¹Cameron University, Lawton, USA. Email: iargyros@cameron.edu

We provide a semilocal convergence analysis for a cubically convergent two-step Newton method (2) recently introduced by H. Homeier [8], [9], and also studied by A. Özban [13]. In contrast to the above works we examine the semilocal convergence of the method in a Banach space setting, instead of the local in the real or complex number case. A comparison is given with a two step Newton--like method using the same information.

Key words: Two-step Newton method, Newton method, Banach space, majorizing sequence, Newton--Kantorovich hypothesis, semilocal convergence, Fréchet-derivative.

Proporcionamos un análisis de convergencia semilocal para un método de Newton de dos pasos, cúbicamente convergente, recientemente introducido por H. Homeier [8], [9], también estudiado por A. Özban [13]. En contraste con esto, examinamos la convergencia local del método en espacios de Banach en lugar del local, en el caso real y complejo. Damos una comparación con el método de Newton de dos pasos usando la misma información.

Palabras clave: Método de Newton de dos pasos, método de Newton, espacio de Banach, secuencia mayorante, hipótesis de Newton--Kantorovich, convergencia semilocal, derivada de Fréchet.

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