Controllability results for multi-order impulsive neutral fuzzy functional integro-differential equations with finite delay

This manuscript focuses on examining the controllability of fuzzy mild solutions for nonlocal impulsive neutral functional integro-differential equations of the first and second order, including systems with finite delay. Furthermore, it explores the characteristics of fuzzy set-valued mappings over real variables, emphasizing important features such upper semi-continuity, convexity, normalcy, and compact support. The key conclusions are obtained by applying the Banach fixed-point theorem. The study makes extensive use of fundamental ideas from functional analysis, fuzzy set theory, and the Hausdorff metric. To demonstrate the practical application of the proposed method, a detailed example is provided.