Kantorovich Theorem On Newton Iteration

Kantorovich established a semilocal convergence theorem for Newton's method in a Banach space, which is now called classical Kantorovich theorem. Since then, many results in convergence for Newton's method were obtained based on Kantorovich theorem. In this paper, we state the most important theoretical results in Kantorovich-type convergence for Newton's method, mainly for Frechet-differentiable equations in Banach spaces.Through improving the Kantorovich-type assumptions on the m-th Frechet-derivative, we find results concerning the convergence of Newton's method, which have more applications. With equivalent improved Kantorovich-type assumptions on the lst,2nd,m-th Frechet-derivative as before, we establish equivalent convergence theorems respectively, which are more easily applied than earlier ones. Besides, we also provide a convergence theorem under a class of weak Lipschtiz-type assumptions on the m-th Frechet-derivative.