Convergence of the relaxed Newton's method (Q2870480)

A solution of a functional equation \(F(x)=0\) in Banach spaces can be approximately calculated by using Newton's method and the generalizations of Kantorovich. The study about convergence is usually based on two types: semilocal and local convergence analysis. The first one uses information around the initial point to give conditions ensuring the convergence of the iterative sequence. The local convergence estimates the radii of convergence balls centered in the solution. The aim of the paper is to develop relaxed Newton's methods. The semilocal convergence is proved under center-Lipschitz and Lipschitz conditions. For some equations of the type \(F(x)+G(x)=0\) with continuous operator \(G\), approximating sequences are generated, too, and sufficient convergence conditions are given. Three examples demonstrate the method (a nonlinear Hammerstein integral equation, a simple polynom equation and a simple system in three variables).