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{U(t,s)}t≥s≥0 acting on a Banach space. It is known that, if a trajectory U(⋅,t0)x0 is asymptotically stable, then its pp-mean tends to zero. We will show here that, if the uniformly weighted pp-means of all the trajectories starting on the unit sphere are bounded, then {U(t,s)}t≥s≥0{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.