A local convergence analysis of Newton's method for solving nonlinear
equations, under a majorant condition, is presented in this paper. Without
assuming convexity of the derivative of the majorant function, which relaxes
the Lipschitz condition on the operator under consideration, convergence, the
biggest range for uniqueness of the solution, the optimal convergence radius
and results on the convergence rate are established. Besides, two special cases
of the general theory are presented as an application.