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.