Qualitative analysis and numerical simulation of ABC fractional impulsive differential systems with implicit nonlinear terms and integral boundary conditions

Abstract

This study presents a comprehensive qualitative analysis and numerical investigation of a novel class of fractional impulsive differential systems. The model incorporates the Atangana–Baleanu–Caputo (ABC) fractional derivative, implicit nonlinear terms, and nonlocal integral boundary conditions within a unified framework. The originality of the work lies in the simultaneous treatment of three challenging features: ABC fractional operators, impulsive effects, and implicit nonlinear structures—a combination not thoroughly addressed in existing literature. First, sufficient conditions for the existence of solutions are established using Krasnoselskii’s fixed-point theorem, while uniqueness is guaranteed via Banach’s fixed-point theorem under appropriate constraints. Furthermore, the Hyers–Ulam stability of the system is rigorously examined, confirming its robustness against small perturbations. Beyond the theoretical analysis, detailed numerical simulations are performed using an L1-type discretization scheme to illustrate and validate the analytical results, demonstrating the practical applicability and computational feasibility of the proposed approach. This work effectively bridges qualitative theory and computational methods, providing a rigorous foundation for studying fractional-order impulsive systems with nonlocal conditions. It offers new insights for modeling real-world phenomena characterized by memory effects, sudden state changes, and complex interdependencies.