Averaging theorems for the large-time behavior of the solutions of nonautonomous systems

We obtain some averaging theorems for the large-time behavior of an evolution family { U ( t , s ) } t ≥ s ≥ 0 acting on a Banach space. It is known that, if a trajectory U ( ⋅ , t 0 ) x 0 is asymptotically stable, then its p -mean tends to zero. We will show here that, if the uniformly weighted p -means of all the trajectories starting on the unit sphere are bounded, then { U ( t , s ) } t ≥ s ≥ 0 is uniformly exponentially stable, while the converse statement is a simple verification. Discrete-time versions of this result are given. Also, variants for the uniform exponential blow-up are obtained. Thus, we generalize some known results obtained by R. Datko, A. Pazy, and V. Pata.