Analytical approach for fractional differential hybrid inclusions involving Ψ-Hilfer fractional derivative

Our approach involved constructing appropriate operator spaces and verifying the conditions necessary for Dhage?s theorem to hold. We demonstrated that our specific ?-Hilfer hybrid inclusion problem meets these criteria, thus guaranteeing the existence of at least one solution. To illustrate and validate our theoretical findings, we presented a concrete example showcasing the application of our theoretical framework, which highlights the practical relevance and effectiveness of our approach in solving fractional differential inclusions. This example serves not only as a proof of concept but also as a practical demonstration of the theoretical results, affirming the robustness and applicability of the ?-Hilfer hybrid fractional differential inclusion framework.