Weak topologies for Carath\'{e}odory differential equations. Continuous dependence, exponential Dichotomy and attractors

TL;DR

This paper develops new weak topologies for Carathéodory differential equations, ensuring continuous dependence of solutions on initial data and vector fields, and explores implications for exponential dichotomies and attractors.

Contribution

It introduces novel weak topologies and spaces for Carathéodory functions, enabling continuous solution dependence and analysis of attractors and spectral properties.

Findings

Continuous dependence of solutions on initial data and vector fields.

Propagation of exponential dichotomy over trajectories.

Existence of bounded pullback and global attractors.

Abstract

We introduce new weak topologies and spaces of Carath\'eodory functions where the solutions of the ordinary differential equations depend continuously on the initial data and vector fields. The induced local skew-product flow is proved to be continuous, and a notion of linearized skew-product flow is provided. Two applications are shown. First, the propagation of the exponential dichotomy over the trajectories of the linearized skew-product flow and the structure of the dichotomy or Sacker-Sell spectrum. Second, how particular bounded absorbing sets for the process defined by a Carath\'eodory vector field $f$ provide bounded pullback attractors for the processes with vector fields in the alpha-limit set, the omega-limit set or the whole hull of $f$. Conditions for the existence of a pullback or a global attractor for the skew-product semiflow, as well as application examples are also given.