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

The subject of fractional calculus has gained considerable popularity and importance during the past three decades, due mainly to its demonstrated applications in fields of science and engineering such as hydrodynamic, material mechanics, astronomy, economics, biology and medical. With the development of the fractional calculus, that fractional order differential equation has been used widely when handling problems and caused more and more attention. More and more documents about this filed appeared. Most of them are devoted to the existence of solution for initial value problems of fractional differential equation, but there are few papers consider the existence of solution for boundary value problem of fractional differential equation. In this paper we give the existence of solutions for some boundary value problems of fractional differential equation.The paper is divided into four chapters:In chapter one, it is a brief introduction of the history and present situation of the study about the boundary value problems of fractional differential equation, and purpose of studied.In chapter two, we divided the space firstly and then give the existence and unique solutions for Caputo fractional order two-point boundary value problem at resonance by Mawhin continuous theorem.In chapter three, we study the Green function firstly and then prove at least exists one positive solutions for the boundary value problem of Riemann-Liouville fractional differential equation by Banach compression mapping theorem, Schauder fixed point theorem and Krassnoselskii's fixed point theorem. At last we obtain three positive solutions by the use of the Leggett-Williams'fixed point.In chapter four, we have considered the boundary value problems involving Caputo fractional derivative on the half line. We give the existence of solution for the problem by using the classical Schauder's fixed point theorem combined with the diagonalization process.