Recurrence relations for Super-Halley's method with Hölder continuous second derivative in Banach spaces

Abstract

The aim of this paper is to study the semilocal convergence of the Super-Halley's method used for solving nonlinear equations in Banach spaces by using the recurrence relations. This convergence is established under the assumption that the second Frëchet derivative of the involved operator satisfies the Hölder continuity condition which is milder than the Lipschitz continuity condition. A new family of recurrence relations are defined based on two constants which depend on the operator. An existence-uniqueness theorem and a proori error estimates are provided for the solution x*. The R-order of the method equals to (2 + p) for p $in$ (0,1] is also established. Three numerical examples are worked out to demonstrate the efficacy of our approach. On comparison with the results obtained for the Super-Halley's method in [3] by using majorizing sequence, we observed improved existence and uniqueness regions for the solution x* by our approach.