Existence And Multiplicity Of Homoclinic Orbits For The First-order Hamiltonian Systems

In this paper,we mainly study the following first-order Hamiltonian systems:(?)(t)=JH_u(t,u)+f(t)(HS)Where J is a standard symplectic matrix,H(t,u)=1/2L(t)u·u+W(t,u).We mainly use the critical point theory of strong indefinite functional of modern variational method, through establishing the variational frame, linking theorem and functional corresponding to the Hamiltonian systems, to find the approximate critical sequence corresponding to the systems, and then get the existence and multiplicity results of homoclinic orbits of the systems. In this paper we consider the situation of homoclinic orbits of the first-order Hamiltonian systems with the following two conditions:The first part mainly studies the related conclusion of the homoclinic orbits of the system(HS) when the disturbance term f(t) =0 and the reduced AR super quadratic conditions satisfied by the nonlinear term W(t,u).The second part mainly studies the related conclusion of the homoclinic orbits of the system(HS) when the disturbance term f(t)≠0 and the reduced AR super quadratic conditions satisfied by the nonlinear term W(t,u).