Concerning the convergence of Newton's method and quadratic majorants

We approximate a locally unique solution of an equation in a Banach space setting using Newton's method. Motivated by optimization considerations, and the elegant studies by Ferreira and Svaiter (in press) but using more precise error estimates along our works in J. Comput. Appl. Math. 169, 315-332, 2004, J. Math. Anal. Appl. 298, 374-387, 2004, Studies in Computational Mathematics, vol. 15, 2007, and under the same computational cost we provide: a larger convergence region; finer error bounds on the distances involved, an at least as precise information on the location of the solution and of the convergence domains on which Newton's method can be repeated indefinitely.