# Topologies of $L^p_{loc}$ type for Carath\'{e}odory functions with applications in non-autonomous differential equations

## TL;DR

This paper introduces new $L^p_{loc}$-type topologies for Carathéodory functions and proves continuous dependence of solutions in these spaces, enabling the development of new linearized semiflows for non-autonomous differential equations.

## Contribution

It develops novel topological structures for Carathéodory functions and establishes continuous dependence results, advancing the analysis of non-autonomous differential equations.

## Key findings

- Established new $L^p_{loc}$-type topologies for Carathéodory functions.
- Proved continuous dependence of solutions on initial data in these topologies.
- Constructed new linearized skew-product semiflows in Carathéodory spaces.

## Abstract

Metric topological vector spaces of Carath\'eodory functions and topologies of $L^p_{loc}$ type are introduced, depending on a suitable set of moduli of continuity. Theorems of continuous dependence on initial data for the solutions of non-autonomous Carath\'eodory differential equations are proved in such new topological structures. As a consequence, new families of continuous linearized skew-product semiflows are provided in the Carath\'eodory spaces.