Asymptotic behavior of discrete evolution families in Banach spaces

Let X be a Banach space, A=(A_n)n∈ℤ₊ be an operator-valued sequence and let u = {∪(n,m) : n ≥ m ∈ 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 (Formula presented.) if and only if it is l¹ ₀(ℤ₊, X) -approximative admissible, i.e. for every sequence f = (f_n) in l¹ ₀(ℤ₊, X) and every positive number Ɛ there exists the sequence g = (g_n) in l¹ ₀(ℤ₊, X) satisfying (Formula presented.) such that the solution of the discrete Cauchy Problem x_n+1 = A_nX_n + g_n+1, n ∈ ℤ₊, x₀= 0, belongs to l¹ ₀(ℤ₊, X). Other types of asymptotic behavior of the family u are also analyzed.