Research On A Class Of Fractional Sturm-Liouville Type Eigenvalue Problems

The Sturm-Liouville problem refers to the studying of the Sturm-Liouville operator's eigenvalues and the expansion of the eigenfunctions.It is inseparable from the establish-ment of the theory of quantum mechanics,and has extensive application in almost every field of our daily production.The solution of many practical problems in the daily life can be transformed into the study on eigenvalues and eigenfunctions of Sturm-Liouville problems and the properties of differential equations,through the study we can predict some characteristics in the process of practical problems in the system,which has made a great contribution to solve practical application problems.Fractional calculus is a step by step generalization of integer calculus,their emer-gence and development are not the same but inseparable.Because the expanding applica-tion context of fractional calculus,this field can develop rapidly.It often provides some practical ideas and methods to solve differential equation and integral equation problems,and makes outstanding contributions to our scientific research innovation.With the rapid development of fractional calculus,fractional differential equation has been widely concerned.Fractional differential equation refers to the equation with fractional derivative.It was first used in physics,biology,chemistry,economics and finance,which is the representative of a large number of mathematical models generated by natural problems with non-local characteristics,so it has a wide range of applications in nowadays society.In this paper,we study a class of fractional Sturm-Liouville type eigenvalue problem with both left and right Riemann-Liouville fractional derivatives and integrals:#12This problem can be regarded as a generalization of the governing equilibrium equation derived from nonlocal mechanics,which is closely related to the practical problem.The main research structure of this paper is as follows:Chapter 2:We introduce the definitions and properties of left and right Riemann-Liouville fractional integrals and derivatives.The eigenvalue theory for second order differential equations and the important theorems.Chapter 3:We prove that when 0<?<1/2,the problem consists of countable num-ber of real eigenvalues and the orthogonal completeness of corresponding eigenfunctions system in the Hilbert spaces.Finally,we present a lower bound for the first eigenvalue.Chapter 4:In order to derive the quantitative conclusion between the geometric multiplicity of eigenvalues,We present a initial value problem:#12We discuss the uniqueness of solutions for the fractional differential equation and prove that the geometric multiplicity of eigenvalues is simple.Finally,we summarize the con-clusions,and put forward three content that can be further studied.The innovation and difficulty of this paper lies in the study of the eigenvalue theory of a class of second order differential equations with left and right fractional derivatives and integrals.We establish the real value,countability,geometric multiplicity problems of eigenvalues and the completeness and orthogonality of eigenfunctions.This kind of problem destroys the uniqueness of the solution of the initial value problem,and it can not use Laplace transform and classical local analysis methods directly.Therefore,in this paper,we use operator theory to solve our problem.