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Uko and Argyros provided in [18] a Kantorovich-type theorem on the existence and uniqueness of the solution of a generalized equation of the form f (u)+g(u) ∋ 0, where f is a Fréchet-differentiable function, and g is a maximal monotone operator defined on a Hilbert space. The sufficient convergence conditions are weaker than the corresponding ones given in the literature for the Kantorovich theorem on a Hilbert space. However, the convergence was shown to be only linear.In this study, we show under the same conditions, the quadratic instead of the linear convergenve of the generalized Newton iteration involved. RESUMENUko y Argyros estudian en [18] un teorema tipo-Kantorovich en el existencia y unicidad de la solución de una ecuación generalizada de la forma f (u) + g(u) ∋ 0, donde f es una función Fréchet-diferenciable, y g es un operador monotono máximo definido en un espacio de Hilbert. Las condiciones de convergencia suficientes son más débiles que los correspondientemente dadas en la literatura para el teorema de Kantorovich en un espacio de Hilbert. Sin embargo, la convergencia ha demostrado ser sólo lineal.En este estudio, mostramos en las mismas condiciones, la ecuación cuadrática en lugar de la lineal convergente de la iteración generalizada de Newton involucradas.

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We extend the applicability of the Gauss-Newton method for solving singular systems of equations under the notions of average Lipschitz-type conditions introduced recently in Li et al. (J Complex 26(3):268-295, 2010). Using our idea of recurrent functions, we provide a tighter local as well as semilocal convergence analysis for the Gauss-Newton method than in Li et al. (J Complex 26(3):268-295, 2010) who recently extended and improved earlier results (Hu et al. ). We also note that our results are obtained under weaker or the same hypotheses as in Li et al. (J Complex 26(3):268-295, 2010). Applications to some special cases of Kantorovich-type conditions are also provided in this study.

A semilocal convergence analysis for directional two-step Newton methods in a Hilbert space setting is provided in this study. Two different techniques are used to generate the sufficient convergence results, as well as the corresponding error bounds. The first technique uses our new idea of recurrent functions, whereas the second uses recurrent sequences. We also compare the results of the two techniques.