Global O(t−α)O(t−α) stability and global asymptotical periodicity for a non-autonomous fractional-order neural networks with time-varying delays

•The concepts of the global O(t−α)O(t−α) stability and the global asymptotical periodicity.•The globally O(t−α)O(t−α) stability for a non-autonomous fractional-order neural networks with delays (FDNN).•The periodic or autonomous FDNN models cannot generate exactly periodic signals under any circumstances.•S-asymptotically periodicity for the periodic FDNN.

The present paper studies global O(t−α)O(t−α) stability and global asymptotical periodicity for a non-autonomous fractional-order neural networks with time-varying delays (FDNN). Firstly, some sufficient conditions are established to ensure that a non-autonomous FDNN is global O(t−α)O(t−α) stable based on a new Lyapunov function method and Leibniz rule for fractional differentiation. Next it is shown that the periodic or autonomous FDNN cannot generate exactly nonconstant periodic solution under any circumstances. Finally, we show that all solutions converge to a same periodic function for a periodic FDNN by using a fractional-order differential inequality technique. Our issues, methods and results are all new.