Periodic Solution Of Fourth Order Nonlinear Differential Equation With Homoclinic Orbits

There have been a lot of results on studying periodic solutions and homo-clinic orbits for differential equations, but rather few achievements on difference equations. However, it is meaningful work either in theory researches or in applications to study periodic solutions and homoclinic orbits for difference equations. In present results, the methods similar to studying periodic solutions and homo-clinic orbits for differential equations are often applied, such as fixed point theory, Green's function, topological degree theory, coincidence degree theory and critical point theory, and so on.In chapter 1, we introduce the critical point theory of the development and application. And, introducing some elementary concepts, some symbols and formulas.In chapter 2, we consider the existence of critical point for this problem by applying critical point theory. Finally, by using Linking Theorem, we obtain a sufficient condition for the existence of periodic solutions of the fourth-order nonlinear p-Laplacian discrete difference equation.In chapter 3, we consider the existence of homoclinic orbits for fourth-order difference equations by applying critical point theorem. We must be find just critical point when the P. S. condition is not satisfied. Finally, by using Mountain Pass Theorem, we obtain a sufficient condition for the existence of homoclinic orbits for fourth-order nonlinear discrete differential equations.In chapter 4, which is the summation of this paper, as well as the prospects for future work.