Existence theorem for the Cauchy's problem on Kuelbs-Steadman spaces of vector-valued functions

Abstract

We present Kuelbs-Steadman spaces designed for vector-valued functions that take values in Banach spaces. Our study focuses on their fundamental properties and their embeddings within $\mathcal{L}^p$ spaces. Additionally, we introduce a fixed point theorem based on the concept of a measure of noncompactness in $\mathcal{KS}^p(\mathcal{X})$. Furthermore, we demonstrate the existence theorem for Cauchy's problem defined by $\hbar^\prime(\eth) = \Im(\eth, \hbar_\eth)$ and the inclusion $\hbar^\prime \in \Im(\eth, \hbar_\eth)$ in $KS^p(\mathcal{X})$, where $\Im$ is a Henstock-Kurzweil integrable function.