Lyapunov Type Inequalities For Some Dynamical Systems On Time Scale

In this thesis,Lyapunov-type inequalities for some dynamical systems are established by using time scale theory and related inequality techniques.The results encourage the theoretical development of Lyapunov inequality,enrich the related theories of the stability of differential equation.The main substances can be divided into chapters as shown below:In chapter one,at first,the results of Lyapunov-type inequality re-search are summarized.Next,introduced the generation,development and research status of time scale theory and the recent research results of Lya-punov-type inequalities for several types of dynamical systems on time scales.And the knowledge of the time scale which will be used in this the-sis are shown.Lastly,the main research results of this thesis are briefly described.In chapter two,combining the relevant conclusions of the integral in-equality on the time scale,using two different methods to study the Lya-punov-type inequality for a class of following second-order dynamic sys-tem on the time scale,as shown below.And we obtained some new results,which expand the existing conclusions.In which,r,pk:T?(0,?)is rd-continuous,and 0=?₁<?₂<L<?_n?1.In chapter three,using the Cauchy Schwartz inequality on the time scale,the partial integral formula and the median value theorem to study a class of three-order dynamic system on the time scale which is as shown below and we have new Lyapunov-type inequalities about it.In chapter four,using Holder's inequality and partial integral formula on the time scale,we study a class of high-order dynamic system on the time scale,as shown below.In which p? 2,p? N,1/p+1/q=1.(f(t)|x^?m-1(t)|^p-2x^?m-1(t))^?+r(t)|x(t)|(p-2)x(t)=0.In chapter five,under certain assumptions,using the definition of Cauchy integral on the time scale combined with the related inequality techniques,we discuss a class of quasi-linear dynamical system on the time scale with certain conditions,as the following:In chapter six,the research work of this text is made a periodic sum-mary,and the future research about the Lyapunov inequality of other sys-tems is prospected.