On the semi-local convergence of Halley's method under a center-Lipschitz condition on the second Fréchet derivative

We expand the applicability of Halley's method for solving nonlinear equations in a Banach space setting. We assume the existence of the center-Lipschitz condition on the second Fréchet-derivative of the operator involved instead of Lipschitz condition used extensively in the literature [1,2,4,5]. The center-Lipschitz condition is satisfied in many interesting cases, where the Lipschitz condition is not satisfied [3,4,6,7,13]. We show that the semi-local convergence theorem established in [X.B. Xu, Y.H. Ling, Semilocal convergence for Halley's method under weak Lipschitz condition, Appl. Math. Comput. 215 (2009) 3057-3067] is not true. A new semi-local convergence theorem is established for Halley's method under the same condition. Our results are illustrated using a nonlinear Hammerstein integral equation of the second kind where our convergence criteria are satisfied but convergence criteria in earlier studies such as [1,2] are not satisfied.