Application of artificial neural networks for existence and controllability in impulsive fractional Volterra-Fredholm integro-differential equations

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This study focuses on impulsive Volterra-Fredholm integro-differential equations (VFIDEs) under specific order conditions, conducting a rigorous analysis involving fractional Caputo derivatives. By applying the Schauder fixed-point theorem, we prove the existence of solutions and examine the interaction between fractional Caputo derivatives and the equation's integro-differential structure. Furthermore, we explore the application of artificial neural networks (ANNs) to predict system states based on input features, demonstrating how these techniques enhance the understanding of control inputs and system responses. These findings improve the theoretical understanding of impulsive VFIDEs and demonstrate their applications in science and engineering, particularly regarding existence conditions and controllability.