Multiple periodic solutions of Hamiltonian systems confined in a box

Abstract

We consider a nonautonomous Hamiltonian system, T -periodic in time, possibly defined on a bounded space region, the boundary of which consists of singularity points which can never be attained. Assuming that the system has an interior equilibrium point, we prove the existence of infinitely many T -periodic solutions, by the use of a generalized version of the Poincaré -Birkhoff theorem.with a i,j ∈ R ∪ {−∞, +∞}. This means that R can be either a rectangle or an unbounded set, like a quadrant, a half-plane, or even the whole plane R 2 .We assume that the Hamiltonian function may be decomposed as2010 Mathematics Subject Classification. 34C25.

Subharmonic solutions of Hamiltonian systems displaying some kind of sublinear growth

“…We will use phase plane analysis methods, combined with a generalized version of the Poincaré-Birkhoff Theorem for Hamiltonian time maps recently proved by the first author and Ureña in [14]. This last theorem has already been used in [2,5,11,12,14] to prove the multiplicity of periodic solutions for different kinds of systems.…”

Subharmonic solutions of bounded coupled Hamiltonian systems with sublinear growth

“…Secondly, in [24], f (t, x) can be indefinite, the typical example in [11] is a partially superlinear equation, for example,…”

“…Using this theorem Fonda and other authors proved a series of results about the multiplicity of periodic solutions of higher dimensional coupled systems. See [13,7,3,11,12,8].…”

Multi-dimensional $ρ$-almost periodic type functions and applications

“…For multi-periodic solutions of various classes of ordinary differential equations and partial differential equations, we also refer the reader to [8,9,22,27,36,37,43,52,53,55,56,58]. Especially, we would like to mention the investigations of G. Nadin [44]- [46] concerning the space-time periodic reaction-diffusion equations, G. Nadin-L.Rossi [47] concerning transition waves for Fisher-KPP equations, L. Rossi [51] concerning Liouville type results for almost periodic type linear operators, the investigation of B. Scarpellini [54] concerning the space almost periodic solutions of reaction-diffusion equations, and the recent investigation of R. Xie, Z. Xia, J. Liu [65] concerning the quasi-periodic limit functions, (ω 1 , ω 2 )-(quasi)-periodic limit functions and their applications, given only in the two-dimensional setting.…”