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 Phi = {Phi(m, n)}((m,n)is an element of Delta) the subspace X-1 = {x is an element of X: Phi(center dot, 0)x is an element of l(infinity) (N, X)}. Supposing that X-1 is closed and complemented, we prove that the admissibility of the pair (l(infinity) (N, X), l (1)(0) (N, X)) implies the uniform dichotomy of Phi. Under the same 0 hypothesis on X-1, we obtain that the admissibility of the pair (l(infinity) (N, X), l(0)(p) (N, X)) with P is an element of (1, infinity] is a sufficient condition for the exponential dichotomy of Phi, which becomes necessary when Phi 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. (c) 2005 Elsevier Inc. All rights reserved.