Enhancing the Applicability of a Parametric Sixth-Order Convergent Method for Solving Nonlinear Equations

The applicability of a highly efficient sixth-order convergent method, originally proposed by Kansal et al., is extended in this study to a Banach space setting. The initial development of this method relied upon Taylor series expansions in Rn and the assumption that the nonlinear operator is sufficiently differentiable. This vague condition implies the existence of high-order derivatives that are not actually utilized by the algorithm. This study transcends these limitations by establishing convergence based solely on generalized continuity conditions of the first Fréchet derivative. By dispensing with these strong smoothness requirements, the domain of applicability is significantly widened. We derive computable radii for the ball of convergence and establish error bounds under local analysis. Furthermore, a rigorous semi-local convergence analysis is presented, a feature previously absent in the literature for this specific scheme, utilizing a majorizing sequence technique to guarantee the existence and uniqueness of the solution. The theoretical results are validated through numerical experiments, which demonstrate that the method converges even when the standard sufficiently differentiable conditions are violated.