Hamilton System Of Homoclinic Solutions And Periodic Solutions

Hamilton system as a nonlinear problem is widely used in the mathematical sci-ences, life sciences, as well as the whole field of social sciences, especially many models in celestial mechanics, plasma physics, space science, and bio-engineering present in the form of Hamilton system(or its disturbance system). The nonlinear analysis main-ly studies all kinds of nonlinear differential equations, the variational method is one of its important study methods. Variational method to differential equations is a method that transfers boundary value problem of differential equations into variational problem in order to prove existence of solutions, determine the number of solutions and seek its approximate solution. In this paper, we study the existence of homoclinic solution of a class of the second order Hamiltonian system by the use of the Mountain Pass Theorem. Then we compare the approaches of solutions of the second order Hamiltonian system with impulsive effects.The thesis is divided into two chapters according to contents.In chapter1, we use a standard version of the Mountain Pass Theorem to study the existence of homoclinic solution and its nontriviality of the following class of second order Hamiltonian system without (AR) condition q+Vq(t,q)=f(t)(HS) where q∈RN and V∈C1(R×RN,R), V(t,q)=-K(t,q)+W(t,q) is T-periodic in t. A map K satisfies the "pinching" condition:b1|q|2(?)K(t,q)(?)b2|q|2.Here the function W∈C1(R×RN,R) needs not to satisfy the global (AR) condition. We give some assumptions weaker than it and prove the existence of homoclinic solution q(t) emanating from0of (HS). A homoclinic orbit is obtained as a limit of2kT-periodic solutions of a certain sequence of the second order differential equations by a standard version of the Mountain Pass Theorem.In chapter2, we use variational methods and critical point theory to discuss the existence and multiplicity of solutions for the following type of second order Hamilton systems with impulsive effects: and where Ï…(ι)=(Ï…1(ι),Ï…1(ι),…,Ï…N(ι))T,A(ι)=[dιm(ι)],ι∈[0,T],dιm∈L∞([0,T]),(?)l,m=1,2,…,N.tj(j=1,2,…,p),and0=t0<t1<t2<…<tp+1=T, Iij:Râ†'R(i-1,2,…,N,j=1,2,…,p)is a continuous function.F:[0,T]×RN R satisfies:(A)(?)υ∈RN,F(t,Ï…)is measurable in tï¼›for almost every t∈[0,T], F(t,Ï…)is continuously differentiable in Ï…,and satisfies the standard condition:(?)b>0,(or(?)α∈C(R+,R+),b∈L1(0,T;R+),such that (?)υ∈RN and almost every t∈[0,T], we have...