Topologies of Llocp type for Carathéodory functions with applications in non-autonomous differential equations

Metric topological vector spaces of Carathéodory functions and topologies of Llocp 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 Carathéodory differential equations are proved in such new topological structures. As a consequence, new families of continuous linearized skew-product semiflows are provided in the Carathéodory spaces.