The Existence And Multiplicity Of Periodic Solutions And Non-periodic Solutions For Hamiltonian Systems

This thesis consists of three parts.In the first part, with the evenness assumption, we prove the result on the existence of infinitely many nontrivial periodic solutions for non-autonomous second order Hamiltonian systems, including the two cases where the potential is respectively asymptotically quadratic and superquadratic near infinity.In the second part, we first prove the result on the existence of infinitely many homoclinic solutions for a class of subquadratic non-autonomous second order Hamiltonian systems under the evenness assumption, then for a class of asymptotically quadratic non-autonomous second order Hamiltonian systems, without the evenness assumption we obtain the existence result, while in the presence of the evenness, we obtain the finite-multiplicity result.In the last part, for a class of first order resonant Hamiltonian systems which may do not satisfy the (PS) condition, we obtain the existence result on a special kind of solutions which start from and end in the same Lagrangian subspace of the standard symplectic space R2N, and we also get the existence result on so-called brake solutions for these systems.