Study Of Lyapunov-type Inequalities And Periodic Solutions

Based on some inequality techniques and some methods of mathematics analysis, this thesis deals with Lyapunov-type inequalities for square convex function, Lyapunov-type inequalities for a class of quasilinear systems and some periodic solutions of a higher-order discrete system. The main content is as follows:In chapter1, it briefly describes the significance of Lyapunov inequality,the research history of Lyapunov inequality and the main contents of this thesis.Based on square convex function, chapter2establishes a new Lyapunov-type inequality for the following nonlinear system:In chapter3, the following quasilinear systems is investigated, Anti-periodic boundary conditions are introduced instead of boundary conditions. Using some techniques of mathematics analysis, a new Lyapunov-type inequality for this quasilinear system is derived. Our results improve the revelant results.In chapter4, we discuss the lower bounds of periods of periodic solutions for the following generalized delay differential equation: the generalized delay function rk(t)with|rk’(t)|<1,k=1,2,...,n, is introduced to replace the constant delay kr.We obtain some results by using some simpler method, which enriches the existing results.In chapter5, we investigate the existence of positive periodic solution of the following equation: x(n+m+k)-a(n+m)x(n+m)-b(n)x(n+k)+a(n)b(n)x(n)+f(n, x(n-τ(n)))=0. We use the function a(n) and b(n) instead of the constant a and b,using fixed point theorem on cone, we get the new sufficient conditions that guarantee the existence of the positive periodic solutions, which improve some existing results.