Abstract
The aim of this paper is to discuss the convergence of a third order Newton-like method for solving nonlinear equations F(x) = 0 in Banach spaces by using recurrence relations. The convergence of the method is established under the assumption that the second Fréchet derivative of F being ω-continuous given by ||F″(x)-F″(y)|| ≤ ω (||x - y||), x, y $\in$ Ω, where ω be a nondecreasing function on R+ and Ω any open set. This ω-continuity condition is milder than the usual Lipschitz/Hölder continuity condition. To get a priori error bounds, a family of recurrence relations based on two parameters depending on the operator F is also derived. Two numerical examples are worked out to show that the method is successful even in cases where Lipschitz/Hölder continuity condition fails but ω-continuity condition is satisfied. In comparison to the work of Wu and Zhao [15], our method is more general and leads to better results.
Citation
Dharmendra K. Gupta. Pradip K. Parida. "A Newton-like method in Banach spaces under mild differentiability conditions." Kodai Math. J. 31 (3) 414 - 430, October 2008. https://doi.org/10.2996/kmj/1225980445
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