| Contents |
| Roughly speaking, we prove that if the solution of the corresponding inhomogeneous variational di fference equation belongs to a sequence space (that has the ideal property and on which the right shift is an isometry) for every inhomogenity from a sequence space (with the same properties), then the continuous-time solutions of the variational homogeneous diff erential equation will exhibit an exponential decay. Converse implications are also pointed out. This approach has many advantages among which we emphasize on the fact that the above conditions are very general (since the class of sequence spaces that we use includes almost all the known sequence spaces, as the classical spaces of p-summable sequences, 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. |
|