A survey on the Ulam-Hyers stability of fractional-order differential equations

Abstract Stability theory plays a central role in the analysis of the behaviour of a solution of a real-order differential equations. In the literature, various stability concepts were introduced from the application point of view. The most popular ones are the Lyapunov and the Ulam-Hyers stability. Here, we first we establish a relation between the Lyapunov and Ulam-Hyers stability concepts for a dynamical system and prove that the concept of Ulam-Hyers is more general than that of Lyapunov. Second, we present a brief overview of recent developments in the Ulam-Hyers stability analysis of fractional-order differential equations (FDEs). These equations include linear FDEs, non-linear FDEs, delay FDEs, fractional-order boundary value problems and impulsive FDEs.