Infinite Dimensional Cohomology Groups and Periodic Solutions of Asymptotically Linear Hamiltonian Systems

In this paper we study the existence of nontrivial 2π-periodic solutions of asymptotically linear Hamiltonian systems. We consider the case of resonance both at zero and at infinity, and we permit time-dependent asymptotic matrices. Our main tools are an infinite dimensional cohomology theory and a corresponding Morse theory recently constructed by W. Kryszewski and the first author. We develop a method to compute the new critical groups.