Analysis of a multifractional hybrid differential system with impulses

This article explores a multifractional hybrid differential system with impulses involving the φ -Hilfer and φ -Caputo fractional derivatives. These derivatives provide greater flexibility in modelling systems with memory effects and non-local behaviours. The primary objective is to establish the existence and uniqueness of solutions to the proposed system. Using Krasnosel’skii’s fixed point theorem, we prove the existence of a solution under suitable growth and compactness conditions. Furthermore, by applying the Banach contraction principle, we establish uniqueness of the solution. An example is constructed to demonstrate the applicability and provide insight into the reliability of the solution. This framework is relevant to systems with multiscale memory effects, such as biological systems and fractional-order control processes, where memory and response evolve dynamically over time. This study provides a foundation for deeper investigations into the analysis of generalised fractional systems with hybrid dynamics.