Minimal Period Problems For Discrete Hamiltonian Systems

This dissertation mainly deals with the existence and multiplicity of subharmonic solutions with prescribed minimal periods to discrete Hamiltonian systems by applying the critical point theory including variational methods, duality variational methods, perturbation argument, dual least action principle, minimax theory and geometrical index theory. It is organized as follows by four parts.In Chapter 1, the historical background of problems to be studied and the significance of this dissertation are introduced. Periodic phenomena occur in various fields of natural science. The problem on periodic solutions attracts important attentions of many famous mathematicians all around the world, especially on periodic solutions with prescribed minimal periods. In this chapter, recent development of periodic solutions to discrete Hamiltonian systems and second-order difference equations is also viewed. Several kinds of methods for studying the minimal period problems of differential equations are outlined. And main results of this dissertation are also summarized.Reducing the problem on periodic solutions to discrete Hamiltonian systems which conform the variational principle to the problem on critical points of the corresponding variational functionals is one of main approaches to study the periodic solution problem. In Chapter 2, the existence of subharmonic solutions with prescribed minimal period of a discrete forced pendulum equation and a class of second order discrete Hamiltonian systems are studied. Some new results on the existence of subharmonic solutions with prescribed minimal periods are obtained.By making use of Clark duality, perturbation technique and dual least action principle, the existence of subharmonic solutions to second order subquadratic discrete Hamiltonian systems is studied in Chapter 3. When the Hamiltonian function satisfies subquadratic growth conditions at infinity, some results on the existence of subharmonics which have any large minimal period are obtained. At the same time, some properties of the subharmonics are also studied. When the Hamiltonian function grows subquadratically both at 0 and at infinity, some new results on the existence of subharmonic solutions with minimal period are given.Chapter 4 mainly deals with the existence and multiplicity of subharmonic solutions with minimal periods to first order and second order subquadratic discrete Hamiltonian systems. It is well-known that the problem concerning periodic solutions with prescribed minimal periods is very difficult by using the direct vari-...