Uniform dichotomy and exponential dichotomy of evolution families on the half-line

The aim of this paper is to give characterizations for uniform and exponential dichotomies of evolution families on the half-line. We associate with a discrete evolution family Φ = {Φ(m,n)}(m,n)∈Δ the subspace X1 = {x ∈ X: Φ(·, 0)x ∈ ∞(N,X)}. Supposing that X1 is closed and complemented, we prove that the admissibility of the pair ( ∞(N,X), 10 (N,X)) implies the uniform dichotomy of Φ. Under the same hypothesis on X1, we obtain that the admissibility of the pair ( ∞(N,X), p0 (N,X)) with p ∈ (1,∞] is a sufficient condition for the exponential dichotomy of Φ, which becomes necessary when Φ is with exponential growth. We apply our results in order to deduce new characterizations for exponential dichotomy of evolution families in terms of the solvability of associated difference and integral equations. © 2005 Elsevier Inc. All rights reserved.