Approximation of fixed points by new iterative schemes

Abstract

This study introduces a set of newly developed iterative schemes designed to approximate the fixed points of contractive-type self and non-self mappings. A novel one-step iterative scheme, incorporating a single parameter, is constructed through a geometric approach using linear approximation. Building upon this, two-step and three-step hybrid iterative schemes are formulated to enhance the efficiency and flexibility of the fixed point approximation process. Theoretical analyses are carried out to establish important convergence properties of these schemes, including data dependence, stability, strong convergence, and order of convergence. To validate the practical applicability of the proposed methods, comprehensive numerical experiments are performed on a variety of problems, including nonlinear scalar equations, nonlinear systems of equations, and differential equations. In addition to the straight-line based approach, another geometrically constructed iterative process is proposed, utilizing two nonlinear approximations. This leads to the development of additional families of two-step and three-step hybrid schemes, which are also thoroughly analyzed for their convergence behaviour. Numerical examples are provided to compare the convergence rates of the proposed methods with those of existing fixed point iterative schemes available in the literature. Finally, a new Jungck-type iterative scheme is introduced for approximating the coincidence points of contractive-type non-self mappings. The scheme is rigorously examined with respect to strong convergence, stability, and data dependence. Numerical experiments confirm that the newly introduced Jungck-type scheme offers an improved rate of convergence compared to the existing Jungck-type iterative schemes available in the literature. newline