Differential problems with Stieltjes derivatives and applications

Abstract

This Thesis is a collection of the research work developed by the author during his predoctoral stage. As the title suggests, this thesis revolves around the concept of Stieltjes derivative and the differential problems associated with it. Roughly speaking, this derivative is a modification of the usual derivative through a nondecreasing and left--continuous map, called derivator. After exploring this concept, we look for conditions ensuring the existence and uniqueness of solution of differential problems with this type of derivative. In particular, we consider differential equations with initial value conditions, differential equations with functional arguments subject to more general boundary conditions, and differential inclusions. For these problems, we look at some classical results for the corresponding problems with the usual derivative, and adapt them to this new setting, accounting for the differences that arise naturally from the Stieltjes derivative. In this case, and since we are basing our results on the classical setting, we only consider one derivator. However, we later explore similar problems in a framework that makes sense in the context of differential problems with Stieltjes derivatives, namely, differential problems with several different derivators. In other words, we consider systems of equations in which each of the equations is differentiated with respect to a different derivator. This, of course, offers a more general setting than the previous case, not only from a theoretical point of view, but also from the applications point of view, as we show with many examples.