Existence Of The Solution Of Several Classes Of Hamiltonian Systems

Most of the content of the nonlinear functional analysis can be traced back to the 1930s. Now generally accepted several aspects, such as the variational method, and the achievements has become played a role from the functional analysis starts to be subject; Topology methods and achievements, fixed point and topological degree theory, and even the analytical method, so is roughly. As subjects which has a long history, for example news of linear partial differential equations theory, often each have their specific purpose, the courses of the nonlinear functional analysis, nature also have their characteristic. Present, in order to achieve the grand goal of building the four modernizations, our country urgent needs to cultivates talents. Requirements that learners who study the nonlinear functional analysis are not confined to be personnel who deal with certain job in functional analysis, such as, various non-linear problem has come up from the fields of physics, chemisty, mathematics, biology, medicine, eco-nomics, engineering, cybernetics, and these problems has aroused people's widespread attention day by day. However, the nonlinear functional analysis offers effective the-oretic tools for these problems, and it is a subject of profound theories and broad applications. The nonlinear functional analysis bases on nonlinear problems of math and science, constructs general theories and methods, and plays an important role in dealing with all kinds of nonlinear integral equations, nonlinear differential equations and partial differential equations. So the nonlinear analysis has become one important research directions in modern mathematics. The nonlinear functional analysis is an important branch in nonlinear analysis, because it can explain well various the natural phenomenon. The boundary value problem of nonlinear differential equation stems from the applied mathematics, the physics, the cybernetics and several kinds of appli-cation discipline. It is one of most active domains of functional analysis at present. The nonlinear differential equation boundary value problem in Banach space is also the hot spot which has been discussed in recent years. So it become a very important domain of differential equation research at present.In this paper, we use the mountain-pass theorem, the variational methods and the generalized linking theorem, to study the homoclinic solutions for some kinds of boundary value problems for nonlinear differential equation and we apply the main results to the boundary value problem for the differential equation. The thesis is divided into three chapters: In Chapter 1, mainly narrated the place of the nonlinear functional analysis andthe problems which to study with the innovation. In Chapter 2, we study the existence of homoclinic solutions of the second-orderHamiltonian systemwhere-K(t, u(t))+W(t, u(t))∈C1(R x RN, R) be T-periodic in t, T> 0, K(t,0)= 0, K(t, u)≥b|u|2, where b> 0 andâ–½W is superlinear at infinity. One homoclinic solution is obtained as a limit of solutions of a sequence of periodic second-order differential equations.In Chapter 3, we deal with the existence of solutions in the Hamiltonian elliptic systemswhere G(x,w) is nonperiodic in x∈RN and superquadratic in w∈R2 as|w|â†'∞with w= (u,v). With certain mild conditions we obtain the solutions via critical point theorem for strongly indefinite problem.