The Stability Of The Solution Of Lyapunov's Inequality And Differential Equations

Differential equations and difference equations are very significant and widely used in real life,because they are difficult to obtain exact solutions,therefore,it is necessary to study the behaviour of their solutions.This paper mainly studies Lyapunov-type inequalities of odd-order and even-order differential equations and the asymptotic stability and boundedness of the equilibrium solutions of two kinds of fourth-order delay differential equations and a class of system of second-order nonlinear difference equation.In chapter one,the historical background and research significance of the stability of equilibrium solutions of differential equations and difference equations are introduced briefly.In chapter two,the Lyapunov-type inequalities for even-order differential equation(?)and odd-order differential equation (?) under five kinds of boundary conditions and their applications are mainly studied.At present,the stability of solutions of fourth-order differential equations has been s-tudied by some scholars.However,there are not many studies on the stability of solutions of fourth-order differential equations with multiple delays.In chapter three,by constructing the Lyapunov functions,the stability and boundedness of zero solutions for fourth-order nonlinear delay differential equations (?) and (?) are discussed,and the sufficient conditions for asymptotic stability of zero solutions are given,furthermore,the uniform ultimate boundedness of the zero solutions and the existence of periodic solutions are studied.In chapter four,we investigate the asymptotic stability of equilibrium solutions for a class of second-order nonlinear difference systems of the form (?)...