On asymptotic stability of linear time-varying infinite-dimensional systems

It is shown that every asymptotically equistable linear time-varying infinite-dimensional discrete-time system x_k+1 = A_kx_k is uniformly asymptotically equistable, if A_k is a collectively compact sequence of bounded linear operators. Next, this result is used to prove that for a broad class of linear retarded functional differential equations, the notions of asymptotic equistability and uniform asymptotic equistability coincide.