Notes on the linear equation with Stieltjes derivatives

摘要

In this paper we continue the study of the linear equation with Stieltjes derivatives in [M. Frigon, R. López Pouso, Adv. Nonlinear Anal. 6(2017), 13–36]. Specifically, we revisit some of the results there presented, removing some of the required conditions as well as amending some mistakes. Furthermore, following the classical setting, we use the connection between the linear equation and the Gronwall inequality to obtain a new version of this type of inequalities in the context of Lebesgue–Stieltjes integrals. From there, we obtain a uniqueness criterion for initial value problems.