Lyapunov-type Inequalities And Stability For Hamilton Systems

By applying the generalized zero and some inequality techniques, the doc-toral dissertation obtains the new Lyapunov-type inequalities for some differential systems. These inequalities are best in a certain sense. It establishes some sta-bility criterions for the first order linear Hamilton system on the scales, unifies, improves and extends some recent results. Furthermore, it gives the better lover bounds of generalized eigenvalues for one dimensional quaslinear elliptic systems. All above results will further reveal the essential characteristic of Hamilton sys-tems, enrich and perfect the relevant theories of Lyapunov-type inequalities and Lyapunov-type stability, and further prompt the development of the qualitative theorems of differential equations.This thesis is made up of the following six sections:In Chapter1, it briefly described the relevant research background of Lya-punov inequality and Hamilton systems, and introduces the main research work of this paper.In Chapter2, it established the new Lyapunov-type inequalities for the fol-lowing first-order linear Hamilton system x△(t)=α(t)x(σ(t))+β(t)y(t), y△(t)=-γ(t)x(σ(t))-α(t)y(t) on time scales T, where α(t),β(t),γ(t) are real valued rd-continuous functions.In Chapter3, it set up the new Lyapunov-type inequalities for some nonlinear discrete differential systemwhere μ, v>1with1/μ+1/v=1, and aαn),β(n),γ(n)are real valued functions on Z.In Chapter4, it built the new Lyapunov-type inequalities for quaslinear elliptic systems andWhere,fi is a real-valued continuous function on R,1<pi<∞,at>0,∑i=1nαi/pi=1.In Chapter5.it established the new Lyapunov-type inequallities for2n order differential equation x(2n)(t)+(-1)n-1q(t)x(t)=0.In Chapter6,it established several stability criteria for first order linear Hamilton system on time scales T by applying Floquet theory and Lyapunov-tyPe inequalities obtained in Chapter2.