Abstract
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 ″ ( x 0 ) ‖ ⩽ 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.
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