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

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.