The Karamata integration theorem on time scales and its applications in dynamic and difference equations

We derive a time scale version of the well-known result from the theory of regular variation, namely the Karamata integration theorem. We show an application of this theorem in asymptotic analysis of linear second order dynamic equations. We obtain a classification and asymptotic formulae for all (positive) solutions, which unify, extend, and improve the existing results. In addition, we utilize these results, in combination with a transformation between equations on different time scales, to study the critical double-root case in linear difference equations. This leads to solving open problems posed in the literature.