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

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 Lᵖ spaces. Additionally, we introduce a fixed point theorem based on the concept of a measure of noncompactness in KSᵖ(X) . Furthermore, we demonstrate the existence theorem for Cauchy?s problem defined by ??(?) = ?(?, ??) and the inclusion ?? ? ?(?, ??) in KSᵖ(X) , where ? is a Henstock-Kurzweil integrable function.