Asymptotic behavior of discrete evolution families in Banach spaces

Let X be a Banach space, A = (A(n))(n is an element of Z+) be an operator-valued sequence and let U = {U(n, m) : n >= m is an element of Z(+)} be the discrete evolution family associated to A. In this paper we prove that the family U is non-uniformly strongly stable (i.e. for every nonnegative integer m and every x is an element of X, lim(n ->infinity) parallel to U(n, m)x parallel to = 0) if and only if it is l(0)(1)(Z(+), X)-approximative admissible, i.e. for every sequence f = (f(n)) in l(0)(1)(Z(+), X), and every positive number epsilon, there exists the sequence g = (g(n)) in l(0)(1)(Z(+), X), satisfying parallel to g - f parallel to(l01(Z+, X)) < epsilon, such that the solution of the discrete Cauchy Problem x(n+1) = A(n)x(n) + g(n+1), n is an element of Z(+), x(0) = 0, belongs to l(0)(1)(Z(+), X). Other types of asymptotic behavior of the family U are also analyzed.