Semilocal convergence for the Super-Halley's method

The semilocal convergence of Super-Halley's method for solving nonlinear equations in Banach spaces is established under the assumption that the second Frëchet derivative satisfies the $\omega$-continuity condition. This condition is milder than the well known Lipschitz and Hölder continuity conditions. The importance of our work lies in the fact that numerical examples can be given to show that our approach is successful even in cases where the Lipschitz and Hölder continuity conditions fail. Difficult computation of the second Frëchet derivative is also avoided by replacing it with a divided difference containing only the first Frёchet derivatives. A number of recurrence relations based on two parameters are derived. A convergence theorem is established to estimate a priori error bounds along with the domains of existence and uniqueness of the solutions. The $R$-order of convergence of the method is shown to be at least three. Two numerical examples are worked out to demonstrate the efficiency of our method. It is observed that in both examples the existence and uniqueness regions of solution are improved when compared with those obtained in [7].