Existence Of Solutions For High-order Fractional Differential Equations With Multi-point Boundary Conditions

Fractional Calculus is the study and application of arbitrary order differential and integral theory, and it is consistent with integer order calculus and it is a natural extension of the integer order calculus. Fractional differential equations is developed accompanied by fractional calculus. In recent years, with the widely applications of fractional calculus in the fields of physical, mechanical, biological, ecological and engineering, the theory of fractional calculus has been paid more and more wide attention from scholars at home and abroad.Especially the study of fractional differential equation abstract from practical problems attracts much attention by many mathematicians.The boundary value problems for fractional differential equations is one of the important issues for the theory of fractional differential equations and it has been a lot of results. And multi-point boundary value problems of differential equation is originated from a variety of applied mathematics and physics, a lot of non-uniform electromagnetic field theory, soil and water and wet soil differential calculus, and the theory of elastic stability problems can be summed up in differential equation with multi-point boundary value conditions. Compared with the two-point boundary value problem of differential equation, multi-point boundary value problem can be more accurately to describe a number of important physical phenomenon, and it has a more extensive practical application background. Such as the issue of population and economy, fluid mechanics, etc. For the further research of this field, it will continue to promote the development of the theory of fractional order differential equations,and it also will continue to its practical application in the fields of many related science to provide a solid theoretical basis.This paper studied the main system under the condition of high-order fractional differential equation three-point boundary value, four-point boundary value, multi-point boundary value, and integral boundary value of the different types of problems. Such as,involving the solution or the existence of positive solution, nonexistence, uniqueness and multiplicity, some innovative results are obtained.In chapter one, we introduce research background, history of development and present situation of fractional calculus, present situation and significance of research of the boundaryvalue problems of fractional differential equation. We list some basic definitions of fractional calculus theory, some related lemmas and the main tool used in this paper. Also we give the main content of this paper.The second chapter will study under different conditions in the singular and nonsingular three-point boundary value problems of a class of fractional differential equations. By using Banach contraction mapping principle, the fixed point index theory, and Leggett-Williams fixed point theorem, to study the existence, uniqueness and multiple positive solutions under the right end function with nonsingular conditions; Height function method was used to study the existence and multiplicity of positive solutions under the right function with singular conditions.The third chapter studies the four-point boundary value problems of a class of coupled system fractional differential equation. Using the operators sum fixed point theorems to obtain the existence of positive solutions and by means of the continuation theorem of coincidence degree developed by Mawhin, a sufficient condition for the existence of solutions is obtained.The fourth chapter studies the three types of multi-point boundary value problems of fractional differential equations. Using Leray-Schauder's nonlinear alternative and the Banach contraction mapping principle to study the existence and uniqueness of solution of a class of fractional differential equations multi-point boundary value problem; Using monotone iterative method to obtain the existence of positive solutions for a class of fractional differential equations multi-point boundary value problems on infinite interval, and give the iterative sequences of maximum solution and minimum solution; we investigate a class of nonlinear Langevin equation involving two fractional orders with infinite-point boundary value conditions, by Leray-Schauder's nonlinear alternative and Leray-Schauder degree theory, several new existence results of solutions are obtained.The fifth chapter, we investigate the nonlinear integral boundary value problem of fractional differential equations with generalized p-Laplacian operator. By means of the monotone iteration method, we obtain the existence of positive solutions and establish the iterative sequence for approximating the solutions. Moreover, the nonexistence of positive solution is also considered. On the other hand, we discuss the problem under singularcondition. We obtain several local existence and multiplicity of positive solutions results by height functions method.Chapter 6 is conclusion and prospect. Sum up in this paper, we study the main work and innovations. At the last, we point out the future research work.