Periodic Solutions And Boundary Value Problems Of Discrete Hamiltonian Systems

This dissertation deals with the existence and multiplicity of periodic solutions and solutions of boundary value problems of discrete Hamiltonian systems by applying Morse theory,degree theory,minimax methods,Z_p-geometrical index theory,etc.These results will motivate the development of qualitative theory of discrete systems.This dissertation is divided into six chapters.The main contents are as follows.Chapter 1 gives a brief introduction to the historical background,status and the up-to-date progress for all the investigated problems together with preliminary tools and main results in this dissertation.Chapter 2 concerns the existence and multiplicity of p periodic solutions of first order asymptotically linear discrete Hamiltonian systems,where p＞2 is a given prime integer.By using both Z_p-geometrical index theory and Morse index theory of first order linear discrete Hamiltonian systems,a sufficient condition for the existence and multiplicity of nontrivial Z_p-orbit of periodic solutions is obtained. This is the first time to deal with the existence of solutions of discrete systems by applying both the above-mentioned theoretical tools.Moreover,our result improves some known ones in the literature in that we obtain a better estimate of the lower bound of the number of periodic solutions.Chapter 3 deals with the existence and multiplicity of periodic solutions of second order discrete Hamiltonian systems with a small forcing term.Firstly,some sufficient conditions are given to guarantee that the corresponding variational functional of the above problem is coercive and hence a minimum critical point is obtained. Then,we prove that if 0 is a non-degenerate periodic solution of the above systems where the forcing term is zero,then the above minimum critical point is non-degenerate.Therefore,by using a three-critical-point theorem of Morse theory, we prove that the original systems have at least three periodic solutions.In particular, some conditions which guarantee that 0 is a non-degenerate periodic solution of the systems where the forcing term is zero are given when the nonlinearity is autonomous and non-autonomous,respectively.All these conditions are clear and easy to be checked.Finally,we give an example to emphasize that our results may not be true when the forcing term is not small enough.All these are significant in the study of perturbed discrete systems. Chapter 4 is devoted to the existence and multiplicity of solutions of boundary value problems of the scalar case.This is the first time to deal with the existence of solutions for boundary value problems of discrete systems by using Morse theory. A series of meaningful results are obtained.For the case that the systems are non-resonant at infinity,we study the existence and multiplicity of solutions of the homogenous problem and the non-homogenous problem by using Morse theory, degree theory and matrix analysis simultaneously,etc.Moreover,some sufficient conditions are obtained for the situation that the non-homogeneous problem has exactly three solutions.Generally,we can only obtain the lower bound of solutions by using critical point theory or fixed point methods.However,we obtain the precise number of solutions in this chapter.Hence,it provides a new method for this kind of problems.For the case that the systems are resonant at infinity,by computing the critical group at infinity,by using both the cut-off technique and the mountain pass lemma,a positive critical point and a negative critical point are obtained together with the corresponding critical groups.This leads to the existence of three nontrivial solutions,with one positive solution and one negative solution.The methods used in this chapter are also applicable to some other kinds of boundary value problems,if only the eigenvalues of the corresponding linear systems are non-zero.In Chapter 5,we consider the existence and multiplicity of solutions of boundary value problems of the high-dimensional case.The nonlinearity in existing literature is generally assumed to be superlinear,sublinear or bounded.To our best knowledge,there are few references dealing with the case that the nonlinearity is asymptotically linear by using critical point theory.In view of these facts,this chapter proposes a generalized asymptotically linear condition on the nonlinearity which includes the asymptotically linear as a special case.By classifying the linear second order discrete Hamiltonian systems,this chapter defines index functions, obtains the properties and the computation of index functions.Then,some new conditions on the existence and multiplicity of solutions of boundary value problems are obtained by combining Leray-Schauder principle and Morse theory,etc. Therefore,this greatly improves the discussion on boundary value problems of discrete systems.The methods used here are also applicable to some other kinds of boundary value problems of discrete systems,for example,boundary value problems of first order discrete Hamiltonian systems.Our results are new even in the case of asymptotically linear.