Enriched Hardy–Rogers F-contractions with applications to nonlinear integral equations involving symmetric feedback

This paper introduces a new contractive condition via the enriched Hardy–Rogers F-contraction, which generalizes and unifies several well-known contraction conditions in normed linear spaces, including those of Banach, Kannan, Reich, and Wardowski. By incorporating a nonlinear control function F from the class F1 into the enriched Hardy–Rogers structure, the proposed condition allows for the analysis of discontinuous, nonlinear, and asymmetric operators. We establish the existence and uniqueness of fixed points for such mappings and prove the convergence of the Krasnoselskij iterative scheme. In contrast to previous formulations, our approach accommodates more complex operator behavior, including mappings with symmetric delay and feedback, which are beyond the scope of classical or enriched contractions alone. To demonstrate the utility of the main result, we apply it to a new class of nonlinear integral equations modeling recurrent neural systems with symmetric feedback, thereby extending fixed point applicability to time-reflective and learning-based systems.