Existence Of Solutions For Several Kinds Of The Boundary Value Problems Of Fractional Differential Equations

The Fractional calculus is a mathematical tool to study the characteristics of any order differential,integral operator and its application,which has obvious advantages over the in-tegral calculus in practical application,and has been successfully applied in the field of signal image processing,Fluid mechanics and electromagnetic field,which has become an impor-tant topic for many scholars.The study about the existence of the solutions to the boundary value problem of fractional differential equations is an important part of the qualitative theory of fractional calculus,which has broad theoretical significance.Based on the theory of fractional differential,the existence of solutions to boundary val-ue problems of third class fractional differential equations are studied by means of functional analysis:Firstly,the existence and uniqueness of a class of nonlinear boundary value problems with fractional order coupled system are discussed in this paper.The corresponding Green functions is obtained from Volterra integral equation,and the corresponding completely con-tinuous operator is constructed.And by using Schauder fixed point theorem,the sufficient conditions for existence of solution to boundary value problems are obtained.The sufficient conditions for uniqueness of solution to boundary value problems are obtained by means of Banach compression mapping principle.Then,the existence of the solution of boundary value problems of fractional order cou-pled system with fractional order boundary value is discussed in this paper.A relatively tight decision criterion for an infinite interval correction is established.By using the Lebesgue control convergence theorem,the defined operator is a fully continuous compact operator.In turn,by using Leray-schauder nonlinear choice,a sufficient condition for existence of the solution to boundary value problems of such a kind of fractional order coupled system is obtained.Finally,the existence of the polar solution of nonlinear Riemann-liouville fractional dif-ferential equations with integral boundary value conditions is studied in this paper.Establish-ing a new comparison principle and its rationality is proved.The sufficient condition of the existence of the boundary value problem is obtained by Banach fixed point theorem.By us-ing the method of upper and lower solutions and monotone iterative technique,the monotone iteration sequence that approximates the unique solution and corresponding error estimator are given.