On Halley-type iterations with free second derivative (Q596220)

The authors prove convergence of the so-called Halley method that is a third-order iterative approximation for solving a nonlinear equation in a Banach space. In contrast to the proof given by the same authors in [Int. J. Pure Appl. Math. 6 (1), 103--114 (2003; Zbl 1026.47056)], here no assumption on the second derivative of the operator is needed. Even the existence of the second Fréchet derivative is not required and it suffices that the first derivative is Lipschitz continuous.