Local and semilocal convergence theorems for Newton's method based on continuously Fréchet differentiable operators (Q2743171)

The paper is concerned with the study of the Newton iteration \(x_{n+1}=x_n-F'(x_n)^{-1} F(x_n)\) for approximating a locally unique solution of equation \(F(x)=0\), where \(F\) is a continuously Fréchet-differentiable operator in a Banach space. Most known results on the convergence of the Newton method are based on the possibility of computing the Lipschitz constant of the operator \(F'(x_0)^{-1} F(x)\). The author presents several local and semilocal convergence results that avoid the computation of this constant and guarantee the quadratic convergence of the method.