Multiple periodic solutions for nonautonomous asymptotically linear Hamiltonian systems.

摘要

In this thesis we study the multiplicity of periodic solutions for the following Hamiltonian system: Jx&d2;+H' t,x=0, * where J is simplectic matrix of order 2 N, H : RxR2N →R is continuous with Ht,˙ convex and differentiable on R2N for each t∈R,H˙ ,u being T-periodic for each u∈R2N , and H't,u continuous. System (*) is called Asymptotically linear if H satisfies ∣H't,x -A0x/x&vbm0;→ 0, H1 as x→0 , and ∣H't,x -Ainfinityx/x&vbm0; →0, H2 as x→infinity , where A0 and Ainfinity are two constant symmetric positive definite matrices of order 2 N x 2N. First, a lower bound for the number of periodic solutions, which depends on the Morse indices for the corresponding linear systems at the origin and infinity, is given by applying Morse theory and a variational theorem by [Costa and Willem(1986)]. The Morse indices for a linear system with constant positive definite matrix A in the well known results such as Ekeland(1986) were computed indirectly via the eigenvalues of JA, which are of pure imaginary. In the second part of the thesis, using the Fourier representation for the periodic solutions, we derive a new formula for the Morse index for linear Hamiltonian systems, which is calculated by the eigenvalues of A. Finally, a generalization of the above results are also given for the time depending matrices A0 and Ainfinity .