In this book the authors use a technique based on recurrence relations to study the convergence of the Newton method under mild differentiability conditions on the first derivative of the operator involved. The authors¿ technique relies on the construction of a scalar sequence, not majorizing, that satisfies a system of recurrence relations, and guarantees the convergence of the method. The application is user-friendly and has certain advantages over Kantorovich¿s majorant principle. First, it allows generalizations to be made of the results obtained under conditions of Newton-Kantorovich type and, second, it improves the results obtained through majorizing sequences. In addition, the authors extend the application of Newton¿s method in Banach spaces from the modification of the domain of starting points. As a result, the scope of Kantorovich¿s theory for Newton¿s method is substantially broadened. Moreover, this technique can be applied to any iterative method.

This book is chiefly intended for researchers and (postgraduate) students working on nonlinear equations, as well as scientists in general with an interest in numerical analysis.

Preface.- The Newton-Kantorovich theorem.- Operators with Lipschitz continuous first derivative.- Operators with Hölder continuous first derivative.- Operators with Hölder-type continuous first derivative.- Operators with w-Lipschitz continuous first derivative.- Improving the domain of starting points based on center conditions for the first derivative.- Operators with center w-Lipschitz continuous first derivative.- Using center w-Lipschitz conditions for the first derivative at auxiliary points.