1007/s00030-012-0181-2 %0 Journal Article %J Differential Integral Equations %D 2012 %T Resonance at the first eigenvalue for first-order systems in the plane: vanishing Hamiltonians and the Landesman-Lazer condition %A Maurizio Garrione %B Differential Integral Equations %I Khayyam Publishing, Inc. %V 25 %P 505–526 %8 05 %G eng %U https://projecteuclid.org:443/euclid.die/1356012676 %0 Journal Article %J Journal of Differential Equations %D 2011 %T Double resonance with Landesman–Lazer conditions for planar systems of ordinary differential equations %A Alessandro Fonda %A Maurizio Garrione %K Double resonance %K Landesman–Lazer conditions %K Nonlinear planar systems %XWe 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.

%B Journal of Differential Equations %V 250 %P 1052 - 1082 %G eng %U http://www.sciencedirect.com/science/article/pii/S0022039610002901 %R https://doi.org/10.1016/j.jde.2010.08.006 %0 Journal Article %J Advanced Nonlinear Studies %D 2011 %T Nonlinear resonance: a comparison between Landesman-Lazer and Ahmad-Lazer-Paul conditions %A Alessandro Fonda %A Maurizio Garrione %XWe 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.

%B Advanced Nonlinear Studies %I Advanced Nonlinear Studies, Inc. %V 11 %P 391–404 %G eng %R 10.1515/ans-2011-0209 %0 Journal Article %J Le Matematiche %D 2011 %T Resonance and Landesman-Lazer conditions for first order systems in R^2 %A Maurizio Garrione %XThe 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].

%B Le Matematiche %V 66 %P 153–160 %G eng %0 Journal Article %J Nonlinear Analysis: Theory, Methods & Applications %D 2011 %T Resonance and rotation numbers for planar Hamiltonian systems: Multiplicity results via the Poincaré–Birkhoff theorem %A Alberto Boscaggin %A Maurizio Garrione %K Multiple periodic solutions %K Poincaré–Birkhoff theorem %K Resonance %K Rotation number %XIn 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.

%B Nonlinear Analysis: Theory, Methods & Applications %V 74 %P 4166 - 4185 %G eng %U http://www.sciencedirect.com/science/article/pii/S0362546X11001817 %R https://doi.org/10.1016/j.na.2011.03.051