Ball Convergence for Second Derivative Free Methods in Banach Space

Abstract

Hueso et al. (Appl Math Comput 211:190–197, 2009) considered a third and fourth order iterative methods for nonlinear systems. The methods were shown to of order third and fourth if the operator equation is defined on the j-dimensional Euclidean space (Hueso et al. in Appl Math Comput 211:190–197, 2009). The order of convergence was shown using hypotheses up to the third Fréchet derivative of the operator involved although only the first derivative appears in these methods. In the present study we only use hypotheses on the first Fréchet-derivative. This way the applicability of these methods is expanded. Moreover we present a radius of convergence a uniqueness result and computable error bounds based on Lipschitz constants. Numerical examples are also presented in this study. © 2015, Springer India Pvt. Ltd.