Existence and stability analysis of impulsive Caputo fractional delay systems

This paper presents a comprehensive investigation into a class of impulsive delay differential equations. We establish the existence of solutions and introduce a refined stability analysis. Using the Caputo fractional derivative and fixed point theory, we rigorously prove the existence of solutions by precisely locating fixed points. Our work addresses the limitations of traditional Hyers-Ulam stability by providing compelling counterexamples, which motivates the development of a new, more robust stability framework. We introduce a novel approach that combines impulsive control with fractional calculus to model complex systems, offering a more accurate representation of phenomena with memory and abrupt changes. A detailed example is provided to validate our theoretical findings, demonstrating the practical applicability of our advanced stability criteria in predicting system behavior.