Applications Of Critical Point Theory To Some Discrete Boundary Value Problems

This dissertation studies discrete mixed boundary value problem, Neumann boundary value problem, periodic boundary value problem, Dirichlet boundary value problem with the one-dimensional p-Laplacian, and higher dimensional boundary value problem. By using critical point theory, we obtain a series of results concerning with the existence of at least one nontrivial solution or multiple solutions and the uniqueness of solution. The obtained results extend and improve some known results in the existing references.In the first Chapter, we introduce the historical background of problems which will be investigated and the main results of this paper.In Chapter 2, we consider a discrete nonlinear mixed boundary value problem. First, by virtue of Green's functions and separation of linear operator, we obtain variational framework. Then, by employing the strongly monotone some conditions including suplinear case, to guarantee that the problem has a unique solution or at least one nontrivial solution.In Chapter 3, by constructing a new variational frame, using the linking theorem and the saddle point theorem in the critical point theory respectively, we study the existence of one solution or two solutions for the discrete two-point Neumann boundary value problem, and some sufficient conditions are obtained.The purpose of Chapter 4 is to study the existence of multiple nontrivial solutions for a class of higher dimensional discrete boundary value problems including the Dirichlet boundary value problems and the mixed boundary value problems. Applying the mountain pass theorem in the critical point theory, we obtain some sufficient conditions to guarantee that these problems have at least two nontrivial solutions. Our results improve and generalize the some known results.In Chapter 5, we deal with a Dirichlet boundary value problem for p-Laplacian difference equations depending on a parameter A. By using the three critical points theorem established by Bonanno, we verify the existence of at least three solutions when A is in two exactly determined open intervals respectively. Moreover, the norms of these solutions are uniformly bounded with respect to A belonging to one of the two open intervals.Chapter 6 mainly consider a convex discrete Hamiltonian system. Basing on the dual theory and a new variational principle, we obtain some criteria for the existence of a periodic solution. Such a solution corresponds to a critical point minimizes the dual action restricted to a subset of some determined space.