Boundary Value Problems Of Nonlinear Fractional Differential Equations With Left And Right Fractional Derivatives

The fractional derivative is an arbitrary real or complex order derivative, it has been used in mathematical models in many branches of mechanics and physics, and the fractional differential equations have appeared in large numbers. Now, the theories of initial and boundary value problems of the equations with left or right fractional derivatives have been elaborated in many papers and books. But there are only a few papers study the differential equations containing both kinds of right and left fractional derivatives. Besides they only considered some very special cases of the linear fractional differential equations. In recent years, due to its importance many scholars began to focus on the differential equations with right and left fractional derivatives, and make it becoming a new hot studying field in fractional differential equation theory.Fractional differential equation models arising from many problems can come down to the definite solution problems finally. This paper investigates the existence and uniqueness of solutions for two classes boundary value problems of nonlinear fractional differential equations with left and right Riemann-Liouville derivatives.Firstly, this paper gives an introduction about the definitions and properties of Riemann-Liouville fractional integral and derivative. Secondly, in Chapter3, we study the Dirichlet boundary value problems of a fractional differential equation which is a foundation model arising from non-local continuum mechanics: In the previous studies, many scholars investigated boundary value problems with single fractional derivative by using the Green function method. However, in this chapter, by combining the perturbation method with Green function method, we reduce the problem to a equivalent intergral equation. By means of the Banach contraction mapping principle, the sufficient conditions of the existence and uniqueness result on a proper space for boundary value problems are obtained; By means of the Schauder’s fixed point theorem, the result of existence on a proper space for the solution are obtained through proofing the integral operator is completely continuous. In chapter4, we investigate the boundary problem for a fractional differential equation model arising from the variational principles with fractional derivatives (Db-αDa+αy)(x)=f(x,y(x),x∈(a,b), under proper boundary condition. Above all, we give a description for self-adjoint boundary condition from the mathematical view:(Da+α-ky)(a)=Γ(α+1-k)ck,k=,2,...,n,(Db-α-kDa+αy)(b)=Γ(α+1-k)dk,k=1,2,...,n. Then, using the connection between fractional integral and derivatives, we reduce the problem to a equivalent Volterra fractional integral equation. And then we construct a integral operator on proper function spaces. By means of the Banach contraction mapping principle, in the case cn=0, the existence and uniqueness of solution on continuous space are obtained, and in the case cn≠0, the existence and uniqueness of solution on a weighted continuous function space are also presented. Finally, we summarize the conclusions, and point out the direction for future research of this paper.