A Newton-like method in Banach spaces under mild differentiability conditions

Abstract

The aim of this paper is to discuss the convergence of a third order Newton-like method for solving nonlinear equations F(x) = 0 in Banach spaces by using recurrence relations. The convergence of the method is established under the assumption that the second Fréchet derivative of F being ω-continuous given by ||F″(x)-F″(y)|| ≤ ω (||x - y||), x, y $\in$ Ω, where ω be a nondecreasing function on R₊ and Ω any open set. This ω-continuity condition is milder than the usual Lipschitz/Hölder continuity condition. To get a priori error bounds, a family of recurrence relations based on two parameters depending on the operator F is also derived. Two numerical examples are worked out to show that the method is successful even in cases where Lipschitz/Hölder continuity condition fails but ω-continuity condition is satisfied. In comparison to the work of Wu and Zhao [15], our method is more general and leads to better results.