Journal of the
Korean Mathematical Society JKMS

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 $0<\lambda< 2$. 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 $\lambda$. 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 $\lambda=1$.

Keywords: relaxed Newton's method, Banach space, Kantorovich hypothesis, majorizing sequence, local convergence, semilocal convergence

MSC numbers: 65J15, 47H17, 65H10, 65G99, 49M15