The Existence Of Solution For High Order Differential Equation With Integral Boundary Value Condition

The theory of differential equation play an important role in the fields of economic and financial,engineering mechanicsthe,and scientific experiment and so on.At present,the stadies of differential equations with boundary value problems have a series of results,but the theoretical studies of many problems is still not perfect.Especially the study of high order differential equations with integral boundary value problem of the results is relatively small,we still needs more discussion.Fractional differential equation as the generalization of the traditional integer differential equation,it has more extensive application in real life than the integer order differential equations.In recent 40 years,the research has attracted great attention of many researchers at home and abroad.With the development of the theory of fractional calculus,fractional order differential equation is widely used in every field in real life.The problem of the existence of solution or positive solution for the relatively more general fractional order differential equation problem with integral boundary conditions has great value in practical application.This article mainly talks about the existence of positive solutions for the high order differential equation with integral boundary conditions and the existence of solutions or positive solutions for high order fractional differential equation with integral boundary conditions.The results of the research are as follows:Firstly,using the Leggett-Williams fixed point theorem,given the sufficient condition of three positive solutions for 2n-order differential equation boundary value problem with integral boundary conditions through by iterated integral structure Green function.Secondly,using the Leray-Schauder fixed point theorem,we research the existence of solution for high order fractional differential equation with integral boundary conditions by structure corresponding Green function and give the properties of the corresponding Green's function.Finally,using the Krasnoselskii fixed point theorem,we research the existence of positive solutions for high order fractional differential equation with integral boundary conditions by structure appropriate Green's function and discuss its properties.