Several Stability Problems For Fractional Differential Equations

Recently, the fractional differential systems have attracted more and more researchers to study, but no matter continuous systems or impulsive systems, the difficulty of the exact solution's study will be relatively large when the system is complex, in order to study these problems, we use the Ulam-Hyers stability to deal with such issues. Ulam-Hyers stability can depict the stability of a system, which is depend on the estimates of the difference between the perturbed and the exact solution.The arrangement of the thesis is as follows:Chapter one introduces the background, the domestic and foreign research situation of our problems and presents the main work of this thesis.Chapter two recalls the preliminaries necessary for the study of this thesis,including some definitions, properties and conclusions of fractional calculus, Gronwall inequality, MittagLeffler function and Hyers-Ulam stability.In chapter three, we deal with the Ulam-Hyers stability of nonlinear fractional differential systems, at present, more of this kind of problem is the first order differential systems. In this dissertation, we show the Ulam-Hyers stability of such differential systems under the fractional order.Chapter four is concerned with the Ulam-Hyers stability of the nonlinear fractional differential systems with positive constant coefficient ?, based on the chapter 3, a positive constant coefficient ? > 0 is added, we have proofed the Ulam-Hyers stability of the nonlinear fractional differential sysytems by utilizing Mittag-Leffler fuction and Gronwall inequality.Chapter five discuss the Ulam-Hyers stability of nonlinear impulsive differential systems, based on the chapter 4, impulsive form is added, in this section, we firstly present the existence of solutions by mean of fractional calculus and fixed points theorem, Next, unlike the assumptions of chapter 3, the nonlinear impulsive differential systems is also Ulam-Hyers stability.Chapter six gives a xonclusion of our present work and introduces some further study ideas in the future.