Lyapunov Inequality And Comparison Theorem For Some Discontinuous Differential Equations

In this paper,we discuss the Lyapunov inequality and the correlation theo-rem of several kinds of discontinuous differential equations.It is consisted of six chapters.As the introduction,in Chapter 1,the background and history of ordinary differential equation and linear impulsive differential systems for the lyapunov inequality,and the main results of this work are stated.In Chapter 2,we discuss the Lyapunov inequality of a second-order differ-ential system with damping term.First of all,we consider two kinds of pulse perturbation,and then provide an example to verify the results.In Chapter 3,we consider the dirichlet boundary value problem for a class of second order linear impulsive differential system.The Lyapunov inequality of such impulsive differential systems is explored by Holder inequality and Riccati trans-form,and the relations between Lyapunov inequalities of two kinds of impulsive differential systems are discussed.In Chapter 4,we discuss the Lyapunov inequality of the Hill equation in the form of the pulse akx(tk)+ bkx'(tk),and the Lyapunov inequality of the impulsive differential equation is discussed by using the Green's formula and introducing the norm.Provides a practical example of the results of the article,the application of MATLAB simulation.In Chapter 5,we consider the comparison theorem of the second order linear differential equation.On the basis of the comparison theorem of the equation of the original second order ordinary differential equation,the impulsive differential equation comparison theorem is more universal by adding the pulse condition and introducing the new method.In Chapter 6,we summarize the full thesis and pointed out the further re-search work.