Ekeland Index Theories Of Periodic Solutions And Symmetric Periodic Solutions Of Convex Hamiltonian Systems

The particle movement of celestial mechanics can be attributed to a nonlinear ordinary differential equation which is called a Hamiltonian system. Since the simplest form of celestial motion-periodic motion-corresponds to the periodic solution of those systems, then the problems of the existence and other properties of periodic orbits of Hamiltonian systems are very important research project concerned by many mathe-maticians. In the past three decades, Variational method and Morse theory were suc-cessfully applied to study Hamiltonian system, and meanwhile, some index theories for Hamiltonian systems were invented, especially the Maslov-type index theory for sym-plectic path and Ekeland index theory have many applications.In this thesis,we mainly study Ekeland indx theory for convex Hamiltonian systems and adopt literature and logical reasoning methods to study the following issues. Firstly, by imitating the index theory of positive definite Hamiltonian system in the classic liter-ature of linear Hamiltonian systems, we show the similar index theory and give a new and simple proof to the fact that the average index is greater than 2 for convex Hamilto-nian systems.Secondly, we mainly study the relationship between two kinds of Ekeland indexes for symmetric periodic solution of convex Hamiltonian systems.