The Existence Of Caputo And Riemann-Liouville Fractional Differential Equations With Boundary Value Problems

The study of boundary values has set off a rush of research in the world with the emergence and development of the fractional differential equations.The integral boundary value problem is a new type of boundary conditions and it has attracted attention from the most researchers.In view of fractional differential equations can describe certain natural laws and phenomena,namely,their practical applicability,fractional differential problems with boundary value conditions are popular among the scholars.This article is intended to investigate the Caputo fractional differential equation with the integral boundary value and the Riemann-Liouville fractional differential equation which contains both the integral and the multi-point boundary value conditions.We draw conclusions by the properties and assumptions of f(x)that is a multiple continuous function and applying a variety of fixed point theorems.In order to make the article easy to understand,we arrange the article as follows:The first chapter introduces the origin and modern development of fractional differential equation by focusing on the status of domestic and foreign researches.The second chapter elaborates the relevant definitions,theorems and lemmas.In the third chapter,the integral boundary value of Riemann-Liouville type fractional differential equation is investigated.The fourth chapter explores the existence results of multi-point boundary value conditions for Caputo fractional differential equations,The existence results,the existence of positive solutions and the existence of multiple positive solutions of certain equations are presented in this paper.