Subharmonic solutions of first-order Hamiltonian systems

Abstract

The aim of this article is to study subharmonic solutions of superquadratic and asymptotically (constant) linear nonautonomous Hamiltonian systems in R2n{{\mathbb{R}}}^{2n} respectively, and to improve the results in Professor Liu’s [Subharmonic solutions of Hamiltonian systems, Nonlinear Anal. 42 (2000), no. 2, 185–198, Ser. A: Theory Methods]. For the TT-periodic Hamiltonian system, we show that for each integer j≥1j\ge 1, there exists a nonconstant jTjT-periodic solution xj{x}_{j} such that xj{x}_{j} and xpj{x}_{pj} (p∈Np\in {\mathbb{N}}) are geometrically distinct provided p>2np\gt 2n; for n=1n=1, xj{x}_{j} and xpj{x}_{pj} (p∈Np\in {\mathbb{N}}) are geometrically distinct provided p>1p\gt 1; if xj{x}_{j} is nondegenerate, then xj{x}_{j} and xpj{x}_{pj} (p∈Np\in {\mathbb{N}}) are geometrically distinct provided p>1p\gt 1.

Keywords

• subharmonic solutions
• maslov-type index
• superquadratic hamiltonian system
• asymptotically linear hamiltonian system
• 70h05
• 34c25
• 58e05