The Solutions For Initial And Boundary Value Problems Of Fractional Differential Equations And Discrete Equations

In recent years, the fractional differential equation have caused people'wides-pread concern,because of extensive applications in the natural science, physics, engineering, and many other areas. Meanwhile, Fractional discrete equations ap-peared ceaselessly in mathematical model and the need of approximation of Frac-tional differential equation approximation. The study of fractional discrete equa-tion have become a new research topic in recent years. In this paper we mainly discuss the fractional differential equation theory, for example the continuation of solutions for initial value problems, the existence and uniqueness of the solution and boundary value problem positive solution. Also the existence and uniqueness of solution of initial value problem and positive solution of boundary value problem to fractional discrete equations. Our main results are as follows:1. We study the continuation problem of fractional differential equations, we find that there are many existence results of fractional differential equations and functional differential equations([10-12],[21-24] and so on), but we also find that in these results solutions are restricted to the given interval. But in practical using, these conclusions bring inconvenience, we always hope the existence interval of solutions the bigger the better.2.We discuss the positive solution existence and uniqueness problems of frac-tional differential equation. Firstly, we consider a class of equations with the poly-nomial with constant coefficient of fractional derivative of unknown function on the left, and on the right side there contain fractional derivative of unknown function in the nonlinear term, the conclusion about the existence and uniqueness of positive solution to these equations are given, we will use the regular fixed-point theorem in cones and contraction mapping principle to get the conclusion. And we discuss the existence of positive solution for boundary value problem to fractional functional differential equation, through exploring the expression of Green function and the property, we get the related theorem by using the fixed point theorem.3. We search for the conditions of the existence of solutions and positive solutions for the fractional discrete equation. We will mainly focus on a class of fractional order differential equation solution, and to prove the existence and uniqueness of the solutions of these equations using the compression of solutions and mapping principle. At last, we study existence the positive solution for boundary value problem of fractional discrete equations.