El método de Newton en espacios de Banach

Abstract

Newton's method is a well known iterative method to solve a nonlinear equation F(x) = 0. We analyze the convergence of this method for operators defined between two Banach spaces, so our results can be applied in a wide range of problems, such as real or complex equations, nonlinear systems of equations, differential or integral equations. Firstly we study Newton's method in terms of the linear operator LFx=F' x-1F ''x F 'x1 Fx. In this sense, new convergence results are given in terms of this operator. Another part of this report is devoted to the study of Newton's method assuming that F satisfies different conditions from the classical ones (Kantorovich). Finally, as an acceleration of Newton's method, a new third order iterative process is obtained. Its basic properties (convergence, unicity of solution, error estimates, etc.) are analysed. This work is mainly theoretical although some results are illustrated with examples.