Existence Of Solutions For Two Classes Of Nonlinear Fractional Differential Equations

This paper uses the Picard successive approximation method,the Banach fixed point theorem and the Schauder fixed point theorem to discuss the initial value problems of two classes of fractional differential equations.These two kinds of equations are nonlinear neutral fractional ordinary differential equations and fractional functional differential equations with infinite delays.In addition,the uniqueness of solutions for a class of nonlinear neutral fractional differential equations is also explained.Finally,examples are given to illustrate the correctness of the results.In this paper,we study fractional differential equations,which is the combination of fractional calculus and two classical differential equations.It is a generalization of classical integer differential equations.The structure of this paper is arranged as follows:In the first chapter,we give a brief introduction to fractional differential,fractional integral,fractional differential equations and fractional functional differential equations.The second chapter is the main part of this paper.In this chapter,the Banach fixed point theorem is used first.Secondly,the Picard successive approximation method is used,and the Schauder fixed point theorem is used again to obtain the existence conditions of the solution of a class of nonlinear neutral fractional ordinary differential equations by the three different methods.Finally,the examples are given to illustrate the results.The result is feasible.In the third chapter,the existence results of the integrable solution of fractional functional differential equations with infinite delay by using Banach fixed point and Schauder fixed point are introduced in this chapter.Then,the applicability of the results is illustrated by an example of application.