Applications Of Variational Methods In The Periodic Boundary Value Problems Of Discrete Hamiltonian Systems

The study to nonlinear Hamiltonian systems has been an important subject for mathematicians and physicists all the time. In recent years, the results obtained in this filed have had a great effect on nonlinear analysis, algebra topology, mathematical physics, differential geometry, etc.Application of variational methods to differential equations is to change the existence of solutions into the existence of critical points of some variational functional. The variational theory has a rich content. In this paper, we mainly use variational methods, including the least action principle, minimax theorem and Morse theory, to deal with the existence of solutions of discrete Hamiltonian systems with periodic boundary value conditions.This Ph.D thesis is organized as follows. In Chapter 1, we give an introduction to the background of the field we concerned.In Chapter 2, by using Morse theory, we study the existence of periodic solutions of first-order discrete systems. Under the assumption that the system has a non-resonance trivial solution, we consider the case when the nonlinearity is asymptotically linear or super-linear at infinity, and obtain some results.In Chapter 3, we use Morse theory and minimax theorem to discuss the existence of periodic solutions of second-order asymptotically linear discrete Hamiltonian systems and obtain a result.For second-order discrete Hamiltonian systems with potential changing sign, Chapter 4 studies the system which is asymptotically super-quadratic or sub-quadratic. The existence results of periodic solutions are achieved. The main tool used here is Morse theory.The existence of solutions of first-order discrete Hamiltonian systems with boundary value conditions is studied in Chapter 5. The corresponding variational functional is constructed. By means of the saddle point theorem and the least action principle, the existence results of solutions are obtained.By combining the Clarke duality with the least action principle and the perturbed technique, Chapter 6 studies the existence of periodic solutions of first-order convex discrete systems with forced term. A dual action functional is introduced. The periodic solution of the system corresponds to the critical point of dual action functional.By using the duality introduced in Chapter 6, Chapter 7 studies the subharmonic so-...