Impulsive fractional delay differential equations with fixed moments and modified Ulam-Hyers-Rassias stability

In this paper, we discuss the existence and uniqueness of solution of Atangana-Baleanu-Caputo impulsive fractional delay differential equations with caratheo-dory function. We further introduce modified Ulam-Hyers-Rassias stability criteria by considering a real-valued function that is Lebesque integrable. This new concept makes the theory more realistic, flexible, and mathematically consistent with modern analysis (fractional calculus, impulsive system and delay equations). It covers unbounded but integrable disturbances, accommodates Caratheordory conditions, and extends applicability to a much larger class of dynamical systems. Extending Ulam-Hyers-Rassias stability to Lebesque integrable perturbations makes it compatible with stronger existence and uniqueness theorems using Banach and Schauder fixed point theorems which often require mappings to be continuous and bounded in L1θ([t0,T]) type norms. The stability of the solution of Atangana-Baleanu-Caputo impulsive fractional delay differential equations with caratheo-dory function is also investigated by using the modified Ulam-Hyers-Rassias stability concept.