The Existence Of Solutions Of Boundary Value Problems For Some Classes Of Fractional Differential Equations

The theory of fractional diferential equations is developping rapidly,and ap-plying widely in physics, chemistry, biology, economic and other various subjects,many mathematical models of these disciplines are described by fractional dif-ferential equations, especially fractional diferential equations under the bound-ary value conditions are of great signifcance in solving practical problems,it hasaroused a great deal of attention from scientists both at home and abroad.In this paper, We discuss mainly three types of nonlinear fractional diferen-tial equation existence uniqueness and multiplicity of BVP solutions.Fulltext isdivided into fve chapters, the main contents as follows:In the frst chapter, First we introduce the research background and currentsituation of the fractional diferential equations, Secondly introducing the majorwork in this paper,Finally we briefly introduce the basic concept of fractional in-tegral and fractional derivative and its properties, functional analysis of the basicknowledge and some fxed point theorems be used in this paper.In the second chapter, we discussed the existence of solutions for a class ofthree-point boundary value problem. First we work out the Green function, con-vert the diferential equation into the equivalent integral equation and talk overits properties.Secongdly, we obtain the existence results of the solution by usingfxed-point theorem of cone expansion and compression of norm type.Finally wegive some examples to illustrate the correctness of the theorem.In the third chapter,we studied the uniqueness for a class of fractional dif-ferential multi-point boundary value problem. frst we convert the diferentialequation into the equivalent integral equation.Secondly we obtain a sufcientcondition of the unique solution by utilizing Banach contraction mapping princi-ple. Finally we give a example to illustrate the correctness of the theorem.In the fourth chapter, we discussed the existence of triple positive solutionsfor a class of fractional diferential equation. First we calculated the Green func-tion and change the diferential equation into the equivalent integral equation anddiscuss its properties, Secondly we take advantage of the Leggett-Williams fxedpoint theorem, and receiving the existence results of triple positive solutions. Fi- nally, We verify the correctness of the theorem.