Convergence and stability analysis of a modified hybrid iterative process with some applications

Abstract

This paper presents a novel modification of the Picard-Noor hybrid iterative process, designed to enhance convergence performance in solving nonlinear equations. The newly proposed scheme is rigorously analyzed and shown to have a superior convergence rate to that of the Picard-S iterative scheme, in Berinde’s sense. We not only prove the strong convergence of the proposed scheme but also establish its stability and data dependency, ensuring the method’s resilience to slight perturbations in the operator or initial approximation. In comparison to classical methods such as Picard, Mann, Ishikawa, Noor, and their hybrid variants, the process achieves improved performance. A detailed theoretical framework supports the convergence claims, and numerical examples are provided to demonstrate its practical efficiency and accelerated convergence behavior. The analysis confirms that this approach offers a reliable and faster alternative in nonlinear functional analysis and related computational applications. In this study, we also demonstrate some applications in differential equation, machine learning and optimization algorithms. Overall, the findings suggest that the modified Picard-Noor hybrid process not only improves convergence speed but also contributes significantly to the development of stable, data-sensitive iterative algorithms. This advancement, coupled with its application in various fields opens up new directions for research in the development of optimized fixed-point iterative schemes tailored for complex nonlinear systems.