ASYMPTOTIC STABILITY OF NONUNIFORM BEHAVIOUR.

This paper is devoted to exponential dichotomies of nonautonomous difference equations. Under the assumptions that (Am)m∈Z is a sequence of bounded operators acting on an arbitrary Banach space X that admits a uniform exponential dichotomy and that (Bm)m∈Z is a sequence of compact operators such that lim|m|→∞||Bm|| = 0, D. Henry proved that either the sequence (Am+Bm)m∈Z admits a uniform exponential dichotomy or there exists a bounded nonzero sequence (xm)m∈Z ⊂ X such that xm+1 = (Am +Bm)xm for each m ∈ Z. In this paper we prove Henry's result in the setting of nonuniform exponential dichotomies. Then we obtain a result on roughness of the nonuniform exponential dichotomy and give stability of Lyapunov exponents. In addition, we establish corresponding results for dynamics with continuous time. [ABSTRACT FROM AUTHOR]