On the convergence region of Newton's method under Hölder continuity conditions

Abstract

We approximate a locally unique solution of an equation in a Banach space setting using Newton's method under Hölder continuity assumptions. Using more precise estimates on the distances involved than before [F. Cianciaruso and E. DePascale, Estimates of majorizing sequences in the Newton–Kantorovich method, Numer. Funct. Anal. Optim. 27(5 and 6) (2006), pp. 529–538], we show that the convergence region is extended with finer error estimates and the location of the solution is precise under the same computational cost.

Keywords:

• Newton's method
• Hölder continuity
• centre-Hölder continuity
• convergence region
• error estimates

2000 AMS Subject Classification :

• 65B05
• 65G99
• 65J15
• 47H17
• 49M15