Problemas de valor inicial en la construcción de sucesiones mayorizantes para el método de Newton en espacios de Banach

Abstract

It is well known that solving nonlinear equations of the form F(x)=0, where F is a nonlinear operator defined between Banach spaces, is a common problem in science and engineering. Usually numerical approximations of the roots of the previous equation are looking for, since finding exact roots is often difficult. To approximate a root of the previous equation is commonly used iterative methods, among which Newton's method is important for its simplicity, easy implementation and efficiency. The first result of semilocal convergence for Newton's method in Banach spaces was given by Kantorovich. In this dissertation it is analyze the semilocal convergence of Newton's method in Banach spaces, whose principal aim is to give greater generality to the problem of approximating the roots of a nonlinear equation by Newton's method, so that it can extend the applicability of this method to situations where the classical theory of Kantorovich cannot. To do this, we use the majorant principle, based on the construction of majorizing sequences, and, under new semilocal convergence conditions of Kantorovich-type, that allows us to generalize the classical conditions of Kantorovich. This plays an important role in the ad hoc construction of majorizing sequences from solving initial value problems. We illustrate the above with different types of nonlinear equations, highlighting the Hammerstein integral equations of mixed type and Bratu's equation, which have their origin in various real-world problems, such as it is evidenced in this dissertation.