On the Dichotomy of the Evolution Families: A Discrete-Argument Approach

摘要

We establish a discrete-time criteria guaranteeing the existence of an exponential dichotomy in the continuous-time behavior of an abstract evolution family. We prove that an evolution family U = {U(t, s)}_(t≥s≥0) acting on a Banach space X is uniformly exponentially dichotomic (with respect to its continuous-time behavior) if and only if the corresponding difference equation with the inhomogeneous term from a vector-valued Orlicz sequence space l~ф(N, X) admits a solution in the same l~ф(N, X). The technique of proof effectively eliminates the continuity hypothesis on the evolution family (i.e., we do not assume that U(·, s)x or U(t, ·)x is continuous on [s,1), and respectively [0, t]). Thus, some known results given by Coffman and Schaffer, Perron, and Ta Li are extended.