Broadening the convergence domain of the modified Weerakoon—Fernando scheme for equations in Banach spaces

Abstract. The effectiveness of iterative schemes in solving nonlinear operator equations heavily depends on the selection of an appropriate initial choice. As a result, accurately estimating the convergence radii and developing theoretical techniques to expand the convergence region are crucial for ensuring the reliability and efficiency of such methods. Local convergence analysis, which focuses on the neighborhood of the exact solution, serves as a vital analytical framework for determining the convergence radii. In this study, we focus on improving the domain of convergence of the modified Weerakoon—Fernando iteration scheme. We conduct a detailed local convergence analysis within the setting of Banach spaces and derive explicit expressions for the convergence radius, error bounds, and the convergence zones associated with this scheme. A key feature of the proposed approach is that it depends only on the first derivative and does not require any extra conditions, which simplifies implementation while substantially enlarging the convergence domain compared to existing techniques. The theoretical advancements are substantiated through a series of rigorous numerical experiments applied to a variety of nonlinear problems. Additionally, the analysis of attraction basins offers further insights into the method's dynamic behavior, stability, and suitability for solving complex polynomial equations.

Key words and phrases. Banach space, local convergence, nonlinear equations, Newton-like methods, generalized Lipschitz—Hölder-type conditions.