Periodic Solutions And Boundary Value Problems For Discrete Hamiltonian Systems

This PH. D. Thesis mainly concerns the applications of critical point theory to investigate periodic solutions, multiple periodic solutions and boundary value problems (BVP for short) for discrete Hamiltonian systems or second order nonlinear difference equations. It is composed of four chapters.In Chapter 1, It is introduced that the historical background and the recent development of problems to be studied, and main results of this paper are also outlined.In Chapter 2, existence and multiplicity of the periodic solutions for a class of general second-order nonlinear difference equations are investigated, which is a generalization of the existence of the periodic solutions for some second-order differences in the literature. By using critical point theory, the variational framework for discrite Hamiltonian sytems or second-order difference equations is established, and the existence of periodic solutions for the given nonlinear second-order difference equations is transfered into that of critical points of some functional. Next, by Saddle Point Theorm and Linking Theorem in critical point theory, existence and multiplicity of periodic solutions and subharmonic solutions for the given second-order difference equations are investigated systematically and some new existence and multiplicity results are obtained.Existence and multiplicity of periodic solutions for some special second order nonlinear difference equations are discussed in Chapter 3. By using Mountain Pass Lemma and Saddle Point Theorem in critical point theory, some new sufficient conditions are obtained. For a special case, a necessary and sufficient condition for the existence of periodic solutions for a given nonlinear difference equation is obtained by algebraic method.Finally in Chapter 4, boundary value problems for a class of second-order semilinear difference equations are studied. By Riccati method, some necessary and sufficient conditions for disconjugacy and disfocality of a second order nonlinear difference equation are obtained. Moreover, by establishing variational structure and applying critical point method, the existence of solutions of boundary value problems for the second order nonlinear difference equations on a finite...