Two Class Of Fractional Differential Equations And Boundary Conditions

With the rapid development of science and technology,fractional differential e-quations have attracted the attention of more and more researchers.It is used more and more in the fields of biology,medicine,physics,chemical engineering,computer sci-ence and so on.In this paper,we study the boundary value problems of two kinds of fractional differential equations by using some fixed point theorems.Details are as follows:The first chapter,introduction,we introduces the background and significance of this topic,as well as the main content of this article.The second chapter,preliminary knowledge,we introduces the relevant definition of fractional differential equations and the relevant lemma required in this paper.The third chapter,we research on a class of fractional differential equations with three-point boundary value problemswhere f₁,f₂ is given continuous function,and ?₁,?₂,?₁,?₂,?₁,?₂ are postive real con-stants with ?₁/?₁>(1/2+?₁/?₁),?₂/?₂>(1/2+?₂/?₂).In this chapter,the existence and uniqueness of the results is first given by the condition (H₁) and the Banach's contraction mapping principle.Next,the existence results is given by the Schaefer-type fixed point theo-rem.Then,the Krasnoselskii fixed point theorem and (H₁),(H₂) give the existence of the results.Finally,the Leray-Schauder nonlinear alternative and (H₃),(H₄) give the existence of results.The fourth chapter,we discuss a class of fractional differential equations with integral boundary value problemswhere J=[t₀,T],^cD_t0^? is the Caputo fractional derivative of order?,n=[?]+1,b_1k,b_2k?R,and f₁,f₂,g_1k,g_2k:J×R×R?R are given continuous func-tions.Firstly,the existence and uniqueness of the solution of the equation are given by the condition (A₁)and Krasnoselskii fixed point theorem,and then the existence and uniqueness of solutions for the problem (4.1) are given by (A₁) and the contraction mapping principle.