Recurrence relations for super-Halley's method with Hölder continuous second derivative in Banach spaces (Q1951669)

The semilocal convergence of a third order super-Halley's method for approximating a locally unique solution of the nonlinear operator equation \(F(x)=0\) in Banach space is discussed. Using a new family of recurrence relations, the authors prove the convergence and the R-order of the method, based on the assumption that the second Fréchet derivatives of the nonlinear operator is Hölder continuous. Numerical examples are used to illustrate the efficiency of the proposed algorithm.