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From Kantorovich’s theory we present a semilocal convergence result for Newton’s methodwhich is based mainly on a modification of the condition required to the second derivativeof the operator involved. In particular, instead of requiring that the second derivativeis bounded, we demand that it is centered. As a consequence, we obtain a modification ofthe starting points for Newton’s method. We illustrate this study with applications tononlinear integral equations of mixed Hammerstein type.

We provided in [14] and [15] a semilocal convergence analysis for Newton’s method on a Banach space setting, by splitting the given operator. In this study, we improve the error bounds, order of convergence, and simplify the sufficient convergence conditions. Our results compare favorably with the Newton-Kantorovich theorem for solving equations.

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A general semilocal convergence result for Newton’s methodunder centered conditions for the second derivative

On Newton’s Method for Solving Nonlinear Equations and Function Splitting

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A general semilocal convergence result for Newton’s methodunder centered conditions for the second derivative

On Newton’s Method for Solving Nonlinear Equations and Function Splitting