Analysis Of Solutions For Fractional Differential Equations

The fractional calculus may be considered as an old and yet novel topic. It is an old topic since, starting from some speculations of G. W. Leibniz (1695, 1697) and L. Euler (1730), it has been developed up to nowadays. However, it is a novel topic as well, since only from a little more than thirty years it has been object of specialized conferences and treatises. Due to its wide application in various fields, it has drawn much attention of investigators.In this thesis we study some problems of solutions for fractional differential equations (FDEs). Firstly, we consider a class of nonlinear fractional differential system, and investigate the existence and uniqueness of solutions. By weakening some of conditions in the known results, the global existence and uniqueness theorem of solutions has been proved. The continuous dependence of the solution on the order has been discussed. Moreover, we prove the existence and uniqueness of positive solutions for a class of nonlinear fractional differential equations by using different ways respectively.Secondly, we study the existence and uniqueness of solutions for fractional functional differential equations (FFDEs), and discuss the relationship between the solutions and the right side functions of equations. By employing the method of steps we study the existence and uniqueness of solutions for the linear fractional retarded differential difference equations. On the other hand, we study the existence and uniqueness of solutions for the related linear fractional time-delay system, and further investigate its finite time stability. Finally, by applying the definition of fractional calculus defined by Nishimoto, we get some explicit solutions of some linear homogeneous and nonhomogeneous fractional differential equations under certain conditions.