Hake’s theorem and its applications for Henstock-Kurzweil-Stieltjes integral of interval-valued functions on time scales

This paper proves a generalization of Hake’s Theorem for the Henstock‑Kurzweil‑Stieltjes (HKS) integral in the context of interval‑valued functions defined on time scales. The developed framework unifies the non‑absolute integration of Henstock‑Kurzweil type with Stieltjes integration on arbitrary time domains, thereby extending classical real analysis to settings that encompass both continuous and discrete dynamics. We provide a comprehensive theoretical extension with potential applications in uncertain dynamical systems modelled by set‑valued functions on hybrid time domains. The research covers fundamental theorems, properties and examples with suitable applications to interval-valued functions, demonstrating the Hake’s theorem significance in handling unbounded functions and infinite time scales.