Existence And Multiplicity Of Periodic And Subharmonic Solutions For Hamiltonian Systems

In this doctoral dissertation, by using variational method, we study the ex-istence and multiplicity of periodic solutions, subharmonic solutions and minimal periodic solutions for Hamiltonian system. This dissertation consists of four chap-ters.In the first chapter, we sketch the historical background, the present situation for our studying problems, and our main work in this dissertation.In the second chapter, by using the generalized Saddle Point Theorem, Dual Variational Method and Linking Theorem, the existence and multiplicity of peri-odic solutions for p-Laplacian systems and are studied. At first, we establish the embedding inequalities when p> 1. Then under sub p-condition, convex condition and super p-condition, respectively, some existence criteria are obtained and our results generalize and improve some known ones.In the third chapter, by using the linking theorem, we study the existence of periodic solutions and subharmonic solutions for the second order Hamiltonian system with linear part At first, under the super-quadratic condition, some existence criteria of periodic solutions are obtained. Then for the case that A(t)?A which is the constant matrix, by using the relation between the Fourier series and the eigenvalues of ma-trix, we well decompose the space on which the functional is defined. Subsequently, under tow different cases, by using the generalized Mountain Pass Theorem, some existence criteria of subharmonic solutions are obtained.In the fourth chapter, at first, we study the existence of periodic solutions for the first order Hamiltonian system Under the case that the linear part is semi-positive definite, some known existence results are improved and when the potential H is even, by using the Fountain Theorem, we prove that system has infinitely many periodic solutions. Subse-quently, we study the multiplicity of minimal periodic solutions for the first order autonomous Hamiltonian system Ju(t)+(?)H(u(t))=0. We point out the common mistakes of those related results in known literature and then obtain a new result.