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The present paper is concerned with the semilocal convergence problems ofHalley's method for solving nonlinear operator equation in Banach space. Undersome so-called majorant conditions, a new semilocal convergence analysis forHalley's method is presented. This analysis enables us to drop out theassumption of existence of a second root for the majorizing function, but stillguarantee Q-cubic convergence rate. Moreover, a new error estimate based on adirectional derivative of the twice derivative of the majorizing function isalso obtained. This analysis also allows us to obtain two important specialcases about the convergence results based on the premises of Kantorovich andSmale types.

Semilocal Convergence Behavior of Halley's Method Using Kantorovich's Majorants Principle