Several Classes Of Lyapunov-Type Inequalities

Abstract:By applying Mathematical analysis method and some inequality techniques, the main research proposal of this paper is to establish and generalize the Lyapunov-type inequalities for some differential and difference systems, the results will help to further explore the essential features of differential equation, and enrich the relevant theory of Lyapunov and functional differential equations.This paper consists of the following five parts:In chapter1, this paper briefly introduces the research history and research status of Lyapunov-type inequality, and a brief overview of the main work in this paper.In chapter2, using the new method, without considering the maximum of｜xi(t)｜,i=1,2,3, a new Lyapunov-type inequality for quasilinear differential system (2.1) based on the periodic boundary value condition and the solution of homogeneous linear equations is given, and it has been generalized to higher order higher dimensional case, the results of this paper generalize some previous results on this topic.In chapter3, changing the differential system into difference system, a new Lyapunov-type inequality for quasilinear difference system (3.19) under the periodic boundary value conditions is given and it has been also generalized to higher order higher dimensional case, improve and popularize some results in the relevant existing literature.In chapter4, two different methods was used to study the three-point boundary value problem (4.4) to establish the corresponding Lyapunov inequality In chapter5, this paper summarizes the main work and looking for follow-up work.