(Lyapunov-type Inequalities And Stability For Discrete Hamilton Systems)

By applying the generalized zero and some inequality techniques, the main research proposal of this doctoral dissertation is to generalize and establish the Lyapunov-type inequalities for some discrete Hamilton systems. As applications, this paper makes use of these obtained inequalities to establish some stability criterions for several discrete Hamilton systems. All above results will help us to further study the essential characteristic of Hamilton systems, enrich the relevant theorems of these Hamilton systems and Lyapunov-type inequalities, and further prompt the development of the qualitative theorems of differential equations.This thesis is made up of the following five sections:In Chapter1, it briefly describes the research history and the present research status of Lyapunov inequality, and introduces the historic background and the research significance of the stability of Hamilton systems. In the end, it briefly summarizes the main work of this paper.In Chapter2, by using the generalized zero and some inequality techniques, it adopts two different research ways to establish the Lyapunov-type inequalities for the following first-order linear discrete Hamilton system from three different cases respectively, which improves and generalizes all the ex-isting revelant results.In Chapter3, based on the existing revelant research, it further generalizes the results of Chapter2to the following first-order nonlinear discrete Hamilton systemIn Chapter4, applying the Floquet theorem of linear discrete periodic sys-tems and the Lyapunov-type inequalities obtained in Chapter2, we eatablish two stablility criterions for the first-order linear discrete periodic Hamilton system. Furthermore, we discuss the stability of the second-order linear difference equa-tion. All the stability conditions obtained in this chapter greatly improve the revelant results of all the existing references. In Chapter5, under some conditions, we prove that the following planar linear discrete Hamilton system with perturbations has the same stability as its corresponding system without perturbation as follows Moreover, it gives some examples to show that these conditions are necessary and sharp.