An Improved Convergence Analysis and Applications for Newton-Like Methods in Banach Space

Abstract

In this study we introduce more general Chen–Yamamoto-type conditions to generate a Newton-like method which converges to a locally unique solution of a nonlinear equation in a Banach space containing a non-differentiable term. Using new and more precise majorizing sequences we provide local and semilocal results, first under the same and secondly, under weaker sufficient convergence conditions than before. In both cases we show that our results can be reduced to the ones by Chen and Yamamoto (Chen, X., Yamamoto, T. (1989). Convergence domains of certain iterative methods for solving nonlinear equations. Numer. Funct. Anal. Optimiz. 10(1&2):37–48.), whereas the error bounds and the information on the location of the solution can be more precise, and under more general conditions. Finally some numerical examples are provided where our results compare favorably with earlier ones in both the local and semilocal case.

Keywords:

• Newton-like method
• Banach space
• Fréchet-derivative
• Majorizing sequence
• Chen–Yamamoto conditions
• Radius of convergence
• Newton–Kantorovich hypothesis
• AMS (MOS) Subject Classification codes: 65J15
• 47H17
• 49M15
• CR:1.5