Halley's method for operators with unbounded second derivative

A new semilocal convergence result of Newton–Kantorovich type for the Halley method is presented, where a new technique is provided to analyze the semilocal convergence. The usual convergence conditions are relaxed, since the second derivative F″ of a nonlinear operator F satisfies ‖F″(x0)‖⩽α instead of ‖F″(x)‖⩽M, for all x in a subset of the domain of F, where α and M are positive real constants.