Abstract.

In the present work, we derive Halanay-type stability theorems for linear delay differential equations on arbitrary time scales when the delay function and the forward jump operator commute. It is shown that depending on the delay term the zero solution is stable, uniformly stable, asymptotically stable, and uniformly asymptotically stable, under the Perron condition. The work sheds light on the discrepancies among particular time scales. We will see that the conditions imposed cannot be satisfied for the uniform asymptotic stability theorem when in the case of quantum difference equations, for which only the stability of the zero solution is achieved

Keywords:

delay, stability, Perron condition, asymptotically stable, differential equation, time scales