Newton-Kantorovich theory and some of its variations

In1948 the Soviet mathematician L.V. Kantorovich proposed the extension of Newton's method for solving nonlinear equations to functional spaces. Perhaps the most relevant aspect of Kantorovich's result is the combination of techniques in Numerical Analysis and Functional Analysis. Another peculiarity of Kantorovich's theorem is that it does not assume the existence of asolution.Sothistheorem canbeseenasanexistenceanduniquenessresultandnotonlyasa convergence theorem for a given method. Kantorovich's theory can be applied to a wide range of nonlinear problems, such as integral equations, ordinary and partial differential equations, variational problems, etc.
There exist numerous versions of Kantorovich's theorem, which differ both in assumptions and results. In this talk we present some of these variations, making especial emphasis on the ones achieved by the speaker and by his research group.