An improvement of the Kurchatov method by means of a parametric modification

Abstract

In this work, a uniparametric generalization of the iterative method due to Kurchatov is presented. The iterative model presented is derivative‐free and approximates the solution of nonlinear equations when the operator is non‐differenciable. As the accessibility of the Kurchatov method is usually a problem in the application of the method, since the set of initial guesses that guarantee the convergence of the method is small, the main objective of this work is to improve the Kurchatov iterative method in its accessibility while maintaining and even increasing its speed of convergence. For this purpose, we introduce a variable parameter in the iterative function of the Kurchatov method that allows us to get a better approximation of the derivative by using a symmetric uniparameteric first‐order divided difference operator. We perform a complex dynamic study that corroborate the improvements in the accessibility region. Moreover, a complete analysis of the local and semilocal convergence is established for the new uniparametric iterative method. Finally, we apply the theoretical results to solve a nonlinear integral equation showing the usefulness of the work.

A Fixed-Point Type Result for Some Non-Differentiable Fredholm Integral Equations

“…Notice that the family (2.1) can be assumed as a combination of the Newton's method (µ = 0) for differentiable case and the Kurchatov method (µ = 1) in both cases, differentiable and non-differentiable for operator K. This uniparametric family maintains the quadratic convergence [14] as the Kurchatov method [2,23] and improves the accessibility of the Kurchatov method by considering values near to µ = 0, which is similar to the Newton's method.…”

“…Hernández et al [14] established the local and semilocal convergence analysis of method (2.1) for non-differentiable operators under ω-conditions.…”