Asymptotic integration under weak dichotomies (Q968574)

This paper provides new important results about the asymptotic integrations of perturbed linear differential systems. The authors obtain a new set of conditions under which the perturbed system and the unperturbed system are strongly asymptotically equivalent. Comparing with the classical Levinson theorem, their new dichotomy conditions are not bounded by constants, instead, by a function \(\beta(s)\). The corresponding requirement of the perturbation \(R(t)\) is not necessarily absolutely integrable, instead, \(\beta(t)R(t)\) is in \(L^1\). Thus, they extend Levinson's theorem to a wider setting. They generalize their new theorem to perturbed Jordan matrices. Also, they obtain a result on general linear differential systems. Furthermore, they get the analogs of their new results in the setting of difference equations. Also, they show some modified known examples which are applicable to their new results in the last part of the paper. This is an important paper in the area of asymptotic analysis of perturbed differential and difference systems. It will be of interest for anyone who is studying asymptotic integration of differential systems, asymptotic summation of difference systems, and their applications.