We prove the existence of periodic solutions of some infinite-dimensional nearly integrable Hamiltonian systems, bifurcating from infinite-dimensional tori, by the use of a generalization of the Poincaré–Birkhoff Theorem.

%B NONLINEAR ANALYSIS %G eng %U https://doi.org/10.1016/j.na.2019.111720 %R 10.1016/j.na.2019.111720 %0 Journal Article %J TOPOLOGICAL METHODS IN NONLINEAR ANALYSIS %D 2019 %T On the topological degree of planar maps avoiding normal cones %A Alessandro Fonda %A Giuliano Klun %XThe classical Poincaré-Bohl theorem provides the existence of a zero for a function avoiding external rays. When the domain is convex, the same holds true when avoiding normal cones.

We consider here the possibility of dealing with nonconvex sets having inward corners or cusps, in which cases the normal cone vanishes. This allows us to deal with situations where the topological degree may be strictly greater than $1$.

We provide a geometric assumption which unifies and generalizes the conditions proposed in [11], [12], so to obtain a higher dimensional version of the Poincaré–Birkhoff fixed point Theorem for Poincaré maps of Hamiltonian systems.

%B Journal of Differential Equations %V 262 %P 1064 - 1084 %G eng %U http://www.sciencedirect.com/science/article/pii/S0022039616303278 %R https://doi.org/10.1016/j.jde.2016.10.002 %0 Journal Article %J Annali di Matematica Pura ed Applicata (1923 -) %D 2016 %T Generalizing the Poincaré–Miranda theorem: the avoiding cones condition %A Alessandro Fonda %A Paolo Gidoni %XAfter proposing a variant of the Poincaré–Bohl theorem, we extend the Poincaré–Miranda theorem in several directions, by introducing an avoiding cones condition. We are thus able to deal with functions defined on various types of convex domains, and situations where the topological degree may be different from \$\$\backslashpm \$\$±1. An illustrative application is provided for the study of functionals having degenerate multi-saddle points.

%B Annali di Matematica Pura ed Applicata (1923 -) %V 195 %P 1347–1371 %8 Aug %G eng %U https://doi.org/10.1007/s10231-015-0519-6 %R 10.1007/s10231-015-0519-6 %0 Journal Article %J Advances in Nonlinear Analysis %D 2016 %T Periodic perturbations of Hamiltonian systems %A Alessandro Fonda %A Maurizio Garrione %A Paolo Gidoni %XWe 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 Nonlinear Analysis: Theory, Methods & Applications %D 2015 %T A permanence theorem for local dynamical systems %A Alessandro Fonda %A Paolo Gidoni %K Lotka–Volterra %K permanence %K Predator–prey %K Uniform persistence %XWe provide a necessary and sufficient condition for permanence related to a local dynamical system on a suitable topological space. We then present an illustrative application to a Lotka–Volterra predator–prey model with intraspecific competition.

%B Nonlinear Analysis: Theory, Methods & Applications %V 121 %P 73 - 81 %G eng %U http://www.sciencedirect.com/science/article/pii/S0362546X14003332 %R https://doi.org/10.1016/j.na.2014.10.011 %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 Advanced Nonlinear Studies %D 2013 %T Periodic bouncing solutions for nonlinear impact oscillators %A Alessandro Fonda %A Andrea Sfecci %B Advanced Nonlinear Studies %I Advanced Nonlinear Studies, Inc. %V 13 %P 179–189 %G eng %R 10.1515/ans-2013-0110 %0 Journal Article %J Journal of Differential Equations %D 2012 %T A general method for the existence of periodic solutions of differential systems in the plane %A Alessandro Fonda %A Andrea Sfecci %K Nonlinear dynamics %K Periodic solutions %XWe propose a general method to prove the existence of periodic solutions for planar systems of ordinary differential equations, which can be used in many different circumstances. Applications are given to some nonresonant cases, even for systems with superlinear growth in some direction, or with a singularity. Systems “at resonance” are also considered, provided a Landesman–Lazer type of condition is assumed.

%B Journal of Differential Equations %V 252 %P 1369 - 1391 %G eng %U http://www.sciencedirect.com/science/article/pii/S0022039611003196 %R https://doi.org/10.1016/j.jde.2011.08.005 %0 Journal Article %J Differential Integral Equations %D 2012 %T Periodic solutions of a system of coupled oscillators with one-sided superlinear retraction forces %A Alessandro Fonda %A Andrea Sfecci %B Differential Integral Equations %I Khayyam Publishing, Inc. %V 25 %P 993–1010 %8 11 %G eng %U https://projecteuclid.org:443/euclid.die/1356012248 %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