Mètode de Newton en espais de Banach: teorema de Kantorovich i aplicacions

Abstract

[en] Let the equation $F x=\overrightarrow{0}$, where $F: X \mapsto Y, X$ and $Y$ are Banach spaces of like dimension, Kantorovich's Theorem gives sufficient conditions for the equation to have a solution $x^{*}$, locally unique, close to a previously picked $x^{0} \in X$, and that the Newton's method with initial condition $x^{0}$ converges to it. Moreover, gives sharp estimations of the regions where $x^{*}$ exists and is unique.

In this thesis we state Kantorovich's Theorem and prove it in two different ways: by recurrence and by majorant sequences. Besides, some variants and like results are stated, along with applications and examples on how to make use of the Theorem.