Discrete-time theorems for global and pointwise dichotomies of cocycles over semiflows

Abstract

In this paper, we consider linear skew-product semiflows on bundles of Banach fibers over a locally compact metric space. Our aim is to give discrete-time theorems for the existence of global and pointwise continuous-time dichotomies with no invariant unstable manifolds. We involve here a concept of exponential dichotomy for skew-product semiflows weaker than the concept used by Sacker and Sell (J Differ Equ 15:429–458, 1974; 22:478–522, 1976) and Magalhaes (in: Chow, Hale (eds) Dynamics of infinite dimensional systems, NATO Advanced Science Institutes Series F: Computer and Systems Sciences, Springer, New York, vol 37, pp 161–168, 1987); our definition (of no past exponential dichotomy) follows roughly the definition given by Chow and Leiva (Proc Am Math Soc 124:1071–1081, 1996) in the sense that we allow the unstable subspace to have infinite dimension. The main improvement is that we go even more general and we do not assume a priori that the cocycle is invertible on the unstable space (actually we do not even assume that the unstable subspace is invariant under the cocycle). Roughly speaking, we prove that if the solution of the corresponding inhomogeneous variational difference equation belongs to any sequence space (on which the right shift is an isometry) for every inhomogeneity from the same class of sequence spaces, then the continuous-time solutions of the variational homogeneous differential equation will exhibit a (no past) exponential dichotomic behavior. This approach has many advantages among which we emphasize on the facts that the above condition is very general (since the class of sequence spaces that we use includes almost all the known sequence spaces, as the classical p-summable spaces, sequence Orlicz spaces, etc.). Since we use a discrete-time technique we are not forced to require any continuity or measurability hypotheses on the trajectories of the exponentially bounded cocycle. Also, it is worth to mention that from discrete-time conditions we get informations about the continuous-time behavior of the solutions of differential variational equations.