The Existence、uniqueness And Controllability Of Several Classes Of Fractional Differential Equation

The fractional differential equation is the abstract form of many physical and engineering problems. We can find the widely application in diffusion of fractal porous media, the electrolytic chemical, semiconductor physics, condensed matter physics, viscoelastic system, biological mathematics, statistics and so on. So the research for fractional differential equations has gained the rapid development. It has become a hot research topic in the differential field. In the past forty years, more and more scholars are interested in this field. In this paper, we mainly study the existence, uniqueness and controllability of fractional differential equations.The paper consists of the following six parts:At the beginning of this paper, we briefly introduce the research background, research status and the application in different fields for the fractional differential equation. Moreover, we have also introduced the main research contents of this paper.In the second chapter of this paper we prove the existence and uniqueness of solutions for a class of the nonlinear fractional differential equation with initial condition and investigate the dependence of the solution on the order of the differential equation and on the initial condition. Then we give an example to demonstrate the main results. The main purpose of the third chapter is to obtain the unique solution of the constant coefficient homogeneous linear fractional differential equations and the constant coefficient nonhomogeneous linear fractional differential equations if P is a diagonal matrix and prove the existence and uniqueness of these two kinds of equations for any P∈L(Rm). Then we give two examples to demonstrate the main results.In the forth chapter of this paper, by using Schauder’s fixed point theorem and the contraction mapping principle, we discuss the antiperiodic fractional boundary value problems for a class of fractional differential equations. Some existence and uniqueness results of solutions are obtained. Two examples are given to illustrate the main results.In the fifth chapter of this paper, we discuss a class of fractional evolution equation with the Riemann-Liouville fractional derivative. Firstly, by using the Laplace transform, we give the definition of the mild solution for this equation. Then, according to some classical fixed point theorem, we obtain the existence and uniqueness of mild solutions. Then we give two examples to demonstrate the main results in this chapter.In the sixth chapter of this paper, by using fractional power of operators and Sadovskii fixed point theorem, we study the complete controllability of fractional neutral differential systems in abstract space without involving the compactness of characteristic solution operators introduced by us. Then we give an example to demonstrate the main result.