Convergence of the iteration of Halley’s family and Smale operator class in Banach space

Smale operator classes of any order for nonlinear operators in Banach space are introduced. For an operatorf in Smale operator class of orderk, a proper condition for the convergence and the exact estimations error for the iteration of Halley’s family {Hj,kn}n=0∞(1≤j≤k) are given. This Halley’s family is a higher order explicit generalization of Newton iteration.