A generalization of the parallelogram law to higher dimensions

We propose a generalization of the parallelogram identity in any dimension N 65 2, establishing the ratio of the quadratic mean of the diagonals to the quadratic mean of the faces of a parallelotope. The proof makes use of simple properties of the exterior product of vectors

Subharmonic solutions of Hamiltonian systems displaying some kind of sublinear growth

We prove the existence and multiplicity of subharmonic solutions for Hamiltonian systems obtained as perturbations of N planar uncoupled systems which, e.g., model some type of asymmetric oscillators. The nonlinearities are assumed to satisfy Landesman\u2013Lazer conditions at the zero eigenvalue, and to have some kind of sublinear behavior at infinity. The proof is carried out by the use of a generalized version of the Poincar\ue9\u2013Birkhoff Theorem. Different situations, including Lotka\u2013Volterra systems, or systems with singularities, are also illustrated

Periodic solutions of weakly coupled superlinear systems

By the use of a higher dimensional version of the Poincar\ue9\u2013Birkhoff theorem, we are able to generalize a result of Jacobowitz and Hartman, thus proving the existence of infinitely many periodic solutions for a weakly coupled superlinear system

Periodic solutions of nearly integrable Hamiltonian systems bifurcating from infinite-dimensional tori

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

An extension of the Poincaré–Birkhoff Theorem coupling twist with lower and upper solutions

In 1983, Conley and Zehnder proved a remarkable theorem on the periodic problem associated with a general Hamiltonian system, giving a partial answer to a conjecture by Arnold. Their pioneering paper has been extended in different directions by several authors. In 2017, Fonda and Ureña established a deeper relation between the results by Conley and Zehnder and the Poincaré–Birkhoff Theorem. In 2020, Fonda and Gidoni pursued along this path in order to treat systems whose Hamiltonian function includes a nonresonant quadratic term. It is the aim of this paper to further extend this fertile theory to Hamiltonian systems which, besides the periodicity-twist conditions always required in the Poincaré–Birkhoff Theorem, also include a term involving a pair of well-ordered lower and upper solutions. Phase-plane analysis techniques are used in order to recover a saddle-type dynamics permitting us to apply the above mentioned results.Ministerio de Economía y Competitividad PRE2018-083803 MINECOEuropean Regional Development Fund MTM2017-82348-C2-1-P ERDFUniversità degli Studi di Trieste UniT

An infinite-dimensional version of the Poincar\ue9-Birkhoff theorem on the Hilbert cube

We propose a version of the Poincar\ue9-Birkhoff theorem for infinite-dimensional Hamiltonian systems, which extends a recent result by Fonda and Ure\uf1a. The twist condition, adapted to a Hilbert cube, is spread on a sequence of approximating finite-dimensional systems. Some applications are proposed to pendulum-like systems of infinitely many ODEs. We also extend to the infinite-dimensional setting a celebrated theorem by Conley and Zehnder

Periodic solutions of discontinuous second order differential equations. The porpoising effect

We construct a mathematical model in order to study the so called porpoising effect in racing cars, and prove that, when adding a small periodic perturbation, large-amplitude subharmonic solutions may aris

On the Dirichlet problem associated with bounded perturbations of positively-(p, q)- homogeneous Hamiltonian systems

The existence of solutions for the Dirichlet problem associated to bounded perturbations of positively-(p; q)-homogeneous Hamiltonian systems is considered both in nonresonant and resonant situations. In order to deal with the resonant case, the existence of a couple of lower and upper solutions is assumed. Both the well-ordered and the non-well-ordered cases are analysed. The proof is based on phase-plane analysis and topological degree theory