New general convergence theory for iterative processes and its applications to Newton-Kantorovich type theorems

Let T: D subset of X -> X be an iteration function in a complete metric space X. In this paper we present some new general complete convergence theorems for the Picard iteration Xn+1 = Tx(n) with order of convergence at least r >= 1. Each of these theorems contains a priori and a posteriori error estimates as well as some other estimates. A central role in the new theory is played by the notions of a function of initial conditions of T and a convergence function of T. We study the convergence of the Picard iteration associated to T with respect to a function of initial conditions E: D -> X. The initial conditions in our convergence results utilize only information at the starting point x(0). More precisely, the initial conditions are given in the form E(x(0)) is an element of J, where J is an interval on R+ containing 0. The new convergence theory is applied to the Newton iteration in Banach spaces. We establish three complete omega-versions of the famous semilocal Newton-Kantorovich theorem as well as a complete version of the famous semilocal alpha-theorem of Smale for analytic functions. (C) 2009 Elsevier Inc. All rights reserved.