Third-order iterative methods for operators with bounded second derivative

We analyse the classical third-order methods (Chebyshev, Halley, super-Halley) to solve a nonlnnear equation F(x) = 0, where F is an operator defined between two Banach spaces. Until now the convergence of these methods is established assuming that the second derivative F″ satisfies a Lipschitz condition. In this paper we prove, by using recurrence relations, the convergence of these and other third-order methods just assuming F″ is bounded. We show examples where our conditions are fulfilled and the classical ones fail.