Existence Of Solutions For Several Classes Boundary Value Problem Of Fractional Diferential Equation

In recent years, the fractional diferential equations have been interestedin studying mathematics workers domestic and international. Mainly it stud-ied on its boundary and initial value problem for the existence and uniquenessof the solution. Generally, its research method is transform the fractional or-der initial or boundary value problem into an equivalent integral equations.Then, using the fxed point theorem to prove existence and uniqueness of so-lutions for the initial or boundary value problems.In this paper, we study the existence and uniqueness of solutions forfractional diferential equation boundary value problem in relaxing the gen-eral restrictive conditions. This paper consists of four chapters as follows:In chapter one, we give a brief description on the research backgroundand development status of fractional order calculus present situation and themain work of this article. In addition, the basic concept and properties offractional derivative, namely Riemann-Liouville,Caputo fractional derivativeof the defnition, properties and some important fxed point theorem is intro-duced.In chapter two, we discusses the existence and uniqueness of solutionsfor boundary value problems of nonlinear fractional integro-diferential equa-tions. To prove this conclusion, mainly by Banach’s fxed point theorem andKrasnoselskii fxed point theorem in some suitable conditions.In chapter three, we study boundary value problems for fractional difer-ential equations involving Riemann-Liouville fractional derivative and Carathe-odory conditions. We obtain the existence of at least one solution using theLeray-Schauder Continuation Principle.