Asymptotic behavior of dynamic equations on time scales

As a way to unify a discussion of many kinds of problems for equations in the continuous and discrete case (but also in order to reveal discrepancies between both cases), a theory of "time scales" was proposed and developed by Aulbach and Hilger. In our paper we investigate the asymptotic behavior of so-called dynamic equations on time scales, and such dynamic equations are differential equations in the continuous case and difference equations in the discrete case. We offer a perturbation result that leads to a time scales version of Levinson's Fundamental Lemma, Crucial are a dichotomy condition and a growth condition on the perturbation. Also, in the case that Levinson's result cannot be applied immediately, we suggest several preliminary transformations that might lead to a situation where Levinson's lemma is applicable. Such transformations have been suggested by Harris and Lutz in the continuous case and by Benzaid and Lutz in the discrete case. Both those cases are covered by our theory, plus cases "in between". Examples for such cases will also be discussed in this paper.