Periodic Solutions And Homoclinic Orbits Of Second Order Discrete Hamiltonian Systems With Potential Changing Sign

The existence of subharmonic solutions, periodic solutions and homoclinic orbits of second order nonlinear discrete Hamiltonian systems with potential changing in sign is studied by using critical point theory in this dissertation. A series of new results are obtained by changing the existence of subharmonic solutions, periodic solutions and homoclinic orbits into the existence of critical points of the corresponding functional on suitable function space. This dissertation is composed of four chapters. The contents of the dissertation are introduced as follows:In chapter 1, the historical background and the recent development of the problems to be studied are introduced, at the same time, the main contents of the dissertation are outlined.The existence of subharmonic solutions of a kind of second order nonlinear discrete Hamiltonian systems is discussed in chapter 2 by using Linking Theorem. In these systems the potential comprises two factors, one is a nonzero periodic function with zero mean, and the other is a function which satisfies certain growth conditions. Some sufficient conditions which can guarantee that these systems have at least one nontrivial subharmonic solution are obtained.The existence of periodic solutions of a class of second order nonlinear discrete Hamiltonian systems with sign-changing potential is considered in chapter 3. In these systems the potential not only has a periodic function with nonzero mean and changing sign, but also has a symmetric matrix-valued periodic function that is indefinite in sign. In some literature the existence of periodic solutions of a special case of these systems with asymptotically quadratic potential is considered by using Morse theory, but in this dissertation these systems without asymptotically quadratic potential are studied by using minimax theorem, and some sufficient conditions of the existence of at least one or two nontrivial periodic solutions of these systems are obtained. The results obtained enrich and develop the theory of discrete Hamiltonian systems.The existence of homoclinic orbits of a kind of second order self-adjoint nonlinear discrete Hamiltonian systems with potential changing in sign is studied in chapter 4 by using Mountain Pass Theorem. In each of these systems, the two symmetric matrix-valued functions are positive definite and the potential function is superquadratic both at zero and at infinity. Some criteria for the existence of homoclinic orbits of these systems with periodic assumptions and without periodic assumptions are worked out, respectively. Our results extend some known results in the literature.