The Existence And Multiplicity Of Periodic Solutions For Asymptomatically Linear Discrete Systems

This paper mainly concerns with the existence and multiplicity of periodic solu-tions for a class of second-order difference equations and first-order discrete Hamiltonsystem with nonlinearities growing asymptotically linear both at zero and at infinity.The main idea is to reduce the problems of finding periodic solutions of second-orderdifference equations and discrete Hamilton system to that of seeking critical points ofthe corresponding functional. The historical background of difference equations anddiscrete Hamilton system and recent development of problems under considerationare introduced in Chapter 1. Some preparatory knowledge is also given.In chapter 2, some sufficient conditions are obtained for the existence of pe-riodic solutions of autonomous system. Firstly, the author reduces the problems offinding periodic solutions of second-order difference equations to that of seeking crit-ical points of the corresponding functional. Then by using the critical point theoryand Zp geometrical index theory, the existence and multiplicity of Zp-orbits of periodsolutions are studied for second-order difference equations, where the nonlinearitygrows asymptotically linear both at zero and at infinity and it is resonant at infinity.In chapter 3, Some sufficient conditions are obtained for the existence of periodicsolutions of non-autonomous system by applying the critical point theory and relativeMorse index theory.Chapter 4 concerns with the existence and multiplicity of period solutions forfirst-order discrete Hamilton system when it is resonant at infinity. The main approachused in this chapter is Morse theory.