P-index theory for linear Hamiltonian systems and multiple solutions for nonlinear Hamiltonian systems

Abstract

For any P ∊ Sp(2n) and B ∊ L∞((0, 1);GLs(R2n)) we establish an index theory for a linear Hamiltonian system and discuss its applications in nonlinear Hamiltonian systems. First we will present two kinds of definitions by making use of the Maslov index and relative Morse index, respectively. Then we will prove the equivalence of these two definitions. Finally, we will make use of these index theories to discuss nontrivial solutions of nonlinear Hamiltonian systems.