Topologies of continuity for Carathéodory differential equations with applications in non-autonomous dynamics

Abstract

The theory developed in this work allows to extend the skew-product formalism to Carathéodory ordinary differential equations and delay differential equations with constant delay through the use of strong and weak metric topologies of integral type. As a result, one obtains a variety of tools from topological dynamics to study the qualitative behavior of the solutions of such classes of differential problems. As an example, the work includes several applications for Carathéodory ODEs such as linearized skew-product flows, propagation of the exponential dichotomy and of the dichotomy spectrum of a linear system and study of pullback and global attractors, as well as some simple motivational examples taken from modelizations of real phenomena, which aim to show the applicability of the theory. Additionally, the thesis provides a rich description of the topological structure of the considered spaces of Carathéodory functions (among which, some are new) presenting, for example, characterizations of the classes of equivalences for functions which differ on negligible subset of the domain, propagation of properties on the so-called m-bounds and l-bounds through the limits in the given topologies, and suffcient conditions of relative compactness for subsets of Lipschitz Carathéodory functions.