Study On The Existence Of Solutions Of Caputo Fractional Differential Equations With Riemann-Stieltjes Integral Boundary Conditions

With the emergence and expansion of calculus,differential equations have gradually developed.It plays an important role in biology,automation,medicine,astronomy and other fields.Moreover,differential equations also play an important role in many branches of mathematics.With the expansion of the application field and the emergence of new problems,it still has its own vitality today.The boundary value problem of differential equation is composed of differential equation and boundary condition.When we consider the solutions of boundary value problems,we usually consider the solutions of differential equations satisfying boundary conditions.In the fields of mathematical model,medicine,image processing and so on,boundary value problems are very common,such as Laplace equation,heat flow equation in image processing and so on.The first chapter is the introduction,which mainly introduces the development background of differential equations,expounds the research status of differential equations,and explains the research value,significance and necessity of this subject.The second chapter is preliminary knowledge,it is mainly introduces that the basic definitions and theorems used in this paper.In chapter three,we study of existence and uniqueness of solutions the complete form of four order two points boundary value problems,the proof method applies The Principle of Compressive Mapping.In chapter four,we investigate the eigenvalue problem for Caputo fractional differential equation with Riemann-Stieltjes integral boundary conditions.By using the Guo-Krasnoselskii's fixed point theorem on cone and the properties of the Green function,some new results on the existence and nonexistence of positive solutions for the fractional differential equation are obtained.In chapter five,we deals with the following Caputo fractional differential equations with Riemann-Stieltjes integral boundary conditions.By mean of coincidence degree theory,we obtain the existence of solutions for the above fractional BVP at resonance.In chapter six,it is mainly the summary and prospect of this paper.