A robust implicit Jungck iterative scheme: stability, data dependency, and applications to nonlinear integral equations

This study introduces a novel implicit Jungck-type iterative scheme for solving nonlinear functional integral equations with double delays. By incorporating weak contractive conditions, the proposed scheme enhances computational efficiency and stability while achieving convergence equivalent to its explicit counterpart. Compared to existing Jungck-type explicit schemes, the implicit iteration exhibits superior stability and faster convergence. This work addresses key limitations in previous studies (Alam, Rohen in J. Appl. Math. Comput., https://doi.org/10.1007/s12190-024-02134-z , 2024; Copur et al. in Math. Comput. Simul. 215:476–497, 2024; Sharma et al. in Contemp. Math. 5:743–760, 2024) by: 1. Establishing fixed-point results for a broader class of mappings with guaranteed existence and uniqueness of fixed point, 2. Refining the stability framework for more reliable theoretical outcomes, 3. Removing restrictive conditions on control sequences to improve convergence, and 4. Deriving sharper estimates for data dependence, ensuring greater accuracy. Theoretical analysis within the weak w2-stability framework is supported by numerical experiments, including boundary value problems, confirming the scheme’s enhanced stability and efficiency. By refining and extending recent developments in fixed-point and coincidence-point theory, this study provides a robust framework for solving complex nonlinear problems.