The Study On Periodic Solution Problems For Some Classes Of Superquadratic Second Order Hamiltonian Systems

In this thesis, we consider the existence of periodic solutions for some classes of superquadratic second order Hamiltonian systems. This paper is constituted with four chapters.In chapter1, we first introduce the background and some known results about the periodic solutions for Hamiltonian systems, and then state the main results we obtained in this paper and some preliminary knowledge.In chapter2, we consider the following second order Hamiltonian systems where V∈C1(R×RN,R) is T-periodic and has the form V(t,u)-1/2<U(t)u,u>+W(t,u) with U(·) a continuous T-periodic symmetric matrix and W(t,u) a even function in u(<·,·> and|·|always denote the standard inner product and the associated norm in RN respectively). Under the following superquadratic conditionwe prove the existence of infinitely many periodic solutions for these second order Hamiltonian systems.In chapter3, we consider the second order Hamiltonian systems with a parameter, where the potential function has the form Under the assumption that W(t, u) satisfies the superquadratic condition and the parameter λ>0, we prove the existence of periodic solutions for these second order Hamiltonian systems.In chapter4, we consider the second order Hamiltonian systems with a perturbation, where the potential function has the form V(t,u)=1/2<U(t)u,u>+W{t,u)+εG{t,u), and eG(t,u) is the perturbation(ε≥0). For every≥0, we show that these second order Hamiltonian systems possess finitely many periodic solutions under the assumption that W(t, u) satisfies the superquadratic condition, but G(t, u) has no growth condition.