Lyapunov-type Inequalities Of Several Types Of Differential Equations

In the last decades,the Lyapunov-type inequality of the differential equation is a valuable tool in many fields of applied sciences.The Lyapunov-type inequalities provide a theoretical basis for the study of the existence of the solution,the oscillation and the eigenvalues of certain differential equations,which lead to more and more people research on it.On the one hand,some authors study the Lyapunov-type inequalities of integer-order linear differential equations,nonlinear differential equations or systems of differential equations.On the other hand,Lyapunov-type inequalities of the fractional differential equations are studied by more and more researchers,such as Riemann-Liouville fractional differential equations.In this paper,the Lyapunov-type inequalities of several kinds of differential equa-tions are studied,and it is divided into four chapters.The first chapter introduces the history and evolution of the Lyapunov-type inequality,and the relevant definitions of fractional integral and derivatives together with some basic properties and lemmas are given.In the second chapter,we consider the Riemann-Liouville fractional differential equation with mixed nonlinearities of order ??(n-1,n]for n ? 3(Da?x)(t)+p(t)| x(t)|?-1 x(t)+q(t)| x(t)|?-1 x(t)= f(t),where p,q,f ? C[t0,+?)and the constants satisfy 0<?<1<?<n(n? 3).The above equation subjects to the following two kinds of boundary conditions,x(a)= x'(a)=x"(a)=…=x(n-2)(a)= x(b)= 0,and x(a)=x'(a)=x"(a)=…=x(n-2)(a)= x'(b)= 0,respectively.This chapter obtains the Lyapunov-type inequalities of the above nonlin-ear differential equation by using Green's functions and new obtained lemmas.The third chapter investigates new types of Lyapunov-type inequalities for the fractional boundary value problems(FBVPs)of the formsandwhere 1<?,? ? 2,Db-? and Da+? are right and left Riemann-Liouville derivatives,k1 ? C([a,b],R)and k2 ? C([a,b],R+),and ?p(x)= |x|p-2x is a p-Laplacian operator with p>1.With the help of new obtained Green's functions and their properties,using the Guo-Krasnoselskii fixed point theorem,we will obtain some new Lyapunov-type inequalities for these kinds of FBVPs.In the forth chapter,the Lyapunov type inequalities for a class of odd order linear differential equationsare studied,where pk(t)?C([a,b],R),k=0,1,…,n+1.In this chapter,according to the conclusions given by the existing literatures,the Lyapunov-type inequality of the above boundary value problem is obtained,which provides a theoretical basis for the existence of the solution.