This monograph examines a variety of iterative methods in Banach spaces with a focus on those obtained from the Newton method. Together with the authors' previous two volumes on the topic of the Newton method in Banach spaces, this third volume significantly extends Kantorovich's initial theory. It accomplishes this by emphasizing the influence of the convexity of the function involved, showing how improved iterative methods can be obtained that build upon those introduced in the previous two volumes. Each chapter presents theoretical results and illustrates them with applications to nonlinear equations, including scalar equations, integral equations, boundary value problems, and more. Convexity in Newton's Method will appeal to researchers interested in the theory of the Newton method as well as other iterative methods in Banach spaces.

Examines a variety of iterative methods in Banach spaces with an emphasis on those obtained from the Newton method Extends Kantorovich's theory for the Newton method by illustrating improved iterative methods developed by the authors Constitutes the third in the authors' trilogy dedicated to the topic of the Newton method in Banach spaces

Inhalt

The degree of logarithmic convexity.- The Newton method and convexity.- Accelerations of the Newton method.- Newton-like methods with high order of convergence.- Optimization of the Chebyshev method.