Abstract
In this work we study the local and semilocal convergence of the relaxed Newton's method, that is Newton's method with a relaxation parameter 0 < � < 2. We give a Kantorovich-like theorem that can be applied for operators defined between two Banach spaces. In fact, we obtain the recurrent sequence that majorizes the one given by the method and we characterize its convergence by a result that involves the relaxation parameter �. We use a new technique that allows us on the one hand to generalize and on the other hand to extend the applicability of the result given initially by Kantorovich for � = 1. In many areas related to the applied sciences one confronts the problem of solving a nonlinear equation of the form f(x) = 0. The solutions of these equa- tions can rarely be found in closed form. That is why most solution methods are iterative. There exist lots of iterative methods with different properties that allow us to solve this kind of equations, but the most well-known and used is the Newton's method, which has the following form:
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