Razumikhin-type stability theorems for functional fractional-order differential systems and applications

•The stability of functional fractional-order differential systems (FFDS) is first studied by using a Lyapunov function.•Razumikhin-type theorems first are given correctly for uniform stability and global uniform asymptotic stability theorems for FFDS.•The work is a major breakthrough on the stability theorems of FFDS that will be more widely used.

In this paper, we studied the stability of functional fractional-order differential systems using a Lyapunov function and develop Razumikhin-type uniform stability and global uniform asymptotic stability theorems for functional fractional-order differential systems, which involve Riemann–Liouville and Caputo derivatives respectively. Two examples are given as application of our theorems.